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+{"text":"\\section{Introduction}\n\nFor a graph $G=(V,E)$ with vertex set $V$ and edge set $E$, a \\textit{binary edge-labeling} is a surjection $f : E \\to \\{ 0,1 \\}$. Let $i \\in \\{ 0,1 \\}$. An edge labeled $i$ is called an \\textit{$i$-edge} and let $e(i)$ denote the total number of $i$-edges in $G$ with respect to a binary edge-labeling $f$. In the case where $|e(1)-e(0)| \\leq 1$, a binary edge-labeling is called \\textit{edge-friendly}. Call the number of $i$-edges incident with a vertex $v$ the \\textit{$i$-degree} of $v$, denoted $\\deg_i(v)$, so that the degree of $v$ is $\\deg(v) = \\deg_1(v) + \\deg_0(v)$. An edge-friendly labeling of $G$ will induce a (possibly partial) \\textit{vertex-labeling} where a vertex $v$ is labeled $1$ when $\\deg_1(v) > \\deg_0(v)$, is labeled $0$ when $\\deg_0(v) > \\deg_1(v)$, and is unlabeled when $\\deg_1(v) = \\deg_0(v)$. Call a vertex labeled $i$ an \\textit{$i$-vertex} and let $v(i)$ denote the total number of $i$-vertices in $G$ with respect to an edge-friendly labeling $f$. The \\textit{edge-balanced index set} of $G$ is defined as\n\\begin{align*}\nEBI(G) = \\big\\{ |v(1)-v(0)|: \\text{over all edge-friendly labelings of $G$} \\big\\}.\n\\end{align*} \nMore information about graph labelings can be found in Gallian's dynamic survey~\\cite{GallianYYYY}.\n\nThe idea of a balanced labeling was introduced in 1992 by Lee, Liu, and Tan~\\cite{LLT1992}. In 1995, Kong and Lee provided results concerning edge-balanced graphs~\\cite{KL1995}. In~\\cite{KWL2009}, Kong, Wang, and Lee introduced the problem of finding the $EBI$ of complete bipartite graphs by solving the cases where the smaller part has cardinality 1, 2, 3, 4, or 5, and the special case where both parts have the same cardinality, but left all other cases open. In~\\cite{KMPR2014}, Krop, Minion, Patel, and Raridan concluded the edge-balanced index set problem for complete bipartite graphs with both parts of odd cardinality. \n\nA natural next step in the problem is to find the $EBI$ of the complete bipartite graphs where at least one part has even cardinality. In this paper, we conclude the problem for complete bipartite graphs where the larger part is of odd cardinality and the smaller is of even cardinality.\n\nFor positive integers $a,b$, where $a \\leq b$, let $[a]$ denote the set of integers $\\{ 1, \\dots, a \\}$ and let $[a,b]$ denote the set of integers $\\geq a$ but $\\leq b$; in the case where $a=1$, $[a]=\\{1\\}$, and in the case where $a=b$, $[a,a]=\\{a\\}$.  \n\nThroughout the rest of this paper, let $K_{m,n}$ be a complete bipartite graph with part $A$ of cardinality $m$ and part $B$ of cardinality $n$, where $m$ is odd, $n$ is even, and $m>n \\geq 2$. Let $q$ be the quotient when $m$ is divided by $\\frac{n}{2}+1$ and let $r$ be the remainder. If $r=0$, then we further partition $A$ into sets $A_i$ and denote the vertices of $A_i$ by $v_j^i$, where $i \\in [q]$ and $j \\in \\left[ \\frac{n}{2}+1 \\right]$. If $r \\geq 1$, then we partition $A$ as in the $r=0$ case with the addition of another partition $A_*$ having vertices denoted $v_i^*$, where $i \\in [r]$. Denote the vertices of $B$ by $u_i$, where $i \\in [n]$. \n\nIn the case where $n=2$, we have $q = \\frac{m-1}{2} \\geq 1$ and $r=1$. If $n \\geq 4$ and $q=1$, then $m = \\frac{n}{2}+1+r$, where $r \\geq 2$; for, if $r=0,1$, then $m \\leq \\frac{n}{2}+2 \\leq n$, a contradiction. In fact, if $n \\geq 4$, $q=1$, and $r \\geq 2$ is even, say $r=2j$ for some $j \\geq 1$, then $\\frac{n}{2}$ is even since $m$ is odd. That is, $n = 4k$ for some $k \\geq 1$. We have $m>n$ if and only if $j \\geq k$ and $r \\leq \\frac{n}{2}$ if and only if $j \\leq k$; hence, $j=k$ and $ r = \\frac{n}{2}$. Similarly, if $n \\geq 4$, $q=1$, and $r \\geq 3$ is odd, we have $r = \\frac{n}{2}$. Thus, for $n \\geq 4$, if $q=1$, then $m=n+1$ and $r = \\frac{n}{2}$. \n\nAny edge-friendly labeling of $K_{m,n}$ has $e(1) = e(0) = \\frac{mn}{2}$. Every vertex in $A$ has even degree so each vertex in this part is a $1$-vertex, a $0$-vertex, or is unlabeled, and every vertex in $B$ has odd degree so each is labeled either $1$ or $0$. Let $v_A(i)$ and $v_B(i)$ represent the number of $i$-vertices in $A$ and $B$, respectively, so that $v(i) = v_A(i)+v_B(i)$, where $i=0,1$. Without loss of generality, we assume that $v_A(1) \\geq v_A(0)$ and $v_B(1) \\geq v_B(0)$, which implies that $v(1) \\geq v(0)$; with this assumption, every element in $EBI(K_{m,n})$ can be computed as $v(1)-v(0)$. Note that not every vertex in $B$ can be a $1$-vertex. If every vertex in $B$ were a $1$-vertex, then the number of $1$-edges incident with these vertices would be at least $n\\left(\\frac{m+1}{2}\\right) > \\frac{mn}{2}$, a contradiction. However, it is possible to have only one $0$-vertex in $B$ since $(n-1)\\left(\\frac{m+1}{2}\\right) < \\frac{mn}{2}$. We have similar results for the vertices in $A$.\n\n\\section{Two Particular Edge-Friendly Labelings}\n\\label{sec:2-e-f}\n\nIn this section, we describe two edge-friendly labelings $f,f'$ of $K_{m,n}$. The labeling $f$ will show that $0 \\in EBI(K_{m,n})$ and $f'$ will show that $n-2 \\in EBI(K_{m,n})$. When $n=2$, the labelings $f$ and $f'$ give the same index $0$, so we do not construct $f'$ when $n=2$. For larger values of $n$, we obtain two distinct edge-friendly labelings that give different indices. \n\n\\subsection{Initializing the Labelings $f,f'$}\n\\label{subsec:initialize-f-f'}\n\nSet $f(v_1^i u_j)=f'(v_1^i u_j) = 1$, where $i \\in [q]$ and $j \\in \\left[ \\frac{n}{2} \\right]$, and label the remaining edges incident with each vertex $v_1^i$ by $0$. Then $v_1^i$ is unlabeled. If $r \\geq 1$, set $f(v_1^* u_j)=f'(v_1^* u_j) = 1$, where $j \\in \\left[ \\frac{n}{2} \\right]$, and label the remaining edges incident with vertex $v_1^*$ by $0$, so that $v_1^*$ is unlabeled.\n\n\\subsection{The Labeling $f$}\n\\label{subsec:f}\n\nAfter initializing $f$ as described above, we continue to label the edges of $K_{m,n}$ as follows to create an edge-friendly labeling $f$: For $i \\in [q]$, $j \\in \\left[ 2, \\frac{n}{2}+1 \\right]$, and $k \\in \\left[ \\frac{n}{2} \\right]$, set $f(v_j^i u_k) = 0$ and label the remaining edges incident with each vertex $v_j^i$ by $1$, so that $v_j^i$ is unlabeled. If $r \\geq 2$, for $j \\in [2, r]$ and $k \\in \\left[ \\frac{n}{2} \\right]$, set $f(v_j^* u_k) = 0$ and label the remaining edges incident with vertex $v_j^*$ by $1$, so that $v_j^*$ is unlabeled. Note that under $f$, all edges in the graph have been labeled either $0$ or $1$ and $f$ is edge-friendly by construction. \n\nAll vertices in $A$ are unlabeled. If $n=2$, then $\\deg_1(u_1) = \\deg_0(u_2) = q+1$ and $\\deg_0(u_1) = \\deg_1(u_2) = q$, so vertex $u_1$ is a $1$-vertex and $u_2$ is a $0$-vertex. For $n \\geq 4$ and $i \\in \\left[ \\frac{n}{2} \\right]$, we have $\\deg_0(u_i) = \\deg_1(u_{i+\\frac{n}{2}}) > \\deg_1(u_i) = \\deg_0(u_{i+\\frac{n}{2}})$, so vertex $u_i$ is a $0$-vertex and $u_{i+\\frac{n}{2}}$ is a $1$-vertex. Thus, $v(1)=v(0)=\\frac{n}{2}$, which gives $0 \\in EBI(K_{m,n})$. \n\n\\subsection{The Labeling $f'$}\n\\label{subsec:f'}\n\nBy previous remarks, we know that it is possible to have an edge-friendly labeling of $K_{m,n}$ where one vertex of $B$, say $u_n$, is a $0$-vertex and the remaining vertices $u_1, \\dots, u_{n-1}$ are $1$-vertices. \n\nAfter initializing $f'$ as described above, we continue to label the edges of $K_{m,n}$, where $n \\geq 4$, as follows to create an edge-friendly labeling $f'$. We begin by labeling the edges (which are not already labeled) incident with vertex $u_n$ by $0$ so that $\\deg_0(u_n)=m$ and $u_n$ is a $0$-vertex. The remaining edge labels will be determined based on the parity of $i \\in [q]$. For $i \\in [q]$ with $i$ odd, $j \\in \\left[2,\\frac{n}{2}+1\\right]$, $k \\in \\left[\\frac{n}{2}-1\\right]$, and $\\ell=j+k-2$, set $f'(v_j^i u_\\ell)=0$ and label the remaining unlabeled edges incident with each vertex $v_j^i$ by $1$, so that $v_j^i$ is unlabeled. For $i \\in [q]$ with $i$ even, $j \\in \\left[2,\\frac{n}{2}\\right]$, $k \\in \\left[\\frac{n}{2}\\right]$, and $\\ell = j+k-1$, set $f'(v_j^i u_\\ell)=1$ and label the remaining unlabeled edges incident with each vertex $v_j^i$ by $0$, so that $v_j^i$ is unlabeled. For $i \\in [q]$ with $i$ even, $j = \\frac{n}{2}+1$, $k \\in \\left[\\frac{n}{2}-1\\right]$, and $\\ell=j+k-1$, set $f'(v_j^i u_\\ell)=1$ and $f'(v_j^i u_1)=1$, and label the remaining unlabeled edges incident with each vertex $v_j^i$ by $0$, so that $v_j^i$ is unlabeled. If $r \\geq 2$, then similar to the even $i$ case, we set $f'(v_j^* u_\\ell)=1$, where $j \\in \\left[2,r\\right]$, $k \\in \\left[\\frac{n}{2}\\right]$, and $\\ell = j+k-1$, and label the remaining unlabeled edges incident with each vertex $v_j^*$ by $0$, so that $v_j^*$ is unlabeled. Under $f'$, all edges in the graph have been labeled either $0$ or $1$, and since $e(0)=e(1)$, the constructed labeling $f'$ is edge-friendly. \n\nAll vertices in $A$ are unlabeled, vertex $u_n$ is a $0$-vertex, and $\\deg_1(u_i) > \\deg_0(u_i)$ for $i \\in [n-1]$, so vertex $u_i$ is a $1$-vertex. Thus, $v(1)=n-1$ and $v(0)=1$, which gives $n-2 \\in EBI(K_{m,n})$. \n\n\\section{Main Result}\n\\label{sec:main-result}\n\nWe are now ready to prove the following:\n\\begin{thm}\n\\label{thm:max-EBI-odd-even}\nLet $K_{m,n}$ be a complete bipartite graph with parts of cardinality $m$ and $n$, where $m$ is odd, $n$ is even, and $m>n \\geq 2$. Then $EBI(K_{m,2}) = \\{0\\}$. For $n\\geq 4$, let $q$ be the quotient when $m$ is divided by $\\frac{n}{2}+1$ and let $r$ be the remainder. Then\n\\begin{align}\nEBI(K_{m,n}) =\n\\begin{cases}\n\\left\\{ 0,1, \\dots, m+n-2q-2 \\right\\}, &\\text{if~$r = 0$}, \\\\\n\\left\\{ 0,1, \\dots, m+n-2q-3 \\right\\}, &\\text{if~$r = 1$}, \\\\\n\\left\\{ 0,1, \\dots, m+n-2q-4 \\right\\}, &\\text{if~$r \\geq 2$}.\n\\end{cases}\n\\end{align}\n\\end{thm}\n\n\\begin{proof}\nLet $n=2$. Then the labeling $f$ given in Section~\\ref{subsec:f} shows that $0 \\in EBI(K_{m,2})$. To see that $0$ is the only index in the edge-balanced index set of $K_{m,2}$, consider switches on pairs of edges incident with a vertex $u \\in B$, say $e=uv$ and $e'=uv'$, where $f(e)=1$ and $f(e')=0$. Such switches will not alter the edge-friendliness of the labeling, nor alter the label on vertex $u$, but each switch will change the unlabeled vertex $v$ to a $0$-vertex and the unlabeled vertex $v'$ to a $1$-vertex. No matter how many switches are performed (up to labeling all but one vertex in part $A$), we will always have $v_A(1)=v_A(0)$ and $v_B(1)=v_B(0)=1$. It is impossible to label all the vertices in part $A$, so $EBI(K_{m,2}) = \\{ 0 \\}$.\n\n\nFor the remainder of the proof, let $n \\geq 4$, let $q$ be the quotient when $m$ is divided by $\\frac{n}{2}+1$, and let $r$ be the remainder.\n\n\nConsider the labeling $f'$ given in Section~\\ref{subsec:f'}, which provides an edge-friendly labeling of $K_{m,n}$ and shows that $n-2 \\in EBI(K_{m,n})$. We perform edge label switches on pairs of $0$-edges and $1$-edges incident with the same vertex in part $B$, noting that such a switch will not alter the edge-friendliness of the labeling. For $i \\in [q]$, switch the label on edge $v_1^i u_1$ with the label on edge $u_1 v_2^i$. These edge label switches will not change the label on vertex $u_1$, but will cause $v_1^i$ to change from an unlabeled vertex to a $0$-vertex and will cause $v_2^i$ to change from unlabeled vertex to a $1$-vertex. After performing these edge label switches, we note that the number of $1$-vertices increased by $q$ and the number of $0$-vertices increased by $q$, so we still have $n-2 \\in EBI(K_{m,n})$. Continuing our $\\{0,1\\}$-edge-pair switches, for $i \\in [q]$ and $j \\in \\left[2,\\frac{n}{2}\\right]$, switch the label on edge $v_1^i u_j$ with the label on edge $u_j v_{j+1}^i$. Each such switch increases the edge-balanced index by one. Moreover, after all of these $\\{0,1\\}$-edge-pair switches, we have that $\\deg_0(v_1^i)=n$, implying that each $v_1^i$ is a $0$-vertex with no incident $1$-edges, and that $\\deg_1(v_j^i) = \\frac{n}{2}+1$, where $j \\in \\left[2,\\frac{n}{2}+1\\right]$, implying that each $v_j^i$ is a $1$-vertex (but just barely). Thus, all vertices in $A_i$, where $i \\in [q]$, are labeled either $0$ or $1$, and we have attained each index from $n-2$ to $n-2+q\\left(\\frac{n}{2}-1\\right)=m+n-2q-2-r$ in the edge-balanced index set. If $r=0$, then we are done as we cannot increase the edge-balanced index further. That is, if $r=0$, then the maximal index in the edge-balanced index set is $m+n-2q-2$. Similarly, if $r=1$, then we do not have any extra $1$-edges incident with vertices $v_j^i$, where $i \\in [q]$ and $j \\in \\left[2,\\frac{n}{2}+1\\right]$, that could be used to change vertex $v_1^*$ into a $1$-vertex, so we cannot increase the edge-balanced index further. That is, for $r=1$, the maximal index is $m+n-2q-3$. For values of $r \\geq 2$, we may perform additional $\\{0,1\\}$-edge-pair switches to force all vertices in part $A$ to be labeled, increasing the number of $0$-vertices in $A$ by one and the number of $1$-vertices by $r-1$. In particular, for $j \\in [r-1]$, switch the label on edge $v_1^* u_j$ with the label on edge $u_j v_{j+1}^*$. Then $v_1^*$ is a $0$-vertex and $v_{j+1}^*$ is a $1$-vertex. In this case, we have that $v_A(0)=q+1$ and $v_A(1)=m-q-1$, which means that the maximal index in the edge-balanced index set is $v(1)-v(0)=v_A(1)+v_B(1)-v_A(0)-v_B(0)=m-q-1+n-1-(q+1)-1=m+n-2q-4$. \n\n\nNow, consider the labeling $f$ given in Section~\\ref{subsec:f}. Performing the same $\\{0,1\\}$-edge-pair switches described above, we find that we are able to achieve subsets of the edge-balanced index set based on the value of $r$. If $r=0$, then we achieve the indices $\\{ 0,1, \\dots, m-2q \\}$. If $r=1$, then we achieve the indices $\\{ 0,1, \\dots, m-2q-1 \\}$. If $r \\geq 2$, then we achieve the indices $\\{ 0,1, \\dots, m-2q-2 \\}$.  \n\n\nFor the last part of the proof, note that if $r=0$, then we have that $\\{ 0,1, \\dots, m-2q \\} \\cup \\{ n-2, \\dots, m+n-2q-2 \\} = \\{ 0,1, \\dots, m+n-2q-2 \\}$, since $q = \\frac{2m}{n+2}$ and $m > n+1$ implies $m-2q = \\frac{m(n-2)}{n+2}> n-3 + \\frac{4}{n+2} > n-3$. That is, if $r=0$, then $EBI(K_{m,n}) = \\{ 0,1, \\dots, m+n-2q-2 \\}$. Now, if $r=1$, then $q = \\frac{2m-2}{n+2}$ and $m \\geq n+3$ implies $m-2q-1 = \\frac{(m-1)(n-2)}{n+2} \\geq n-2$, and $EBI(K_{m,n}) = \\{ 0,1, \\dots, m+n-2q-3 \\}$. Finally, for values of $r \\geq 2$, we consider three cases: (i)~$m=n+1$, (ii)~$m=n+3$, and (iii)~$m \\geq n+5$. For case~(i), if $m=n+1$, then $q=1$ and $m-2q-2=n-3$. For case~(ii), if $m=n+3$, then $q=2$ and $m-2q-2=n-3$. For case~(iii), if $m \\geq n+5$, then $m-2q-2 = \\frac{(m-2)(n-2)}{n+2} + \\frac{4r-8}{n+2} \\geq \\frac{(n+3)(n-2)}{n+2} > n-2$. Thus, if $r \\geq 2$, then $m-2q-2 \\geq n-3$ and $EBI(K_{m,n}) = \\{ 0,1, \\dots, m+n-2q-4 \\}$.\n\\end{proof}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:intro}\nEnd of 1998 the concept of ``Grid computing\" was introduced in the monograph ``The Grid: Blueprint for a New\nComputing Infrastructure\" by I. Foster and C. Kesselman \\cite{Foster:1998:GBN}. Two years earlier, in 1997, the\ndevelopment of the UNICORE - Uniform Interface to Computing Resources - system was initiated to enable German\nsupercomputer centers to provide their users with a {\\em seamless, secure, and intuitive access} to their\nheterogeneous computing resources. Like in the case of the Globus Toolkit\\textsuperscript{\\textregistered}\n\\cite{Foster:1997:GMI} UNICORE was started before ``Grid Computing\" became the accepted new paradigm for\ndistributed computing.\n\nThe UNICORE vision was proposed to the German Ministry for Education and Research (BMBF) and received funding. A\nfirst prototype was developed in the UNICORE\\footnote{funded in part by BMBF grant 01 IR 703, duration: August\n1997 - December 1999} project \\cite{Erwin:2000:UNI}. The foundations for the current production version were laid\nin the follow-up project UNICORE Plus\\footnote{funded in part by BMBF grant 01 IR 001 A-D, duration: January 2000\n- December 2002} \\cite{Erwin:2003:UNI}, which was successfully completed in 2002. Since then UNICORE was used in\noperation at German supercomputing centers and became a solid basis for numerous European projects. In this paper\nwe will describe the evolution of UNICORE from a prototype software developed in research projects to a Grid\nmiddleware used today in the daily operation of production Grids.\n\nAlthough already set out in the initial UNICORE project proposal in 1997, the goals and objectives of the UNICORE\ntechnology are still valid:\n\\bi\n\\item Foremost, the aim of UNICORE is to hide the rough edges resulting from different hardware architectures, vendor\nspecific operating systems, incompatible batch systems, different application environments, historically grown\ncomputer center practices, naming conventions, file system structures, and security policies -- just to name the\nmost obvious.\n\\item Equally, security is a constituent part of UNICORE's design relying on X.509 certificates for the\nauthentication of users, servers, and software, and the encryption of the communication over the internet.\n\\item Finally, UNICORE is usable by scientists and engineers without having to study vendor or site-specific\ndocumentation. A Graphical User Interface (GUI) is available to assist the user in creating and managing jobs.\n\\ei\n\nAdditionally, several basic conditions are met by UNICORE: the Grid middleware supports operating systems and\nbatch systems of all vendors present at the partner sites. In 1997 these were for instance large Cray T3E systems,\nNEC and Hitachi vector machines, IBM SP2s, and smaller Linux clusters. Nowadays the spectrum is even broader, of\ncourse with modern hardware, such as IBM p690 systems. The deployed software has to be non-intrusive, so that it\ndoes not require changes in the computing centers hard- and\/or software infrastructure. Maintaining site autonomy\nis still a major issue in Grid computing, when aspects of acceptability and usability in particular from the\nsystem administrator's point of view are addressed. In addition to UNICORE's own security model, site-specific\nsecurity requirements (\\eg firewalls) are supported.\n\nNear the end of the initial funding period of the UNICORE Plus project, a working prototype was available, which\nshowed that the initial concept works. By combining innovative ideas and proven components over the years, this\nfirst prototype evolved to a {\\em vertically integrated} Grid middleware solution.\n\nThe remainder of this paper is structured as follows. In section 2 the architecture of UNICORE and its core\nfeatures are described. European funded projects, which use UNICORE as a basis for their work are described in\nSection 3, and in Section 4 the usage of UNICORE in production is described. Section 5 gives an outlook on the\nfuture development of UNICORE. The paper closes with conclusions and acknowledgements.\n\n\n\n\\section{The Architecture of UNICORE}\n\\label{sec:arch}\n\\figref{fig:UNICORE-arch} shows the layered Grid architecture of UNICORE consisting of user, server and target\nsystem tier \\cite{Romber:2002:UGI}. The implementation of all components shown is realized in Java. UNICORE meets\nthe Open Grid Services Architecture (OGSA) \\cite{Foster:2003:TPG} concept following the paradigm of 'Everything\nbeing a Service'. Indeed, an analysis has shown that the basic ideas behind UNICORE already realizes this paradigm\n\\cite{Snelling:2002:UGI,Snelling:2003:UOG}.\n\n\\subsection{User Tier}\n\\label{sec:arch-userT}\nThe UNICORE Client provides a graphical user interface to exploit the entire set of services offered by the\nunderlying servers. The client communicates with the server tier by sending and receiving Abstract Job Objects\n(AJO) and file data via the UNICORE Protocol Layer (UPL) which is placed on top of the SSL protocol. The AJO is\nthe realization of UNICORE's job model and central to UNICORE's philosophy of abstraction and seamlessness. It\ncontains platform and site independent descriptions of computational and data related tasks, resource information\nand workflow specifications along with user and security information. AJOs are sent to the UNICORE Gateway in form\nof serialized and signed Java objects, followed by an optional stream of bytes if file data is to be transferred.\n\\begin{figure}[htb]\n\\centering\n\\includegraphics[width=\\textwidth]{UNICORE-architecture.eps}\n\\caption{The UNICORE architecture.} \\label{fig:UNICORE-arch}\n\\end{figure}\n\nThe UNICORE client assists the user in creating complex, interdependent jobs that can be executed on any UNICORE\nsite (Usite) without requiring any modifications. A UNICORE job, more precisely a job group, may recursively\ncontain other job groups and\/or tasks and may also contain dependencies between job groups to generate job\nworkflows. Besides the description of a job as a set of one or more directed a-cyclic graphs, conditional and\nrepetitive execution of job groups or tasks are also included. For the monitoring of jobs, their status is\navailable at each level of recursion down to the individual task. Detailed log information is available to analyze\npotential error conditions. At the end of the execution of the job it is possible to retrieve the {\\sf stdout} and\n{\\sf stderr} output of the job. Data management functions like import, export, and transfer are available through\nthe GUI as explicit tasks. This allows the user to specify data transfer from one target system to another (\\eg\nfor workflows), from or to the local workstation before or after the execution of a job, or to store data\npermanently in archives.\n\nThe previously described features already provide an effective tool to use resources of different computing\ncenters both for capacity or capability computing, but many scientists and engineers use application packages. For\napplications without a graphical user interface, a tool kit simplifies the development of a custom built UNICORE\nplug-in. Over the years many plug-ins were developed, so that plug-ins already exist for many standard scientific\napplications, as \\eg for CPMD (Car-Parrinello Molecular Dynamics) \\cite{Huber:2001:SCP}, Fluent or MSC Nastran.\n\n\\subsection{Server Tier}\n\\label{sec:arch-serverT}\nThe server tier contains the Gateway and the Network Job Supervisor (NJS). The Gateway controls the access to a\nUsite and acts as the secure entry point accepting and authenticating UPL requests. A Usite identifies the\nparticipating organization (\\eg a supercomputing center) to the Grid with a symbolic name that resolves into the\nURL of the Gateway. An organization may be part of multiple Grids offering the same or different resources to\ndifferent communities. The Gateway forwards incoming requests to the underlying Network Job Supervisor (NJS) of a\nvirtual site (Vsite) for further processing. The NJS represents resources with a uniform user mapping scheme and\nno boundaries like firewalls between them.\n\nA Vsite identifies a particular set of resources at a Usite and is controlled by a NJS. A Vsite may consist of a\nsingle supercomputer, \\eg a IBM p690 System with LoadLeveler, or a Linux cluster with PBS as resource management\nsystem. The flexibility of this concept supports different system architectures and gives the organization full\ncontrol over its resources. Note that, there can be more than one Vsite inside each USite as depicted in\n\\figref{fig:UNICORE-arch}.\n\nThe NJS is responsible for the virtualization of the underlying resources by mapping the abstract job on a\nspecific target system. This process is called ``incarnation\" and makes use of the Incarnation Database (IDB).\nSystem-specific data are stored in the IDB describing the software and hardware infrastructure of the system.\nAmong others, the available resources like software, incarnation of abstract commands (standard UNIX command like\nrm, cp, ...) and site-specific administrative information are stored. In addition to the incarnation the NJS\nprocesses workflow descriptions included in an AJO, performs pre- and post-staging of files and authorizes the\nuser via the UNICORE User Database (UUDB). Typically the Gateway and NJS are running on dedicated secure systems\nbehind a firewall, although the Gateway could be placed outside a firewall or in a demilitarized zone.\n\n\\subsection{Target System Tier}\n\\label{sec:arch-targetsysT}\nThe Target System Interface (TSI) implements the interface to the underlying supercomputer with its resource\nmanagement system. It is a stateless daemon running on the target system and interfacing with the local resource\nmanager realized either by a batch system like PBS \\cite{openPBS:url} or CCS \\cite{Hovestadt:2003:SHR}, a batch\nsystem emulation on top of \\eg Linux, or a Grid resource manager like Globus' GRAM\n\\cite{GRAM:url,Menday:2003:GEU}.\n\n\\subsection{Single Sign-On}\n\\label{sec:arch-signon}\nThe UNICORE security model relies on the usage of permanent X.509 certificates issued by a trusted Certification\nAuthority (CA) and SSL based communication across `insecure' networks. Certificates are used to provide a single\nsign-on in the client. The client unlocks the user's keystore when it is first started, so that no further\npassword requests are handed to the user. All authentication and authorization is done on the basis of the user\ncertificate. At each UNICORE site user certificates are mapped to local accounts (standard UNIX uid\/gid), which\nmay be different at each site, due to existing naming conventions. The sites retain full control over the\nacceptance of users based on the identity of the individual -- the distinguished name -- or other information that\nmight be contained in the certificate. UNICORE can handle multiple user certificates, \\ie it permits a client to\nbe part of multiple, disjoint Grids. It is also possible to specify project accounts in the client allowing users\nto select different accounts for different projects on one execution system or to assume different roles with\ndifferent privileges.\n\nThe private key in the certificate is used to sign each job and all included sub-jobs during the transit from the\nclient to sites and between sites. This protects against tampering while the job is transmitted over insecure\ninternet connections and it allows to verify the identity of the owner at the receiving end, without having to\ntrust the intermediate sites which forwarded the job.\n\n\n\n\\section{UNICORE Based Projects}\n\\label{sec:projects}\nDuring the evolutionary development of the UNICORE technology, many European and international projects have\ndecided to base their Grid software implementations on UNICORE or to extend the growing set of core UNICORE\nfunctions with new features specific to their project focus. The goals and objectives of projects using UNICORE\nare not limited to the computer science community alone. Several other scientific domains such as bio-molecular\nengineering or computational chemistry are using the UNICORE technology as the basis of their work. In the\nfollowing we present short overviews of goals and objectives of UNICORE-based projects and describe additional\nfunctions and services contributed to the UNICORE development.\n\n\\subsection{EUROGRID -- Application Testbed for European Grid Computing}\n\\label{sec:projects-eurogrid}\nIn the EUROGRID\\footnote{funded in part by EC grant IST-1999-20247, duration: November 2000 - January 2004}\nproject \\cite{eurogrid:url} a Grid network of leading European High Performance Supercomputing centers was\nestablished. Based on the UNICORE technology application-specific Grids were integrated, operated and\ndemonstrated: \\bi\n\\item Bio-Grid for biomolecular science\n\\item Meteo-Grid for localized weather prediction\n\\item CAE-Grid for coupling applications\n\\item HPC-Grid for general HPC end-users\n\\ei\n\nAs part of the project, the UNICORE software was extended by an efficient data transfer mechanism, resource\nbrokerage mechanisms, tools and services for Application Service Providers (ASP), application coupling methods,\nand an interactive access feature \\cite{Mallmann:2001:EAT}. Efficient data transfer is a important issue, as Grids\ntypically rely on public parts of Internet connections. The available limited bandwidth has to be used efficiently\nto reduce the transfer time and the integrity of the transferred data has to be maintained, even if the transfer\nis interrupted. Depending on the application domain, additional security and confidentiality concerns need to be\nconsidered. This UNICORE high performance data transfer also uses X.509 certificates for authentication and\nencryption. To achieve not only a fast and secure transfer of data, but also high-performance capabilities,\nnetwork Quality of Service (QoS) aspects, overlapping of streamed data transfers, and packet assembling and\ncompression techniques are included.\n\nIn order to optimize the selection of resources -- either done by the users manually or by a metascheduler\nautomatically -- resource brokerage mechanisms and detailed resource description abilities are important. Within\nthe EUROGRID project, mechanisms were added to UNICORE, which allow users to specify their jobs in an abstract way\nimproving the overall resource selection and accounting. In particular for the benefit of the industrial user\naspects of security, convenience, and cost efficiency were addressed. To this end, the already existing security\nconcepts of UNICORE were thoroughly evaluated and assessed as being adequate, hence no additional development had\nto be done. The task of the developed resource broker is to match the abstract specification of the users jobs and\ntheir requirements with the available resources in the Grid. The resource broker reports the best match back to\nthe user including an estimate of the costs, which than allows the user to assign the appropriate resources to the\njob. For the suppliers of Grid resources (\\eg supercomputing centers) the resource broker allows to specify\ninformation about computational resources, architectures, processing power, storage and archiving facilities,\npost-processing facilities like visualization equipment, available software packages, and security guarantees. All\nthis data is enhanced by billing information.\n\nSupercomputing centers converge from pure providers of raw supercomputing power to Application Service Providers\n(ASP) running relevant scientific applications. For accounting and billing purposes the ASP needs to know the\nexact resources consumed by each customer in each run. For measuring the usage of supercomputers standard\nmechanisms provided by the resource management and operating system can be used, but measuring the usage of\nlicenses requires a sophisticated approach. For some applications, \\eg from the Computer Aided Engineering (CAE)\ndomain, this includes a connection to the applications licence manager. Establishing a link to the above mentioned\nresource broker is required to influence their decisions.\n\nFor solving complex problems applications from different domains, \\eg fluid-structure or\nelectromagnetism-structure, need to be coupled. This is established by using the EUROGRID resource broker\nfunctionality and combining it with the available Metacomputing functionality developed in the UNICORE Plus\nproject (\\cf \\secref{sec:intro}), which allows different schedulers of compute and application resources to\ncooperate. Finally, an interactive access to control and steer running application is needed for many scientific\napplications. The interactive use includes an interactive shell to actually login to computing resources using the\nUNICORE technology and security infrastructure.\n\nEUROGRID used the UNICORE technology to provide the above described services and functionalities by developing new\ncomponents. After the project ended, the developed components were revised and useful additions to the core\nUNICORE functions are now part of the available UNICORE software.\n\n\n\\subsection{GRIP -- Grid Interoperability Project}\n\\label{sec:projects-grip}\nGrid computing empowers users and organizations to work effectively in an information-rich environment. Different\ncommunities and application domains have developed distinct Grid implementations some based on published open\nstandards or on domain and community specific features. GRIP\\footnote{funded in part by EC grant IST-2001-32257,\nduration: January 2002 - February 2004} \\cite{GRIP:url} had the objective to demonstrate that the different\napproaches of two distinct grids can successfully complement each other and that different implementations can\ninteroperate. Two prominent Grid systems were selected for this purpose: UNICORE and Globus\\texttrademark\n\\cite{globus:url}, a toolkit developed in the United States. In contrast to UNICORE, Globus provides a set of APIs\nand services which requires more in-depth knowledge from the user. Globus is widely used in numerous international\nprojects and many centers have Globus installed as Grid middleware.\n\nThe objectives of GRIP were: \\bi\n\\item Develop software to enable the interoperation of independently developed Grid solutions\n\\item Build and demonstrate prototype inter-Grid applications\n\\item Contribute to and influence international Grid standards\n\\ei\n\nDuring the runtime of the GRIP project the Open Grid Service Architecture was proposed by the Global Grid Forum\n(GGF) \\cite{GGF:url}. The arrival of OGSA also was an opportunity to influence the standards directly which were\nto be created and to start developments that allow UNICORE to interoperate not only with Globus but with services\non the Grid in general, once the definition of the services and their interfaces became mature. OGSA did not\nchange the overall objectives of GRIP, however, it influenced directly some of the technical results.\n\nA basic requirement of GRIP was that the Grid interoperability layer should not change the well-known UNICORE user\nenvironment. As developers from both communities cooperated in the GRIP project, this goal was reached with only\nlittle changes of the UNICORE server components and no changes of the Globus Toolkit. This was achieved by the\ndevelopment of the so called Globus Target System Interface (Globus TSI), which provides UNICORE-access to\ncomputational resources managed by Globus. The Globus TSI was integrated into a heterogeneous UNICORE and Globus\ntestbed.\n\nTo achieve the main objective of GRIP, the interoperability between UNICORE and Globus and initial OGSA services,\nthe following elements had to be implemented: \\bi\n\\item The interoperability layer between UNICORE and Globus Version 2\n\\item The interoperability layer between UNICORE and Globus Version 3\n\\item The Access from UNICORE to simple Web services as a first step towards full integration of Web services\n\\item The Interoperability of the certificate infrastructures of UNICORE and Globus\n\\item A resource broker capable of brokering between UNICORE and Globus resources\n\\item The Ontology of the resource description on an abstract level\n\\ei\n\nIn GRIP, two important application areas were selected to prove that the interoperability layers work as\nspecified: \\bi\n\\item Bio-molecular applications were instrumented in such a way that they are Grid-aware in any\nGrid environment and capable to seamlessly use UNICORE and Globus managed resources. The techniques developed in\nGRIP were designed and implemented in a generalized way to ensure that they can be used in other application\ndomains as well.\n\\item A meteorological application, the Relocatable Local Model (RLM), was decomposed in such\na way that the components could execute on the most suitable resources in a Grid, independent of the middleware.\n\\ei\n\nThe results of the GRIP project are important for understanding general interoperability processes between Grid\nmiddleware systems. The experience and knowledge of the GRIP partners allowed to work in many relevant areas\nwithin GGF, like security, architecture, protocols, workflow, production management, and applications, and to\ninfluence the work in GGF.\n\n\n\\subsection{OpenMolGRID -- Open Computing Grid for Molecular Science and Engineering}\n\\label{sec:projects-openmolgrid}\nThe OpenMolGRID\\footnote{funded in part by EC grant IST-2001-37238, duration: September 2002 - February 2005}\nproject \\cite{OpenMolGrid:url} was focused on the development of Grid enabled molecular design and engineering\napplications. {\\em In silico} testing \\cite{Sild:2005:OAW} has become a crucial part in the molecular design\nprocess of new drugs, pesticides, biopolymers, and biomaterials. In a typical design process $O(10^5)$ to\n$O(10^6)$ candidate molecules are generated and their feasibility has to be tested. It is not economical to carry\nout experimental testing on all possible candidates. Therefore, computational screening methods provide a cheap\nand cost effective alternative to reduce the number of candidates. Over the years Quantitative Structure\nActivity\/Property Relationship (QSAR\/QSPR) methods have been shown to be reliable for the prediction of various\nphysical, chemical, and biological activities \\cite{Karelson:2000:MDQ}.\n\nQSPR\/QSAR relies on the observation that molecular compounds with similar structure have similar properties. For\neach specific application a set of molecules is needed for which the target property is known. This requires\nsearching globally distributed information resources for appropriate data. For the purpose of exploring molecular\nsimilarity, descriptors are calculated from the molecular structure. Thousands of molecular descriptors have been\nproposed and are used to characterize molecular structures with respect to different properties. Their calculation\nputs high demands on computer resources and requires high-performance computing.\n\nBased on this complex application the objectives of the OpenMolGRID project were defined as:\n\\bi\n\\item Development of tools for secure and seamless access to distributed information and computational methods\nrelevant to molecular engineering within the UNICORE frame\n\\item Provision of a realistic testbed and reference application in life science\n\\item Development of a toxicity prediction model validated with a large experimental set\n\\item Provision of design principles for next generation molecular engineering systems.\n\\ei\nIn particular this included to use UNICORE to automatize, integrate, and speed-up the drug discovery pipeline.\n\nThe OpenMolGRID project addressed the objectives above by defining abstraction layers for data sources (databases)\nand methods (application software), and integrating all necessary data sources (\\eg ECOTOX \\cite{ecotox:url}) and\nmethods (\\eg 2D\/3D Molecular Structure Conversion and Optimization, Descriptor Calculation, Structure Enumeration)\ninto UNICORE. The project developed application specific user interfaces (plug-ins) and a mechanism to generate a\ncomplete UNICORE Job from an XML workflow specification. This so called Meta-Plug-in takes care of including all\nauxiliary steps like data format transformation and data transfers into the job, distributing data parallel tasks\nover available computational resources, and allocating resources to the tasks. Thereby the molecular design\nprocess was significantly improved as the time to build QSAR\/QSPR models, the probability for mistakes, and the\nvariability of results was reduced. In addition a command line client (CLC) for UNICORE was developed to enable\nthe data warehouse to use Grid resources for its data transformation processes. The CLC offers the generation of\nUNICORE jobs from XML workflow description as well as the job submission, output retrieval, status query, and job\nabortion. The CLC consists of commands, an API, and a queuing component.\n\nBesides the technical achievements of OpenMolGRID and the added value for pharmaceutical companies its results\nwill contribute to the standardization of QSAR models.\n\n\n\\subsection{VIOLA -- Vertically Integrated Optical Testbed for Large Applications}\n\\label{sec:projects-viola}\nThe aim of the VIOLA\\footnote{funded in part by BMBF grant 01AK605F, duration: May 2004 - April 2007} project\n\\cite{VIOLA:url} is to build up a testbed with the latest optical network technology (multi 10 Gigabit Ethernet\nlinks). The goals and objectives of VIOLA are: \\bi\n\\item Testing of new network components and network architectures\n\\item Development and testing of software for dynamic bandwidth management\n\\item Interworking of network technology from different manufacturers\n\\item Development and testing of new applications from the Grid and Virtual Reality (VR) domain\n\\ei\n\nThe performance of the new network technology is evaluated with different scientific applications that need a very\nhigh network performance and network flexibility. UNICORE is used to build up the Grid on top of the hardware\nwithout taking fundamental software modifications. Only an interface to the meta-computer software library\nMetaMPICH \\cite{METAMPICH:url} needs to be integrated into UNICORE. Grid applications from the High Performance\nSupercomputing and Virtual Reality domain are enhanced for an optimized usage of the available bandwidth and the\nprovided Quality of Service classes. In this context a Meta-Scheduler framework is developed, which is able to\nhandle complex workflows and multi-site jobs by coordinating supercomputers and the network connecting them.\n\n\\begin{figure}[htb]\n\\centering\n\\includegraphics[width=\\textwidth]{VIOLA-superscheduler.eps}\n\\caption{The VIOLA Meta-Scheduler architecture.} \\label{fig:VIOLA-supersched}\n\\end{figure}\n\nVIOLA's first generation Meta-Scheduler architecture focuses on the scheduling functionality requiring only\nminimal changes to the UNICORE system. As depicted in \\figref{fig:VIOLA-supersched}, the system comprises the\nAgreement Manager, the Meta-Scheduler itself \\cite{Quecke:2000:MAR}, and a Meta-Scheduling plug-in (which is part\nof the client and not pictured separately). Before submitting a job to a Usite (\\cf \\secref{sec:arch-serverT}),\nthe Meta-Scheduling plug-in and the Meta-Scheduler exchange the data necessary to schedule the resources needed.\nThe Meta-Scheduler is then (acting as an Agreement Consumer in WS--Agreement terms \\cite{GRAAP:url}) contacting\nthe Agreement Manager to request a certain level of service, a request which is translated by the Manager into the\nappropriate resource management system commands. In case of VIOLA's computing resources the targeted resource\nmanagement system is the EASY scheduler. Once all resources are reserved at the requested time the Meta-Scheduler\nnotifies the UNICORE Client via the Meta-Scheduling plug-in to submit the job. This framework will also be used to\nschedule the interconnecting network, but potentially any resource can be scheduled if a respective Agreement\nManager is implemented and the Meta-Scheduling plug-in generates the necessary scheduling information. The\nfollow-on generation of the Meta-Scheduling framework will then be tightly integrated within UNICORE\/GS (\\cf\n\\secref{sec:future-unigrids-GS}).\n\n\n\\subsection{NaReGI -- National Research Grid Initiative}\n\\label{sec:projects-naregi}\nThe Japanese NaReGI project \\cite{NAREGI:url} includes the UNICORE technology as the basic middleware for research\nand development. NaReGI is a collaboration project between industry, academia, and government. The goals and\nobjectives are: \\bi\n\\item Establishment of a national Japanese research Grid infrastructure\n\\item Revitalization of the IT industry through  commercialization of Grid middleware and strengthened\ninternational competitiveness\n\\item Dissemination of Grid environments throughout industry\n\\item Trailblazing the standardization of Grid technology\n\\item Cultivation of human resources specializing in IT technology for Grids\n\\ei\n\nSimilar to the GRIP project (\\cf \\secref{sec:projects}) where an interoperability layer between UNICORE and Globus\nToolkit 2 and 3 was developed, the NaReGI project plans to implement such a layer between UNICORE and Condor\n\\cite{CONDOR:url}, called UNICONDORE. This interoperability layer will allow to submit jobs from the UNICORE\nclient to Condor pools and to use Condor commands to submit jobs to UNICORE managed resources.\n\nIn the first phase of the NaReGI testbed UNICORE provides access to about 3000 CPUs in total with approximately 17\nTFlops of peak performance. It is expected to increase the integrated peak performance to 100+ TFlops by the end\nof the project in 2007.\n\n\n\n\\section{UNICORE in Production}\n\\label{sec:production}\nFrom its birth in two German BMBF-funded projects to its extensive use and further development in a variety of EU\nand BMBF research projects, the UNICORE technology ran through an evolutionary process transforming from an\ninitial prototype software to a powerful production Grid middleware.\n\n\\subsection{UNICORE@SourceForge}\n\\label{sec:production-SF}\nSince May 2004, the UNICORE technology with all its components is available as open source software under the BSD\nlicense. It can be downloaded from the SourceForge repository. Besides the core developers of UNICORE (namely\nFujitsu Laboratories of Europe, Intel Germany and the Research Center J\\\"{u}lich), there are numerous contributors\nfrom all over the world, \\eg Norway, Poland, China and Russia. The Web site \\cite{UNICORE-sourceforge:url} offers\na convenient entry point for interested users and developers. In the download section the UNICORE software is\nbundled in different packages, \\eg the client package and individual packages for the different server components\nGateway, NJS, TSI\/IDB, UUDB (\\cf \\secref{sec:arch}), and plug-ins. Until January 2005 more than 2800 downloads of\nUNICORE are counted.\n\nA tracker section linked on the Web site establishes a communication link to the core developer community. The\ncorresponding mailing lists allow users to report bugs, to request new features, and to get informed about bug\nfixes or patches. For the announcement of new software releases a separate mailing list was created. The Grid team\nat the Research Center J\\\"{u}lich is responsible for UNICORE@SourceForge. Its work includes coordinating and driving\nthe development effort, and producing consolidated, stable, and tested releases of the UNICORE software.\n\n\\subsection{Production System on Jump}\n\\label{sec:production-Jump}\nSince July 2004 UNICORE is established as production software to access the supercomputer resources of the John\nvon Neumann-Institute for Computing (NIC) at the Research Center J\\\"{u}lich. These are the 1312-processor IBM p690\ncluster (Jump) \\cite{Jump:url}, the Cray SV1 vector machine, and a new Cray XD1 cluster system. As an alternative\nto the standard SSH login, UNICORE provides an intuitive and easy way for submitting batch jobs to the systems.\nThe academic and industrial users come from all over Germany and from parts of Europe. The applications come from\na broad field of domains, \\eg astrophysics, quantumphysics, medicine, biology, chemistry, and climate research,\njust to name the largest user communities. A dedicated, pre-configured UNICORE client with all required\ncertificates and accessible Vsites is available for download. This alleviates the installation and configuration\nprocess significantly. Furthermore, an online installation guide including a certificate assistant, an user\nmanual, and example jobs help users getting started.\n\nTo provide the NIC-users with adequate certificates and to ease the process of requesting and receiving a\ncertificate, a certificate authority (CA) was established. User certificate requests are generated in the client\nand have to be send to the CA. Since introduction of UNICORE at NIC, more than 120 active users requested a\nUNICORE user certificate.\n\nA mailing list serves as a direct link of the users to UNICORE developers in the Research Center J\\\"{u}lich. The list\nallows to post problems, bug reports, and feature requests. This input is helpful in enhancing UNICORE with new\nfeatures and services, in solving problems, identifying and correcting bugs, and influences new releases\nof UNICORE available at SourceForge.\n\n\\subsection{DEISA -- Distributed European Infrastructure for Scientific Applications}\n\\label{sec:production-DEISA}\nTraditionally, the provision of high performance computing resources to researchers has traditionally been the\nobjective and mission of national HPC centers.On the one hand, there is an increasing global competition between\nEurope, USA, and Japan with growing demands for compute resources at the highest performance level, and on the\nother hand stagnant or even shrinking budgets. To stay competitive major investments are needed every two years --\nan innovation cycle that even the most prosperous countries have difficulties to fund.\n\nTo advance science in Europe, eight leading European HPC centers devised an innovative strategy to build a\nDistributed European Infrastructure for Scientific Applications (DEISA) \\cite{DEISA:url}. The centers join in\nbuilding and operating a tera-scale supercomputing facility. This becomes possible through deep integration of\nexisting national high-end platforms, tightly coupled by a dedicated network and supported by innovative system\nand grid software. The resulting virtual distributed supercomputer has the capability for natural growth in all\ndimensions without singular procurements at the European level. Advances in network technology and the resulting\nincrease in bandwidth and lower latency virtually shrink the distance between the nodes in the distributed\nsuper-cluster. Furthermore, DEISA can expand horizontally by adding new systems, new architectures, and new\npartners thus increasing the capabilities and attractiveness of the infrastructure in a non-disruptive way.\n\nBy using the UNICORE technology, the four core partners of the projects have coupled their systems using virtually\ndedicated 1 Gbit\/s connections. The DEISA super-cluster currently consists of over 4000 IBM Power 4 processors and\n416 SGI processors with an aggregated peak performance of about 22 teraflops. UNICORE provides the seamless,\nsecure and intuitive access to the super-cluster.\n\nThe Research Center J\\\"{u}lich is one of the DEISA core partners and is responsible for introducing UNICORE as Grid\nmiddleware at all partner sites and for providing support to local UNICORE administrators.\n\nAll DEISA partners have installed the UNICORE server components Gateway, NJS, TSI, and UUDB to access the local\nsupercomputer resources of each site via UNICORE. \\figref{fig:DEISA-architecture} shows the DEISA UNICORE\nconfiguration. For clarity only four sites are shown. At each site, a Gateway exists as an access to the DEISA\ninfrastructure. The NJSs are not only registered to their local Gateway, but to all other Gateways at the partner\nsites as well. Local security measures like firewall configurations need to consider this, by permitting access to\nall DEISA users and NJSs. This fully connected architecture has several advantages. If one Gateway has a high\nload, access to the high performance supercomputers through DEISA is not limited. Due to the fully connected\narchitecture, no single point of failure exists and the flexibility is increased.\n\\begin{figure}[htb]\n\\centering\n\\includegraphics[width=0.95\\textwidth]{DEISA-architecture.eps}\n\\caption{The DEISA architecture.} \\label{fig:DEISA-architecture}\n\\end{figure}\n\nThe DEISA partners operate different supercomputer architectures, which are all accessible through UNICORE.\nInitially all partners with IBM p690 clusters are connected to one large virtual supercomputer. In a second step\nother supercomputers of different variety are connected to DEISA, making the virtual supercomputer heterogeneous.\nUNICORE can handle this, as it is designed to serve such heterogeneous architectures in a seamless, secure, and\nintuitive way.\n\nIn December 2004 a first successful UNICORE demonstration between the four DEISA core sites FZJ (Research Center\nJ\\\"{u}lich, Germany), RZG (Computing Center Garching, Germany), CINECA (Italian Interuniversity Consortium, Italy) and\nIDRIS (Institute for Development and Resources in Intensive Scientific Computing, France) was given. Different\nparts of a distributed astrophysical application were generated and submitted with UNICORE to all four sites.\n\nThe experience and knowledge of the researchers, developers, users, and administrators in working with UNICORE in\nthe DEISA project on a large production platform will be used as useful input for future developments of the\nUNICORE technology. A close synchronization with the UniGrids project (\\cf \\secref{sec:future-unigrids}) is\nforeseen.\n\n\n\n\\section{Future of UNICORE}\n\\label{sec:future}\nThe current UNICORE software implements a vertically integrated Grid architecture providing seamless access to\nvarious resources. Every resource is statically integrated into the UNICORE Grid by providing an interface to the\nappropriate resource manager.\n\nOne of the benefits Web services will bring to Grid computing is the concept of loosely coupled distributed\nservices. Merging the idea of ``everything being a service'' with the achievements of the Grid community led to\nGrid services, enabling a new approach to the design of Grid architectures. The adoption of XML and the drive for\nstandardization of the Open Grid Service Architecture provide the tools to move closer to the promise of\ninteroperable Grids. A demonstrator validated the correspondence of UNICORE's architectural model with the\nOGSA\/OGSI (Open Grid Service Infrastructure \\cite{Tuecke:2003:OGSI}) approach, which encouraged the development of\nan OGSA\/OGSI compliant UNICORE Grid architecture in the GRIP project (\\cf \\secref{sec:projects-grip}).\n\nIn \\cite{Menday:2003:GEU} UNICORE is examined for the evolution of a Grid system towards a service oriented Grid,\nprimarily focussing on architectural concepts and models. Based on the current architecture and the enhancements\nprovided by GRIP, first steps already integrate Web services into UNICORE. This included the provision of OGSI\ncompliant port types parallel to the proprietary ones as well as the design of XML based protocols. This work was\ncontinued in the UniGrids project.\n\nAs mentioned above the development of a Grid middleware is an continuous process of integrating new features,\nservices, and adapting to emerging standards, and UNICORE is no exception. In the following we present new\ndevelopments, some technical details, and report on projects, which enhance the UNICORE technology to serve the\ndemands of the Grid in the future \\cite{Jeffrey:2004:NGG}.\n\n\n\\subsection{UniGrids -- Uniform Interface to Grid Services}\n\\label{sec:future-unigrids}\nThe strength of the UNICORE architecture is well-proven as described above. The rapid definition and adoption of\nOGSA allow the UNICORE development community to re-cast and extend the concepts of UNICORE through the use of Web\nservices technologies. The goal of the UniGrids\\footnote{funded in part by EC grant IST-2002-004279, duration:\nJuly 2004 - June 2006} project \\cite{UniGrids:url} is to lift UNICORE on an architecture of loosely-coupled\ncomponents while keeping its 'end-to-end' nature.\n\nThus, the integration of Web services techniques and UNICORE, which already started in the GRIP project (\\cf\n\\secref{sec:projects-grip}), will continue in the \\mbox{UniGrids} project. Interoperability, through adopting and\ninfluencing standards, form the philosophical foundation for UniGrids. The project aims to transform UNICORE into\na system with interfaces that are compliant with the Web Services Resource Framework (WS-RF) \\cite{WSRF:url} and\nthat interoperate with other WS-RF compliant software components.\n\nSuch an approach offers great advantages both for the ease of development of new components by aggregation of\nservices and through the integration of non-UNICORE components into the standards-based infrastructure.\n\nIn this sense, work is continuing in the following areas: \\bi\n\\item Development of a compliant WS-RF hosting environment used for publishing UNICORE job and file services as\nWeb services.\n\\item Support of dynamic virtual organizations by enhancing the UNICORE security infrastructure to allow different\nusage models such as delegation and collective authorization.\n\\item Development of translation mechanisms, such as resource ontologies, to interoperate with other OGSA compliant\nsystems. Support for Grid economics by developing a Service Level Agreement (SLA) framework and cross-Grid\nbrokering services.\n\\item Development and integration of generic software components for visualization and steering of simulations (VISIT\n\\cite{visit:url}), device monitoring and control, and tools for accessing distributed data and databases.\n\\ei\n\nApplications from the scientific and industrial domain, like biomolecular and computational biology, geophysical\ndepth imaging by oil companies, automotive, risk-management, energy, and aerospace are used to prove the\ndevelopments in UniGrids.\n\nThe development in the UniGrids project will lead to UNICORE\/GS, which follows the architecture of OGSA through\nthe standardization of WS-RF and related work like \\eg the Web Services Notification technology \\cite{WSN:url}.\nThe results will be made available under an open source BSD license.\n\n\\subsubsection{UNICORE\/GS}\n\\label{sec:future-unigrids-GS}\nWeb service technology, and in particular the WS-RF, forms the basis for the UNICORE\/GS software. WS-RF is the\nfollow-on to OGSI, but more in line with mainstream Web services architecture \\cite{WSARCH:url}. Based on this new\ntechnology, UNICORE\/GS will retain its key characteristics of seamlessness, security, and intuitiveness from both\nthe user and administrative perspective, but will be built on a service oriented framework. This means that there\nis a loosening of the coupling between the components of the system. UNICORE\/GS keeps the classical UNICORE\ntopology of Usites, each containing a number of Vsites, but provides a new framework for integrating other\nservices and providing common infrastructure functionality as services. This has the implication that new services\nwill be easily integrated into the UNICORE\/GS environment. Conversely, UNICORE\/GS will be well-prepared to make\nuse of external services.\n\nThe WS-RF technology is used to model core functionalities such as job submissions and file transfers as\nWS--Resources. These services are accessible via web service interfaces and thus establishing the UniGrids atomic\nservices layer. This layer will be realized making extensive use of existing UNICORE server components.\n\nAll services in a Usite are accessible through the UniGrids Gateway that provides a secure entrance into the\nUNICORE\/GS infrastructure. The principal is exactly the same as for classic UNICORE, however, the Gateway now\nroutes messages according to Web Services Addressing (WS--Addressing) \\cite{WSA:url}. Authentication is based on\ntransport level HTTPS security, although the intention is to move to Web Services Security (WS--Security)\n\\cite{WSS:url}. Regarding authorized access to resources, the UNICORE User Database (UUDB) will be available as a\nservice to other services in the Usite, and will form the basis for future work concerning virtual organizations\nand fine-grained authorization schemes.\n\nThe underlying UniGrids atomic services layer will provide an excellent framework to deploy higher-level services\nsuch as co-allocation schedulers, workflow engines, and services for provision and easy access to data-intensive,\nremotely-steerable simulations.\n\n\\subsection{NextGrid -- Architecture for Next Generation Grids}\n\\label{sec:future-nextgrids}\nIn comparison to the UniGrids project which evolves the existing UNICORE Grid system to a service-oriented one,\nthe NextGRID\\footnote{funded in part by EC grant IST-2002-511563, duration: September 2004 - August 2007}\n\\cite{nextgrid:url} project aims for the future: The goal is to provide the foundations for the next generation of\nGrids. NextGRID is not a project based on the UNICORE architecture or Grid system as-is, but institutions and\npeople involved in the UNICORE development from the beginning on contribute expertise and experience to NextGRID.\n\nSince it is obvious that there is no such thing as the one and only next generation Grid, and experts envisage the\nco-existence of multiple Grids with well-defined boundaries and access points, NextGRID is going to define a Grid\narchitecture which can be seen as building blocks for Grids. It does not only provide interoperability by-design\nbetween entities which exist within one instantiation of such an architecture, but it also facilitates the\ninteroperability between different Grids developed according to the NextGRID architecture.\n\nAlthough developing a Grid one generation ahead, NextGRID is not starting from scratch. Properties to incarnate\nand functions to realize future Grids are expertly described in \\cite{Priol:2003:NGG} and \\cite{Jeffrey:2004:NGG}.\nThese reports frame NextGRID's architectural development while the Open Grid Services Architecture is going to\ndefine Grid services and their interactions and does therefore make up a staring point for the conceptualization\nand design of NextGRID. In addition, regarding the underlying technology and architectural model, NextGRID\npropagates the usage of Web Services and the adoption of Service-Oriented Achitecture (SOA) \\cite{Erl:2004:SOA}\nconcepts and models.\n\nNextGRID focuses on security, economic sustainability, privacy\/legacy, scalability and usability. The following\nproperties have the highest priorities when carrying out the following work: \\bi\n\\item Developing an architecture for next generation Grids\n\\item Implementing and testing prototypes aligned with the concepts and design of the NextGRID architecture\n\\item Creating reference applications which make use of the NextGRID prototypes\n\\item Facilitating the transition from scientific- to business-oriented Grids by integrating the means to negotiate\na certain Quality of Service (QoS) level\n\\item Specifying the methods, processes, and services necessary to dynamically operate Grids across multiple\norganizations which comprise heterogeneous resources\n\\ei\n\nSince the ongoing UNICORE development in projects like UniGrids shares resources as well as the technological\nfoundation with NextGRID there is a high chance that the outcome of NextGRID will also represent the next step of\nUNICORE's evolution.\n\n\n\n\\section{Conclusion}\n\\label{sec:conclusion}\nIn this paper we presented the evolution of the UNICORE technology from a Grid software with prototype character\ndeveloped in two German projects to a full-grown, well-tested, widely used and accepted Grid middleware. UNICORE\n-- Uniform Interface to Computing Resources -- provides a {\\em seamless, secure and intuitive} access to\ndistributed Grid resources. Although the UNICORE vision was already coined in 1997, the then stated goals and\nobjectives of hiding the seams of resource usage, incorporating a strong security model, and providing an easy to\nuse graphical user interface for scientists and engineers are still valid today: to achieve these goals and\nobjectives, UNICORE is designed as a vertically integrated Grid middleware providing components at all layers of a\nGrid infrastructure, from a graphical user interface down to the interfaces to target machines.\n\nInitially developed in the German projects UNICORE and UNICORE Plus, UNICORE was soon established as a promising\nGrid middleware in several European projects. In the GRIP project an interoperability layer between UNICORE and\nthe Globus Toolkit 2 and 3 was developed to demonstrate the interoperability of independently developed Grid\nsolutions, allowing to build and to demonstrate inter-Grid applications from the bio-molecular and meteorological\ndomain. In the EUROGRID project, European high performance supercomputing centers joined to extend UNICORE with an\nefficient data transfer, resource brokerage mechanisms, ASP services, application coupling methods, and an\ninteractive access. In addition, a Bio-Grid, Meteo-Grid, CAE-Grid, and HPC-Grid were established to integrate a\nvariety of application domains. The main objective of the OpenMolGRID project is to provide a unified and\nextensible information-rich environment based on UNICORE for solving problems from molecular science and\nengineering. In the VIOLA project a vertically integrated testbed with the latest optical network technology is\nbuilt up. UNICORE is used as the Grid middleware for enabling the development and testing of new applications in\nthe optical networked testbed, which provides advanced bandwidth management and QoS features.\n\nWith these developments UNICORE grew to a software system usable in production Grids. In this context UNICORE is\ndeployed in the large German supercomputing centers to provide access to their resources. At the John von\nNeumann-Institute for Computing, Research Center J\\\"{u}lich, many users submit their batch jobs through UNICORE to the\n1312-processor 8.9 TFlop\/s IBM p690 cluster and the Cray SV1 vector machine. Leading European HPC centers joined\nin the project DEISA to build a distributed European infrastructure for scientific applications based on UNICORE\nto build and operate a distributed multi tera-scale supercomputing facility.\n\nThe future of UNICORE is promising and follows the trend of ``Everything being a Service\" by adapting to Open Grid\nService Architecture (OGSA) standards. In this context, the UniGrids project continues the effort of the GRIP\nproject in integrating the Web Services and UNICORE technology to enhance UNICORE to an architecture of\nloosely-coupled components while keeping its ``end-to-end\" nature. To this end UNICORE\/GS will be developed, which\nmakes UNICORE compliant with the Web Services Resource Framework (WS-RF).\n\nToday the UNICORE software is available as open source under a BSD licence from SourceForge for download. This\nenables the community of core UNICORE developers to grow and makes future development efforts open to the public.\n\n\n\n\\section{Acknowledgments}\n\\label{sec:ack}\nThe work summarized in this paper was done by many people. We gratefully thank them for their past, present, and\nfuture contributions in developing the UNICORE technology. Most of the work described here was supported and\nfunded by BMBF and different programmes of the European Commission under the respective contract numbers mentioned\nabove.\n\n\n\\newcommand{\\noopsort}[1]{}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nLet $X$ be a scheme over an algebraically closed field $k$ of\ncharacteristic $p$, with $p\\, >\\, 0$. Fix a $k$-point $x$ of $X$. Nori\nintroduced the notion of fundamental group scheme $\\pi(X,x)$ in \\cite{N1}\nand further developed it in \\cite{N2}. Since then the\nfundamental group scheme is being\nstudied and in the process has turned into\nan important tool in algebraic geometry of positive\ncharacteristic. In \\cite{N2} Nori proves that\n$\\pi(X,x)$ is trivial for proper rational normal varieties. More generally,\n$\\pi(X,x)\\,=\\, 0$ if $X$ is separably rationally connected \\cite{Bi}.\nZhu proves that a general Fano (proper, smooth, connected with\nample anticanonical bundle) hypersurfaces in projective spaces are\nseparably rationally connected \\cite{Zh}. Therefore, the fundamental\ngroup schemes of general Fano hypersurfaces in a projective space\nare trivial.\n\nIn \\cite{CL} Chambert-Loir proves that every proper rationally\nchain connected normal variety has finite \\'etale\nfundamental group, and its order is coprime to $p$\n(the characteristic of $k$) \\cite{CL2}. This result can also be obtained as a\nconsequence of \\cite[Theorem 1.5]{Ko} and \\cite[Theorem 1.6]{Ko}. Shioda gave an example of a rationally\nconnected variety over a field of characteristic $p\\,\\neq\\, 5$ whose \\'etale\nfundamental group is $\\mathbb{Z}\/5\\mathbb{Z}$ \\cite{Sh}. Examples of rationally chain connected varieties whose local \nfundamental group scheme is not trivial are also known.\nFor example, a supersingular Enriques surface $E$ over an algebraically \nclosed field of characteristic $2$ is unirational (see\n\\cite[Corollary 1.3.1]{CD}), hence it is rationally \nchain connected. It is known that there exists a nontrivial\n$\\alpha_2$--torsor over $E$ (see \\cite[Chapter I, \\S 3]{CD}). \n\nWe prove the following (see Theorem \\ref{teoMAIN} and Remark \\ref{remFANO}):\n\n\\begin{theorem}\nLet $k$ be an algebraically closed field and $X$\na proper normal, rationally chain connected $k$--scheme. Let $x\\,\\in\\, X(k)$ be a point.\nThen the fundamental group scheme $\\pi(X,x)$ is finite.\n\\end{theorem}\n\nThe strategy of the proof is similar to that in \\cite{CL}, adapted to the \nnew setting.\n\n\\section{Preliminaries}\\label{sez:PREM}\n\nWe will write $\\pi(X)$ instead of $\\pi(X,x)$ to simplify the notation. \nHowever all the schemes for which we will compute the fundamental group \nscheme are meant to be pointed and all the morphisms between them take the \nmarked point in the domain space to the marked point in the target space. \nThe same convention will be applied to torsors: we assume \nthey are pointed and morphisms between them take the marked point in the \ndomain space to the marked point in the target space.\n\nLet $k$ be an algebraically closed field of any characteristic.\nA proper variety $X$ over $k$ is said to be rationally chain connected if for every algebraically\nclosed field $\\Omega$ containing $k$, for any two points in $X(\\Omega)$ there is a\nproper and connected curve passing through them such that its normalization is a\ndisjoint union of projective lines. If this union consists of only one projective line\nwe say that $X$ is rationally connected.\n\nLet $X$ be a rationally chain connected variety over $k$.\nWe recall \\cite[Lemma 1]{CL} and we sketch its proof:\n\n\\begin{lemma}\\label{lemCL1}\nLet $k\\,\\subseteq\\, \\Omega$ be a field extension\nwhere $\\Omega$ is algebraically closed. Let $$F_{\\Omega}\\,:\\, \\mathbb{P}_{\\Omega}^1\n\\,\\longrightarrow\\, X_{\\Omega}$$ be a rational curve of $X_{\\Omega}$. Let\n$x_0\\,:=\\,F_{\\Omega}(0)$ and $x_{\\infty}\\,:=\\,F_{\\Omega}(\\infty)$ be points of\n$X_{\\Omega}$ then let $V_0$ and $V_{\\infty}$ be their Zariski closure in $X$.\nThen there exist a normal integral $k$-scheme $T$, a morphism\n$$F\\,:\\, \\mathbb{P}_{T}^1\\,\\longrightarrow\\, X$$ such that the morphisms defined as\n$$F_0(t)\\,:=\\,F(0,t)\\,:\\,T\\,\\longrightarrow\\, X\\ ~ \\text{ and }~\\ F_{\\infty}(t)\\,:=\\,\nF(\\infty,t)\\,:\\,T\\,\\longrightarrow\\, X$$\nare dominant over $V_0$ and $V_{\\infty}$ respectively.\n\\end{lemma}\n\n\\begin{proof}\nThere exists a finitely generated $k$-algebra $k\\,\\subseteq\\, A$ contained\nin $\\Omega$, and there is a morphism $$F_{A}\\,:\\,\n\\mathbb{P}_{A}^1\\,\\longrightarrow\\, X_{A}$$ such that $F_{\\Omega}\\,=\\,F_A\n\\otimes_A\n\\Omega$. We set $T\\,:=\\,{\\rm Spec}(A)$ that we assume to be normal (otherwise we replace it with a finite extension). We now consider the morphism\n$F\\,:\\, \\mathbb{P}_{T}^1\\,\\longrightarrow\\, X$ obtained from $F_{A}$ after composing with the projection $X_A\\,\\longrightarrow\\, X$. In \\cite[Lemma 1]{CL} it has been proved that $F_0$ and $F_{\\infty}$ are\ndominant over $V_0$ and $V_{\\infty}$; we briefly recall this last part for the\nconvenience of the reader: We study $F_{0}$ (it will\nbe the same for $F_{\\infty}$). The\nimage by $F_{0}$ of the generic point of ${\\rm Spec}(A)$ is the generic\npoint of $V_0$. Since $V_0$ is closed in $X$, the inverse image\n$F_{0}^{-1}(V_0)$ is closed in ${\\rm Spec}(A)$ and dense. Thus\n$F_{0}^{-1}(V_0)$ coincides with\n${\\rm Spec}(A)$, and hence the image of $F_{0}$ is contained in $V_0$ and contains\nits generic point. Therefore it contains an open dense subset of $V_0$. \n\\end{proof}\n\n\\section{The main theorem}\n\nThe following lemma is well-known. We include\na short proof of it for the convenience of the reader.\n\n\\begin{lemma}\\label{lemGROUPS}\nLet $G$ be a finite $k$-group scheme and let $G^{\\text{\\'et}}$ and $G^{\\rm loc}$ be\nrespectively the maximal \\'etale quotient and the maximal connected quotient. Then the\nnatural morphism $$\\alpha\\,:\\,G\\,\\longrightarrow\\, G^{\\text{\\'et}}\\times G^{\\rm loc}$$\nis faithfully flat. \n\\end{lemma}\n\n\\begin{proof}\nThe field being perfect the reduced subscheme $G_{\\text{red}}$ is a subgroup\nscheme of $$N^{\\text{loc}}\\,:=\\,\\text{kernel} (G\\to G^{\\text{loc}})\\, ,$$ while the\nconnected component $G^0$ of $G$ is $\\text{kernel}(G\\to \nG^{\\text{\\'et}})$. If $\\alpha$ is not faithfully flat we can factor it as in\nthe following diagram:\n$$\\xymatrix{ & G\\ar@{->>}[ldd]\\ar@{->>}[d]^{q}\\ar@{->>}[rdd] & \\\\ &\nG'\\ar@{->>}[ld]\\ar@{^{(}->}[d]^{j}\\ar@{->>}[rd] & \\\\G^{\\text{\\'et}} &\nG^{\\text{\\'et}}\\times G^{\\text{loc}}\\ar@{->>}[l]\\ar@{->>}[r] & G^{\\text{loc}} }\n$$\nwhere $q\\,:\\,G\\,\\twoheadrightarrow\\, G'$ is faithfully flat and $j\\,:\\,G'\\,\n\\hookrightarrow\\, G^{\\text{\\'et}}\\times G^{\\text{loc}}$ is a closed immersion. Clearly\n$G^{\\text{\\'et}}$ and $G^{\\text{loc}}$ are still the maximal \\'etale\nquotient and the maximal connected quotient of $G'$ respectively. So we can\nassume $\\alpha$ is a closed immersion. Therefore, the lemma is equivalent to the\nassertion that $\\alpha$ is an isomorphism.\n\n{}From \\cite[\\S~6, Ex. 9]{WW} it follows that $G^{\\text{\\'et}}$\nis isomorphic to a subgroup-scheme of $G$ which we identify with\n$G^{\\text{\\'et}}$, so in particular $G^{\\text{\\'et}}\\,\\leq\\, \nG_{\\text{red}}\\,\\leq\\, N^{\\text{loc}}$. Therefore, we have \n$$\\vert G^{\\text{\\'et}}\\vert\\vert G^{\\text{loc}}\\vert\\,=\\,\n\\vert G^{\\text{\\'et}}\\vert\\frac{\\vert G\\vert }{\\vert N^{\\text{loc}}\\vert\n}\\,\\leq\\,\\vert G^{\\text{\\'et}}\\vert\\frac{\\vert G\\vert }{\\vert G^{\\text{\\'et}}\n\\vert } \\,=\\, \\vert G\\vert $$\nwhich implies that $\\alpha$ is an isomorphism.\n\\end{proof}\n\nWe recall that when $X$ is a reduced and connected scheme over a field $k$\nthen the fundamental group scheme can be defined. In this case a finite $G$-torsor $Y\\,\\longrightarrow\\, X$ is called Nori-reduced if the canonical morphism $\\pi(X)\n\\,\\longrightarrow\\, G$ is faithfully flat. \n\n\\begin{lemma}\\label{lemTORS}\nLet $X$ be a connected and reduced scheme over $k$. Let $G$ (respectively,\n$H$) be a finite local (respectively, finite \\'etale) $k$--group\nscheme. Let $$Y\\,\\longrightarrow\\, X$$ and $T\\,\\longrightarrow\\, X$ be a $G$--torsor and\nan $H$--torsor respectively. We assume that both $Y$ and $T$ are Nori-reduced. Then\nthe $H\\times G$--torsor $$T\\times_X Y\\,\\longrightarrow\\, X$$ is also Nori-reduced. \n\\end{lemma}\n\n\\begin{proof}\nIf $\\pi(X)\\,\\longrightarrow\\, H\\times G$ is not faithfully flat then there\nexists a triple $(M\\, , Z\\, ,\\iota)$, where\n\\begin{itemize}\n\\item $M\\,\\hookrightarrow\\, H\\times G$ is a subgroup-scheme,\n\n\\item $Z\\,\\longrightarrow\\, X$ is a $M$--torsor, and\n\n\\item $\\iota\\, :\\, Z\\,\\hookrightarrow\\, T\\times_X Y$ is a reduction of structure\ngroup-scheme, to $M$, of the $H\\times G$--torsor $T\\times_X Y$.\n\\end{itemize}\nLet\n$$T'\\,\\longrightarrow\\, X~ \\ \\text{ and }~\\ Y'\\,\\longrightarrow\\, X$$ be the\n$M^{\\text{\\'et}}$ and $M^{\\text{loc}}$--torsors respectively,\nobtained from the $M$--torsor $Z\\,\\longrightarrow\\, X$ using the projections\nof $M$ to $M^{\\text{\\'et}}$ and $M^{\\text{loc}}$ respectively (the notation\nis as in Lemma \\ref{lemGROUPS}).\nWe have a closed immersion $$Z\\,\\hookrightarrow\\, T'\\times_X Y'$$ induced by\nthe closed immersion $M\\,\\hookrightarrow \\,M^{\\text{\\'et}}\\times M^{\\text{loc}}$.\nThe latter is an isomorphism by Lemma \\ref{lemGROUPS}, so the same is true for\n$Z\\,\\hookrightarrow\\, T'\\times_X Y'$.\n\nThe projection $M\\,\\twoheadrightarrow\\, H$ (respectively, $M\\,\\twoheadrightarrow\n\\, G$) clearly factors through $M^{\\text{\\'et}}$ (respectively, $M^{\\text{loc}}$).\nNote that the projections $$M\\,\\longrightarrow\\, H \\ \\ \n\\text{ and } \\ \\ M\\,\\longrightarrow \\,G$$\nare faithfully flat morphisms because the two torsors $Y$ and $T$ are\nNori-reduced. Consequently, the two homomorphisms $M^{\\text{\\'et}}\\,\\longrightarrow\\,\nH$ and $M^{\\rm loc}\\,\\longrightarrow\\, G$ are isomorphisms. Now using\nLemma \\ref{lemGROUPS} it follows that the inclusion $M\\,\\hookrightarrow\\, H\\times G$\nis an isomorphism. Consequently, the $H\\times G$--torsor $T\\times_X Y\\,\\longrightarrow\\,\nX$ is Nori-reduced.\n\\end{proof}\n\nThe following result was proved in \\cite[Proposition 3.6]{EPS} under the assumption\nthat $X$ is proper.\n\n\\begin{corollary}\\label{corTORS}\nLet $X$ be a connected and reduced scheme over $k$. Let $G$ (respectively, $H$)\nbe a finite local (respectively, finite \\'etale) $k$-group\nscheme. Let $Y\\,\\longrightarrow\\, X$ be a $G$--torsor and\n$T\\,\\longrightarrow\\, X$ an $H$--torsor. We assume that both the torsors are\nNori-reduced. Then the $G$--torsor\n$$T\\times_X Y \\,\\longrightarrow\\,T$$ is also Nori-reduced. In particular, the morphism\n$\\pi^{\\rm loc}(T)\\,\\longrightarrow\\, \\pi^{\\rm loc}(X)$ is faithfully flat.\n\\end{corollary}\n\n\\begin{proof}\nLet us assume that there is a finite local $k$-group\nscheme $G_1\\,\\subset\\,G$, and $G_1$--torsor $U\\,\\longrightarrow\\, T$ and a\nreduction\n$$i\\,:\\,U\\,\\hookrightarrow\\, T\\times_X Y$$ of structure group to $G_1$.\nLet $S$ be any $k$-scheme. For any $x\\,\\in\\, X(S)$ we choose $u_x\\,\\in\\, U(S)$ whose\nimage in $X(S)$ is $x$. We set $(t_x\\, ,y_x)\\,:=\\,i(u_x)$, then $$T(S)\\times_{X(S)}Y(S)\n\\,=\\,\\{(ht_x,gy_x)\\, ,~\\ \\forall ~x\\,\\in\\, X(S)\\, ,~\\ \\forall ~ g\\,\\in\\, G(S)\\, ,\n~\\ \\forall ~ h\\,\\in\\, H(S) \\}$$ so the image\nof $U(S)$ by $i_S$ can be identified with the subset \n$$\\{(ht_x\\, ,gy_x)\\, ,~\\ \\forall ~x\\,\\in\\, X\\, ,~\\ \\forall ~g\\,\\in\\,\nG_1\\, ,~\\ \\forall ~h\\,\\in\\, H \\}\\, ;$$ this gives $U$ the structure of an $H\\times\nG_1$--torsor over $X$, contained in the $H\\times G$--torsor $T\\times_X Y$. This\nimplies that $G_1\\,=\\,G$ by Lemma \\ref{lemTORS}.\n\\end{proof}\n\n\\begin{corollary}\\label{corTORS2}\nLet $X$ be a connected reduced scheme over $k$ and $G$ a finite local $k$-group\nscheme. Let $T\\,\\longrightarrow\\, X$ be a finite \\'etale cover, and let $Y\n\\,\\longrightarrow\\, X$ be a $G$--torsor. If\n$Y\\,\\longrightarrow\\, X$ is Nori-reduced and $T$ is connected, then the $G$--torsor\n$$T\\times_X Y\\,\\longrightarrow\\, T$$ is also Nori-reduced. In particular the homomorphism\n$\\pi^{\\rm loc}(T)\\,\\longrightarrow\\, \\pi^{\\rm loc}(X)$ is faithfully flat.\n\\end{corollary}\n\n\\begin{proof} This follows from Corollary \\ref{corTORS} and the fact that there exist a\nfinite \\'etale $k$--group scheme $H'$ and an $H'$--torsor $T'\\,\\longrightarrow\\, X$\nthat dominates $T\\,\\longrightarrow\\, X$.\n\\end{proof}\n\n\\begin{remark}\\label{remNORI}\nIn \\cite{N2}, Nori proved that if\n$i\\,:\\,U\\,\\longrightarrow\\, Y$ is an open immersion between connected and\nreduced schemes with $Y$ normal, then the morphism $\\pi(U)\\,\\longrightarrow\\,\n\\pi(Y)$ induced by $i$ is faithfully flat (see \\S~II, Proposition 6 and\nits corollaries). Consequently, the homomorphism\n$\\pi^{\\rm loc}(U)\\,\\longrightarrow\\, \\pi^{\\rm loc}(Y)$ induced by $i$ is\nalso faithfully flat. \n\\end{remark}\n\n\\begin{notation}\\label{notFINDEX}\nLet $k$ be a field and $u\\,:\\,M\\,\\longrightarrow\\, G$ a $k$-group scheme\nhomomorphism. We say that $u$ is of finite index if the following\nproperty is satisfied: for any $k$-group scheme $Q$ and any faithfully\nflat morphism of $k$-group schemes $G\\,\\longrightarrow\\, Q$, if the group\nscheme image of $M\\to Q$ is finite then $Q$ is also finite.\n\\end{notation}\n\n\\begin{lemma}\\label{lemCLinsep}\nLet $f\\,:\\,X\\,\\longrightarrow\\, Y$ be a finite purely inseparable morphism between normal\nintegral schemes. Then the homomorphism $$\\pi(X)\\,\\longrightarrow\\,\n\\pi(Y)$$ is of finite index, while $\\pi^{\\rm \\acute{e}t}(X)\\,\\longrightarrow\\,\n\\pi^{\\rm \\acute{e}t}(Y)$ is faithfully flat. So in particular $\\pi^{\\rm loc}(X)\\,\\longrightarrow\\,\n\\pi^{\\rm loc}(Y)$ is of finite index. \n\\end{lemma}\n\n\\begin{proof}\nWe assume that $char(k)\\,=\\,p\\,>\\,0$. We observe that under the above assumptions the morphism\n$f$ is surjective (see \\cite[Ex. 5.3.9]{Li}). Let us first consider the case where $Y\\,=\\,X$ with $f\\,:=\\,F_Y$ being \nthe absolute Frobenius morphism of $Y$. Let $T \\,\\longrightarrow\\, Y$ be the universal \n$\\pi(Y)$--torsor of $Y$ (it is a scheme, as all the transition morphisms\nare affine), where $\\pi(Y)$ is the fundamental group scheme of Nori. We set\n$$T^{(p)}\\,:= \\, T\\times_Y Y$$ via the Frobenius $F_Y$ of $Y$. As\nusual, $F_{T\/Y}\\,:\\, T\\,\\longrightarrow\\, T^{(p)}$ is the \nrelative Frobenius. The relative Frobenius commutes with base change, so if we \npull back over $x\\,:\\,{\\rm Spec}(k) \\,\\longrightarrow\\, Y$ (a fixed closed point) what we obtain is the relative\nFrobenius\n$$ F_{\\pi(Y)\/{\\rm Spec}(k)}\\,:\\, \\pi(Y) \\,\\longrightarrow\\, \\pi(Y)^{(p)}\n\\,\\simeq\\, \\pi(Y)\\, ,$$\nwhere the last isomorphism clearly follows from the fact that $k$ is algebraically closed, whence perfect; thus,\nin particular, $$F_{T\/Y}\\,:\\, T \\,\\longrightarrow\\, T^{(p)}$$\nis the natural morphism from the universal torsor to\nthe pro-finite torsor obtained after pulling back. The same holds for any\ntorsor: so let $P$ be a $Q$-torsor, where $\\pi(Y)\\,\\longrightarrow\\, Q$ is a\nfaithfully flat $k$-group scheme homomorphism; then we have the\nrelative Frobenius\n$$ F_{Q\/{\\rm Spec}(k)}\\,:\\, Q \\,\\longrightarrow\\, Q^{(p)}\\,\\simeq\\, Q\\, ,$$ which\nfactors as $Q\\,\\longrightarrow\\, F\\,\\longrightarrow\\, Q$ (where $Q\\,\n\\longrightarrow\\, F$ is faithfully flat and $F\\,\n\\longrightarrow\\, Q$ is a closed immersion). Since the kernel is finite, if\nwe assume $F$ to be finite then $Q$ is also finite thus\n$F_{\\pi(Y)\/{\\rm Spec}(k)}$ is of finite index.\n\nNow $F_{\\pi(Y)\/{\\rm Spec}(k)}$ is a finite endomorphism and this is sufficient to conclude that it is of finite index. As \\'etale\ntorsors are not \nmodified by the Frobenius, it follows that $ F_{\\pi^{\\rm \\acute{e}t}(Y)\/{\\rm\nSpec}(k)}$ is an isomorphism. What has been \nproved for $f\\,=\\,F_Y$ still holds, of course,\nfor $f\\,=\\,F_Y^m$, the Frobenius iterated $m$ times. So now we consider the general case where $f\n\\,:\\,X\\,\\longrightarrow\\,Y$ is the given purely \ninseparable morphism. Then there exist a positive integer $m$ and a morphism $h\\,:\\, Y\\,\\longrightarrow\\, X$ such that \n$$f\\circ h \\,=\\,F_Y^m \\,:\\, Y \\,\\longrightarrow\\, Y$$\n(the absolute Frobenius morphism iterated $m$ times). We consider the pullback \n$$T_X \\,:=\\, T\\times_Y X$$ and the universal $\\pi(X)$--torsor $P\\,\n\\longrightarrow\\, X$ on $X$. There are natural morphisms\n$P\\,\\longrightarrow\\, T_X$ and $$u\\,:\\,\\pi(X)\\,\\longrightarrow\\, \\pi(Y)\\, .$$\nPulling back further to $h\\,:\\, Y\n\\,\\longrightarrow\\,X$, the following factorization is obtained:\n$$\\xymatrix{\\pi(Y) \\ar@\/_1pc\/[rr]_{F^m_{\\pi(Y)\/{\\rm Spec}(k)}} \\ar[r] & \\pi(X) \\ar[r]^u & \\pi(Y).}$$\n\nThe previous discussion yield the following:\n\n\\begin{itemize}\n \\item if we assume that $\\pi(Y)$ and $\\pi(X)$ are both \\'etale, this implies\nthat $$u\n\\,:\\, \\pi(X)\\,\\longrightarrow\\, \\pi(Y)$$ is faithfully flat;\n \\item otherwise we can only conclude that $u$ is of finite index, which is all\nthat we can expect.\n\\end{itemize}\nThis is enough to conclude the proof.\n\\end{proof}\n\n\\begin{lemma}\\label{lemCL2}\nLet $f\\,:\\,X\\,\\longrightarrow\\, Y$ be a dominant morphism between normal\nintegral schemes. Then the homomorphisms\n$$\\varphi^{\\rm loc}:\\pi^{\\rm loc}(X)\\,\\longrightarrow\\,\n\\pi^{\\rm loc}(Y)\\ \\ \\text{ and } \\ \\ \\varphi^{\\rm \\acute{e}t}:\\pi^{\\rm \\acute{e}t}(X)\\,\n\\longrightarrow\\,\n\\pi^{\\rm \\acute{e}t}(Y)$$ induced by $f$ are of finite index. \n\\end{lemma}\n\n\\begin{proof}\nThis is inspired by \\cite[Lemme 2]{CL} (see also \\cite[Lemme 4.4.17]{De}\nfor the zero characteristic case). \n\nLet $t$ be a closed point of the generic fiber of $f$, and let $T$ denote its Zariski\nclosure in $X$. The morphism $f$ induces a generically finite morphism $f_{\\vert T}\\,\n:\\,T \\longrightarrow\\, Y$: indeed its generic fiber has relative dimension\nzero and it this thus a finite number of points; therefore, there exists an\nopen dense subscheme $U\\,\\subseteq\\, Y$ such that\n$$f'\\,:\\,V\\,\\longrightarrow\\, U\\, ,$$\nwhere $V\\,:=\\,T\\times_Y U$, is a finite morphism. Hence there exist a scheme $W$ and two\nfinite morphisms\n$$e\\,:\\,V\\,\\longrightarrow\\, W\\ \\ \\text{ and }\\ \\ i\\,:\\,W\\,\\longrightarrow\\, V$$ such that $i\\circ e\\,=\\,f'$, where $i$ is purely inseparable\nand $e$ is generically \\'etale. This implies\nthat there exists an open dense subscheme $W'\\,\\subseteq\\, W$ such that\n$$e'\\,:\\,V'\\,\\longrightarrow\\, W'\\, ,$$ where $V':=V\\times_W W'$, is a finite \\'etale cover. In what follows \nwe study the morphism\n$$\\varphi^{\\rm loc}\\,:\\,\\pi^{\\rm loc}(X)\\,\\longrightarrow\\,\n\\pi^{\\rm loc}(Y)\\, ,$$ the (similar) details for \n$\\varphi^{\\rm \\acute{e}t}$ are left to the reader. By Corollary\n\\ref{corTORS2} the morphism $\\pi^{\\rm loc}(V')\\,\\longrightarrow\\,\n\\pi^{\\rm loc}(W')$ induced by $e'$ is faithfully flat while the morphism \n$$\\pi^{\\rm loc}(W)\\,\\longrightarrow\\, \\pi^{\\rm loc}(U)$$\ninduced by $i$ is of finite index by Lemma \\ref{lemCLinsep} and clearly \n$\\pi^{\\rm loc}(W')\\,\\longrightarrow\\, \\pi^{\\rm loc}(W)$ is faithfully flat (see Remark \\ref{remNORI})\nso the composition $$u:\\pi^{\\rm loc}(V')\\,\\longrightarrow\\, \\pi^{\\rm loc}(U)$$ is of finite index. Now\nwe have the diagram of homomorphisms\nof local fundamental group schemes \n$$\\xymatrix{\\pi^{\\rm{loc}}(V')\\ar[r]\\ar[d]_{u} & \\pi^{\\rm{loc}}(X)\\ar[d]^{\\varphi^{\\rm loc}}\n\\\\ \\pi^{\\rm{loc}}(U)\\ar[r]^{v} & \\pi^{\\rm{loc}}(Y). }$$\nNow $u$ is of finite index and the homomorphism $v$ is\nfaithfully flat (see, again, Remark \\ref{remNORI}). Hence $\\varphi^{\\rm loc}$ is finite index.\n\\end{proof}\n\nIn \\cite{MS} Mehta and\nSubramanian proved that $$\\pi(X\\times Y)\\,=\\, \\pi(X)\\times \\pi(Y)$$ for two connected, proper and reduced schemes \n$X$ and $Y$. If one of the two schemes ($X$ or $Y$) is not proper anymore then the previous formula may not hold. \nHowever a weaker result will be sufficient for our purposes; the following proposition can be found in \n\\cite{N2}, Chapter II, Proposition 9, here we suggest a different approach :\n\n\\begin{proposition}\\label{propNEWPROD}\nLet $Y$ be an integral scheme over $k$, then the homomorphism $$\\varphi\\,:\\,\n\\pi(\\mathbb{P}^1\\times Y)\\,\\longrightarrow\\, \\pi(Y)$$ induced by \nthe projection $p_2\\,:\\,\\mathbb{P}^1\\times Y\\,\\longrightarrow\\, Y$ is an\nisomorphism.\n\\end{proposition}\n\n\\begin{proof}\nIt is clear that \n$\\varphi$ is faithfully flat as $p_2$ has a section, so we only need to prove\nthat given a finite $k$--group scheme \n$G$ and a $G$--torsor $T\\,\\longrightarrow\\, \\mathbb{P}^1\\times Y$, there exists a\n$G$--torsor $$T'\\,\\longrightarrow\\,\nU$$ whose pullback to $\\mathbb{P}^1\\times Y$ is the given one. First we briefly recall that the fundamental group scheme of $Y$ at a $k$-point $y$ is the automorphism group scheme of the fiber functor $y^{\\ast}$ on the category of essentially finite vector\nbundles, as described in \\cite{N1}. Let $Rep_k(G)$ \ndenote the category of $k$--linear finite dimensional representations of $G$. Then \nassociated to our $G$--torsor $T\\,\\longrightarrow\\, \\mathbb{P}^1\\times Y$ there is a\nfiber functor \n$$F_T\\,:\\,Rep_k(G)\\,\\longrightarrow\\, \\mathcal{Q}coh(\\mathbb{P}^1\\times Y)$$ by a\nfundamental result in \nTannakian theory (recalled for instance in \\cite[Proposition (2.9)]{N1}). From this \nwe will construct a functor $$F\\,:\\,Rep_k(G)\\,\\longrightarrow\\,\n\\mathcal{Q}coh(Y)\\, .$$ For \nany $G$--module $V$, set $$F(V)\\,=\\,(p_2)_*(F_T(V))\\, .$$ We first observe that $F(V)$ is a \nvector bundle: when restricted to $\\mathbb{P}^1$, clearly $F_T(V)$ is an essentially finite \nvector bundle over the projective line, thus trivial, whence $H^1(\\mathbb{P}^1, \nF_T(V))\\,=\\,0$, and the evaluation homomorphism $$(p_2)^*(p_2)_* (F_T(V))\n\\,\\longrightarrow\\, (F_T(V))$$ is an isomorphism.\nMoreover $F$ is compatible with the operations of taking tensor \nproducts, direct sums and duals. Hence $F$ is a fiber functor and we can \nassociate to it a $G$--torsor $T'\\,\\longrightarrow\\, U$ which is the desired one\nsince its pullback to $\\mathbb{P}^1\\times Y$ is isomorphic to\n$T\\,\\longrightarrow\\, \\mathbb{P}^1\\times Y$.\n\\end{proof}\n\nWe now recall that for any $0\\, \\leq\\, \\,\\nu\\, \\leq\\, \\, \\dim(X)$, \nthere is a point in $X(\\Omega)$, where $\\Omega$ in the\nalgebraic closure of the function field of $X$, whose Zariski closure in $X$\nis of dimension $\\nu$.\n\n\\begin{theorem}\\label{teoMAIN}Let $k$ be an algebraically closed field and $X$\na normal, rationally chain connected $k$--scheme. Then $\\pi^{\\rm loc}(X)$ is finite.\n\\end{theorem}\n\n\\begin{proof}\nSince $X$ is rationally chain connected, there exists a chain of rational curves\nconnecting a rational point $x_0\\,\\in\\, X(k)$ to a generic point $x_m\\,\\in\\, X(\\Omega)$,\nwhere $\\Omega$ is the algebraic closure of the function field of $X$. According to Lemma\n\\ref{lemCL1} there exists a sequence of integral subvarieties $V_0\\, , \\cdots\\, , V_m$\nof $X$ where $V_0\\,=\\,x_0$ and $V_m\\,=\\,X$ and for every integer $i\\,\\in\\,\n\\{0, \\cdots , m-1\\}$ a family of rational curves\n$$\nF^i\\,:\\,\\mathbb{P}^1_k\\times T_i \\,\\longrightarrow\\, X\n$$\nwith $T_i$ normal and projective, such that the morphisms\n$$F^i_0\\,:\\,T_i\\,\\longrightarrow\n\\, X \\ \\ \\text{ and }\\ \\ F^i_{\\infty}\\,:\\, T_i\\,\\longrightarrow\\, X\\, ,$$\ndefined by $F^i_0(t)\\,:=\\,\nF^i(0,t)$ and $F^i_{\\infty}(t)\\,:=\\,F^i(\\infty,t)$, are dominant on $V_i$ and $V_{i+1}$\nrespectively. If $V_i$ is not normal then we can consider an open normal\nsubscheme $V_i'\\,\\subset\\, V_i$ and the pullback \n$$\\xymatrix{T_i'\\ar[r]\\ar[d] & V_i'\\ar[d] \\\\ T_i\\ar[r] & V_i. }$$\nIn a similar way, if $V_{i+1}$ is not normal then we can consider an open normal\nsubscheme $V_{i+1}'\\,\\subset\\, V_{i+1}$ and its pullback, as before, that we will call\n$T_i^{''}$. This will not affect $V_0$ and $V_m$ of course. So this induces the\nfollowing commutative diagram on local group schemes:\n$$\\xymatrix{ & & \\pi^{\\text{loc}}(T_i')\\ar[ld]_{\\alpha}\\ar[rd]^{u}\n & & \\\\ & \\pi^{\\text{loc}}(T_i)\\ar[ld]_{\\beta} & &\n\\pi^{\\text{loc}}(V_i')\\ar[dr]^{v} & \\\\ \\pi^{\\text{loc}}(\\mathbb{P}^1_k\\times\nT_i) \\ar[rrrr]^{\\pi(F^i)} & & & & \\pi^{\\text{loc}}(X) \\\\ &\n\\pi^{\\text{loc}}(T_i)\\ar[lu]^{\\gamma} & & \\pi^{\\text{loc}}(V_{i+1}')\\ar[ru]_{w}\n& \\\\ & & \\pi^{\\text{loc}}(T_i^{''})\\ar[lu]^{\\delta}\\ar[ru]_{z} & & }$$\nWe avoid to put the index $i$ on the morphisms not to make notation too\nheavy. We know that $\\pi^{\\text{loc}}(V_0)\\,=\\,0$, both $u$ and $z$ are of finite index by Lemma \\ref{lemCL2}, \nboth $\\alpha$ and $\\delta$ are faithfully flat by \nRemark \\ref{remNORI} and both $\\beta$ and $\\gamma$ are isomorphisms by Proposition \n\\ref{propNEWPROD}. So at each step we prove that the image of\n$\\pi^{\\text{loc}}(V_{i+1}')$ in $\\pi^{\\text{loc}}(X)$ is finite. The last step\nwill finally prove that $\\pi^{\\text{loc}}(X)$ is finite.\n\\end{proof}\n\n\\begin{lemma}\\label{lemLAST}Let $X$ be a rationally chain connected variety and let $f : Y \\longrightarrow X$ be an \\'etale Galois cover. Then $Y$ is rationally chain connected.\n\\end{lemma}\n\n\\begin{proof}\nFix a point $y$ of $Y$. Let $U_y$ be the subset of $Y$ that is rationally chain connected to the point $y$. Let $\\Omega$ be any algebraically closed field containing $k$, then any morphism  $g : \\mathbb{P}^1_{\\Omega}\\longrightarrow X_{\\Omega}$ lifts to a morphism\n$g' : \\mathbb{P}^1_{\\Omega}\\longrightarrow Y_{\\Omega}$ using the homotopy lifting property  because $\\mathbb{P}^1_{\\Omega}$ is simply connected. Since $X$ is also rationally chain connected, these two together imply that\n$U_y$ is both open and closed. Hence $U_y = Y$, and $Y$ is rationally chain connected.\n\\end{proof}\n\n\\begin{remark}\\label{remFANO}\nLet notations be as in Theorem \\ref{teoMAIN}, then from \\cite[Th\\'eor\\`eme]{CL} and Lemma \\ref{lemLAST} we also obtain that $\\pi(X)$ is finite. If $char(k)\\,=\\,p\\,>\\,0$, and $X$ is moreover smooth and proper, then $\\vert\\pi(X)^{\\rm \\acute{e}t}\\vert$ is coprime to $p$, as proved in \\cite{CL2}. In particular all this holds when $X$ is a Fano variety since in this case it is rationally chain connected (cf. \\cite{Ca} and \\cite{KMM}). Furthermore when $X$ is a general hypersurface of a projective space then then it is separably\nrationally connected, and by \\cite{Bi} this implies that $\\pi(X)\\,=\\,0$.\n\\end{remark}\n\n\\section*{Acknowledgments}\n\nWe thank Antoine Chambert-Loir for a useful communication. We thank the two\nreferees for comments that helped us in improving the paper. The first-named author would like to thank T.I.F.R. for \nits hospitality and Cinzia Casagrande for useful discussion. The second-named author \nthanks Universit\\'e Lille 1 and Niels Borne for hospitality. He also acknowledges the \nsupport of a J. C. Bose Fellowship.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nLife processes such as cellular signaling, control and regulation\narise from complex interactions and reactions between biomolecules.\nA fundamental challenge of understanding and controlling life processes\nis that they are inherently multiscale \u2013 cellular signaling alone\ninvolves $6$ orders of magnitude in lengthscales ($0.1$ nanometers\nto $100$ micrometers) and $18$ orders of magnitude in timescales\n(femtoseconds to hours). Unfortunately, these scales are tightly coupled\n\u2013 a single-point mutation in a protein can disturb the biochemical\ninteractions such that this results in disease or death of the organism.\nNo single experimental or simulation technique can probe all time-\nand lengthscales at a resolution required to understand such a process\ncomprehensively. \n\nIn computer simulations, this dilemma can be mitigated by multiscale\ntechniques \u2013 different parts of the system are described by a high-resolution\nand a low-resolution model, and these parts are coupled to give rise\nto a hybrid simula\\textcolor{black}{tion. A famous example of such\na multiscale model in biophysical chemistry is the coupling of quantum\nmechanics and molecular mechanics (QM\/MM) \\citep{WarshelLevitt_JMB76_QMMM}.\nHere we lay the foundations for a hybrid simulation technique that\ncouples two scales that are particularly useful to model intracellular\ndynamics: a Markov state model (MSM) of the molecular dynamics (MD)\nscale that describes structural changes of biomolecules and their\ncomplexes, and the reaction-diffusion scale that describes diffusion,\nassociation and dissociation on the lengthscale of a cell. We call\nthis approach MSM\/RD, due to the combination of the simulation models\nchosen at these scales:}\n\\begin{enumerate}\n\\item \\textcolor{black}{MSMs of the molecular scale: MD simulation allows\nus to probe molecular processes at atomic detail, but its usefulness\nhas long been limited by the sampling problem. Recently, the combination\nof hard- and software for high-throughput MD simulations \\citep{ShirtsPande_Science2000_FoldingAtHome,BuchEtAl_JCIM10_GPUgrid,Shaw_Science10_Anton,DoerrEtAl_JCTC16_HTMD}\nwith MSMs \\citep{PrinzEtAl_JCP10_MSM1,BowmanPandeNoe_MSMBook,SarichSchuette_MSMBook13}\nhas enabled the extensive statistical description of p}rotein folding\nand conformation changes \\citep{NoeSchuetteReichWeikl_PNAS09_TPT,Bowman_JCP09_Villin,LindorffLarsenEtAl_Science11_AntonFolding,KohlhoffEtAl_NatChem14_GPCR-MSM},\nas well as the association of proteins with ligands \\citep{BuchFabritiis_PNAS11_Binding,SilvaHuang_PlosCB_LaoBinding,PlattnerNoe_NatComm15_TrypsinPlasticity,deSancho2015identification,kubas2016mechanism}\nand even other proteins \\citep{PlattnerEtAl_NatChem17_BarBar}. Using\nmulti-ensemble Markov models (MEMMs) \\citep{WuMeyRostaNoe_JCP14_dTRAM,RostaHummer_DHAM,WuEtAL_PNAS16_TRAM,MeyWuNoe_xTRAM},\nMSMs can be derived that even capture the kinetics of ultra-rare events\nbeyond the seconds timescale at atomistic resolution \\citep{PaulEtAl_PNAS17_Mdm2PMI,CasasnovasEtAl_JACS17_UnbindingKinetics}.\nMSM approaches can thus model the long-lived states and transition\nrates of molecular detail interactions, but the cost of atomistic\nMD sampling limits them to relatively small biomolecules and complexes. \n\\item Reaction-diffusion (RD) scale: While atomic detail is relevant for\nsome processes that affect the cellular scale, it is neither efficient\nnor insightful to maintain atomic resolution at all times for cellular\nprocesses. We choose particle-based reaction-diffusion (PBRD) dynamics\nkinetics as a reference model for the cellular scale. PBRD simulates\nparticles, representing individual copies of proteins, ligands or\nother metabolites. Particles move in space via diffusion and reactive\nspecies will react with a probability according to their reaction\nrate when being clo\\textcolor{black}{se by. Here, a reaction may represent\nmolecular processes such as binding, dissociation, conformational\nchange, or actual enzymatic reactions. PBRD acknowledges that chemical\nreactions are inherently discrete and stochastic in nature \\citep{qian2010cellular},\nand that diffusion in cells is often not fast enough to justify well-stirred\nreaction kinetics \\citep{erban2009stochastic,fange2010stochastic,takahashi2010spatio}.\nA large number of recent software packages and codes implement some\nform of PBRD \\citep{andrews2004stochastic,BiedermannEtAl_BJ15_ReaddyMM,donev2010first,donev2017efficient,hattne2005stochastic,SchoenebergNoe_PlosOne13_ReaDDy,van2005green,ZonTenWolde_PRL05_GFRD},\nsee also the reviews \\citep{mereghetti2011diffusion,SchoenebergUllrichNoe_BMC14_RDReview}.\nHydrodynamic interactions at this scale could be incorporated by particle-based\ncoupling terms \\citep{ermak1978brownian,geyer2009n}. The effect of\ncrowders and complicated boundaries such as membranes on the particle\ndiffusion can be represented by including interaction forces on the\nRD scale \\citep{SchoenebergNoe_PlosOne13_ReaDDy}.}\n\\end{enumerate}\n\\textcolor{black}{In the limit that the conformational transitions\nof all molecules are fast, the MSM dynamics of each molecule effectively\naverages, and the interaction between the molecules (e.g. association)\noccurs with suitably averaged rates, reducing the problem to PBRD.\nHowever, when the lifetimes of some conformations are long compared\nto the typical time between two molecular interactions, or even the\ntime between successive rebinding events of two molecules, the conformation\ndynamics of molecules described by the MSM part couples with the RD\ndynamics. MSM\/RD opens up the possibility to simulate and analyze\nsuch effects quantitatively. For example, bimolecular binding rates\nfrom MD-derived MSMs can be inaccurate due to periodic boundary effects\nand a short-lived dissociated state in comparison to the MSM lag-time\n\\citep{PlattnerEtAl_NatChem17_BarBar}. MSM\/RD can overcome these\nissues by extending the diffusion domain available lessening the periodic\nboundary effects and increasing the lifetime of the dissociated state. }\n\n\\textcolor{black}{The ultimate aim of MSM\/RD is to produce an efficient\nmultiscale simulation that reproduces the essential statistical behavior\nof a practically unaffordable large-scale MD simulation by employing\nonly statistics obtained from simulations of the constituent biomolecules\nin small solvent boxes. As developing a full theory involving rotational\ndiffusion, three- or more-body interactions, hydrodynamics will be\nhighly complex, we here aim to make a first step towards this goal\nby coupling MSM and RD scales for bimolecular systems without large-scale\nhydrodynamic interactions.}\n\n\\textcolor{black}{We first derive a theory of MSM\/RD for bimolecular\nsystems, as depicted in Fig. \\ref{fig:schemeMSMRD-myoglobin}. When\nthe two molecules are far from each other, they both undergo a diffusion\nprocess. When they come close to each other, molecular interactions,\nmodeled with MD-derived MSMs, need to be taken into account. We further\ndevelop an algorithm to couple the MSM and RD scales for the special\ncase of a protein interacting with a ligand, which is one of the main\nadvances in this paper. This is not a trivial undertaking since one\nneeds to solve two problems: to couple the MSM and RD part in such\na way that the correct macroscopic rates and equilibrium probabilities\nare recovered, and to develop a suitable MSM discretization such that\nthis coupling can be made. We demonstrate the validity of our theory\nand algorithms on a toy model of p}rotein-ligand interaction and on\nbinding of carbon monoxide to myoglobin.\n\n\\begin{figure}\n\\centering \n\n\\includegraphics[width=1\\columnwidth]{figs\/multiscale_scheme_simple}\n\n\\caption{Sketch of the MSM\/RD scheme. When molecules $A$ and\\textcolor{black}{{}\n$B$ are not in close proximity, they diffuse freely. When $A$ and\n$B$ are close, they merge into a complex particle $C$ which itself\ndiffuses and whose internal dynamics are encoded by coupled MSM state\ntransitions. When the molecules transition into a dissociated state,\nthey are again separated into two separately diffusing particles $A$\nand $B$ with initial positions depending on the last MSM state. Note\nthat in the dissociated state, molecules $A$ and $B$ could also\npotentially undergo conformational changes encoded in independent\nMSM state transitions.}}\n\n\\label{fig:schemeMSMRD-myoglobin} \n\\end{figure}\n\nIn related work, \\citep{sbailo2017efficient,vijaykumar2015combining}\nhave coupled MD with a diffusion scheme. The work \\citep{VotapkaAmaro_SEEKR_SPCB17}\nfurther incorporates milestoning theory \\citep{faradjian2004computing}\nto compute the local kinetic information in terms of transitions between\nmilestones via short MD runs. In contrast with their work, we do not\nemploy direct MD simulations at the ``small'' scale, but represent\nthe small scale by an MSM as this allows us to operate on roughly\nthe same timesteps for the small and the large scales. Other works\nhave proposed alternative schemes to couple random walks (MSMs) with\nBrownian diffusion schemes, some examples can be seen in \\citep{del2016discrete,flegg2012two,flegg2015convergence}.\nHowever, these works focus on specific contexts that are not directly\napplicable for coupling MD-derived MSMs with reaction-diffusion schemes.\n\n\\section{MSM\/RD: coupling Markov state models and reaction-diffusion}\n\n\\textcolor{black}{We develop a theoretical description for MSM\/RD.\nThe relevant scenarios for MSM\/RD can be classified by the number\nof interacting particles, or the related reaction order:}\n\\begin{enumerate}\n\\item \\textcolor{black}{First-order reactions: isolated diffusing particles\ncan be modeled by an MSM obtained from MD simulations in a solvent\nbox. The MSM directly translates into a set of unimolecular reactions\nthat can be implemented in standard PBRD software. As long as the\nparticles don't interact, the only effect of different states on the\ndynamics are changes between different diffusion constants\/tensors.}\n\\item \\textcolor{black}{Second-order reactions: interactions between two\nmolecules that can be modeled as bimolecular reactions including protein-ligand\nor protein-protein association ($A+B\\rightarrow C$). As soon as the\ncomplex $C$ has been formed, its dynamics may be described by state\ntransitions of an MSM of the complex.}\n\\item \\textcolor{black}{Higher-order reactions: simultaneous interactions\nbetween more than two molecules. }\n\\end{enumerate}\n\\textcolor{black}{In this work, we will focus on second-order reactions.\nFirst-order reactions are trivial state changes of a particle that\nare occurring as part of the MSM dynamics. Consistent with current\nconventions in PBRD frameworks, we follow the convention of breaking\ndown higher-order reactions to second-order reactions, although in\nSec. \\ref{sec:Conclusion} we suggest possible extensions to treat\nthese explicitly.}\n\n\\textcolor{black}{In order to derive the theory for second-order reactions,\nwe concentrate on the dynamics of two molecules, $A$ and $B$. For\nthe sake of simplicity, we assume the two molecules do not have conformational\nchanges of their own, so they can only diffuse and interact with each\nother. However, it is straightforward to extend MSM\/RD to include\nconformational changes (first-order reactions) coupled with second-order\nreactions.}\n\n\\subsection{The \\emph{ground truth} model with full dynamics}\n\n\\textcolor{black}{\\emph{Ground truth}}\\textcolor{black}{{} is a term\noften used in machine learning that refers to a reference model with\nrespect to which modeling errors are measured. In the present context,\nthe }\\textcolor{black}{\\emph{ground truth}}\\textcolor{black}{{} model\ncontains the two (or more) solute molecules whose interactions will\nbe later approximated by an MSM in a }\\textcolor{black}{\\emph{large-scale}}\\textcolor{black}{{}\nsimulation, i.e. a simulation box that is not truncated after a small\nsolvent boundary as customary for MD simulation. Importantly, there\nis no universally correct ground truth, but this model employs the\nMD simulation setup and dynamical model chosen by the user for the\nmodeling task at hand. This choice includes the MD force field, solvation\nconditions and ion concentration, the protonation state at the pH\nof interest or even constant-pH simulations \\citep{DonniniEtAl_JCTC2011_ConstantPH},\nthe treatment of electrostatics, the thermostat, the integrator and\ntime step, etc.}\n\n\\textcolor{black}{If such a large-scale model were simulated for a\nlong time or with many trajectories, it would give rise to statistical\nproperties of the solute molecules that we want to reproduce, such\nas their equilibrium constants and association rates. However, such\na simulation is in general inefficient or infeasible, and our aim\nis that to reproduce its statistical properties using an MSM\/RD model\nthat is parametrized only using small MD simulations of the constituent\nsolute molecules and complexes.}\n\n\\textcolor{black}{For simplicity, we derive the MSM\/RD theory using\nall-atom explicit solvent MD simulations with a Langevin thermostat\nas the ground truth, as this setup is frequently used for MD simulations.\nHowever, the MSM\/RD results apply more generally, e.g. to different\nchoices of thermostats or integrators, as the MSM limit for long-time\ndescription of the dynamics and the overdamped limit for long-time\nand large-scale description of the solute transport are achieved from\na large family of ground truth models.}\n\n\\textcolor{black}{Langevin dynamics evolve as:}\n\\begin{equation}\nm_{k}\\frac{d^{2}}{dt^{2}}x_{k}(t)=-\\nabla_{k}U(\\mathbf{x}_{t})-\\gamma_{k}\\frac{d}{dt}x_{k}+\\sqrt{2k_{B}T\\gamma_{k}}\\boldsymbol{\\xi}_{k}(t),\\label{eq:Langevin}\n\\end{equation}\nwhere $x_{k}$ represents the three-dimensional position of the $k^{th}$\natom in the system (including the solvent), $\\mathbf{x}_{t}=[x_{1}(t),\\dots,x_{k}(t),\\dots,x_{N}(t)]$,\n$N$ the total number of atoms, $U$ is the potential energy and $-\\nabla_{k}U$\nis the force acting on the $k^{th}$ particle, $m_{k}$ is the $k^{th}$\nparticle mass, $\\gamma_{k}$ is the $k^{th}$ damping coefficient,\nand $\\boldsymbol{\\xi}_{k}(t)$ is a Gaussian random force such that\nthe expectations of its components satisfy $E[\\xi_{k,i}(t)]=0$ (zero\nmean) and $E[\\xi_{k,i}(t)\\xi_{k,j}(s)]=\\delta_{ij}\\delta(t-s)$ (white\nnoise) with $k_{B}T$ being the thermal energy. In simulations, we\nuse finite-time-step approximations of (\\ref{eq:Langevin}) and use\nit to generate stochastic trajectories. For the theoretical analysis,\nit is more useful to look at the ensemble dynamics, i.e., the propagation\nof probability densities in time. For this, we can ask: If we start\nthe dynamical system in phase space point $\\mathbf{y}$ and let it\nrun, with which probability will we find it in a point $\\mathbf{x}$\na time $\\tau$ later? We call this probability the transfer probability\n$p(\\mathbf{y}\\rightarrow\\mathbf{x};\\,\\tau)$, and we will use it to\ndescribe the action of the ground truth dynamics \\citep{SchuetteFischerHuisingaDeuflhard_JCompPhys151_146}.\nThe transfer probability $p(\\mathbf{y}\\rightarrow\\mathbf{x};\\,\\tau)$\nsubsumes the full complexity of the MD model, in\\textcolor{black}{cluding\ninteraction energies of the molecules with each other and external\nfields, and it can be constructed regardless of which thermostat or\nintegrator is used. The propagation of prob}ability densities $\\rho(\\mathbf{x};\\,t)$\nin time is formally described by the propagator $\\mathcal{P}_{\\tau}$:\n\\begin{align}\n\\rho(\\mathbf{x};\\,t+\\tau) & =\\mathcal{P}_{\\tau}\\rho(\\mathbf{x};\\,t)\\nonumber \\\\\n & =\\int p(\\mathbf{y}\\rightarrow\\mathbf{x};\\,\\tau)\\rho(\\mathbf{y};\\,t)\\,\\mathrm{d}\\mathbf{y}\\label{eq:propagation_propagator}\n\\end{align}\nWe want to find an efficient algorithm to approximate these dynamics.\nMore specifically we want to approximate certain aspects of these\ndynamics, such as the long-time behavior.\n\nIt is often useful to consider densities relative to the stationary\ndensity $\\pi(\\mathbf{x})$ given by\n\\[\nu(\\mathbf{x};\\,t)=\\frac{\\rho(\\mathbf{x};\\,t)}{\\pi(\\mathbf{x})},\n\\]\nwhich defines the propagator relative to the stationary density, or\ntransfer operator \\citep{SchuetteFischerHuisingaDeuflhard_JCompPhys151_146}:\n\\begin{align}\nu(\\mathbf{x};\\,t+\\tau) & =\\mathcal{T}_{\\tau}u(\\mathbf{x};\\,t)\\nonumber \\\\\n & =\\int\\frac{\\pi(\\mathbf{y})}{\\pi(\\mathbf{x})}p(\\mathbf{y}\\rightarrow\\mathbf{x};\\,\\tau)u(\\mathbf{y};\\,t)\\,\\mathrm{d}\\mathbf{y}\\nonumber \\\\\n & =\\int p(\\mathbf{x}\\rightarrow\\mathbf{y};\\,\\tau)u(\\mathbf{y};\\,t)\\,\\mathrm{d}\\mathbf{y}\\label{eq:propagation_transfer_operator}\n\\end{align}\nThe third row follows from detailed balance. For reversible systems,\nwhere detailed balance is fulfilled, $\\mathcal{T}_{\\tau}$ is often\ncalled backward propagator, as it appears to evolve densities backward\nin time.\n\nWe will now introduce a scale separation by treating molecules $A$\nand $B$ different when they are close (interacting) and far apart\n(non-interacting). More specifically these scales are defined by the\ndistance between the centers of mass of $A$ and $B$, $r_{AB}$:\n\\begin{enumerate}\n\\item MSM domain: molecules are in the \\emph{interaction} region $I=\\{\\mathbf{x}\\mid r_{AB}(\\mathbf{x})<R\\}$. \n\\item RD domain: molecules are in the \\emph{outside} region $O=\\{\\mathbf{x}\\mid r_{AB}(\\mathbf{x})\\ge R\\}$.\n\\end{enumerate}\nThe definition of $R$ will be investigated later. Next, we take a\ncloser look at the dynamics valid in these respective domains.\n\n\\subsection{Markov state models for interacting molecules\\label{subsec:innerMSM}}\n\nWe consider molecules that are closer than $R$ to be interacting,\nhence we call the corresponding subset of state space $I$. The kinetics\nin $I$ are fully described by Eq. (\\ref{eq:propagation_transfer_operator}),\nwhich can be approximat\\textcolor{black}{ed by an MSM derived from\na MD simulation that fully includes $I$ (usually plus some extra\nspace, because MD simulations typically employ periodic rather than\nspherical boundary conditions). We implicitly assume that the interaction\nforces between  proteins or protein-ligand pairs have decayed to zero\nat distances $R$ or greater. Note that this assumption requires that\nthe MD simulation conducted to parametrize an MSM\/RD model has a sufficiently\nlarge simulation box, suitable electrostatics treatment and solvation\nconditions (ions etc.) such that in the dissociated state the solutes\ncan be in any orientation without significantly interacting with each\nother or with their periodic images.}\n\n\\textcolor{black}{The interaction region $I$ will here be approximated\nby an MSM. We perform a spectral decomposition of (\\ref{eq:propagation_transfer_operator}),\nassuming that there exists a unique stationary density $\\pi$ and\nthe dynamics obey detailed balance. Furthermore, we truncate the spectral\ndecomposition after a finite number of $k$ terms:\n\\begin{equation}\n\\rho(\\mathbf{x};\\,t+\\tau)\\approx\\pi(\\mathbf{x})\\sum_{i=1}^{k}\\lambda_{i}(\\tau)\\langle\\psi_{i}(\\mathbf{x}),\\rho(\\mathbf{x};\\,t)\\rangle\\psi_{i}(\\mathbf{x})\\label{eq:transfSpectral}\n\\end{equation}\nHere, $\\langle,\\rangle$ denotes the scalar product with respect to\n$\\pi(\\mathbf{x})$, $\\psi_{i}$ are the eigenfunctions of $\\mathcal{T}_{\\tau}$\nand its leading eigenvalues have the form\n\\[\n\\lambda_{i}(\\tau)=\\mathrm{e}^{-\\tau\/t_{i}},\n\\]\nwhere $t_{i}$ is a characteristic relaxation timescale. The truncation\nafter $k$ terms in Eq. (\\ref{eq:transfSpectral}) assumes that $\\tau$\nis long compared to $t_{k+1}$, where $3t_{k}<\\tau$ is sufficient\nfor practical purposes. Most of the solvent dynamics correspond to\nthe fast coordinates that are averaged out (large-scale hydrodynamics\nare not part of the MSM term, while solvent molecules with long-lived\ninteractions with the solute molecule can be considered to be part\nof that solute molecule). Now we can perform a Galerkin projection\nof the transfer operator by discretizing the phase space using basis\nfunctions $\\chi_{i}(\\mathbf{x}),\\,i=1,...,n$. In MSMs, these are\ncharacteristic functions \n\\[\n\\chi_{i}(\\mathbf{x})=\\begin{cases}\n1 & \\mathbf{x}\\in S_{i}\\\\\n0 & \\mathbf{x}\\notin S_{i}\n\\end{cases}\n\\]\nwhere the $S_{i}$ form a complete partition of phase space, i.e.\n$\\Omega=\\{S_{1}\\cup S_{2}\\cup\\cdots\\cup S_{n}\\}$. The phase space\nhas now been discretized into a finite state space. The local densities\nbecome vectors simply given by\n\\[\n\\pi_{j}=\\int_{x\\in S_{j}}\\pi(\\mathbf{x})d\\mathbf{x},\\qquad\\rho_{j}(t)=\\int_{x\\in S_{j}}\\rho(\\mathbf{x};\\,t)d\\mathbf{x}.\n\\]\nFurthermore, we want the transfer operator to be approximated by a\nmatrix. We can obtain this matrix by noting that the eigenfunctions\nof the transfer operator also become vectors in state space}\n\n\\textcolor{black}{\n\\[\n\\psi_{i}^{j}=\\frac{1}{\\int_{\\Omega}\\chi_{j}(\\mathbf{x})d\\mathbf{x}}\\int_{x\\in S_{j}}\\psi_{i}(\\mathbf{x})d\\mathbf{x}.\n\\]\nInserting these equations into (\\ref{eq:transfSpectral}), and rewriting\nit in matrix form, we obtain the Chapman-Kolmogorow equation}\n\n\\textcolor{black}{\n\\begin{equation}\n\\boldsymbol{\\rho}(t+\\tau)=\\mathbf{T}^{\\top}(\\tau)\\,\\boldsymbol{\\rho}(t),\\label{eq:MSM}\n\\end{equation}\nwith $\\lambda_{i}(\\tau)$ and $\\boldsymbol{\\psi_{i}}=[\\psi_{i}^{1},\\dots,\\psi_{i}^{n}]$\nthe $i^{th}$ eigenvalue and eigenvector of the transition probability\nmatrix $\\mathbf{T}(\\tau),$ respectively, and with $\\boldsymbol{\\rho}(t)=[\\rho_{1}(t),\\dots,\\rho_{n}(t)]$\nthe probability mass function \\citep{PrinzEtAl_JCP10_MSM1}.}\n\n\\textcolor{black}{Estimating a high-quality MSM from MD simulation\ndata can be quite complex. It typically involves (i) mapping the MD\ncoordinates to a set of features, such as residue distances, contact\nmaps or torsion angles, (ii) reducing the dimension to slow collecti}ve\nvariables (CVs), often based on the variational approach or conformation\ndynamics \\citep{NoeNueske_MMS13_VariationalApproach,NueskeEtAl_JCTC14_Variational}\nor its special case time-lagged independent component analysis (TICA)\n\\citep{PerezEtAl_JCP13_TICA,SchwantesPande_JCTC13_TICA} \u2013 see \\citep{NoeClementi_COSB17_SlowCVs,KlusEtAl_JNS17_DataDriven}\nfor an overview, (iii) optionally, embedding the resulting coordinates\nin a metric space whose distances correspond to some form of dynamical\ndistance \\citep{NoeClementi_JCTC15_KineticMap,NoeClementi_JCTC16_KineticMap2},\n(iv) discretizing the result space using data-based clustering \\citep{BowmanPandeNoe_MSMBook,SchererEtAl_JCTC15_EMMA2,HarriganEtAl_BJ17_MSMbuilder},\ntypically resulting in 100-1000 discrete states, and (v) estimating\nthe transition matrix $\\mathbf{T(\\tau)}$ or a transition rate matrix\n$\\mathbf{K}$ with $\\mathbf{T}(\\tau)=\\exp\\left(\\tau\\mathbf{K}\\right)$\nat some lag time $\\tau$, and validating it \\citep{Bowman_JCP09_Villin,PrinzEtAl_JCP10_MSM1,BucheteHummer_JPCB08,TrendelkampSchroerEtAl_InPrep_revMSM}.\nFinally, the MSM may be coarse-grained to few metastable states \\citep{KubeWeber_JCP07_CoarseGraining,HummerSzabo_JPCB15_CoarseGraining,OrioliFaccioli_JCP16_CoarseMSM,NoeEtAl_PMMHMM_JCP13}.\nThe MSM software packages PyEMMA \\citep{SchererEtAl_JCTC15_EMMA2}\nand MSMbuilder \\citep{HarriganEtAl_BJ17_MSMbuilder} can greatly help\nto simplify this process and make it reproducible.\n\nIn the case where there are well-defined meta-stable regions in phase\\textcolor{black}{{}\nspace, we can greatly reduce the number of states in the MSM. One\nway to simplify the MSM construction process above and to directly\nend up with a few-state MSM is to employ VAMPnets, where the complex\nMSM construction pipeline is replaced by a neural network that is\ntrained using the variational approach for Markov processes \\citep{MardtEtAl_VAMPnets}.\nAlternatively, one can replace the discretization step (iv) above\nby employing a core set approach that was derived in \\citep{BucheteHummer_JPCB08}\nand further analyzed in \\citep{schutte2011markov}. The essential\nidea is to define the states as cores around the metastable regions.\nDue to the metastability, the probability of finding the system outside\nof the metastable regions is very small, so to a good approximation\nthe kinetics can be described as a core-to-core jump process \\citep{schutte2011markov}.\nThis approach will be employed throughout this paper and and explained\nin more detail in Sec. \\ref{sec:MSM\/RD-implementation}.}\n\n\\subsection{Reaction-diffusion dynamics for noninteracting molecules}\n\nWhen molecules are far apart, and thus in the RD domain defined by\n$r_{AB}(\\mathbf{x})\\ge R$, they are not directly interacting. As\nthe dynamics of the two molecules are independent, it is convenient\nto only track the net diffusion of the centers of mass, $\\boldsymbol{r}_{A}$\nand $\\boldsymbol{r}_{B}$. Furthermore, we assume that the dynamics\nin the RD domain can be tracked by coarse timesteps of at least $\\Delta t$\nwhich exceeds the typical velocity autocorrelation time (picoseconds).\nAt such timescales, the fast dynamics corresponding to the solvent\nare averaged out. It is possible that even longer timesteps are made\nusing an event-based integration scheme such as first-passage kinetic\nMonte Carlo (FPKMC) algorithm, Green's function reaction dynamics\n(GFRD) or MD-GFRD \\citep{donev2010first,takahashi2010spatio,vijaykumar2015combining,van2005green,ZonTenWolde_PRL05_GFRD}.\nAt such timesteps, the Langevin equation (\\ref{eq:Langevin}) becomes\nan overdamped Langevin equation for the centers of mass of the two\nmolecules, i.e. the motion is governed by pure diffusion: \n\n\\begin{equation}\n\\frac{d\\boldsymbol{r}_{A}(t)}{dt}=\\sqrt{2D_{A}}\\boldsymbol{\\xi}_{A}(t),\\quad\\frac{d\\boldsymbol{r}_{B}(t)}{dt}=\\sqrt{2D_{B}}\\boldsymbol{\\xi}_{B}(t),\\label{eq:overdampMassCenters}\n\\end{equation}\nwhere $\\boldsymbol{\\xi}_{A}(t)$ and $\\boldsymbol{\\xi}_{B}(t)$ are\nindependent white noise vectors with each of their components satisfying\n$E[\\xi_{K,i}(t)]=0$ and $E[\\xi_{K,i}(t)\\xi_{K,j}(s)]=\\delta_{ij}\\delta(t-s)$.\n$D_{A}$ and $D_{B}$ are the net diffusion coefficients for the centers\nof mass, which can be obtained from MD simulations. In general, as\nwe are tracking the center of mass, we also need to track the rotational\ndiffusion of the molecules. However, as rotational diffusion is not\nrelevant for the examples discussed in this manuscript, we refer to\n\\citep{SchluttigEtAl_JCP08_PatchyParticleAssociation,VijaykumarEtAl_Arxiv16_AnisotropicMultiscaleGFRD}. \n\nIn the present case, we can simply fix the frame of reference in $\\boldsymbol{r}_{A}(t)$,\nassume the rotation of $A$ is slower than the diffusion of $B$,\nwhich is true for protein-ligand systems, and fix the orientation\nof the axis to that of molecule $A$. We further assume that $B$\nis a small molecule such that its orientation is not very relevant,\nas it will be the case in our implementation of the scheme. This simplifies\nEqs. \\ref{eq:overdampMassCenters} into a simple diffusion in $\\boldsymbol{r}_{B}$\nonly \n\n\\begin{equation}\n\\frac{d\\boldsymbol{r}_{B}(t)}{dt}=\\sqrt{2(D_{B}+D_{A})}\\boldsymbol{\\xi}(t),\\label{eq:COM_diffusion}\n\\end{equation}\nwith the components of $\\boldsymbol{\\xi}(t)$ satisfying $E[\\xi_{i}(t)]=0$\nand $E[\\xi_{i}(t)\\xi_{,j}(s)]=\\delta_{ij}\\delta(t-s)$. \n\n\\subsection{MSM\/RD coupled dynamics\\label{subsec:MSM\/RD-coupled}}\n\nThe present coupled model only considers interactions between up to\ntwo molecules. This is a frequent assumption in PBRD \\citep{ZonTenWolde_PRL05_GFRD,donev2010first,SchoenebergNoe_PlosOne13_ReaDDy,BiedermannEtAl_BJ15_ReaddyMM}\nbut may be restrictive from a molecular standpoint. We assume that\nsimultaneous reactions between three or more molecules such as $A+B+C\\rightarrow D$\ncan always be broken down into $A+B\\rightarrow AB;\\,AB+C\\rightarrow D$\nor other bimolecular pathways, and therefore focus on MSM\/RD involving\ntwo molecules. In order to do the coupling, as the dynamics in the\n$I$ and $O$ region are given in terms of states and coordinates\nrespectively, we need to recognize that $\\mathbf{x}$ and $\\mathbf{y}$\nin the transfer density $p(\\mathbf{y}\\rightarrow\\mathbf{x};\\,\\tau)$\ncan be either coordinates $\\mathbf{c}$ (center-of-mass position and\nperhaps orientation of the molecule) or states $s$ (metastable regions\nin the coordinate space). In order to implement the coupling, we suggest\ndefining two quantities: \n\\begin{itemize}\n\\item \\texttt{$p_{\\mathrm{entry}}\\left[\\mathbf{c}_{t}\\rightarrow\\mathbf{x}_{t+\\Delta};\\,\\Delta\\right]$},\ntransfer probability of starting in coordinates $\\mathbf{c}_{t}$\njust inside the MSM domain ($r_{AB}(\\mathbf{c}_{t})<R$) conditioned\non hitting only one state $\\mathbf{x}_{t+\\Delta}=s_{t+\\Delta}$ in\nthe MSM domain (transition event) OR on exiting once the MSM domain\n$\\mathbf{x}_{t+\\Delta}=\\mathbf{c}_{t+\\Delta}$ (return event)\n\\item $p_{\\mathrm{exit}}\\left[s_{t}\\rightarrow\\mathbf{x}_{\\mathrm{t}+\\Delta};\\,\\Delta\\right]$,\ntransfer probability of starting in state $s_{t}$ conditioned on\nexiting once the MSM domain $\\mathbf{x}_{t+\\Delta}=\\mathbf{c}_{t+\\Delta}$\n(exit event) OR hitting once any other state $(\\mathbf{x}_{t+\\Delta}=s_{t+\\Delta})$. \n\\end{itemize}\nOnce we know these transfer probabilities, we can introduce the basic\nalgorithm where $\\tau_{\\mathrm{RD}}$ and $\\tau_{\\mathrm{MSM}}$ correspond\nto the diffusion and MSM time-step, respectively: \n\n\\texttt{\\noindent}\\texttt{\\textbf{Input}}\\texttt{: Initial mode (RD\nor MSM), initial condition (coordinates $\\mathbf{c}_{0}$ or state\n$s_{0}$, respectively) and $t=0$:}\n\n\\texttt{While $t\\leq t_{\\mathrm{final}}:$}\n\\begin{enumerate}\n\\item \\texttt{If in RD mode:}\n\\begin{enumerate}\n\\item \\texttt{Propagate $\\mathbf{c}_{t}\\rightarrow\\mathbf{c}_{t+\\tau_{\\mathrm{RD}}}$\nby diffusion}\n\\item \\texttt{Update time $t\\mathrel{{+}{=}}\\tau_{\\mathrm{RD}}$}\n\\item \\texttt{If $r_{AB}(\\mathbf{c}_{t})<R$ (enter MSM domain):}\n\\begin{itemize}\n\\item \\texttt{Sample next event ($\\mathbf{x}_{t+\\Delta}$,$\\Delta$)from}~\\\\\n\\texttt{ $p_{\\mathrm{entry}}[\\mathbf{c}_{t}\\rightarrow\\mathbf{x}_{t+\\Delta};\\,\\Delta]$.}\n\\item \\texttt{If transition event:}~\\\\\n\\texttt{Map to state }$s_{t+\\Delta}=\\mathbf{x}_{t+\\Delta}$\\texttt{}~\\\\\n\\texttt{Update time $t\\mathrel{{+}{=}}\\Delta$ }~\\\\\n\\texttt{Switch to MSM mode}\n\\item \\texttt{Else (return event):}~\\\\\n\\texttt{Map to coordinates $\\mathbf{c}_{t+\\Delta}=\\mathbf{x}_{t+\\Delta}$}~\\\\\n\\texttt{Update time $t\\mathrel{{+}{=}}\\Delta$}\n\\end{itemize}\n\\end{enumerate}\n\\item \\texttt{Else (MSM mode):}\n\\begin{enumerate}\n\\item \\texttt{If $s_{t}\\neq s_{t-\\tau_{MSM}}$or previous mode $\\neq$ MSM\nmode:}\n\\begin{itemize}\n\\item \\texttt{Sample next event ($\\mathbf{x}_{t+\\Delta}$,$\\Delta$) from\n}~\\\\\n$p_{\\mathrm{exit}}\\left[s_{t}\\rightarrow\\mathbf{x}_{\\mathrm{t}+\\Delta};\\,\\Delta\\right]$\\texttt{. }\n\\item \\texttt{If exit event:}~\\\\\n\\texttt{Map to coordinates $\\mathbf{c}_{t+\\Delta}=\\mathbf{x}_{t+\\Delta}$}~\\\\\n\\texttt{Update time $t\\mathrel{{+}{=}}\\Delta$}~\\\\\n\\texttt{Switch to RD mode}\\texttt{\\textcolor{blue}{{} }}\\texttt{\\textcolor{black}{and\nbreak current loop iteration}}\n\\end{itemize}\n\\item \\texttt{Propagate $s_{t}\\rightarrow s_{t+\\tau_{MSM}}$ using the MSM}\n\\item \\texttt{Update time} $t\\mathrel{{+}{=}}\\tau_{MSM}$\n\\end{enumerate}\n\\end{enumerate}\nThere are additional issues in specific scheme implementations, such\nas estimating the unknown conditional transfer probabilities, and\nchoosing the MSM discretization and $R$ such that the overall discretization\nerror is small, among others. These issues are non-trivial and could\npotentially be tackled with different approaches. In order to quantify\nthe accuracy of a given approach, we quantify how well is our scheme\napproximating the ground truth by comparing relevant macroscopic observables.\nWe present one possible implementation of the scheme in Sec. \\ref{sec:MSM\/RD-implementation}.\n\n\\section{An MSM\/RD implementation for protein-ligand systems\\label{sec:MSM\/RD-implementation}}\n\nNow we develop an implementation of the MSM\/RD scheme for a special\nclass of systems: th\\textcolor{black}{e binding of a small ligand\nto a protein \u2013 a case that is relevant in the study of protein-drug\nbinding kinetics \\citep{schuetz2017kinetics}. While the theory described\nbefore is more general, implementations to more challenging systems\nsuch as protein-protein interaction will be treated in future contributions.\nWe begin by considering the macromolecule $A$ fixed at the origin\nwith fixed orientation and the ligand $B$ freely diffusing around\nit with an overall diffusion constant $D=D_{A}+D_{B}$. The macromolecule\nhas several possible binding sites given by some interaction po}tential.\nIn order to present the MSM\/RD scheme in detail, we distinguish three\ndifferent simulations: \n\\begin{enumerate}\n\\item \\textbf{Reference simulation} (ground truth, if available): MD simulation\nof $B$ and its interaction with $A$ in a large spherical domain\nwith radius $R_{s}$. Unfortunately, reference simulations of realistic\nsystems are in general not computationally feasible due to the time\nand lengthscales of the simulation. Nonetheless, reference simulations\nof simple systems are used to verify the MSM\/RD scheme and validate\nits use in more complex systems. \n\\item \\textbf{Small-scale simulation} (MD simulation): analogous to the\nreference simulation with the difference that $B$ is constrained\nto a small box with periodic boundary conditions, see Fig. \\ref{fig:trajectoriesIllustration}a.\nAs the potential is negligible outside this box, the main interaction\ndynamics are extracted from this simulation's data into an MSM. This\nsimulation is used to parametrize the MSM\/RD model.\n\\item \\textbf{MSM\/RD simulation} (hybrid model): couples the MSM for short-range\ninteractions derived from the MD simulation 2. with a diffusion scheme\nfor the long-range, see Fig. \\ref{fig:trajectoriesIllustration}c.\nThe goal of the scheme is to approximate the ground truth dynamics\ngiven by the reference simulation 1. \n\\end{enumerate}\n\\begin{figure*}\n\\centering\n\n(a)\\includegraphics[width=0.3\\textwidth]{figs\/cartoons\/trajectory_types_black}\n(b)\\includegraphics[width=0.3\\textwidth]{figs\/cartoons\/trajectory_types_color}\n(c) \\includegraphics[width=0.3\\textwidth]{figs\/cartoons\/MSMRD_cartoonV3}\n\n\\caption{Illustrations of a MD trajectory, its classification to extract the\nrelevant dynamics and the MSM\/RD scheme.\\textbf{ a)} Illustration\nof a trajectory of a ligand in a MD simulation within a box with periodic\nboundaries (small-scale simulation). Note that there are two metastable\nregions, e.g. binding sites on a protein, where the ligand stays for\na longer time.\\textbf{ b)} Illustration of truncation and classification\nof the trajectory. The MSM domain is chosen so the interaction potential\nis effectively zero outside this region (bath state); the cores $X$\nand $Y$ are chosen to represent the metastable regions in phase space.\nThe truncated trajectories are classified into entry trajectories\n(green), transition trajectories (red) and exit trajectories (blue),\nwhich are used for the coupling in the MSM\/RD scheme. In order to\nobtain the MSM for the MSM\/RD scheme, the system is also classified\ninto three states, the bath state and the two cores $X$ and $Y$;\nit is also shown when the transition between these states occur along\na trajectory. \\textbf{c)} Representation of the MSM\/RD scheme. The\nfull trajectories from the MD-simulation are used to derive an MSM\nto model the dynamics in the MSM domain. The entry and exit trajectories\nfrom the MD-simulation are used to couple the Brownian dynamics in\nthe diffusion domain with the dynamics in the MSM domain. }\n\n\\label{fig:trajectoriesIllustration} \n\\end{figure*}\n\n\\subsection{Estimation}\n\nIn order to parametrize the MSM\/RD scheme, we need to estimate quantities\nfrom the small-scale simulation 2. that characterize the state-to-state\ndynamics and the coupling between the MSM and RD domain. The state\nto state dynamics are estimated using an MSM, and the coupling is\ngiven in terms of entry and exit events from the MSM domain. These\nmight happen on different timescales, so we would like to be free\nfrom the fixed time-step the MSM requires to be well equilibrated.\nTherefore, we use trajectory statistics for entry and exit events.\n\n\\subsubsection{MSM}\n\nAs a first step for the construction of the Markov model and MSM\/RD\nparametrization, we need to find a discrete representation of the\nunderlying data. In this work we use the core MSM approach \\citep{schutte2011markov},\nwhich requires the definition of cores as metastable regions of phase\nspace. Cores are given by spherical domains around the metastable\nregions in the MD simulation and can be found using a clustering algorithm.\nIn the core MSM approach a discrete trajectory is constructed by assigning\nthe last visited state-index to each point in the trajectory. Note\nthe trajectory may leave the core of a given state and re-enter multiple\ntimes without transitioning to other states. Using this discretization\ntechnique we truncate the discrete trajectory into three types of\ntrajectories as shown in Fig. \\ref{fig:trajectoriesIllustration}b:\n$i)$ entry trajectories that start just inside the MSM domain and\neither leave the domain next or hit a core inside the domain. $ii)$\ntransition trajectories that start in a state and hit another state\nas next event and $iii)$ exit trajectories that start in a state\nand leave the MSM domain as next event. These trajectories are used\nto estimate the transfer densities and to parametrize the MSM\/RD simulation.\n\nThe MSM for the short-range interactions is built using the full discrete\ntrajectories and the exit trajectories (Fig. \\ref{fig:trajectoriesIllustration}a).\nWe follow the methods from \\citep{PrinzEtAl_JCP10_MSM1} to estimate\na transition matrix $\\mathbf{T}(\\tau)$, where the entries are the\ntransition probabilities $T_{i,j}$ from state $i$ to $j$. Using\nthe discrete trajectories, we create count matrices $C_{i,j}^{\\mathrm{full}}(\\tau)$\nfrom the complete data set and $C_{i,j}^{\\mathrm{exit}}(\\tau)$ from\nthe exit trajectories, which count all the transitions from state\n$i$ to $j$ at a lag time $\\tau$ observed in the respective datasets.\nAs the coupling between the MSM and RD domain is handled separately,\nthe MSM dynamics only accounts for transitions amongst the cores and\ntherefore the counts arising from exit trajectories have to be subtracted\n\\begin{align}\nC_{i,j}(\\tau)=C_{i,j}^{\\mathrm{full}}(\\tau)-\\delta_{i,j}C_{i,j}^{\\mathrm{exit}}(\\tau),\\label{eqn:countMatrix}\n\\end{align}\nwhere $\\delta_{i,j}$ denotes the Kronecker delta. We then use a maximum\nlikelihood estimator to obtain a transition matrix from the given\ncounts $C_{ij}$. Note that here we have chosen an irreversible estimator,\nas we can no longer assume that detailed balance holds for this count\nmatrix.\n\n\\subsubsection{Entering the MSM domain\\label{subsec:EnterMSM}}\n\n The protocol to enter the MSM domain from the RD domain is constructed\nwith the entry trajectories as defined above. It consists of generating\na list $L_{\\mathrm{entry}}=\\{\\mathbf{c}_{\\mathrm{entry}},\\mathbf{x}_{\\mathrm{end}},\\Delta\\}$\nof all start coordinates $\\mathbf{c}_{\\mathrm{entry}}$(just inside\nthe MSM domain) and endpoints $\\mathbf{x}_{\\mathrm{end}}$ of entry\ntrajectories and their corresponding times $\\Delta$. The endpoints\nmay be either MSM states or coordinates in the RD domain, see Fig.\n\\ref{fig:trajectoriesIllustration}b. The ensemble of trajectories\nin this list estimates the conditional transfer probability \\texttt{$p_{\\mathrm{entry}}\\left[\\mathbf{c}_{t}\\rightarrow s_{t+\\Delta};\\,\\Delta\\right]$}\n(Sec. \\ref{subsec:MSM\/RD-coupled}) for several times $\\Delta$. In\nthe MSM\/RD simulation samples are drawn from this list of entry points.\n\n\\subsubsection{Exiting the MSM domain\\label{subsec:ExitMSM}}\n\nFor each state $\\mathbf{s}$ of the MSM, we collect all exit and transition\ntrajectories and save their end coordinate or state along with their\nrespective exit time in the lists $L_{\\mathrm{exit},s}=\\{\\mathbf{c}_{exit},\\Delta\\}$\nand $L_{\\mathrm{trans},s}=\\{s_{\\mathrm{trans}},\\Delta\\}$. The ensemble\nof trajectories in these list estimates the conditional transfer probability\n\\texttt{$p_{\\mathrm{exit}}\\left[s_{t}\\rightarrow\\mathbf{c}_{\\mathrm{t}+\\Delta};\\,\\Delta\\right]$\n}(Sec. \\ref{subsec:MSM\/RD-coupled}) for several times $\\Delta$.\nThe probability of an exit event $P_{\\mathrm{exit},s}$ is simply\nestimated as the ratio of exiting trajectories over the total numbers\nof trajectories,\n\n\\begin{align}\nP_{\\mathrm{exit},s}=\\frac{\\#\\text{\\text{ of trajectories in} }L_{\\mathrm{exit},s}}{\\#\\text{\\text{ of trajectories in} }L_{\\mathrm{exit},s}\\text{ and }L_{\\mathrm{trans},s}}.\\label{eqn:exitProbability-1}\n\\end{align}\n\nThe MSM\/RD scheme for this implementation is based on the scheme from\nSec. \\ref{subsec:MSM\/RD-coupled} and is shown in the Appendix.\n\n\\subsection{Verification of the MSM\/RD scheme}\n\nIn order to verify the MSM\/RD scheme, we use systems where a reference\nsimulation is available. We verify the internal dynamics by comparing\nthe first passage times (FPTs) distributions and mean first passage\ntimes (MFPTs) for each pair of metastable states within region $I$\nbetween the MSM\/RD and reference simulations. We estimate the ground\ntruth MFPTs by computing the FPTs $t_{i,j}^{\\textrm{ref}}$, where\nthe initial conditions are chosen as the minima $\\mu_{i}$ and the\nsystem is propagated following the reference simulation until hitting\nstate $j$ (conditioned on not leaving the MSM domain). For the MSM\/RD\nscheme, we compute the FPTs $t_{i,j}^{\\textrm{MSM}}$ by placing the\nparticle in state $i$ and propagating the system following the MSM\/RD\nscheme until state $j$ is hit. If the particle exits the MSM domain\nbefore reaching state $j$, the trajectory is not taken into account.\nWhen a sufficiently large sample is generated, we can estimate the\ndistributions of FPTs by histograms. The MFPTs are estimated as $\\tau_{ij}^{\\textrm{ref}}$=$\\overline{t_{ij}^{\\textrm{ref}}}$\nand $\\tau_{ij}^{\\textrm{MSM}}$=$\\overline{t_{ij}^{\\textrm{MSM}}}$,\nrespectively. The MFPT relative error between the MSM\/RD and the reference\nsimulations is estimated as \n\\begin{align}\n(E_{\\text{rel}})_{\\mathrm{ij}}=\\frac{\\tau_{ij}^{\\textrm{ref}}-\\tau_{ij}^{\\mathrm{MSM}\\mathscr{}}}{\\tau_{ij}^{\\textrm{ref}}}.\\label{eqn:relativeError}\n\\end{align}\nIn order to verify the coupling between the RD and MSM domain, we\nalso estimate and compare the unbinding rate, binding rate and equilibrium\nconstant. The two latter are calculated for different particle concentrations\n$c$ by fixing the radius $R_{s}$ of the simulation domain such that\n$c=1\/V_{\\text{RD}},$with $V_{\\mathrm{RD}}$ the volume of the RD\ndomain.\n\n\\section{Results\\label{sec:results}}\n\nIn this section, we implement the MSM\/RD scheme from Sec. \\ref{sec:MSM\/RD-implementation}\nin two systems. The first is a simple model of a ligand diffusing\nin a potential landscape, which is used to verify that the MSM\/RD\nscheme reproduces the correct dynamics. The second corresponds to\na more realistic MD system, where we study the binding of carbon monoxide\nto myoglobin.\n\n\\subsection{Ligand diffusion in potential landscape\\label{subsec:3D-diff-pot}}\n\nWe implement the MSM\/RD scheme in a simple model, where the reference\nsimulation is available. The model consists of a ligand $B$ under\nover-damped Langevin dynamics in a three-dimensional potential landscape\n\\begin{equation}\n\\frac{d\\mathbf{x}(t)}{dt}=-\\frac{1}{\\gamma}\\nabla U(\\mathbf{x})+\\sqrt{2D}\\boldsymbol{\\xi}(t),\\label{eq:overdLangevin}\n\\end{equation}\nwith $U$ the interaction potential with some macromolecule $A$ fixed\nat the origin, $\\gamma$ the damping, and each component of the noise\nsatisfies $E[\\xi_{i}(t)]=0$ and $E[\\xi_{i}(t)\\xi_{j}(s)]=\\delta_{ij}\\delta(t-s)$\nwith $D=k_{B}T\/\\gamma$ the diffusion coefficient. A trajectory density\nplot of the potential landscape chosen is shown in Fig. \\ref{fig:diff3Dpot}a,\nand it consists of nine Gaussians with different depths and widths\n\\begin{align}\nU(\\mathbf{r})=-\\sum_{i=1}^{9}s_{i}\\mathcal{N}(\\mathbf{\\boldsymbol{\\mu}}_{i},\\mathbf{\\boldsymbol{\\Sigma}}_{i}),\\label{eqn:potentialLandscape}\n\\end{align}\nwhere $\\mathcal{N}(\\boldsymbol{\\mu}_{i},\\boldsymbol{\\Sigma}_{i})$\ndenotes a Gaussian centered at minimum $\\boldsymbol{\\mu}_{i}$ with\ncovariance matrix $\\boldsymbol{\\Sigma}_{i}$, $s_{i}$ denotes a scale\nfactor. The small-scale simulation consists of Euler-Maruyama numerical\nrealizations of Eq. (\\ref{eq:overdLangevin}) under this potential\nconstrained to a box with an edge length of $6$ units with periodic\nboundary conditions. The reference simulation is analogous to the\nsmall-scale simulation with the difference that it uses a larger spherical\ndomain with reflective boundary conditions at a range of radii corresponding\nto simulations at different ligand concentrations. \n\n\\subsubsection{Parametrization of the MSM\/RD scheme}\n\nWe use a radiu\\textcolor{black}{s of $R=2.5\\:\\mathrm{nm}$ for the\nMSM domain ($I$ region) since outside this domain the potential (Eq.\n\\ref{eqn:potentialLandscape}) is essentially zero. We generate $120$\nsmall-scale simulation trajectories, each with a length of $10^{7}$\nsteps, a time-step of $\\Delta t=10^{-4}\\:\\mathrm{ns}$, and sampled\nevery tenth step. This results in a total simulation time of $t=1.2\\cdot10^{5}\\:\\mathrm{ns}$. }\n\n\\textcolor{black}{The cores are defined as spheres with radius $0.2\\:\\mathrm{nm}$\naround the minima $\\boldsymbol{\\mu}_{i}$, and the count matrix of\ntransition between cores is generated from the trajectories following\nEq. (\\ref{eqn:countMatrix}). A maximum likelihood estimator (implemented\nin PyEMMA \\citep{SchererEtAl_JCTC15_EMMA2}) is then applied to the\ncount matrix to yield the MSM. From the trajectories, we also generate\nthe lists $L_{\\mathrm{entry}}$, $L_{\\mathrm{exit},s}$, $L_{\\mathrm{trans},s}$\nand $P_{\\mathrm{exit},s}$, introduced in Secs. \\ref{subsec:EnterMSM}\nand \\ref{subsec:ExitMSM}. We then estimate the timescales of the\neigenmodes for different MSM lag times to test how well the underlying\nprocess is estimated by the MSM. The timescales have small variations\nfor different lag times (Fig. \\ref{fig:diff3Dpot}d), which means\nthe system can be considered Markovian for all lag times. However,\nwe have to be careful not to choose the lag time too large, such that\nrelevant fast timescales are neglected resulting in significant errors.\nFor all further analyses, we consider a lag time of $\\tau_{MSM}=500\\Delta t=0.05\\:\\mathrm{ns}$\nto be a}n optimal compromise.\n\n\\begin{figure*}\n\\centering\n\n(a) \\includegraphics[width=0.3\\textwidth]{figs\/cartoons\/meatballs_potential}\n(b)\\includegraphics[width=0.3\\textwidth]{figs\/asym3D\/internalDynamicsWhiteCenter_notitle}\n\\textcolor{blue}{(c)}\\includegraphics[width=0.3\\textwidth]{figs\/asym3D\/internalDynamicsHistograms_units}\n\n\\textcolor{blue}{(d)} \\includegraphics[width=0.3\\textwidth]{figs\/asym3D\/impliedTimescales_resamplingError_units}\n\\textcolor{blue}{(e)}\\includegraphics[width=0.3\\textwidth]{figs\/asym3D\/Kon_units}\n\\textcolor{blue}{(f)}\\includegraphics[width=0.3\\textwidth]{figs\/asym3D\/Keq_units}\n\n\\caption{Visualization and verification results for the simple model of ligand\ndiffusing in a potential landscape.\\textbf{ (a)} Density plot of the\nposition of the ligand in the three dimensional potential. Red indicates\nregions of higher density while blue indicates regions of lower density.\n\\textbf{(b)} Relative error of MFPTs conditioned on not leaving the\nMSM domain between the MSM\/RD and the reference simulation. \\textbf{(c)\n}Comparison of first passage times distribution histograms for the\ntransitions with the highest error in (b). The left pane corresponds\nto transitions with negative relative error, and the right pane to\ntransitions with positive relative error. \\textbf{(d)} Implied timescales\nof the MSM. The shaded area represents the standard deviation of the\nbootstrapping sample. We observe well converged timescales for all\nconsidered lag times.\\textbf{ (e)} The rate $k_{on}$ as function\nof the concentration of the system for the MSM\/RD and reference simulations.\n\\textbf{(f)} Same plot as e) but for the logarithm of the equilibrium\nconstant $\\log(K_{eq})$. The error bars in (e) and (f) represent\nthe 95\\% confidence interval using a bootstrapping approach.}\n\n\\label{fig:diff3Dpot}\n\\end{figure*}\n\n\\subsubsection{Comparison of dynamic properties}\n\nIn order to compute the binding rate, we calculate the first passage\ntime from a uniforml\\textcolor{black}{y sampled location close to\nthe boundary $r=R_{S}-\\delta$ to any MSM state. We choose $\\delta=0.05\\:\\mathrm{nm}$\nand use $10^{4}$ simulations to average and estimate the $\\text{MFPT}_{\\mathrm{on}}$,\nfrom which we calculate the binding rate as $k_{\\mathrm{on}}^{*}=1\/\\text{MFPT}_{\\mathrm{on}}$.\nThis procedure is performed for both the MSM\/RD and the reference\nsimulation, and we observe excellent agreement between the two (Fig.\n\\ref{fig:diff3Dpot}e).}\n\n\\textcolor{black}{For the unbinding rate, we consider the inverse\nprocess by starting in an MSM state and propagating the dynamics until\ncrossing a boundary defined by a sphere with radius $2.7\\:\\mathrm{nm}>R$.\nWe obtain a reference value of $0.402_{0.400}^{0.404}\\:\\mathrm{ns^{-1}}$\n(Sub- and superscript indicate lower and upper bound of the $95\\%$\nprecentile) and an MSM\/RD simulation value of $k_{\\mathrm{off}}=0.400_{0.398}^{0.402}\\:\\mathrm{ns^{-1}}$.\nWe further compute the logarithm of the equilibrium constant $\\log(K_{\\mathrm{eq}})=\\log(k_{\\mathrm{off}}\/k_{\\mathrm{on}}^{*})$\nfor both models and for the chosen values of concentrations, resulting\nin accurate reproduction of the reference values by the MSM\/RD scheme(Fig.\n\\ref{fig:diff3Dpot}f). Thus we verify that the coupling between the\nMSM domain and the RD domain works consistently in the MSM\/RD simulation\nscheme.}\n\n\\textcolor{black}{Next, we want to ensure that also the dynamics between\nthe states inside the MSM are reproduced to a high accuracy. We compare\nMFPTs between all pairs of states conditioned on not leaving the MSM\ndomain. In the reference simulation this is done by placing the particle\nat position $\\boldsymbol{\\mu}_{i}$ and propagating the system until\nstate $j$ is reached. If the particle leaves the MSM domain before\nreaching state $j$, this trajectory is discarded. For the MSM\/RD\nsimulation, we simply start in state $i$ and propagate until state\n$j$ is hit, while discarding trajectories that leave the MSM domain.\nThis procedure is repeated until $10^{4}$ successful trajectories\nare found for both simulations, w}hich are averaged to obtain the\nMFPTs. The relative errors are calculated with Eq. (\\ref{eqn:relativeError});\nall relative errors are below $9\\%$ (Fig. \\ref{fig:diff3Dpot}b).\nWe further observe that negative errors arise for state pairs that\nare close together and thus have short passage times. For these transitions,\nwe tend to overestimate the MFPT in the MSM\/RD simulation as short\nprocesses are truncated in the MSM estimation. Moreover, we observe\nthat the highest positive errors arise for transitions which are far\napart. These are the hardest to sample since for these transitions\nthere are a very high number of possible long and non-direct transition\ntrajectories, which are less likely to be observed . We chose the\nfour transitions with the highest relative error and compared their\nFPTs distribution histograms (Fig. \\ref{fig:diff3Dpot}c). Even though\nthese transitions have the highest errors, we observe the distributions\nmatch well. Therefore, we verify MSM\/RD scheme also describes the\ninternal dynamics accurately.\n\n\\subsection{Binding of CO to myoglobin\\label{subsec:BindingCO}}\n\nAs an application of the MSM\/RD scheme, we study the binding of carbon\nmonoxide (CO) to myoglobin. Myoglobin is a globular protein which\nis responsible for the transport of oxygen in muscle tissue. The binding\nprocess of CO to myoglobin has recently be\\textcolor{black}{en studied\nby de Sancho et al. \\citep{deSancho2015identification}, whose data\nwe use to parametrize the MSM\/RD scheme. The dataset consist of MD\ntrajectories of $20$ CO molecules and one myoglobin protein for a\ntotal simulation time of $500\\:\\mathrm{ns}$. The MD simulation is\nconfined to a periodic box with edge length of $5\\:\\mathrm{nm}$.\nDespite the fact that only one CO molecule can reside in the binding\npocket, the error of treating $20$ CO molecules as being statistically\nindependent is small within statistical uncertainty (see  \\citep{deSancho2015identification}\nfor details). We therefore extract$20$ independent CO trajectories,\neffectively  increasing the total simulation time to $10\\:\\mu\\mathrm{s}$.}\n\n\\subsubsection{Parametrization of MSM\/RD scheme}\n\nIn order to parametrize the scheme, all frames are first \\textcolor{black}{aligned\nusing the $C_{\\alpha}$ atoms of the myoglobin as reference. On the\naligned data, we run the density-based spatial clustering of applications\nwith noise algorithm (DBSCAN) \\citep{ester1996density}, which finds\na total of 16 metastable regions\/cores. The positions and size of\nthe cores are shown in Fig. \\ref{fig:Myoglobin}a, where it can be\nobserved that the algorithm correctly identifies regions of high ligand\ndensity, including the myoglobin bound state indicated in red. The\nradius of the spherical cores is the radius at which $80\\:\\%$ of\nthe datapoints that were assigned to the respective state are inside\nthe core. Four states are discarded as they are not part of the largest\nconnected set. As the simulation box had been set up to just contain\nthe protein and a $1\\:\\mathrm{nm}$ solvent layer, we choose the largest\nMSM domain that still fits inside the box $(R=2.5\\:\\mathrm{nm)}$.\nAnalogous to the previous example, we follow Sec. \\ref{sec:MSM\/RD-implementation}\nto estimate an MSM for the close-range dynamics and generate $L_{\\mathrm{entry}}$,\n$L_{\\mathrm{exit},s}$, $L_{\\mathrm{trans},s}$ and $P_{\\mathrm{exit},s}$\nto couple the dynamics in the two domains. }\n\n\\textcolor{black}{We compute the implied timescales for the MSM and\nchoose a lag time of $150\\:\\mathrm{ps}$ where timescales are sufficiently\nconverged (Fig. \\ref{fig:Myoglobin}b). The diffusion constant is\ncomputed using the mean squared displacement (MSD) of the parts of\nthe CO trajectories that are far from the protein, with $D=\\Delta\\text{MSD}(t)\/6\\Delta t$.\nWe find a diffusion constant of $\\ensuremath{D_{\\text{CO}}=2.5\\:\\mathrm{nm}^{2}\\mathrm{ns}^{-1}}$,\nwhich is comparable to the experimental value which is in the range\nof $D_{\\text{CO}}=2.03\\:\\mathrm{nm}^{2}\\mathrm{ns}^{-1}$ (at $20\\:C\\text{\\textdegree}$)\nto $D_{\\text{CO}}=2.43\\:\\mathrm{nm}^{2}\\mathrm{ns}^{-1}$ (at $30\\:C\\text{\\textdegree}$)\n\\citep{wise1968diffusion}.}\n\n\\begin{figure}\n\\centering\n\n(a) \\includegraphics[width=0.8\\columnwidth]{figs\/myoglobin\/myoglobin_centers_pbc_high_contrast}\n\n(b)\\includegraphics[width=0.8\\columnwidth]{figs\/myoglobin\/impliedTimescales_myoglobin_pbc}\n\n(c)\\includegraphics[width=0.8\\columnwidth]{figs\/myoglobin\/reactionRateFit_pbc}\n\n\\caption{\\textcolor{black}{Discretization and results of the CO-myoglobin system.}\\textbf{\\textcolor{black}{{}\n(a)}}\\textcolor{black}{{} Definition of the cores (wire frame spheres)\nwithin the myoglobin. The red sphere indicates the bound state. The\ngray spheres correspond to the states that were not in the connected\nset and therefore discarded. The blue dots are positions of the CO\nmolecules for every 50th frame in the vicinity of the protein. }\\textbf{\\textcolor{black}{b)}}\\textcolor{black}{{}\nImplied timescales of the dynamics of the CO myoglobin system. The\ndatapoints and shaded area denote the sample mean and standard deviation\nof the bootstrapping sample over the trajectories: from the 20 given\ntrajectories we resample 20 with replacement. Over this sample we\nrun our discretization process which returns a sample of timescales.\nThe trajectory-samples which are not ergodic or do not lead to a connected\ncount matrix are considered invalid and discarded. Solid lines are\nfound using the full dataset. }\\textbf{\\textcolor{black}{c)}}\\textcolor{black}{{}\nReaction rate as estimated from multiple simulations at different\nconcentrations.}}\n\n\\label{fig:Myoglobin}\n\\end{figure}\n\n\\subsubsection{Comparison of dynamic properties}\n\n\\textcolor{black}{As in the previous example, we compute the binding\nrate by sampling positions sampled uniformly in the RD domain and\nsimulating the MSM\/RD model until it reaches the bound state. For\neach concentration, 200 trajectories are run to estimate the binding\nrate $k_{\\mathrm{on}}^{*}$. These rates are plotted against the concentration\nand shown in Fig. \\ref{fig:Myoglobin}c. The reaction rate $k_{\\mathrm{on}}=57_{52}^{62}\\:\\mu\\mathrm{M}^{-1}\\mathrm{s}^{-1}$\nis obtained as the slope of the linear fit. For the unbinding rate,\nwe start simulations in the bound state and collect MFPTs for leaving\nthe MSM domain; we find a rate of $k_{\\mathrm{off}}=19.0_{18.8}^{19.2}\\:\\mu\\mathrm{s}^{-1}$.\nThe resulting equilibrium constant $K_{\\mathrm{eq}}=k_{\\mathrm{on}}\/k_{\\mathrm{off}}=3.0_{2.7}^{3.3}\\:\\mathrm{M}^{-1}$\nis similar to $3.6\\,\\mathrm{M}^{-1}$ found by de Sancho et al. \\citep{deSancho2015identification},\nboth of which are close to the experimental value of $2.2\\:\\mathrm{M}^{-1}$\\citep{carver1990analysis}\n(see Tab. \\ref{tab:Myoglobin_kinetics} for comparison). The binding\nrate and unbinding rate found by de Sancho et al. \\citep{deSancho2015identification},\nalthough yielding a similar equilibrium constant, are both nearly\nan order of magnitude faster than the ones obtained with MSM\/RD (Tab.\n\\ref{tab:Myoglobin_kinetics}). The first indication that the present\nrates are an improved estimate is the fact that the kinetics (both\nthe MSM relaxation timescales and $k_{\\mathrm{on}}$) are independent\nof the lag time (Fig. \\ref{fig:Myoglobin}b, c). }\n\n\\textcolor{black}{To validate that the MSM\/RD estimates of $k_{\\mathrm{off}}$\nand $k_{\\mathrm{on}}$ have been estimated without significant bias,\nit must be shown that they are statistically consistent with the ground\ntruth (in this case a sufficiently large and sufficiently long MD\nsimulation). Here, $k_{\\mathrm{off}}$ can be estimated directly by\ncounting the frequency of ligand dissociation events from the binding\npocket in the underlying MD simulations. Since there are not sufficient\nfull dissociation pathways from the bound to the dissociated states\nin the MD data in order to make a statistically relevant comparison,\nwe obtain a more precise estimate by computing the MFPT using an MSM\ndirectly constructed from the original MD data with the same discretization\nas used in the MSM\/RD model. This resulted in a reference estimate\nof $23.4_{11.6}^{46.6}\\:\\mu\\mathrm{s}^{-1}$ (95\\% percentile computed\nwith 1000 bootstrap samples), which is consistent with the MSM\/RD\nestimate (Tab. \\ref{tab:Myoglobin_kinetics}).}\n\n\\textcolor{black}{Unfortunately, this method is not as accurate for\nthe binding rate $k_{\\mathrm{on}}$, which is notoriously difficult\nto estimate from small MD simulation boxes, where the length of trajectory\nsegments in which the ligand stays in the dissociated state without\ntouching the protein or crossing the periodic boundary are short compared\nto lagtimes $\\tau$ used in an MSM approach, resulting in biased estimates\n\\citep{PlattnerEtAl_NatChem17_BarBar}. Therefore, we performed another\nMyoglobin MD simulation in an eightfold larger periodic box (edge\nlength $10\\,\\mathrm{nm}$) with the same CO concentration as in the\nsmall MD simulation (resulting in $160$ CO molecules) for a total\nsimulation time of $405\\:\\mathrm{ns}$. For this data, a direct MSM\nestimate of the binding rate yields $74.7_{29.9}^{130.9}\\:\\mu\\mathrm{M}^{-1}\\mathrm{s}^{-1}$\n(95\\% percentile computed with 1000 bootstrap samples). As a result,\nthe MSM\/RD binding and dissociation rates are consistent with standard\nestimates computed directly from MD simulation, and the MSM\/RD modeling\nerror can be concluded to be statistically insignificant.}\n\n\\textcolor{black}{Given the consistency of the model,  we also compare\nthe results to experimental measurements, which is essentially a test\nof the MD model (e.g. force field, thermostat, integrator). These\nare yet a factor 4-5 slower than our estimates ($k_{\\mathrm{on}}=12\\:\\mu\\mathrm{M}^{-1}\\mathrm{s}^{-1}$\nand $k_{\\mathrm{off}}=5.3\\:\\mu\\mathrm{s}^{-1}$ found in \\citep{carver1990analysis}),\nconfirming that the major part of the difference between the estimates\nin \\citep{deSancho2015identification} and theexperimental values\ncould be removed by the fact that MSM\/RD is a significantly more accurate\nmodel of the binding kinetics. }\n\n\\begin{table*}\n\\begin{tabular}{|c|>{\\centering}p{2cm}|>{\\centering}p{3cm}|c|>{\\centering}p{2cm}|>{\\centering}p{2cm}|}\n\\hline \n & MSM\/RD  & Reference (approx. ground truth) & MSM in \\textcolor{black}{\\citep{deSancho2015identification}} & Experiment \\textcolor{black}{\\citep{carver1990analysis}} & Unit\\tabularnewline\n\\hline \n\\hline \n$k_{\\mathrm{on}}$ & $57.0_{52.0}^{62.0}$ & $74.7_{27.9}^{130.9}$ & 647 & $12$ & $\\mathrm{M}^{-1}\\mu\\mathrm{s}^{-1}$\\tabularnewline\n\\hline \n$k_{\\mathrm{off}}$ & $19.0_{18.8}^{19.2}$ & $23.4_{11.6}^{46.6}$ & 179 & $5.3$ & $\\mu s^{-1}$\\tabularnewline\n\\hline \n$K_{\\mathrm{eq}}$ & $3.0_{2.7}^{3.3}$ & $3.19_{2.6}^{3.8}$ & $3.6$ & $2.2$ & $\\mathrm{M}^{-1}$\\tabularnewline\n\\hline \n\\end{tabular}\n\n\\caption{\\textcolor{black}{\\label{tab:Myoglobin_kinetics}Rates and equilibrium\nconstants for Myglobin-CO estimated from different methods. The reference\nvalues approximate the ground truth by conducting a standard MSM-based\nMFTP estimate from the MD simulation (for $k_{\\mathrm{on}}$ a larger\nsimulation box was used to allow for a generous definition of the\ndissociated state).}}\n\\end{table*}\n\n\\section{Conclusion\\label{sec:Conclusion}}\n\nWe introduced and developed the MSM\/RD scheme, which couples MD-derived\nMSMs with RD simulations. We showed an implementation for protein-ligand\nsystems and applied it to two simple systems. The main advantage of\nthe algorithm is that it can simulate large time- and lengthscales\nwhile conserving molecular resolution and computational efficiency.\nThis is achieved by extracting the characteristic features of the\ndynamics fro\\textcolor{black}{m several short MD simulations into\nan MSM, which can produce new data with great accuracy and at a much\nfaster rate than the original MD simulations. This is a clear advantage\nin comparison to previous works, like \\citep{vijaykumar2015combining,VijaykumarEtAl_Arxiv16_AnisotropicMultiscaleGFRD},\nsince it does not require running MD simulations every time two particles\nare close to each other. It can further yield more accurate binding\nrates than traditional MSM methods by extending the diffusion domain\navailable, lessening the periodic boundary effects and increasing\nthe lifetime of the dissociated state. The scheme can be, in principle,\ncoupled to any RD scheme, like over-damped Langevin dynamics, Langevin\ndynamics, GFRD \\citep{van2005green,ZonTenWolde_PRL05_GFRD} and FPKMC\nalgorithm \\citep{donev2010first}, which could yield additional efficiency\nand accuracy or even incorporate long-range hydrodynamic interactions.}\n\n\\textcolor{black}{We first implemented the MSM\/RD scheme for a simple\nligand diffusion model (Sec. \\ref{subsec:3D-diff-pot}), which served\nto verify the scheme. It reproduced the expected dynamics and binding\/unbinding\nrates of the reference simulation. It was also able to generate an\naccurate MSM for the internal dynamics with a relatively small amount\nof data, which hints that it is feasible to extract the characteristic\ndynamics of a computationally feasible amount of MD simulations. Moreover,\nwe implemented the MSM\/RD scheme for the binding of CO to myoglobin\nsystem. After successfully extracting a self-consistent MSM and a\ncoupling scheme, we found that the equilibrium constant is consistent\nwith previous experimental and computational results \\citep{carver1990analysis,deSancho2015identification}.\nWe also showed that the MSM\/RD estimates are consistent with the underlying\nMD simulations \u2013 in particular our estimated association rate is consistent\nwith the association rate estimated from a reference MD simulation\nconducted in a large simulation box that was not used to parametrize\nthe MSM\/RD model. This is a significant improvement over Ref. \\citep{deSancho2015identification},\nwhere tenfold higher rates were estimated.}\n\n\\textcolor{black}{The MSM\/RD theory we introduced provides the framework\nupon which schemes for more complex systems can be constructed. In\nparticular, the next steps are to include association of two macromolecules,\nwhich may require to account for rototranslational diffusion, and\nthe coupling between protein-ligand association and conformational\nchanges. With the addition of these features, biologically relevant\nscenarios can be simulated. For example, if conformational changes\nof the protein are rare events and have different ligand association\n\/ dissociation rates, then the conformational dynamics and the ligand\nbinding dynamics are nontrivially coupled at high ligand concentrations\n\u2013 see \\citep{PlattnerNoe_NatComm15_TrypsinPlasticity} for the example\nof Trypsin and Benzamidine. A biological relevant example is the activation\nof the Calcium sensor Synaptotagmin in neuronal synapses \\citep{Suedhof_Neuron13_Neurotransmission}.\nHere, a locally very high Calcium concentration is created by the\nopening of voltage-gated Calcium channels as a response to an electric\nsignal. Synaptotagmin then binds up to five Calcium ions while going\nthrough different conformations, while the local Calcium concentration\nis reduced by diffusion. If Synaptotagmin successfully binds enough\nCalcium ions and transitions into an active conformation, it can catalyze\nthe fission of neuronal vesicles, which transduces the signal to the\npostsynaptic side. Such scenarios can be simulated with MSM\/RD simulations,\nin which the channels, the Synaptotagmin proteins and the ions are\nresolved as individual particles, and the binding\/dissociation kinetics\nand conformational changes of Synaptotagmin is encoded in an MSM.}\n\n\\textcolor{black}{MSM\/RD could be extended to deal with higher-order\nreactions. The most direct approach is to treat interactions of order\n2, 3, etc., by different MSMs which are then coupled in a regular\nMSM\/RD framework. The question then is how the higher-order MSMs are\nobtained. The brute-force approach would be to simulate the dynamics\nbetween three or more molecules with MD \u2013 e.g. with the help of enhanced\nsampling methods \u2013 and to extract corresponding higher-order MSMs.\nA cheaper, but approximate approach would be to ignore coupling between\ndifferent states and assume that multiple ligands can bind and transition\nbetween binding sites independently, perhaps except for multiple occupation\nof the same binding site. Based on such an assumption, higher-order\nMSMs could be constructed by tensor products of MSMs with one protein\nand one ligand. In practice, conducting }\\textcolor{black}{\\emph{some}}\\textcolor{black}{{}\nbut not all higher-order simulations and combining them to a generative\nmodel via machine learning methods may present a feasible pathway.}\n\n\\textcolor{black}{Finally, when considering protein interactions at\nhigh concentrations, the diffusion dynamics and long-range interactions\nof proteins are expected to be more complicated and involve hydrodynamic\neffects and anomalous diffusion. To include such effects, appropriate\ndynamical schemes should be included in the RD part.}\n\n\\textcolor{black}{In future developments, we will extend the MSM\/RD\nscheme to address these issues; however, it should be acknowledged\nthat some of these extensions come with their own set of challenges\nthat are not trivial to address.}\n\n\\section*{Acknowledgments}\n\nWe gratefully acknowledge support by the Deutsche Forschungsgemeinschaft\n(grants SFB1114, projects C03 and A04), the Einstein Foundation Berlin\n(ECMath grant CH17) and the European research council (ERC starting\ngrant 307494 \\textquotedbl{}pcCell\\textquotedbl{}). David De Sancho\nwas supported by grants CTQ2015-65320- R and RYC-2016- 19590 from\nthe Spanish Ministry of Economy, Industry and Competitiveness (MINECO).\nWe also thank Tim Hempel and Nuria Plattner for helpful discussions\nand software tutorials.\n\n\\section*{Appendix: MSM\/RD scheme for Sec. \\ref{sec:MSM\/RD-implementation}\\label{sec:Appendix:MSM\/RDscheme}}\n\n\\textcolor{black}{Based on the estimated quantities defined in the\nSec. \\ref{sec:MSM\/RD-implementation}, we introduce an implementation\nof the MSM\/RD algorithm from Sec. \\ref{subsec:MSM\/RD-coupled}.}\n\n\\texttt{\\textcolor{blue}{\\noindent}}\\texttt{\\textcolor{black}{Input: Initial\nmode (RD or MSM), initial condition (coordinates $\\mathbf{c}_{0}$\nor state $s_{0}$, respectively) and $t=0$:}}\n\n\\texttt{\\textcolor{black}{While $t\\leq t_{\\mathrm{final}}:$}}\n\\begin{enumerate}\n\\item \\texttt{\\textcolor{black}{If in RD mode:}}\n\\begin{enumerate}\n\\item \\texttt{\\textcolor{black}{Propagate $\\mathbf{c}_{t}\\rightarrow\\mathbf{c}_{t+\\tau_{\\mathrm{RD}}}$\nby diffusion }}\n\\item \\texttt{\\textcolor{black}{Update time $t\\mathrel{{+}{=}}\\tau_{\\mathrm{RD}}$}}\n\\item \\texttt{\\textcolor{black}{If $r_{AB}(\\mathbf{c}_{t})<R$ (enter MSM\ndomain):}}\n\\begin{itemize}\n\\item \\texttt{\\textcolor{black}{Select trajectory from }}\\textcolor{black}{$L_{\\mathrm{entry}}=\\{\\mathbf{c}_{\\mathrm{entry}},\\mathbf{x}_{\\mathrm{end}},\\Delta\\}$}\\texttt{\\textcolor{black}{{}\nwith $\\mathbf{c}_{\\mathrm{entry}}$ closest to $\\mathbf{c}_{t}$}}\n\\item \\texttt{\\textcolor{black}{If $\\mathbf{x}_{\\mathrm{end}}$ is a state:}}~\\\\\n\\texttt{\\textcolor{black}{Map to state $s_{t+\\Delta}=\\mathbf{x}_{\\mathrm{end}}$}}~\\\\\n\\texttt{\\textcolor{black}{Update time $t\\mathrel{{+}{=}}\\Delta$}}~\\\\\n\\texttt{\\textcolor{black}{Switch to MSM mode}}\n\\item \\texttt{\\textcolor{black}{If $\\mathbf{x}_{\\mathrm{end}}$ are coordinates:}}~\\\\\n\\texttt{\\textcolor{black}{Map to coordinates $\\mathbf{c}_{t+\\Delta}=\\mathbf{x}_{\\mathrm{end}}$}}~\\\\\n\\texttt{\\textcolor{black}{Update time $t\\mathrel{{+}{=}}\\Delta$}}\n\\end{itemize}\n\\end{enumerate}\n\\item \\texttt{\\textcolor{black}{Else (MSM mode):}}\n\\begin{enumerate}\n\\item \\texttt{\\textcolor{black}{If $s_{t}\\neq s_{t-\\tau_{MSM}}$or previous\nmode $\\neq$ MSM mode:}}\n\\begin{itemize}\n\\item \\texttt{\\textcolor{black}{Sample exit event with }}\\textcolor{black}{$P_{\\mathrm{exit},s}$}\\texttt{\\textcolor{black}{{} }}\n\\item \\texttt{\\textcolor{black}{If exit event:}}~\\\\\n\\texttt{\\textcolor{black}{Uniformly select trajectory from }}\\textcolor{black}{$L_{\\mathrm{exit},s}=\\{\\mathbf{c}_{\\mathrm{exit}},\\Delta\\}$}\\texttt{\\textcolor{black}{}}~\\\\\n\\texttt{\\textcolor{black}{Map to coordinates $\\mathbf{\\mathbf{c}_{\\mathrm{t+\\Delta}}=c}_{\\mathrm{exit}}$}}~\\\\\n\\texttt{\\textcolor{black}{Update time $t\\mathrel{{+}{=}}\\Delta$}}~\\\\\n\\texttt{\\textcolor{black}{Switch to RD mode and break current loop\niteration}}\n\\end{itemize}\n\\item \\texttt{\\textcolor{black}{Propagate $s_{t}\\rightarrow s_{t+\\tau_{\\mathrm{MSM}}}$\nusing}}\\textcolor{black}{{} $\\mathbf{T}(\\tau_{\\mathrm{MSM}})$}\n\\item \\texttt{\\textcolor{black}{Update time}}\\textcolor{black}{{} $t\\mathrel{{+}{=}}\\tau_{\\mathrm{MSM}}$.}\n\\end{enumerate}\n\\end{enumerate}\n\\textcolor{black}{The diffusion in the RD domain is done using a Euler-Maruyama\ndiscretization of Eq. (\\ref{eq:COM_diffusion}) \\citep{higham2001algorithmic}.\nNote the diffusion step can be simulated more efficiently with event-based\nalgorithms, like FPKMC or eGFRD \\citep{donev2010first,van2005green,takahashi2010spatio,vijaykumar2015combining}\nfor systems with low particle concentrations. In order to optimize\nthe efficiency of the algorithm, the entry points of entry trajectories\nare classified into equal area bins on the sphere. This allows the\nalgorithm to find the closest trajectory to a given entry point more\nefficiently. The partition of the sphere was done following \\citep{leopardi2006partition}.}\n\n\\bibliographystyle{unsrtnat}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{S:1}\n\nAs early as the 2008 Beijing Olympic Games, the concept of Smart Sports Hall was put forward, which is the choice of sports modernization development and the choice of deepening scientific and technological innovation. Increasing the construction of intelligent stadiums can better meet the needs of sports competitions on the site. In 2014, China's State Council issued a No. 46 documents to encourage national fitness and vigorously build public sports infrastructure. With the continuous improvement of people's requirements for the quality of life, the traditional stadiums and gymnasiums only used for competition and sports can no longer meet people's needs. The emergence of intelligent stadiums and gymnasiums is an inevitable trend. An intelligent stadium system has a variety of functions, such as live events, sports data collection, and analysis, spectator navigation, and spectator statistics \\cite{wang2016study}. Therefore, the intelligent stadium system based on position tracking can collect athletes' data faster, which significantly promotes sports information. It is of great practical significance to construct an intelligent gymnasium that can improve the scientific nature of physical education. First of all, the use of intelligent facilities can improve the scientific nature of sports. Secondly, sports venues are important places for all kinds of athletic activities. Compared with traditional sports venues, intelligent sports venues have inherent advantages in order management, risk protection, and other aspects, which can improve the order and safety of sports. Finally, intelligent technology can improve the use efficiency of sports venues, the utilization rate of physical education resources, reduce the waste of resources, and provide students with personalized physical exercise guidance and guide students to carry out sports activities with higher efficiency.\n\n\nLocation tracking is an essential problem in intelligent stadiums. At present, there are a lot of indoor positioning technologies based on different algorithms and technologies that have been proposed, such as ultra-wide bandwidth (UWB) positioning, Bluetooth positioning, WiFi positioning, etc. \\cite{zhu2020indoor}. Ubisense proposed a UWB indoor positioning technology that is not affected by line-of-sight (LoS) and has high positioning accuracy \\cite{s16050707}, but the hardware cost is very high. IBeacon is a protocol that can be used for indoor positioning systems proposed by Apple. It has low power consumption and comprehensive coverage, but it is easy to receive interference, leading to RSSI instability \\cite{jianyong2014RSSI}. Microsoft proposed location fingerprint positioning technology based on WiFi. With the help of wireless local area network (WLAN) infrastructure, mobile intelligent terminal positioning can be realized without additional special equipment and only needs positioning software. \n\n\n\nAfter years of construction, most large stadiums have completed the construction of a WiFi wireless network. WiFi technology provides an environmental basis for the application and development of intelligent sports construction. WiFi signal is undoubtedly a cheap and effective solution to complete positioning compared with other indoor positioning methods, such as sensors or Bluetooth, which often require additional equipment \\cite{yang2012locating}. However, there are still many problems with intelligent stadium WiFi location technology. When serving athletes with fast movement speed and uncertain direction, receiving signal strength indicator (RSSI) of the access point (AP) with long-distance and unstable signal tends to fluctuate greatly \\cite{jun2013social}. At the same time, in order to ensure positioning accuracy, the equipment often needs a faster sampling speed \\cite{shu2015gradient} to ensure that there will be no noticeable calculation delay in the process of rapid data collection. \n\n\n\n\n\\subsection{Related work}\n\\label{S:2}\nAt present, indoor positioning technology based on WiFi is most suitable for large-scale use in intelligent stadiums. Wireless infrastructure has been widely deployed in all indoor places. The two key technologies of existing WiFi fingerprint locations are fingerprint database construction and fingerprint matching. The location design and collection of the initial fingerprint database usually adopt two modes. One is to locate reference points, and the corresponding fingerprint information through intensive collection of fingerprints in the location scene \\cite{serodio2012lightweight}. Second, the location and fingerprint information of representative and relatively sparse reference points are collected. Then the fingerprint database is reconstructed by interpolating or predicting fingerprint information of other locations in the positioning scene through a particular model \\cite{guo2019robust,nikitaki2013localization}. A method using user feedback and rating mechanism to update the fingerprint database is proposed, but it can not effectively identify valid information and error feedback \\cite{luo2017smartphone}. Moreover, a crowd-sourcing fingerprint database updating method selectively extracts reliable information through a clustering algorithm to update fingerprint data, which is an effective method to improve the quality of the fingerprint database \\cite{niu2015wicloc}.\n\n\nFor WiFi indoor positioning, the mainstream positioning method is the classical k-nearest neighbor (KNN), and its variants, such as weighted k-nearest neighbor (WKNN) \\cite{miao2014novel,lin2014group}. The location effect of these algorithms depends on data acquisition. In addition, even for a very accurate database, these methods tend to rudely discard all data except the predicted location and its neighbors, leading to inaccurate positioning. Although these methods ensure low computation and real-time, the utilization efficiency of the database is very low, and the accuracy is also limited. \n\\par\nSince the nature of the localization problem still belongs to the classification problem, the neural network will become potential direct competitors \\cite{abdullah2016k,chen2017indoor}. The intuitive idea is that a three-dimensional image can be constructed from the X-axis, Y-axis, and WiFi signal intensity within the physical plane. By processing the database in the way of image processing \\cite{zhang2017discriminative} and taking the positioning position as the classification result, each classification output can make use of other data of the database, thus ensuring high precision \\cite{zhang2017elastic-net}. The self-organizing map SOM neural network further eliminates the randomness of the mapping, provides the stability of the algorithm, and at the same time ensures the low delay of the algorithm \\cite{zhang2018deterministic}. Although there is no application of SOM in indoor positioning, it will be a very potential solution if a sufficient amount of special database information in specific scenarios such as smart stadium is provided to enhance SOM's generalization ability. But for the trained neural network, the dimension of input and output is fixed, which determines the upper limit of the algorithm to a certain extent. Input imprecise RSSI will reduce accuracy, also difficult to reduce the amount of computation. The positioning accuracy entirely depends on the output dimension. For a large stadium, the system must ensure a very high output dimension and a very high delay or accept a very low accuracy.\n\\par\nUnlike the previous two algorithms, I-WKNN combines the specific signal distribution model and proposes asymmetric Gaussian filtering to improve the generalization ability. As an algorithm that can meet the requirements of real-time and high precision, it obtains lower delay by discarding some useless information and improving the database's utilization rate to ensure higher accuracy.\n\n\n\n\\subsection{Contributions}\nThe main contributions of the paper can be summarized as follows:\n\\begin{itemize}\n    \\item This paper puts forward the application scheme of WiFi location in the sports field. As far as we know, this was the first time that indoor real-time positioning technology was applied to real-time positioning in intelligent stadiums.\n    \\item The scheme of AP signal selection is established. The scheme filters out the AP with an unstable signal value and corrects the relatively stable AP with a Gaussian filter.\n    \\item This paper proposes a new algorithm I-WKNN to improve the positioning accuracy and speed. This algorithm improves the speed and reliability of positioning, mainly by optimizing fingerprint collection methods and reducing the search space for fingerprint matching. \n    \\item The algorithm's performance is analyzed and compared from the perspective of delay and positioning accuracy. The comparison algorithm is the triangulation algorithm, the conventional WKNN algorithm, which is based on a fingerprint database.\n\\end{itemize}\n\n\\section{System Model}\n\\subsection{WiFi fingerprint positioning principle}\nThe principle of WiFi fingerprint positioning technology is mainly to match the test point's WiFi signal strength information with the geographic coordinates \\cite{8255985}. The strength of the WiFi signal, described by RSSI, monotonically decreases with distance. There are three stages in WiFi indoor positioning, the preparation phase, the offline training stage, and the real-time positioning stage. Note that the AP signal selection mechanism is introduced in the latter two stages.\n\nPreparation stage: Divide the stadium into blocks, a total of $M$ blocks, one block as a reference point and corresponding to a geographic coordinate. Moreover, $N$ WiFi devices are fixed indoors, and each WiFi device is an AP.\n\nOffline training stage: Developers select reference points and collect multiple RSSI values from different APs at each reference point. Then record the geographic coordinates of test points, the MAC addresses of APs, and the RSSI values as fingerprints \\cite{hu2018toward,hu2018experimental}. The RSSI value should be processed before real-time positioning \\cite{bisio2014trainingless}. More specifically, $\\overrightarrow{\\text{RSSI}}_{m}\\left( n \\right)$  represents the RSSI value of the ${{m}^{{th}}}$ reference point corresponding to the ${{n}^{{th}}}$ AP and $\\overrightarrow{r}_{m,n}$  represents the RSSI value at $S$ the time slot obtained by sampling the ${{n}^{{th}}}$ AP at the ${{m}^{{th}}}$ reference point. In the $S$ time slot, the WiFi signal strength RSSI values $\\overrightarrow{r}_{m,n}$ of $N$ AP signals are received, and then AP signals are selected in this stage at each reference point. In the $S$ time slot, the WiFi signal strength RSSI values $\\overrightarrow{r}_{m,n}$ of $N$ AP signals are received, and then AP signals are selected in this stage at each reference point. In the offline training stage, the AP signals with a high signal loss rate and strong signal jitter are eliminated first, and then the AP signals are corrected by asymmetric Gaussian filtering. After the selection of AP signals, the RSSI value $\\overrightarrow{\\text{RSSI}}_{m}\\left( n \\right)$  of $N$ APs corresponding to $M$ reference points in the stadium is obtained, and the value range of $N$ is $\\left[ 1,N \\right]$, and the value range of $M$ is $\\left[ 1,M\\right]$. The WiFi fingerprint of the ${{m}^{{th}}}$ reference point is stored in the fingerprint database together with $N$  RSSI value $\\overrightarrow{\\text{RSSI}}_{m}\\left( n \\right)$  \n and corresponding APs' MAC addresses. \n\nReal-time positioning stage: Users measure the RSSI value of $N$ AP signals in real-time at the current position in the stadium and then select the AP signals in the real-time positioning stage. In this stage, the AP signals with a high signal loss rate and strong signal jitter are eliminated first, and then the AP signals are corrected by asymmetric Gaussian filtering. After that, the RSSI value of the AP signal selected online is matched with the WiFi fingerprint recorded in the database. Then, the location estimation algorithm is run according to the matching results to estimate the user's geographic coordinates. The specific process is shown in the following subsection.\n\n\\subsection{A typical fingerprint location algorithm: WKNN}\n\nThe principle of WiFi fingerprint positioning technology matches the WiFi signal strength received by the test point with the geographic coordinates. The received WiFi signal strength is described by RSSI, which decreases monotonically with increasing distance. The WiFi indoor positioning is divided into two stages, and the specific process is shown in Figure~\\ref{f1}.\n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width=0.8\\linewidth]{f1.png}\n    \\caption{The Principle of WiFi Fingerprint Positioning.}\n    \\label{f1}\n\\end{figure}\n\nOffline training stage: The developers divide the indoor area into blocks, and there are $M$ blocks in total. One block is a reference point and corresponding to a geographic coordinate $\\left( {{x}_{i}},{{y}_{i}} \\right),\\text{ }i\\text{ }=1,2,\\cdots ,M$. Fixed $N$ WiFi devices indoors, and each WiFi device is an AP.  Multiple RSSI values for different APs are collected at each reference point. Then, the geographic coordinates of the reference point, the MAC address of APs, and the RSSI value are taken as the WiFi fingerprint of the reference point $\\left( {{x}_{i}},{{y}_{i}} \\right)$ is $\\{{{\\text{MAC}}_{1}},{{\\text{RSSI}}_{1}};\\ldots ;\\text{ }{{\\text{MAC}}_{N}},{{\\text{RSSI}}_{N}}\\}$, which is stored in the fingerprint database. \n\nReal-time positioning stage: The user (the object to be positioned) measure the RSSI  \\\\$\\{{{\\text{MAC}}_{1}}^{(1)},{{\\text{RSSI}}_{1}}^{(1)};\\ldots ;\\text{ }{{\\text{MAC}}_{N}}^{(1)},{{\\text{RSSI}}_{N}}^{(1)}\\}$, $\\{{{\\text{MAC}}_{1}}^{(2)},{{\\text{RSSI}}_{1}}^{(2)};\\ldots ; \\text{ }{{\\text{MAC}}_{N}}^{(2)},{{\\text{RSSI}}_{N}}^{(2)}\\}$, $\\cdots $, $\\{{{\\text{MAC}}_{1}}^{(t)},{{\\text{RSSI}}_{1}}^{(t)};\\ldots ;\\text{ }{{\\text{MAC}}_{N}}^{(t)},{{\\text{RSSI}}_{N}}^{(t)}\\}$ at time slot $t$ in real-time at the current position. After filtering, the RSSI will carry out Matching Algorithm with the WiFi fingerprint recorded in the database. The location estimation algorithm is run to estimate the user's geographic coordinates based on the Matching results.\n\nThen the WKNN algorithm is described in detail. At the time slot  $t$, analyze the geographical location of the stadium according to the positioning results of the last few moments. From the ${{i}^{{th}}}$ reference point in the fingerprint database, all of the physically adjacent reference points are selected, and the Euclidean geometric distance from them is calculated. The Euclidean geometric distance between the t moment and the ${{i}^{{th}}}$ reference point is denoted as $D(i,t)$,\n\\begin{equation}\nD(i,t)=||{{\\overrightarrow{\\text{RSSI}}}_{i}}-\\overrightarrow{\\text{RSSI}}_{u}^{^{(t)}}|{{|}_{2}},\n\\end{equation}\nwhere ${{\\left\\| \\centerdot  \\right\\|}_{2}}$ represents the 2-norm of the vector, ${{\\overrightarrow{\\text{RSSI}}}_{i}}$ is the signal strength value of the ${{i}^{{th}}}$ reference point, which is an n-dimensional vector with an immutable order. $\\overrightarrow{\\text{RSSI}}_{u}^{^{(t)}}$ is the signal intensity value collected by the user at time slot $t$, and the vector is also n-dimensional, and the order should correspond to the order of APs in the database. If some APs' RSSI values can not be measured, a small value $\\text{RSSI}_{\\min}$ is substituted (e.g. $-100$~dB). \n\nAfter obtaining the Euclidean geometric distance between the user and the ${{i}^{{th}}}$ reference point, we order the Euclidean distance values, leaving $K$ minimum values. Generally, the idea of weighted Euclidean algorithm in the Landmarc algorithm is adopted. The reciprocal of Euclidean distance square is used as the weighted factor, and the weight $w(k,t)$ of the reference point $i$ is denoted as,\n\n\\begin{equation}\nw(k,t)=\\left\\{ \\begin{matrix}\n   \\frac{1}{{{D}^{2}}(k,t)}\/\\sum\\limits_{i}{\\frac{1}{{{D}^{2}}(i,t)}}, & i\\text{ is selected},  \\\\\n   0, & \\text{otherwise},  \\\\\n\\end{matrix} \\right.\n\\end{equation}\nthe final positioning result is,\n\\begin{equation}\n({{x}^{(t)}},{{y}^{(t)}})=\\sum\\limits_{i}{\\text{w}(i,t) \\cdot ({{x}_{i}},{{y}_{i}})},\n\\end{equation}\nwhere $({{x}^{(t)}},{{y}^{(t)}})$ and $({{x}_{i}},{{y}_{i}})$ are all binary groups, representing the horizontal and vertical coordinate values in physical locations. However, in large venues, intelligent functions such as real-time positioning based on WKNN algorithm still face the problems of poor positioning accuracy and large positioning delay.\n\n\\subsection{Asymmetric Gaussian filtering}\n\nFor WiFi signal, if the distance between AP and user remains constant, the user receives a stable RSSI signal. However, when positioning in the stadium, the WiFi signal will be subject to any interference such as noise and blocking by athletes, and the intensity of the AP signal will also fluctuate. Assuming that the noise at each moment is independent of each other, and has a similar distribution, the stability of the RSSI time average can be approximately described by the Law of Large Numbers,\n\\begin{equation}\n    \\underset{n\\to \\infty }{\\mathop{\\lim }}\\,\\mathbb{P} \\left[ \\left| \\frac{1}{n}\\sum\\limits_{i=1}^{n}{{\\overrightarrow{\\text{RSSI}}}_{\\text{i}}}-\\mu  \\right|<\\sigma  \\right]=1,\n\\end{equation}\nwhere $E({\\overrightarrow{\\text{RSSI}}})=\\mu ,\\text{ }D({\\overrightarrow{\\text{RSSI}}})={{\\sigma }^{2}}\\text{ }(i=1,2,\\cdots )$.  Under the finite sample, it can be approximately considered that if the sample value is less than $\\mu -3\\sigma $ or greater than $\\mu +3\\sigma $, it can not be accepted. At the same time, the occurrence probability of this event is only around 0.3 percent.\n\nRSSI is enhanced only because of the protocol in the measurement of the stadium signal, and it is not significant. However, because of complex scenes inside the stadium, depth decline often occurs, so it can be considered that the signal does not meet the Symmetry Gaussian distribution. The signal which is less than $\\mu -{{g}_{\\inf }}\\sigma $ or greater than $\\mu +{{g}_{\\text{sup}}}\\sigma $ should be eliminated, with g as the parameters to be determined. Therefore, the problem of Gaussian Filtering can be translated into,\n\\begin{equation}\n    \\left\\{ \\begin{matrix}\n   \\arg \\min  & {{g}_{\\text{inf}}}+{{g}_{\\text{sup}}}  \\\\\n   \\text{s}.t. & \\int_{\\mu -{{g}_{\\text{inf}}}\\sigma }^{\\mu +{{g}_{\\text{sup}}}\\sigma }{f(x)dx>1-\\text{ }\\!\\!\\epsilon\\!\\!\\text{ }}  \\\\\n\\end{matrix} \\right.,\n\\end{equation}\nwhere $\\epsilon$ is the allowable deviation of the algorithm, $f(.)$ is the probability density function of the sample.\n\nIn other words, the goal of asymmetric Gaussian filtering is to filter out as few signals as possible on the premise of tolerating errors. Since the problem is non-convex and requires extensive computation, this optimization problem is only used in the offline phase. In the online phase, we directly utilize the offline phase's solution results to reduce the solution time. \n\n\\section{I-WKNN Algorithm}\n\\subsection{Offline stage of the Algorithm}\n\nAssuming that there are S values of RSSI, they are obtained from sampling the ${{n}^{{th}}}$ AP at the $m$ reference point. They are denoted as ${{\\overrightarrow{r}}_{m,n}}$. In the offline stage, if the signal is lost seriously or fluctuates violently, the data of the source will be removed.\nWhen the signal cannot be measured, it will be replaced by a smaller value $\\text{RSSI}_{\\min}$. If the signal has not received data, the value will be zero after subtracting this value. The proportion of unreceived data can describe the degree of signal loss, that is to say, the ratio of 0-norm value to vector length in step (B). The elimination of the signal is replaced by $\\text{RSSI}_{\\min}$, which is equivalent to the data value when the signal is not received. The normalized variance in step (C) is used to describe the fluctuation. Asymmetric Gaussian filtering in step (D) eliminates partial signals that deviate entirely from the expected value. Then the AP selection algorithm in the offline stage is:\n{\n\\begin{algorithm}[h]\n    \\caption{AP selection in offline phase}\n   \n    \\begin{algorithmic}\n    \\STATE \\textbf{(A) Initialize:} $m \\leftarrow 1,n \\leftarrow 1$, initialize ${{\\Theta}_{1}}$ and ${{\\Theta }_{2}}$.\n    \n    \n    \\FOR{m = 1 \\textbf{to} M}\n    \\FOR{n = 1 \\textbf{to} N}\n    \n    \\STATE \\textbf{(B) Eliminated part of AP by the signal loss rate:} \n    \\IF{$\\left \\|{{\\overrightarrow{\\text{r}}}_{m,n}}+\\text{RSSI}_{\\min} \\right \\|_{0} \\geq S{{\\Theta }_{1}}$}\n    \\STATE ${{\\overrightarrow{\\text{RSSI}}}_{m}}(n) \\leftarrow \\text{RSSI}_{\\min}$.\n    \\ENDIF\n    \n    \\STATE \\textbf{(C) Eliminated part of AP by fluctuation:}\n    \\IF{$\\left \\|\\text{S}\\cdot {{\\overrightarrow{\\text{r}}}_{m,n}} - \\left \\| {\\overrightarrow{\\text{r}}_{m,n}} \\right \\|_{1}\\right \\|_{2} \\geq {{\\Theta }_{2}} \\left \\|{{\\overrightarrow{\\text{r}}}_{m,n}}\\right \\|_{1}$}\n    \\STATE ${{\\overrightarrow{\\text{RSSI}}}_{m}}(n) \\leftarrow \\text{RSSI}_{\\min}$.\n    \\ENDIF\n    \n    \\STATE \\textbf{(D) Perform asymmetric Gaussian filtering:}\n    \\IF{${{\\overrightarrow{\\text{RSSI}}}_{m}}(n)\\ne \\text{RSSI}_{\\min}$}\n    \\STATE ccording to criterion of asymmetric Gaussian filtering, eliminate the RSSI value which is less than $\\mu -{{g}_{\\inf }}\\sigma $ or larger than $\\mu +{{g}_{\\text{sup}}}\\sigma $ in ${{\\overrightarrow{\\text{r}}}_{m,n}}$, then take the mean of the subsectors as ${{\\overrightarrow{\\text{RSSI}}}_{m}}(n)$.\n    \\ENDIF\n    \n    \\ENDFOR\n    \\ENDFOR\n    \n\\end{algorithmic}\n\\label{a1}\n\\end{algorithm}\n\\par\nIn step (A), ${{\\Theta}_{1}}$ is the miss rate threshold,  and ${{\\Theta }_{2}}$ is the jitter peak average ratio threshold. $\\text{RSSI}_{\\min}$ in step (B) is a minimum preset value. ${{\\left\\| \\centerdot  \\right\\|}_{p}}$ is the p-norm of a vector, where $p=0,1,2$.\n}\n\n\\subsection{I-WKNN Algorithm with Its Online stage}\n\nIn the offline stage, the parameter selection of the Gaussian filter is related to the data distribution in the database. To some extent, it requires human intervention. If the filtering effect is not effective, the parameters need to be changed. In the offline stage, each reference point will have a set of parameters of asymmetric Gaussian filtering for each AP. For the sake of saving time, this set of parameters will be directly used in the online stage. \n\n{\n\\begin{algorithm}[h]\n    \\caption{AP selection in offline phase}\n   \n    \\begin{algorithmic}\n    \\STATE \\textbf{(A) Initialize:} $n \\leftarrow 1$, initialize the threshold ${{\\Theta }_{1}}$ and ${{\\Theta }_{2}}$.\n    \n    \\FOR{n = 1 \\textbf{to} N}\n    \\STATE \\textbf{(B) Eliminated part of AP by the signal loss rate:} \n    \n    \\IF{$\\left \\|   \\sum\\limits_{\\tau =t-T+1}^{t} {\\overrightarrow{\\text{RSSI}_{\\text{u}}^{\\left( \\tau  \\right)}}\\left( n \\right)}-\\text{RSSI}_{\\min}   \\right \\|_{0}\\leq T{{\\Theta }_{1}}$,$\\tau \\in [t-T+1,t]$}\n    \n        \\STATE $\\overrightarrow{\\text{RSSI}_{\\text{u}}^{\\left( t \\right)}}\\left( n \\right)=\\frac{1}{T}\\sum\\limits_{\\tau =t-T+1}^{t}{\\overrightarrow{\\text{RSSI}_{\\text{u}}^{\\left( \\tau  \\right)}}\\left( n \\right)}$,\n        \n        \\ELSE\n        \\STATE Abandoned the $n^{th}$ AP information in time $\\tau$.\n        \n    \\ENDIF\n    \\ENDFOR\n    \n    \\STATE \\textbf{(D) obtain the RSSI of $N$ APs:} \\\\ $\\overrightarrow{\\text{RSSI}_{\\text{u}}^{\\left( t \\right)}} \\leftarrow \\left( \\overrightarrow{\\text{RSSI}_{\\text{u}}^{\\left( t \\right)}}\\left( 1 \\right),\\overrightarrow{\\text{RSSI}_{\\text{u}}^{\\left( t \\right)}}\\left( 2 \\right),\\cdots ,\\overrightarrow{\\text{RSSI}_{\\text{u}}^{\\left( t \\right)}}\\left( N \\right) \\right)$.\n\\end{algorithmic}\n\\label{a2}\n\\end{algorithm}\nThe algorithm assumes that the measurements at $T$ moments before the current moment are still valid. At time $T$, the AP selection algorithm in the online stage is as follows: In step (B), $T$ is the impact time on the measurement results at the time slot $t$, $\\overrightarrow{\\text{RSSI}}_{\\text{u}}^{(\\tau )}\\left( n \\right)$ is the signal strength received by the user at the time moment $\\tau$, and the value range of $\\tau$ is $[t-T+1,t]$. Step (B) filters out part of the signal values according to the signal strength values, filter the remaining signal according to the criterion of asymmetric Gaussian filtering. The ${{n}^{{th}}}$ AP information received by the user at time slot $t$ after asymmetric Gaussian filtering is the mean value of filtered results after $T$ time slots before $t$ time slot.\n\n\n\\par\nIn addition to standard WKNN, the algorithm introduces the AP selection mechanism and asymmetric Gaussian filtering algorithm. The overall flow chart of the I-WKNN algorithm is shown in Figure~\\ref{f2}, containing the offline stage and online stage. \n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.8\\linewidth]{f2.png}\n    \\caption{Flow Chart for Positioning.}\n    \\label{f2}\n\\end{figure}\n}\n\n\n\\section{The Application in Intelligent Stadiums}\n\n\\subsection{Experimental Scene}\n\nIn this article, the intelligent stadium is based on the client\/server (C\/S) architecture, uses JAVA as the server development language, and MySQL database as the fingerprint database. Our server implements three main functions: it can efficiently complete the database storage and socket communication. \n\n\nThe typical attenuation of WiFi signal is caused by the superposition of signal propagation in space. RSSI can present the effect of dynamic distribution, and the dynamic change and the nature of the most common Gaussian distribution are exceptionally similar under the free space. In addition, when pedestrians are excluded, the collection of RSSI presents a Gaussian distribution, as shown in Figure~\\ref{f3}.\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.8\\textwidth]{f3.png}\n    \\caption{The Distribution of RSSI in An Ideal Situation.}\n    \\label{f3}\n\\end{figure}\n\nIn an ideal channel, the data obtained by RSSI sampling at the same place at different times is approximately Gaussian. Under multipath channels, the distribution with maximum information entropy is Gaussian distribution. Under the condition of known mean value and variance, the Gaussian distribution model is first introduced. However, in the actual scene, the RSSI of the WiFi signal will fluctuate, superposition, and disappear due to the influence of shielding, personnel movement, and multipath in the sampling process. Thus, instead of presenting as a Gaussian distribution, it is the double-peak situation in the actual sampling.\n\nDatabase and real-time measured RSSI information and geographical location are uploaded and processed by MATLAB software on the computer. We set up 250 points in the test area as clustering points, then numbered the position of each training point from right to left and from top to bottom. Run the WiFi location client program at each training point, measured each address and signal strength 1000 times. The data was processed, and after Gaussian filtering, the remaining records were written to the MySQL database.\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.8\\linewidth]{f5.jpg}\n    \\caption{The Result of Gaussian Filtering for One AP.}\n    \\label{f5}\n\\end{figure}\n\nThe result of Gaussian filtering for one AP is shown in Figure~\\ref{f5}. The blue data should be abandoned, and the red data will remain. From the signal distribution of the AP, the fading ratio is significantly higher than the enhancement ratio, and occasionally the signal cannot be detected, which is represented by $-70$~dB in this example.\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.8\\linewidth]{f6.jpg}\n    \\caption{The Proportion of RSSI for One AP.}\n    \\label{f6}\n\\end{figure}\n\nMoreover, we show the corresponding histogram of the proportion of RSSI in Figure~\\ref{f6}. It can be seen that through the Gaussian filter, the signal becomes noticeably more concentrated. Moreover, the adaptive Gaussian filter can better describe the current RSSI distribution of WiFi signals.\n\n\\subsection{The Performance of Accuracy}\n\nIn the online stage, we set $K=5$ and placed 10 APs. The device stores information for the last 20 slots. The following two comparison algorithms are given.\n\\begin{itemize}\n    \\item WKNN algorithm: No extra processing for fingerprint database, just find five largest RSSI values, and get the gravity center of the corresponding five points in every moment.\n    \\item KNN algorithm: A sample belongs to a category if most of the K most similar (that is, closest to each other in the feature space) samples in the feature space belong to that category.\n\\end{itemize}\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.8\\linewidth]{f7.jpg}\n    \\caption{The CDF Graph of Accuracy for Three Algorithms.}\n    \\label{f7}\n\\end{figure}\n\nThe CDF graph is used to observe the accuracy of each positioning since the result of each positioning is equally important and the positioning error of each positioning needs to be known. It can be seen from the Figure~\\ref{f7}, that the algorithm proposed in this paper is superior to the other two. Although the elimination of some APs seems to discard some information, the accuracy is greatly improved. In addition, it can be seen from the figure that the proportion of deviation of the four positioning algorithms below 2m is 95\\% (I-WKNN), 79\\% (WKNN), 39\\% (WKNN), and 46\\% (KNN), respectively. Similarly, the mean positioning deviations were 1.14~m (I-WKNN), 1.42~m (WKNN), and 2.32~m (KNN), respectively.\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.8\\linewidth]{f8.jpg}\n    \\caption{The PDF Graph of Accuracy for Three Algorithms.}\n    \\label{f8}\n\\end{figure}\n\nThe corresponding PDF of location error is shown in Figure~\\ref{f8}. The proposed method has the highest probability of location error of about 1~m, and the maximum error will not exceed 3~m. The error of WKNN is slightly larger than that of our method. The performance of KNN is the worst, with a maximum error of almost 6~m.\n\n\\subsection{The Performance of Time Delay}\n\nTo reduce the influence of various factors such as device heterogeneity and time uncertainty on the positioning accuracy, we used a single device to run the two algorithms and completed 40 positioning. The overall time-consuming and average time-consuming of the experiment are shown in Figure~\\ref{f9}. Because positioning speed is related to hardware and software performance, this chart is for reference only.\n\nUnlike the positioning accuracy, the time delay only needs to be lower than an acceptable value, and the samples with a significant delay should be paid special attention. So the bar chart shows the average delay, the best 20\\% experiment, and the worst 20\\% experiment, rather than every sample.\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.8\\linewidth]{f9.png}\n    \\caption{The Comparison of Average Time Delay for Three Algorithms.}\n    \\label{f9}\n\\end{figure}\n\nAs can be seen from the figure, the KNN method, which has a significant delay, is entirely unsuitable for the rapidly changing scene such as an intelligent stadium. For the best cases, the delay of the I-WKNN algorithm has the smallest time delay. In terms of average delay, I-WKNN has obvious improvement over traditional WKNN. For the worst 20 percent, I-WKNN is still a small advantage over WKNN.\n\n\\section{Conclusion}\n\nAn improved WKNN algorithm is proposed in this paper, called I-WKNN. The improved AP selection algorithm and asymmetric Gaussian filter algorithm optimize the offline and online stage of fingerprint location. In the experiment, the triangulation algorithm, the traditional WKNN algorithm, is used to compare the proposed one. The accuracy of I-WKNN has obvious advantages compared with the other three algorithms. Its average deviation is 1.14~m, and the proportion of the deviation lower than 2~m is 95\\%. I-WKNN algorithm is only worse than triangulation in the time delay, thus ranks second, with an average delay of 326~ms and a maximum delay of 432~ms, which can meet the requirements of rapid positioning of the stadium. It can be seen that in complex environments, it has high precision and fast positioning speed, which will make it suitable for the scene of the stadium.\n\n\n\n\n\\input{ccs.bbl}\n\\bibliographystyle{IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}