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+{"text":"\\section*{Background}\nIn the last decade or so there have been significant advances in developing new\nhigh throughput technologies such as microarray, ChiP-Chip or ChiP-seq~\\cite{lee2002}\nto uncover large scale cellular networks. Understanding these\nnetworks has been facilitated by decomposing these networks into smaller\nparts, so-called motifs, with known structure and function~\\cite{milo2002}.\nTwo major strategies currently exist for accomplishing this task. The\nfirst one, so called top-down analysis involves mining databases\nfor recurring patterns of interacting genes~\\cite{milo2002, shen2002, yeger2004}, or in the\ncase of signaling networks, proteins-protein interactions~\\cite{gerstein2006}. The emerging motifs are\nanalyzed for function, especially the specific roles they might serve in\na particular network. One interesting result from this work is the discovery of an enrichment in\nso-called feedforward motifs~\\cite{mangan2003a}, Fig. 1 (a), a three gene network.\nAlon and colleagues~\\cite{mangan2003a,mangan2003b,kalir2005,zaslaver2006}\nsubsequently demonstrated the functionality of these motifs using both\ntheory and experiment.\n\nAnother approach to understanding biological networks is the\nbottom-up approach~\\cite{guido2006}. Here knowledge of specific\nprotein-DNA interactions and signaling proteins has allowed\nresearchers to synthetically design gene regulatory networks. The\ndesign of these networks has resulted in the construction of a wide\nvariety of network motifs such as oscillators, switches, and logic\ngates~\\cite{andrianantoandro2006, kaernweissbook, entus2007}. One of the goals\nof synthetic biology is to design simple networks that can behave\nas modules. In recent years\nthere has been a resurgence of interest in the quantitative\nproperties of gene regulatory networks~\\cite{bolouri2002,kaern2003,alon2007}.\nIn addition, the ability to carry out relatively precise\nmeasurements of gene activity using synthetic biology\ntechniques~\\cite{andrianantoandro2006} and single cell\nmeasurements~\\cite{lahav2004,gevazatorsky2006} has led to considerable progress\nin this field.\n\nThe top-down and bottom-up approaches are two ways to generate\nfunctional modules. Based on a few network\nmotifs motivated by bottom-up approaches, we generated several novel functional network\nmotifs by exploring additional transcriptional control mechanisms and combinations of feedforward\nnetworks.\n\nOne of the principles of gene regulation is regulated recruitment,\nwhere transcription factors bind to the promoter regions of a gene,\nrecruiting other transcription factors and RNA Polymerase\n(RNAP)~\\cite{ptashne}. Furthermore, signaling molecules often\nactivate some of these transcription factors through binding\nor phosphorylation.\nThere are multiple ways by which these interactions\ntake place, such as blocking the promoter region, twisting\nthe DNA, Histone modification, and DNA looping, {\\it\netc}.~\\cite{bintu2005a}. In addition, the combinatorial complexity\nof multiple transcription factors that bind to the same promoter region\nwill also determine the transcription rate ~\\cite{buchler2003}.\nGiven all this complexity, a very useful approach to model these\nkinds of interactions in gene regulatory networks is the\nShea-Ackers approach~\\cite{sheaackers1982}. In this work, we used this method to\nconstruct several different kinds of network motifs.\n\nThe network motifs we will discuss can be broadly classified into\nthree categories. (1) Feedforward related motifs:\nobtained by additional feedbacks on the basic incoherent (type I)\nfeedforward network \\cite{milo2002}. (2) Tunable motif: This\nsubnetwork can either behave as an oscillator or a bistable switch\ndepending upon the concentration of a transcription factor. (3)\nAdjustable gates: This motif can switch between an AND and an OR gate\ndepending upon the concentration of a transcription factor. All\nthree network motifs arise from multiple transcription factor regulation\nat a particular gene. In designing these {\\it in silico} networks,\nwe didn't concern ourselves very much with exact mechanism such as DNA\nlooping {\\it etc.}, but rather focused on the architecture of the resulting\nnetwork.\n\nIn the next section, we briefly review the Shea-Ackers approach, which is used\nto compute the transcription rate of genes. Then, we will describe three different kinds\nof models, amplitude filters,\ntunable motifs, and adjustable gates. The latter two behave functionally\nvery differently depending upon the concentration of an input transcription\nfactor. A discussion and summary can be found in the last section.\n\n\n\n\\section*{Results and Discussion}\n\n\\subsection*{Modeling Gene Regulatory Networks Using Shea-Ackers Method}\n\nA common method for modeling protein-gene interactions is by using\nMichaelis-Menten or Hill function kinetics. However, in\nthe models we will describe in this paper, all kinetics will be\nbased on the Shea-Ackers\nformalism\\cite{sheaackers1982,bintu2005b}. This method estimates\nthe probabilities for different transcription states from\nwhich an overall rate of transcription is derived. The approach is\nless empirical and more flexible than the standard methods\nand enables one to easily incorporate repressers and activators\ninto the equation. In addition the formalism has a relatively\nstraightforward relationship to the stochastic representations of\ngene expression models. Finally, the free energy terms in the\nformalism can be easily changed by altering the promoter region\nwhich enables the models based on Shea-Ackers to be tested\nexperimentally.\n\nWe assume that the occupancy of the binding sites on promoters is\ngoverned by equilibrium statistical thermodynamics\nprobabilities~\\cite{sheaackers1982,bintu2005b,hill1960}. The\nprobabilities arise from the binding free energies, the free\nenergies of interaction between transcription factors bound to\nadjacent sites, and the concentrations of the participating\ntranscription factors and the RNA polymerase. This assumption\nholds when binding and unbinding\nto the promoter is rapid~\\cite{bintu2005b}. Slow binding\nhas been shown to result in stochasticity due to operator\noccupancy fluctuations~\\cite{kepler2001, hornos2005}, whereas for\nsmall concentrations, the individual birth and death of proteins\nintroduce noise into the system~\\cite{kaern2005}.\n\nWe illustrate the Shea-Ackers method with a given\ngene regulated by two transcription factors, $T_1$ and $T_2$.\nWe assume that these three proteins bind at three distinct sites on\nthe promoter, providing nine possible combinations of binding. Each combination\nis associated with a free energy that\nis proportional to the probability of this state\nand can be represented by the following equation ~\\cite{bintu2005b}:\n\\begin{equation}\nf = \\exp (-\\Delta G_f\/{k_B T})\\ {[T_1]}^{n_1} {[T_2]}^{n_2} [R]\n\\end{equation}\nwhere $\\Delta G_f$ is the free energy difference of the\nbound and unbound states, and the factors $[T_1]$ and [$T_2]$ are\nthe concentrations of transcription factors, $R$ is the RNAP\nconcentration and $n_1$ and $n_2$ are the number of monomers\nwhich combine into higher order multimers. $k_B$ and T are the\nBoltzmann constant and absolute temperature respectively. From\nEq (1) the normalized probability of a given state is then\ngiven by the ratio\n\\begin{equation}\n  Z_f = \\frac{\\exp (-\\Delta G_f\/{k_B T}) {[T_1]}^{n_1} {[T_2]}^{n_2} [R]}{\\sum_1^{9} \\exp (-\\delta G_f\/{k_B T}) \\ {[T_1]}^{n_1} {[T_2]}^{n_2} [R]}\n\\end{equation}\nIf we assume that the rate of transcription is proportional to the\nrelative probability when the polymerase is bound to the gene, then\nwe can partition the states into the\npolymerase bound state ($Z_{on}$) and the\npolymerase unbound state ($Z_{off}$)~\\cite{buchler2003}. The probability of gene expression is then:\n\\begin{equation}\nP=\\frac{Z_{\\mbox{\\scriptsize on}}}{(Z_{on}+Z_{\\mbox{\\scriptsize off}})}.\n\\end{equation}\nIf a transcription factor is an activator, its free energy to\nbind with the RNAP will be very low, and hence this\ninteraction will be favored. Whereas for an inhibitor, it would\nbe highly improbable for RNAP to be recruited for transcription.\nTherefore, the interactions between the transcription factors and RNAP\ndetermine the regulatory rules, which we will explore in the next\nsection for several different kinds of dynamical networks.\n\n\n\\subsection*{Feedforward Related Motifs}\n\nFirst we assume that the\ndynamics lumps together transcription and translation into one\nprocess. Although explicit modeling and experiment has been shown\nto give rise to interesting effects such as protein\nbursts~\\cite{mcadams1998, kaern2005} or oscillations as a result of transciption delay~\\cite{Hasty2008}, we believe that the main\nfeatures of our models are captured by protein-DNA interactions.\n\n\\subsubsection*{Incoherent Type I Feedforward Related Networks}\n\n\nHigh throughput approaches have uncovered several re-occurring\nmotifs termed the feedforward motif~\\cite{milo2002,shen2002}.\nFig. 1(a) shows a common transcription\nfactor for two genes, where the third gene is regulated in a\nfeedforward fashion.\nFig. 1(b) shows a\nprotein-protein interaction in addition to this scheme.\nFeedforward motifs can lead to two types of dynamics depending on\nthe nature of the two signals that converge on the third\ngene. Fig. 2 illustrates a functional\nrepresentation of the schematic shown in\nFig. 1(a)~\\cite{mangan2003a, mangan2003b}.\nIn this representation there are three different kinds of\ninteractions: protein degradation, protein synthesis and gene\nregulation. In Fig. 2 the input transcription\nfactor $p_1$ modulates the activity of a target gene (G2)\ndirectly and indirectly through the gene product $p_2$ of another\ngene (G1), which is also a transcription factor. The\ninteraction between $p_1$ and $p_2$ at the promoter region of G2\nand their ability to recruit RNAP determines the rate of\ntranscription of Gene G2. This type of architecture has been\nshown~\\cite{mangan2003b,kalir2005,zaslaver2006} to lead to two types\nof dynamics depending on the nature of the regulation that occurs\nat the target gene G2. If $p_1$, $p_2$ are both activators, the\ngene circuit acts as a low pass filter, i.e.\\ it is able to filter\nout transient signals and transcribe only when the input signal is\nlong lived~\\cite{mangan2003a}. If $p_1$, $p_2$ regulate G2 as an\nactivator and repressor respectively then the system can act as a\nbandpass filter, since the delayed response of $p_1$ through $p_2$\ntends to suppress activity of G2~\\cite{mangan2003b}. This has been\nsuggested to be a general mechanism for speeding up response times in\ntranscriptional networks~\\cite{mangan2003a, zaslaver2006}. Recently,\nit has also been argued that the steady state characteristics of\nsuch incoherent feedforward loops could be very important in\nestablishing spatial stripes and pulsed temporal expression profiles\nof transcription factors involved in developmental\nprocesses~\\cite{ishihara2005}.\n\n\n\n\\subsubsection*{Simple Feedforward}\n\nWe assume that $p_1$ activates G1 and G2, $p_2$, the\ngene product of G1, is recruited by $p_1$, and the protein complex\n$p_1 p_2$ acts as a repressor of G2 (we assume that $p_2$ cannot\nbind to G2 by itself). These assumptions lead to the following\nrate equations,\n\\begin{eqnarray}\n\\label{simpleFeedforwardEquation}\n\\frac {d [p_2]}{dt}&=& \\frac{a_1 + a_2 [p_1] }{1+ a_3 + a_4 [p_1]}-\\gamma_1 [p_2] \\\\ \\nonumber\n\\frac {d [p_3]}{dt}&=&\\frac{b_1 + b_2 [p_1] }{1+ b_3 + b_4 [p_1] + b_5 [p_1][p_2]}-\\gamma_2 [p_3],\n\\end{eqnarray}\nwhere the transcriptional rates, and the ``lumped'' parameters are\nderived in the appendix. $a_1, b_1$ represent leaky\ntranscription, which is due to the probability that RNAP can bind to\nthe operator region of the gene in the absence of a recruiting\ntranscription factor. At steady state, before G1 is saturated with\n$p_1$, $p_2$ is proportional to $p_1$. The transcription rate\nfor G2 (assuming negligible leaky transcription) can be approximated\nas,\n\\begin{equation}\nT \\propto \\frac{{b_1}^{\\prime} [p_1] }{1+ {b_2}^{\\prime} [p_1]+\n{b_3}^{\\prime} {[p_1]}^2}\n\\end{equation}\nThe transcriptional rate rises in proportion to $p_1$ for small $p_1$, and\nfalls in proportion to $1\/p_1$ for large values of $p_1$. Hence it\nreaches a maximum at some intermediate value of $p_1$.\n\n\nIn Fig. 3, we show the steady state values of\n$p_2$ and $p_3$ with respect to the input $p_1$. We note that the\nconcentration of $p_3$ has a maximum for a given value of $p_1$.\n\nAt low input concentration, $p_1$ transcribes G2 and G1, and hence\nas its input level increases, $p_2$ tends to grow. Recruitment of\n$p_2$ by $p_1$ at G2 makes it possible for the $p_1 p_2$ complex\nto halt further transcription of G2. This module is aptly named an\n``amplitude filter'' (originally called a ``band detector'',\n\\cite{basu2004, basu2005, kaernweissbook}, since its output is\nmaximal for a specific range of input. Such biphasic response has\nalso been discussed in other systems~\\cite{wolf2003, mayya2005}.\nRecently, Ishihara et. al.~\\cite{ishihara2005} discussed the band properties of\nsuch networks and used this to explain pulsed behavior and\npatterning in {\\it Drosophila} developmental processes.\n\n\\textsl{\\\\The effect of dimerization} \\pb\n\nIn this section we will consider the effect of dimerization of\n$p_2$, before it binds to the regulatory region of G2.\nDimerization has been shown to be important to\nreduce the effects of stochastic fluctuations~\\cite{bundschuh2003}\nin a feedforward regulatory scheme. Here we discuss the steady\nstate behavior of a network regulated by $p_1$, which\nrecruits a dimer of $p_2$, and this complex is a repressor at G2.\nThe only changes that need to be made to\nEq.~\\ref{simpleFeedforwardEquation} are to include the\ndimerization equations. These modified equations are,\n\n\\begin{eqnarray}\n\\frac {d [p_2]}{dt}&=& \\frac{e_1 + e_2 [p_1] }{1+ e_3 + e_4 [p_1]} -\\gamma_1 [p_2] -2 (k_{d1} {[p_2]}^2-k_{d2} [p_d]) \\\\ \\nonumber\n\\frac {d [p_3]}{dt}&=&\\frac{d_1 + d_2 p_1 }{1+ d_3 + d_4 p_1 +  d_5 p_1 p_d}-\\gamma_2 p_3 \\\\ \\nonumber\n\\frac {d p_d}{dt}&=& k_{d1} {[p_2]}^2 -k_{d2} [p_d] -k_{d3} [p_d] \\\\ \\nonumber\n\\end{eqnarray}\n\nwhere in addition to the formation and dissociation of the dimer\ncomplex, the dimer can also degrade. With these equations it is\neasy to compute the steady state values of $p_3$ with respect to\nthe input $p_1$. In Fig. 4,\nwe plot the steady state value of $p_3$ as a function of\n$p_1$ in the upper panel, which shows a much steeper fall off of the amplitude filter\ncurve, compared to the simple feedforward case discussed earlier.\nThe amplitude filter in this case, has a narrower bandwidth, and\nhence its filtering capabilities of the input transcription factor\n$p_1$ are much more enhanced. The sharp fall off is due to the\nquadratic dependence of the amount of dimer with respect to the\ninput; as the input $p_1$ increases the amount of available dimer\n$p_d$ to be recruited by $p_1$ at G2 increases, thus increasing\nthe amount of repression at G2. Dimerization of $p_1$ has the\neffect of increasing both the width and shifting the peak of the\namplitude filter curve.\n\n\n\\textsl{\\\\Effects of mutations at G1 and G2} \\pb\nWe now discuss the effects of two types of mutations at the binding\nsites at G1 and G2, for the simple feedforward with dimerization of\n$p_2$. Similar qualitative results can be obtained for the simple\nfeedforward and mixed feedforward. In general, a mutation at the binding\nsite tends to change the free energy of binding of the transcription factor,\ngenerally decreasing the binding affinity. In\nFig. 4(a), a mutation at G1 reduces the ability of $p_1$ to bind to it. Hence larger\namounts of $p_1$ are required to achieve the same transcription rate\nthereby shifting the amplitude filter peak to the right. In\nFig. 4(b), a mutation at G2 reduces the ability of $p_2$ to be recruited to G2.\nThis leads to a slower fall off, since repression does not\ntake place very efficiently. The two types of mutations can be used\nto engineer the shape of the amplitude filter, by changing its\nbandwidth and peak value~\\cite{entus2007}.\n\n\\subsubsection*{Mixed Feedforward Motifs}\n\nIn Fig. 5, we show a slightly different\nfeedforward scheme, which is a functional interpretation of\nFig. 1b. The gene product $p_2$ is\nshown as an activator of G2, whereas $p_1$ is a repressor of G2.\nAlso indicated in the figure is a protein-protein interaction,\nwhereby the input $p_1$ binds to $p_2$, and targets it for\ndegradation.\n\n\nWe assume that $p_1$ activates G1 leading to the production of\n$p_2$; $p_2$ can individually bind to G2 and act as an activator.\nFurthermore $p_1$ can be recruited by $p_2$ at the operator region\nof G2, and together this complex acts as a repressor. $p_1$ binds\nto $p_2$ and actively degrades it. One possible mechanism by which this can\noccur is if $p_1$ labels $p_2$ with ubiquitin molecules for\nproteolytic degradation~\\cite{alberts}. The above assumptions lead\nto the following rate equations,\\\n\\begin{eqnarray}\n\\frac {d [p_2]}{dt}&=& \\frac{c_1 + c_2 [p_1] }{1+ c_3 + c_4\n[p_1]}-\\gamma_c [p_1] [p_2] -\\gamma_1 [p_2] \\\\ \\nonumber \\frac {d\n[p_3]}{dt}&=&\\frac{d_1 + d_2 [p_2] }{1+ d_3 + d_4 [p_1] + d_5\n[p_2] + d_6 [p_1] [p_2]}-\\gamma_2 [p_3],\n\\end{eqnarray}\nwhere the transcriptional rates, and the ``lumped'' parameters are\nderived in the appendix. In the equation for $p_2$, the extra\ndegradation term is due to the protein-protein interaction between\n$p_1$ and $p_2$. For this system of equations the plots shown in\nFig. 6 show the behavior of the\nsteady state value of $p_3$, demonstrating the amplitude filter\neffect.\n\n\n\nAs $p_1$ increases, $p_2$ begins to grow and transcribe G2;\nhowever, two key factors prevent G2 from being continually\ntranscribed with further increases in $p_1$. The first is that\n$p_2$ is targeted by $p_1$ for degradation, and the second is that\n$p_2$ binds to G2 and recruits $p_1$, which turns off the\ntranscription.\n\nIn both the simple incoherent Type I feedforward, with\/without\ndimerization, as well as the mixed feedforward, we have shown how\nit is possible to obtain a basic amplitude filter. The property of\nsuch an amplitude filter can now be further exploited to generate\nnew types of networks when additional feedbacks are added to this\nbasic motif.\n\n\\textsl{\\\\Application 1: Time Ordering}\\pb\n\nThe amplitude filter or ''concentration detector'' has been shown\nto perform temporal processing functions such as pulse\ngeneration~\\cite{basu2004}. Pulse-like behavior has been simulated\nin~\\cite{ishihara2005}, with a series of cascaded feedforward\nloops, which in fact use the amplitude filter property of the\nnetworks to generate pulsatile behavior. We use a similar idea\nwhere a single transcription factor could serve as an\ninput to several amplitude filters, each of which has a different\nshape. In particular we assume that for each amplitude filter\nmodule, the peaks of $p_3$ occur at different\ninput transcription factor ($p_{1opt}$) concentration values.\nThen, as the input concentration crosses $p_{1opt}$, the amplitude filters get\nactivated in a sequence, depending on how far apart $p_{1opt}$ are\nfor different amplitude filters~\\cite{mangan2003b}.\n\n\nA single input can therefore activate several genes in a sequence. One\nexample could be the sequential release of different\nproteins required in a certain developmental process.\n\n\nFig. 8a plots $p_3$ as a function of the\ninput transcription factor concentration, showing a shifted peak for\nthe two amplitude filters with respect to the same input\ntranscription factor. In Fig. 8b, the\ntemporal profile for the outputs are shown assuming that the input\nconcentration is ramped up as a function of\ntime. This leads to an ordered protein production in time.\n\n\n Kashtan et. al.~\\cite{kashtan2004}, have explored the\nconsequences of multi-output networks regulated by\nfeedforward networks. They show through simulations that in some of\nthese cases feedforward loops with a common input can regulate\ngenes in a temporal order. Such temporal order can occur in\nmulti-output feedforward loop systems such as in the {\\it E. coli}\nflagellar synthesis regulation systems, where proteins need to\nassembled in a timed fashion to make up the flagellar basal-body\nmotor~\\cite{kalir2001}. It has also been found\nin~\\cite{eichenberger2004}, that the logic of the program of gene\ntranscription during differentiation in {\\it Bacillus Subtilis}\nsporulation involves a series of feedforward loops that generate\ngene transcription in a pulse like manner.\n\n\n\\textsl{\\\\Application 2: Homeostatic Networks}\\pb\n\nHomeostatic networks are important in several biological systems.\nOne example in which homeostasis has been shown to occur through\nintegral feedback control is in the chemotaxis network in {\\it E.\ncoli} (\\cite{yidoyle2000}). Our motivation was to design a network\nusing the amplitude filter which would display homeostasis to\ninput perturbations.\n\nUsing the output of the amplitude filter module as an input to\nitself, one can construct the motif shown in\nFig. 9(a), where the filter module's output\n$p_3$ participates as a transcription factor for gene G1, whose\nprotein product $p_1$ serves as the input\nto the filter module. The feedback could be positive or negative, depending on whether the\namplitude filter output $p_3$ is an activator or repressor of G1.\nIn addition, an external input protein A can bind as an\nactivator to gene G1. Depending on the interaction between\nthe input control A, the feedback $p_3$, and RNAP, we obtain\ndifferent types of behavior. Consider $p_3$ as an activator of\nG1 in Fig. 9(a), we assume that the\ninteraction between $A$, $p_3$ and RNAP is such that we obtain an\n{\\bf AND} gate at G1, {\\it i.e.} G1 expresses only when both $A$\nand $p_3$ are present. The equation that describes the\nadditional variable, $p_1$, is given by\n\\begin{eqnarray}\n\\frac{d p_1}{d t}&=&\\frac{u_1+u_2 [A][p_3]}{1+u_3 + u_4 [A]+u_5\n[p_3]+u_6 [A][p_3]}-\\gamma_3 p_1\n\\end{eqnarray}\nwhich is the equation for the input of the amplitude filter, and the\nequation for $p_3$, the output of the amplitude filter, is the same as in\nEq 6. The steady state plot for the output $p_3$ as a function of the\ninput A, is shown in Fig. 10(a).\n\n\n\n\nFigure 10 shows that the steady state values of\n$p_3$ are constant even for large inputs. As the input A of\nthe filter module increases, the output $p_3$ decreases (assuming\nthat $p_3$ is maximal at the initial value of A). This is because\nthe input transcription factor concentration moves away from\n$p_{1opt}$, at which the maximal value of transcription occurs.\nTherefore, this decreases the transcription of the filter module,\nand since its output feeds back as an activator to G1, its input\nlevel tends to decrease the transcription of G1. Essentially the\nfilter module balances the increase in the input A to G1 by\ndecreasing its output. As seen in the lower plot of\nFig. 10(a), $p_3$ stabilizes to an almost\nfixed value even though the input control $A$ increases in time.\nThis is an example of a homeostatic gene network which fixes its\nresponse to input transcription factor concentrations. The other\ncase shown in the scheme in Fig. 10(b),\nwhere $p_3$ is a repressor for G1.\nIn the regulation at G1, A as an activator and $p_3$ is a repressor. The\nnegative feedback of $p_3$ into G1 suppresses the input to the\namplitude filter. If A decreases, the input to the\nfilter module $p_1$ also decreases, this reduces the transcription rate\n(assuming that at steady state the value of $p_3$ is maximal).\nThis then lifts the repression from G1, and the input to the\nfilter module increases, thereby balancing the effect due to the\nreduced input A, and achieving the same steady state as before.\nThe above two circuits produce a fixed amount of output proteins,\neven though the input might vary by a large amount. In the first\ncase it is homeostatic to an increase in the concentration value\nof the input transcription factor, in the second case it is\nhomeostatic with respect to a decrease in the input concentration.\n\n\n\\subsection*{Tunable Motifs}\n\n\nIn this section we discuss an example of a gene network that\nexhibits oscillatory dynamics at small values of an input\ntranscription factor, and bistability for high input values.\nThis network therefore implements two\ndifferent types of motif functionality, depending upon the\nconcentration of an input transcription factor. Recently Voigt et.\nal.~\\cite{voigt2005} discussed a model of a network in the\nBacillus sporulation pathway,  which was shown to exhibit one of\ntwo alternatives, i.e.\\ either a bistable switch or oscillatory\nbehavior, depending on the environmental conditions.  A\nsynthetically constructed network exhibiting multifunctionality in\n{\\it E. coli}~\\cite{atkinson2003} was demonstrated to be able to flip\nfunction from an oscillator to a switch, by removing a particular\ninteraction.\n\nThere are now several examples of synthetically designed genetic\nnetworks such as switches~\\cite{gardner2000,becskei2001} and\noscillators~\\cite{elowitz2000,andrianantoandro2006} that use the\ncommon rules of mutual inhibition and activation to achieve a\ndesired functional behavior. The network we will describe is\nmotivated by the work of Gardner et. al.~\\cite{gardner2000}, where\nthe authors designed a toggle switch. In ~\\cite{hasty2001}, it was\ndiscussed how a toggle switch could be converted into a relaxation\noscillator by suitably manipulating the basic toggle switch\nnetwork by adding extra regulation. We further extend this design\nby introducing a new type of regulation, which involves an\nexternal transcription factor whose concentration can flip the\nsystem function between an oscillator and a switch.\n\nFig. 11 shows two mutually repressing genes, G1 and G2.\nThe repression is assumed to occur through\ntetramer binding of each of their gene products, $p_1$ and $p_2$,\nthe gene products of G1 and G2, ($p_1$ binds to\nG2, and $p_2$ binds to G1). Hence, if these were the only\ninteracting genes in the network, the system could be in one\nof two stable states, i.e., G1 is fully expressed, and G2 is\nsilent or {\\it vice versa}. Gene G3 is activated by $p_2$, and\nits product $p_3$ further activates G1. The feedback of $p_3$ to\nG1 has the effect of turning the bistable switch into a relaxation\noscillator~\\cite{hasty2001}. Assuming that initially G2 is {\\bf ON},\nwhich causes G3 to get transcribed and $p_3$ to grow, $p_3$ then\nactivates gene G1, leading to the growth of $p_1$. Thus G1\nswitches to a high state, which ultimately shuts down gene G2 due\nto its repressive effects. This leads to a decrease in $p_3$ which\nsubsequently turns G1 {\\bf OFF}, thereby completing one cycle of\nrelaxation oscillations.\n\nConsider now an extra piece of regulation, an external input $A$,\nas an activator of gene G1. $A$ can bind to the promoter region and\nactivate G1, but can also cooperatively bind with the tetramer of\n$p_2$, which represses G1. $p_3$ can also bind cooperatively\nwith the tetramer of $p_2$, which has the effect of repressing G1.\nThe two activators $A$ and $p_3$, however, are assumed to be\nmutually exclusive, i.e.\\ both $A$, and $p_3$ cannot bind\ntogether, but each can individually bind to the DNA. As described\nin the Appendix, these regulatory rules make G1 behave like an {\\bf\nOR} gate with respect to the inputs $A$ and $p_3$. From the above\nregulatory mechanisms, the following equations for the protein\ndynamics emerge,\n\n\\begin{eqnarray}\n\\frac {dp_1}{dt}&=& \\frac{m_1 [A]+m_2 [p_3]}{1+m_3 [A]+m_4\n[p_2]^4+m_5 [A][p_2]^4+m_6 [p_3] +m_7 [A] [p_3]} -\\gamma_1 p_1 \\\\ \\nonumber\n\\frac {dp_2}{dt}&=&\\frac{n_1}{1+n_2+n_3 [p_1]^4} -\\gamma_2 p_2, \\\\ \\nonumber\n\\frac{dp_3}{dt}&=&\\frac{o_1+o_2 [p_2]}{1+ o_3 + o_4 [p_2]}\n-\\gamma_3 p_3,\n\\end{eqnarray}\n\nWe now consider approximating the transcription rate for G1 for\nsmall and large values of the control transcription factor, $A$. For\nsmall $A$,\n\\begin{equation}\nT \\simeq \\frac{m_2 [p_3]}{1+m_4 [p_2]^4 + m_6 [p_3] +m_7 [p_2]^4\n[p_3]}\n\\end{equation}\nG1 is activated by $p_3$, and repressed by $p_2$. As\ndiscussed earlier $p_3$ toggles the bistable switch\nformed between the gene products of G1 and G2. For large $A$,\n\\begin{equation}\nT \\simeq \\frac{1}{\\frac{m_3}{m_1}+\\frac{m_5}{m_1}[p_2]^4}\n\\end{equation}\nwhich is the transcription rate, one would obtain a toggle\nswitch, between G1 and G2.  When $A$ is large, it is more likely\nfor $A$ to bind to the DNA than its competitor $p_3$ and the\nsystem is put into the bistable regime. The input $A$ can\ntherefore be used to tune the system into a relaxation\noscillator, or a switch. Fig. 12 shows the\nbifurcation plot for the steady state values of $p_1$ as a\nfunction of the input transcription factor $A$. The plot shows a\nsubcritical Hopf bifurcation~\\cite{strogatz,kuznetsov} for $p_1\n\\simeq 9$, and a saddle-node bifurcation at $p_1 \\simeq 35$. As\ndiscussed earlier, the system exhibits oscillations for $A< \\simeq\n11$ and bistability for $A> \\simeq 35$. In the right panel of\nFig. 12, the time series plots for $p_1$,\n$p_2$, $p_3$, are displayed, which show oscillations, for the case\nwhere $A=1$. In the lower right panel of\nFig. 12 the bistable behavior of $p_1$ is\ndisplayed (for the input $A=50$), taking one of two values,\ndepending on the initial conditions.\n\n\n\\subsection*{Adjustable Gates}\n\nThe transcriptional interaction between two transcription factors\nand RNAP has previously been shown to generate several instances of\nlogic such as {\\bf AND, OR, XOR}, {\\it etc.}~\\cite{buchler2003}.\nMultiple transcription factors can regulate the gate properties of a\nnetwork through their input concentrations. Alon and\ncolleagues~\\cite{setty2003} have studied the gate properties of the\nregulation of the {\\it lacZYA} operon in {\\it E. coli}; and by using a\nmathematical model, their work shows that the regulation can be made to behave\nas a fuzzy AND, a pure AND, and an OR logic gate. In this section we\nconsider gene regulation by three transcription factors which have the\nproperty of switching between two different logical functions,\ndepending on one of the input transcription factor concentrations.\nFurther we propose that such a circuit with appropriate feedback can\nbe made to exhibit other kinds of functionality such as homeostasis\nor oscillatory behavior.\n\nConsider gene G1 transcribed by the interaction of three\ntranscription factors, $p_1$, $p_2$ and $p_3$. $p_3$ is the control\nthat determines the output logic. We make the following\nassumptions: binding of all three transcription factors and\nRNAP to the gene is unlikely; individual binding of $p_1, p_2,\np_3$ and RNAP is also unfavorable; we also assume that\ntranscription is not leaky. With these assumptions, transcription\noccurs due to the binding of the following complexes $p_1 p_2 P,\np_1 p_3 P, p_2 p_3 P$. The transcription rate, takes the following\nform,\n\\begin{equation}\nT=\\frac{r_1 p_1 p_2 + r_2 p_1 p_3 +r_3 p_2 p_3}{1+ r_4 p_1+ r_5\np_2 + r_6 p_3  +r_7 p_1 p_2 + r_8 p_1 p_3 +r_9 p_2 p_3+r_{10} p_1\np_2 p_3}\n\\end{equation}\nThe above formula can be simplified for two cases, i.e.\\ high\nand low values of the concentration of the control transcription\nfactor, $p_3$. For low values of $p_3$,\n\\begin{equation}\nT \\simeq \\frac{r_1 p_1 p_2}{1+ r_4 p_1+ r_5 p_2+r_7 p_1 p_2 }\n\\end{equation}\nwhich implies that the transcription is activated only when both\n$p_1$ and $p_2$ are present, implementing an {\\bf AND}\ngate. For high values of $p_3$, the transcription rate is,\n\\begin{equation}\nT=\\frac{{r_2}^{\\prime} p_1  +{r_3}^{\\prime}p_2}{1  +{r_8}^{\\prime}\np_1 +{r_9}^{\\prime} p_2 +{r_{10}}^{\\prime} p_1 p_2}\n\\end{equation}\nwhere ${r_i}^{\\prime}=\\frac{r_i}{r_6}$, for $i=2,3,8,9,10$. Here the\ntranscription is activated when either $p_1$ or $p_2$ is present,\nimplementing an {\\bf OR} gate. The control $p_3$ is able to switch from an\n{\\bf AND } to {\\bf OR} gate. In Fig. 13 we plot the\ntranscription rates as a function of the input transcription factors, $p_1$ and\n$p_2$.\n\n\nThe ability to switch from one kind of logical function to another\nby varying the control $p_3$ opens up the possibility to use such\na motif in a gene network with other interacting genes. Consider\nFig. 14(i), a regulatory circuit\nwith G1 having three inputs $p_1, p_2$ and the control $p_3$,\nwhich emerges from a long negative feedback loop from its gene\nproduct $p_5$ in the following way: The gene G2 constitutively\nproduces protein $p_4$, which forms a protein complex with $p_5$.\n$p_4$ is also a transcription factor for the gene G3, which\nproduces protein $p_3$, and hence this is how a feedback into G1\nis achieved. We further assume that $p_4$ binds to the regulatory\nregion of G3 as a tetramer. The transcription factors $p_1$ and\n$p_2$ are assumed to be external to the system, and can be\neither set to a constant or a time dependent value.\nIn our case we shall fix the value of $p_2$, but allow\n$p_1$ to fall to low levels, starting at some fixed value.\nHowever, the dynamics of the network is symmetric with respect to\n$p_1$ and $p_2$, and hence we could have equally well chosen\nto vary $p_2$ and keep $p_1$ fixed. Fig. 15 shows\nthe logical structure for the adjustable gate network. Note that\nthe output from the gate determines its behavior. The\ndifferential equations for the rates of production of the various\nspecies are,\n\\begin{eqnarray}\n\\frac{d[p_5]}{dt} &=& \\frac{r_1 [p_1] [p_2] + r_2 [p_1] [p_3] +r_3\n[p_2] [p_3]}{1+ r_4 [p_1]+ r_5 [p_2] + r_6 [p_3]  +r_7 [p_1] [p_2]\n+ r_8 [p_1] [p_3] +r_9 [p_2]\n[p_3]+r_{10} [p_1] [p_2] [p_3]} \\\\ \\nonumber \\\\ \\nonumber\n&-& k_1 [p_4] [p_5] + k_2 [C] - \\gamma_{p_5} [p_5] ,\\\\ \\nonumber \\\\ \\nonumber \\frac{d[C]}{dt}\n&=& k_1 [p_4] [p_5] - k_2 [C] ,\\\\ \\nonumber \\frac{d [p_4]}{dt} &=&\nc_0 - k_1 [p_4] [p_5] + k_2 [C] -\\gamma_{p_4} [p_4],\\\\ \\nonumber\n\\frac{d [p_3]}{dt} &=&\n\\frac{v_2 [p_4] ^4}{1 + v_4 [p_4] ^4} - \\gamma_{p_3} [p_3], \\\\\n\\nonumber\n\\end{eqnarray}\nFor gene G1, the input $p_2$ is held constant, but the input $p_1$\nis made to decay from some initial value.\nGene G2 produces $p_4$ constitutively, and\nis sequestered by the output of G1, {\\it i.e.} $p_5$, into the\ncomplex $C$. Since $p_4$ is a transcription factor for G3, its\nsequestration away from G2 results in a lower value of the protein\n$p_3$. Hence the control $p_3$ is changed, the gate properties\nof G1 switch between {\\bf AND}\/{\\bf OR}. The negative feedback\narising from G1 onto itself is inhibitory, due to the complex\nformation. In Fig. 14(ii), the upper plot shows\nthe steady states of the value of the control $p_3$ as a function\nof the input $p_1$. A supercritical Hopf bifurcation is seen to\noccur at $p_1 \\simeq 2.5$. Fig. 14(ii) lower\nright-hand plot shows steady oscillations of $p_3$. The oscillations\narise due to the gate properties of G1. Initially, when the system\nis at steady steady state, and the inputs $p_1$ and $p_2$ are fixed,\n$p_3$ has a small value which makes G1 behave as an {\\bf AND} gate.\nsince both inputs are present, the output $p_5$ is high. There\nis thus considerable sequestration of $p_4$ due to the complex\nformation. This implies a reduced production of $p_3$, which is\nconsistent with G1 being in the {\\bf AND} state. Now if one of the\ninputs to the system is removed ({\\it e.g} by making $p_1$ decay),\nsince initially the gate is in the {\\bf AND} state, the output $p_5$\ndecreases. This results in the release of $p_4$, which begins\nincreasing the transcription of G3; then $p_3$ increases, and the\ngate G1 switches to the {\\bf OR} state. But in this state, G1 can\nbe transcribed by $p_2$, and the output $p_5$ increases. This once\nagain results in sequestration of  $p_4$, and finally reduces\nthe control $p_3$, and in this way we complete one cycle. The system\ntherefore switches back and forth between the two states and hence\nthis leads to oscillations.\n\n\nWe now describe another application of an adjustable gate. However\nin this case, the gate properties are reversed, i.e., the gate\nimplements an {\\bf OR} gate for low input control transcription\nfactor concentration, and an {\\bf AND} gate for high input control\ntranscription factor concentration. We discuss the nature of the\nregulation and its consequences without simulations, since this\ncase is very similar to the previously described model. For this\ncase we assume the following regulatory rules: all three\ntranscription factors and RNAP bind to the gene;\neach of the transcription factors $p_1, p_2$ and\nRNAP bind individually to the gene; and the complex $p_1$ $p_2$ and\nRNAP can bind to the gene. These assumptions lead to the following\nrate law,\n\\begin{equation}\nT=\\frac{s_1 p_1 p_2+ s_2  p_1 +s_3  p_2 +s_4 p_1 p_2 p_3}{1+ s_5\np_1+ s_6  p_2 +s_7 p_3  +s_8 p_1 p_2 +s_9 p_1 p_3+s_{10} p_2\np_3+s_{11} p_1 p_2 p_3}\n\\end{equation}\nBy inspection it is clear that for low $p_3$, G1 behaves like an\n{\\bf OR} gate with respect to $p_1$ and $p_2$, whereas for high $p_3$, it\nbehaves like an {\\bf AND} gate. If we now consider the same network\nas described above but substitute this motif into G1, then the\nsystem shows an almost homeostatic behavior with respect to a change\nin its input. This is easy to understand since initially, when both\ninputs are present, the output $p_5$ must be large. Due to the\nnature of the feedback, this determines the value of $p_3$ to be at\na low level. The system is therefore initially in the {\\bf OR}\nstate. If now one of the inputs is suddenly decreased, since the\nsystem is initially in an {\\bf OR} state, the output would continue\nto be high. There would be a small transient due to the sudden\nchange in input, but the system would reach a steady state as\nbefore. This then allows this network to be fairly unperturbed to\nchanges in one of its input transcription factors.\n\n\nWhat could be the function of such networks with adjustable gates?\nIn the first case (with {\\bf AND} gate properties for low $p_3$),\na sudden drop in a transcription factor concentration could set up\noscillatory patterns in the network which would then signal the\nnext program to be carried out by the genetic network. In the\nsecond case (with the gate properties reversed), it is clearly\nuseful to have a homeostatic network which works in such a way so\nas to counteract any sudden changes in input transcription factor\nconcentration levels.\n\nAlthough we have not found explicit examples for many of the\nnetworks we have discussed, which display such complex behavior,\nwe believe that such an endeavor is worth exploring.\n\n\n\n\n\\section*{Conclusions}\n\nAs a first step towards recognizing and understanding large\ncomplicated pathways, we have discussed in this work the modular\ndesign of several functional network motifs. Each of the networks\nconsists of genes which are regulated by multiple transcription\nfactors. The combinatorial regulation was explored in each case, and\nthe networks which emerged were found to have very distinct\nproperties. Our modeling procedure used the Shea-Ackers method~\\cite{bintu2005b}, which allowed us to derived the rates of\ntranscription which were then used to explore the network dynamics.\nThe networks could broadly be classified into: networks which are\nderived from incoherent feedforward motifs; and networks which can\nchange their gating properties based upon an external input.\n\nWe first discussed the steady state properties of feedforward\nnetworks, which can be used as amplitude filters. We found that both\nthe Type I simple and mixed feedforward networks led to a similar\ndesign, i.e.\\ filtering out the input transcription factor\nconcentration, although both networks worked through different types\nof interactions. Furthermore, we discussed the effects of\ndimerization for a simple Type I feedforward, which has the effect\nof narrowing the bandwidth of the amplitude filter. To study the\nfilter characteristics of these networks, we simulated the effects\nof mutations which would change the protein-DNA binding strengths.\nThese generally have the effect of shifting the amplitude filter\ncurve and modulating its bandwidth. Furthermore we described how\nthese motifs can be applied to a biological setting. By having a\ncommon transcription factor as the input to two amplitude filter\nmodules, but with shifted filter characteristics, it was possible to\nobtain a time ordered response of protein production. Homeostatic\nnetworks emerged if the output of the amplitude filter was fed back\nto itself. This network was found to be resilient in its output to\neither increasing or decreasing values of an external input\ntranscription factor, depending on whether the feedback was assumed\nto be positive or negative respectively.\n\nWe next described a tunable motif network, where the regulation at\none of the genes made it possible for the networks to exhibit\nbistability or oscillatory behavior if one of the external input\ntranscription factor concentration was made to increase\/decrease\nrespectively. Finally we discussed gate properties of regulation at\na gene, which can be made to switch between an {\\bf AND\/OR}\ndepending on the one of the input transcription factor\nconcentrations.\n\n\n\\section*{Methods}\n\n\\subsection*{Simulations}\n\nAll simulations were carried out using the Systems Biology Workbench\n(SBW) tools~\\cite{Sauro:Omics}: the network designer,\nJDesigner, the simulation engine Jarnac~\\cite{Sauro:2000}.\nBifurcation diagrams were computed using SBW with an interface to\nMATLAB~\\cite{Cameron:MATLAB}, and a bifurcation discovery\ntool~\\cite{ChickarmaneBifTool}. Bifurcation plots were also computed\nand cross checked using\nOscill8~\\footnote{http:\/\/sourceforge.net\/projects\/oscill8}, an\ninteractive bifurcation software package which is linked to\nAUTO~\\cite{Do81}, and SBW~\\cite{Sauro:Omics}. In all our simulations\nthe species concentrations are regarded as dimensionless, whereas\nthe kinetic constants have dimensions of inverse time, with\ndimensionless Michaelis-Menten constants. All models are available\nas Jarnac scripts (supplied in the supplement) which can be easily translated to\nSBML~\\cite{hucka:2002d} using the JarnacLite tool that is part of\nthe SBW suite~\\cite{Sauro:Omics}.\n\n\n\n\n\\section*{Author contributions}\n    HS helped conceive and fund the project; SY contributed to manuscript preparation and final data analysis.\n\n\n\\section*{Acknowledgements}\n  \\ifthenelse{\\boolean{publ}}{\\small}{}\nThis work was supported by grants from the National Science Foundation (0432190 and FIBR 0527023) to HMS. The authors wish to thank Carsten Peterson for useful discussions and most significantly to Vijay Chickarmane for technical assistance in carrying out the simulations and deriving the transcription factor binding kinetics. VC declined to be a co-author on the manuscript but was also supported by NSF FIBR 0527023.\n\n\n\n\n\n{\\ifthenelse{\\boolean{publ}}{\\footnotesize}{\\small}\n \\bibliographystyle{Fundamental_Dynamic_Units_Sauro} \n  ","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{s1}\nA $\\ensuremath{\\mathcal{PT}}$-symmetric quantum theory is defined by a Hamiltonian that is invariant\nunder combined space reflection (parity) $\\ensuremath{\\mathcal{P}}$ and time reversal $\\ensuremath{\\mathcal{T}}$. An early\nclass of non-Hermitian $\\ensuremath{\\mathcal{PT}}$-symmetric Hamiltonians that has been studied in\ndepth is $H=p^2+x^2(ix)^\\varepsilon$ ($\\varepsilon$ real). These Hamiltonians are $\\ensuremath{\\mathcal{PT}}$\ninvariant because $x\\to-x$ under $\\ensuremath{\\mathcal{P}}$ and $i\\to-i$ under $\\ensuremath{\\mathcal{T}}$. It was observed\nin Refs.~\\cite{R1,R2} that the eigenvalues of this class of Hamiltonians are\nreal, discrete, and positive for all $\\varepsilon\\geq0$ and the reality of these\neigenvalues was attributed to the $\\ensuremath{\\mathcal{PT}}$ symmetry of $H$. Subsequently, the\nreality of the spectrum was established at a mathematically rigorous level in\nRefs.~\\cite{R3,R4}.\n\nThe eigenvalues of a $\\ensuremath{\\mathcal{PT}}$-symmetric Hamiltonian are either real or come in\ncomplex-conjugate pairs. If the eigenvalue spectrum is entirely real, the $\\ensuremath{\\mathcal{PT}}$\nsymmetry of the Hamiltonian is said to be {\\it unbroken}, but if some of the\neigenvalues are complex, the $\\ensuremath{\\mathcal{PT}}$ symmetry of the Hamiltonian is said to be\n{\\it broken}. Many studies of model quantum systems whose Hamiltonians are\n$\\ensuremath{\\mathcal{PT}}$ invariant have been published (see Refs.~\\cite{R5,R11}). $\\ensuremath{\\mathcal{PT}}$-symmetric\nHamiltonians often exhibit a transition from a parametric region of unbroken\n$\\ensuremath{\\mathcal{PT}}$ symmetry to a region of broken $\\ensuremath{\\mathcal{PT}}$ symmetry. This $\\ensuremath{\\mathcal{PT}}$ transition\noccurs in both the classical and in the quantized versions of a $\\ensuremath{\\mathcal{PT}}$-symmetric\nHamiltonian \\cite{R2} and this transition has been observed in numerous\nlaboratory experiments \\cite{R5,R11}.\n\nMany papers on $\\ensuremath{\\mathcal{PT}}$-symmetric Hamiltonians having discrete spectra have been\npublished, but there have been only very few studies of $\\ensuremath{\\mathcal{PT}}$-symmetric\nHamiltonians having continuous spectra. Therefore, in this paper we consider\nHamiltonians $H=p^2+V(x)$ whose potentials possess continuous spectra. The\npotentials $V(x)$ that we discuss here decay to $0$ as $|x|\\to\\infty$. Thus, it\nis not surprising that we find that in general the real part of the spectrum is\ncontinuous and that these eigenvalues range continuously from 0 to $+\\infty$.\nSince the potential $V(x)$ is {\\it odd} under $x\\to-x$ and is pure imaginary,\n$V(x)$ is $\\ensuremath{\\mathcal{PT}}$ invariant, and hence complex eigenvalues must appear as\ncomplex-conjugate pairs. On the basis of our study we believe that short-range\npotentials typically have a finite number of discrete complex eigenvalues and\nthat long-range potentials have an infinite number of discrete complex\neigenvalues.\n\nTo be specific, in this paper we study five one-dimensional Hamiltonians of the\nform $H_n=p^2+V_n(x)$, where\n\\begin{eqnarray}\nV_1(x)&=&iA_1\\,{\\rm sech}(x)\\tanh(x),\\nonumber\\\\\nV_2(x)&=&iA_2\\,x\/(1+x^4),\\nonumber\\\\\nV_3(x)&=&iA_3\\,x\/(1+|x|^3),\\nonumber\\\\\nV_4(x)&=&iA_4\\,{\\rm sgn}(x)\\theta(2.5-|x|),\\nonumber\\\\\nV_5(x)&=&iA_5\\,x\/(1+x^2),\n\\label{e1}\n\\end{eqnarray}\nand the strength parameters $A_n$ are real. In all cases $V_n(x)$ is odd in $x$\nand is pure imaginary, so the $H_n$ are all $\\ensuremath{\\mathcal{PT}}$ symmetric. Furthermore, these\npotentials all vanish as $|x|\\to\\infty$. In all these cases the Hamiltonians $H_n$\nhave real eigenvalues that range continuously from $0$ to $+\\infty$. However,\nthe universal property of the five Hamiltonians studied here is that in addition\nto the real continuous part of the spectra, there are complex-conjugate pairs of\ndiscrete eigenvalues for sufficiently large strength parameters $A_n$. Thus, in\nall these cases the $\\ensuremath{\\mathcal{PT}}$ symmetry of these Hamiltonians $H_n$ is broken.\n\nThe five potentials $V_n(x)$ vanish at different rates for large $|x|$: The\nScarf-II potential $V_1(x)$ decays exponentially for large $|x|$\n\\cite{R6,R7,R8,R9}, the rational potentials $V_2(x)$ and $V_3(x)$ decay\nalgebraically like $|x|^{-3}$ and $|x|^{ -2}$ for large $|x|$, and the\nstep-function potential $V_4(x)$ has compact support \\cite{R10}. These are all\n{\\it short-range} potentials and we find that these potentials confine a {\\it\nfinite} number of discrete complex bound states. The number and size of these\ncomplex eigenvalues increase as the strength parameters $A_n$ increase.\n\nThe potential $V_5(x)$ is special; it vanishes slowly like $1\/|x|$ for large\n$|x|$. Because it vanishes slowly,\/and to\nit is a {\\it long-range} potential like the\nCoulomb potential. Even though this potential is bounded, the property that it\nis long range allows it to confine {\\it infinitely many} discrete complex bound\nstates. Like the Balmer series for the hydrogen atom, the sequence of\ncomplex-conjugate pairs of eigenvalues converges to a limit point, which happens\nto be at $0$, and the $k$th pair of eigenvalues approaches $0$ like $1\/k^2$.\n\nIn previous studies of $\\ensuremath{\\mathcal{PT}}$-symmetric Hamiltonians it was found that accurate\nnumerical calculations of real discrete eigenvalues could be done by using the\nshooting method \\cite{R1}. However, if the discrete eigenvalues are complex, the\nshooting method becomes unwieldy. Therefore, alternative techniques based on the\nfinite-element and variational methods were used. A numerical technique known as\nthe {\\it Arnoldi algorithm} was used in Ref.~\\cite{R12}.\n\nLet us summarize the numerical technique used in this paper: For numerical\ncalculations of eigenvalues one cannot work directly on the infinite $x$ axis,\nso one reduces the problem to solving the Schr\\\"odinger equation on a large but\n{\\it finite} interval. Consequently, the numerical techniques used to calculate\neigenvalues can only return discrete values and one must determine whether a\ngiven eigenvalue belongs to a discrete or a continuous part of the spectrum. To\ndistinguish between these two possibilities we examine the associated\neigenfunctions and observe how they satisfy the boundary conditions. As\nexplained in detail in Ref.~\\cite{R12}, the eigenfunctions associated with\ndiscrete eigenvalues are localized in space (like bound states) and decay to 0\nsmoothly and exponentially as $x$ approaches the boundary points of the\ninterval. However, the eigenfunctions for eigenvalues that belong to the\ncontinuous part of the spectrum drop abruptly to $0$ at one or both endpoints\nof the finite interval.\n\nThe technique used here to compute the eigenvalues of $H_n$ is called {\\it\nChebyshev spectral collocation}. This technique relies on the properties of\nChebyshev polynomials and Chebyshev series and is explained in detail in\nRef.~\\cite{R13}. To calculate the spectra of the Hamiltonians $H_n$ by using\nChebyshev spectral collocation we replace the infinite $x$ axis by the finite\ninterval $-L\\leq x\\leq L$. We then decompose the interval $[-L,L]$ into $N$\nsubintervals bounded by grid points at $x_j$, where $j=0,~1,~2,~3,~\\dots,~N$.\nThese subintervals are not of equal length; rather, the subintervals shorten\nas we approach the endpoints at $x=-L$ and $x=L$. To determine the positions of\nthe grid points, we construct a semicircle of radius $L$ centered at the origin\n$x=0$ and divide the circle into equal sectors. We then project onto the $x$\naxis. Thus, the grid points are located at $x_j=L\\cos(\\pi j\/N)$. For all\ncomputations done in this paper we take $N=2^{14}-1$. The first and last grid\npoints lie at $x=\\pm L$, but since $N$ is odd there is no grid point at the\norigin. We make this choice because the potential $V_4(x)$ is discontinuous at\n$x=0$. To find the eigenvalues we impose homogeneous boundary conditions at the\nendpoints $\\pm L$ and finally let $L$ tend to infinity. In all cases we take\n$L=10$, $100$, and $1000$, and we find that the eigenvalues converge rapidly to\ntheir $L=\\infty$ values.\n\nThis paper is organized very simply. In Secs.~\\ref{s2}-\\ref{s6} we describe\nin turn the spectra of $V_n(x)$ for $n=1$, 2, 3, 4, and 5, and then in\nSec.~\\ref{s7} we give some concluding remarks.\n\n\\section{Eigenvalues for the Scarf-II potential $V_1(x)$}\n\\label{s2}\nFor the potential $V_1(x)$ in (\\ref{e1}) we have chosen $A_1=30$. In\nFig.~\\ref{F1} we plot the eigenvalues in the complex plane for $L=10$ in panel\n(a) and for $L=100$ in panel (b). We observe two kinds of eigenvalues,\nbound-state eigenvalues, which are indicated by circles, and continuum\neigenvalues, which are indicated by crosses in (a) and by dots in (b). As we\nwill see in Fig.~\\ref{F4}, we can distinguish between bound-state and continuum\neigenvalues by examining the corresponding eigenfunctions. This plot of the\nabsolute values of the eigenfunctions as functions of $x$ shows that the\neigenfunctions for bound-state eigenvalues decay smoothly and exponentially to 0\nas $x$ approaches the boundaries at $\\pm L$ while the continuum eigenfunctions\nabruptly drop to 0 at one or both boundaries. Note that as $L$ increases from\n$10$ in (a) to $100$ in (b), the positions of the bound-state eigenvalues\nstabilize rapidly but the continuum eigenvalues approach the real axis.\n\n\\begin{figure}[b!]\n\\begin{center}\n\\includegraphics[scale=0.50]{Fig1a.png}\\hspace{-.1cm}\n\\includegraphics[scale=0.50]{Fig1b.png}\n\\end{center}\n\\caption{[Color online] Eigenvalues in the complex plane for $V_1(x)$ in\n(\\ref{e1}) with the strength parameter $A_1=30$ for the case $L=10$ in panel\n(a) and for $L=100$ in panel (b). Continuum eigenvalues are indicated by red\ncrosses in panel (a) and by green dots in panel (b). Discrete bound-state\neigenvalues are indicated by black circles. Note that as $L$ increases, the\nlocations of the bound states stabilize but the continuum eigenvalues all\ncollapse onto the real axis. As they do so, a new complex-conjugate pair of \nbound states is uncovered. Observe that the continuum eigenvalues come in\ncomplex-conjugate pairs until the real parts of these eigenvalues exceeds about\n$28$. Above this critical value the continuum eigenvalues are all real.}\n\\label{F1}\n\\end{figure}\n\nWe observe two kinds continuum eigenvalues in Fig.~\\ref{F1}. Above a critical\nvalue near $28$ the continuum eigenvalues are real, but below this critical\nvalue the continuum eigenvalues come in complex-conjugate pairs that lie\nslightly above and below the real axis. These pairs of eigenvalues approach the\nreal axis as $L$ increases but the position of the critical value near $x=28$\ndoes not change. With increasing $L$ each member of the complex-conjugate pair\nof eigenvalues approaches the real axis vertically, but as they reach the real\naxis, one member of the pair moves slightly rightward and the other moves\nslightly leftward along the real axis, thus doubling the density of points.\n(This behavior of the continuum eigenvalues is in exact analogy to the motion of\nthe roots of the famous Wilkenson polynomial \\cite{R14}.) For $L=1000$, the\ncontinuum eigenvalues below $x=28$ are extremely close to the real axis, but the\npositions of the complex bound states do not move, as we can see in\nFig.~\\ref{F2}.\n\n\\begin{figure}[t!]\n\\begin{center}\n\\includegraphics[scale=0.50]{Fig2.png}\n\\end{center}\n\\caption{[Color online] Eigenvalues in the complex plane for $V_1(x)$ in\n(\\ref{e1}) with the strength parameter $A_1=30$ for the case $L=1000$. The\ndiscrete bound-state eigenvalues (black circles) have not moved but the\ncontinuum eigenvalues (blue dots) are now very close to the real axis.}\n\\label{F2}\n\\end{figure}\n\n\\begin{figure}[h!]\n\\begin{center}\n\\includegraphics[scale=0.50]{Fig3.png}\n\\end{center}\n\\caption{[Color online] Logarithmic plot of the eigenvalue data in\nFigs.~\\ref{F1} and \\ref{F2}. This plot presents dramatic evidence that the\ntransition from slightly complex continuum eigenvalues to exactly real\ncontinuum eigenvalues near $x=28$ is sharp. There is a jump at the dashed line\nof about {\\it ten orders of magnitude} at this transition point. Observe\nthat the location of the transition does not change as $L$ is increased from\n$10$ to $100$ to $1000$.}\n\\label{F3}\n\\end{figure}\n\nFigure~\\ref{F3} is a logarithmic plot of the eigenvalues shown in Figs.~\\ref{F1}\nand \\ref{F2}. Observe that the critical point at $x=28$ at which the continuum\neigenvalues jump from being complex-conjugate pairs to real numbers does not\nmove as $L$ is increased.\n\nTo distinguish between discrete and continuum eigenvalues we investigate the\nbehavior of the associated eigenfunctions. Six possible behaviors of the\neigenfunctions are displayed in Fig.~\\ref{F4}. In general, the eigenfunctions of\ndiscrete eigenvalues decay smoothly to 0 at the endpoints of the interval but\nthe eigenfunctions of continuum eigenvalues abruptly drop to 0 at one or both\nendpoints.\n\n\\begin{figure}[t!]\n\\begin{center}\n\\includegraphics[scale=0.50]{Fig4a.png}\\hspace{-.1cm}\n\\includegraphics[scale=0.50]{Fig4b.png}\\\\\n\\includegraphics[scale=0.50]{Fig4c.png}\\hspace{-.1cm}\n\\includegraphics[scale=0.50]{Fig4d.png}\\\\\n\\includegraphics[scale=0.50]{Fig4e.png}\\hspace{-.1cm}\n\\includegraphics[scale=0.50]{Fig4f.png}\n\\end{center}\n\\caption{[Color online] Plots of the absolute value of the eigenfunctions\nassociated with some eigenvalues in Fig.~\\ref{F1}. Panels (a), (b), and (c)\ndisplay the eigenfunctions of the three discrete bound-state eigenvalues in\npanel (b) in Fig.~\\ref{F1}: Panel (a) shows the eigenfunction for the eigenvalue\n$2.374999999702702+12.272301129148877\\,i$ for $L=10$; panel (b) shows the\neigenfunction for the eigenvalue $5.875000000021835 +6.817945071620461\\,i$ for\n$L=10$; panel (c) shows the eigenfunction for the eigenvalue $7.374999997301000+\n1.363589013462076\\,i$ for $L=100$. The next three plots show the behavior of the\neigenfunctions for some continuum eigenvalues: Panel (d) shows the eigenfunction\nfor the continuum eigenvalue $7.361943725638523+2.501634415578858\\,i$ for $L=\n10$; panel (e) shows the eigenfunction for the continuum eigenvalue\n$0.277713365597523+0.009073644654185\\,i$ for $L=100$; panel (f) shows the\neigenfunction for the continuum eigenvalue $30.234638465149410+0.000000000526705\n\\,i$ for $L=10$. Bound-state eigenfunctions decay smoothly and exponentially as\n$x$ approaches the endpoints but the continuum eigenfunctions abruptly and\nsharply drop to $0$ at one or both endpoints.}\n\\label{F4}\n\\end{figure}\n\n\\section{Eigenvalue behavior of $V_2(x)$}\n\\label{s3}\nWhile the potential $V_1(x)$ decays exponentially for large $|x|$, the\npotential $V_2(x)$ decays algebraically like $|x|^{-3}$ for large $|x|$.\nNevertheless, the spectral properties of $V_2(x)$ are strikingly similar to\nthose of $V_1(x)$. For $V_2(x)$ the analog of Fig.~\\ref{F1} is Fig.~\\ref{F5}.\nAgain, we have taken the strength parameter $A_2$ to be $30$ and we observe one\ncomplex-conjugate pair of bound-state eigenvalues for $L=10$ in panel (a) and\ntwo complex-conjugate pairs of bound-state eigenvalues for $L=100$ in panel (b).\nThe new pair of bound-state eigenvalues is uncovered as the continuum\neigenvalues collapse towards the real axis. \n\n\\begin{figure}[t!]\n\\begin{center}\n\\includegraphics[scale=0.50]{Fig5a.png}\\hspace{-.1cm}\n\\includegraphics[scale=0.50]{Fig5b.png}\n\\end{center}\n\\caption{[Color online] Energy spectra for $V_2(x)$ with $A_2=30$. Panel (a)\nshows the eigenvalues for $L=10$ and panel (b) shows the eigenvalues for\n$L=100$. Like the case for the Scarf-II potential $V_1(x)$, the continuum\npart of the spectrum is slightly complex until the critical point near $21$,\nafter which the continuum eigenvalues are real. One pair of bound-state\neigenvalues (black circles) is visible in panel (a) but as $L$ is increased\nto $100$ in panel (b), a new pair of bound-state eigenvalues is uncovered.}\n\\label{F5}\n\\end{figure}\n\nIf we increase $L$ to $1000$, we observe in Fig.~\\ref{F6} that the bound-state\neigenvalues do not move. However, the continuum eigenvalues lie very close to\nthe real axis.\n\n\\begin{figure}[h!]\n\\begin{center}\n\\includegraphics[scale=0.50]{Fig6.png}\n\\end{center}\n\\caption{[Color online] Energy eigenvalues for the $V_2(x)$ potential for\n$A_2=30$. For this calculation $L=1000$. Note that the discrete bound-state \nenergies have not changed from their values in Fig.~\\ref{F5}(b). However, the \ncontinuum part of the spectrum has moved closer to the real axis.}\n\\label{F6}\n\\end{figure}\n\nAs with the Scarf-II potential $V_1(x)$, there is a transition in the continuum\npart of the spectrum that for this model occurs near $21$. To examine this\ntransition, we plot the eigenvalues on a logarithmic scale in Fig.~\\ref{F7}.\nObserve that near $21$ the continuum eigenvalues undergo an abrupt jump in\ntheir imaginary parts of 10 orders of magnitude.\n\n\\begin{figure}[t!]\n\\begin{center}\n\\includegraphics[scale=0.50]{Fig7.png}\n\\end{center}\n\\caption{[Color online] Logarithmic plot of the energy eigenvalues for the\npotential $V_2(x)$ plotted for $L=10$, $100$, and $1000$. Like the eigenvalues\nin Fig.~\\ref{F3}, the continuum eigenvalues here undergo an abrupt jump at the\ndashed line near the critical value close to $21$, where the imaginary parts of\nthe continuum eigenvalues suddenly drop by about 10 orders of magnitude.\nThe location of this line is insensitive to the value of $L$.}\n\\label{F7}\n\\end{figure}\n\n\\section{Eigenvalue behavior of $V_3(x)$}\n\\label{s4}\nThe spectral structure associated with the potential $V_3(x)$ is qualitatively\nsimilar to that of $V_2(x)$. We take the strength parameter $A_3=30$ and\nplot the eigenvalues for $L=10$ in Fig.~\\ref{F8}, panel (a), and for $L=100$\nin Fig.~\\ref{F8}, panel (b).\n\n\\begin{figure}[h!]\n\\begin{center}\n\\includegraphics[scale=0.50]{Fig8a.png}\\hspace{-.1cm}\n\\includegraphics[scale=0.50]{Fig8b.png}\n\\end{center}\n\\caption{[Color online] Energy eigenvalues for $V_3(x)$ with $A_3=30$. In panel\n(a) we display the data for $L=10$ and in panel (b) we display the data for\n$L=100$. Observe that as $L$ increases, the continuous eigenvalues approach the\nreal axis. However the discrete complex-conjugate bound-state eigenvalues\n(black circles) remain fixed.}\n\\label{F8}\n\\end{figure}\n\nObserve that as $L$ is increased from $10$ to $100$, the discrete bound-state\neigenvalues do not move but the continuum part of the spectrum rapidly\napproaches the real axis. In Fig.~\\ref{F9} we increase the value of $L$ to 1000.\nThis higher-accuracy calculation shows that the continuum eigenvalues are\nextremely close to the real axis. However, there is still a critical point where\nthe continuum eigenvalues go from having a small imaginary part to a vanishing\nimaginary part. This transition point is near $27$ and the transition is\nindicated in Fig.~\\ref{F10} by a dashed line. Once again, the logarithmic plot\nshows that at the transition the imaginary parts of the continuum eigenvalues\nabruptly drop by about ten orders of magnitude.\n\n\\begin{figure}[t!]\n\\begin{center}\n\\includegraphics[scale=0.50]{Fig9.png}\n\\end{center}\n\\caption{[Color online] Energy eigenvalues for $V_3(x)$ with $A_3=30$\ncalculated at $L=1000$.}\n\\label{F9}\n\\end{figure}\n\n\\begin{figure}[h!]\n\\begin{center}\n\\includegraphics[scale=0.50]{Fig10.png}\n\\end{center}\n\\caption{[Color online] Logarithmic plot of the data in Figs.~\\ref{F8} and\n\\ref{F9}. As in Figs.~\\ref{F3} and \\ref{F7} we see once again that there is a\ntransition point, in this case near 27, at which the imaginary parts of the\ncontinuum eigenvalues suddenly drop by about ten orders of magnitude. The\nlocation of this transition, which is indicated by a dashed line, appears to be\nindependent of the choice of $L$.}\n\\label{F10}\n\\end{figure}\n\n\\section{Eigenvalue behavior of $V_4(x)$}\n\\label{s5}\nThe pattern of eigenvalues associated with $V_4(x)$ is similar to that of\n$V_1(x)$, $V_2(x)$, and $V_3(x)$. For this potential we take the strength\nparameter $A_4=3$ and plot the spectra for $L=10$ and $L=100$ in Fig.~\\ref{F11}\nand for $L=1000$ in Fig.~\\ref{F12}. These figures show no qualitatively new\nfeatures.\n\n\\begin{figure}[h!]\n\\begin{center}\n\\includegraphics[scale=0.50]{Fig11a.png}\\hspace{-.1cm}\n\\includegraphics[scale=0.50]{Fig11b.png}\n\\end{center}\n\\caption{[Color online] Energy eigenvalues associated with $V_4(x)$ for $L=10$\nin panel (a) and for $L=100$ in panel (b). The strength parameter $A_4=3$. One\npair of bound-state energies (black circles) can be seen in panel (a) but a\nnew pair becomes visible in Panel (b).}\n\\label{F11}\n\\end{figure}\n\n\\begin{figure}[h!]\n\\begin{center}\n\\includegraphics[scale=0.50]{Fig12.png}\n\\end{center}\n\\caption{[Color online] Energy eigenvalues for $V_4(x)$ with $A_4=3$ for\n$L=1000$. Note that the bound-state eigenvalues have not changed from their\nposition from those in panel (b) of Fig.~\\ref{F11}.}\n\\label{F12}\n\\end{figure}\n\nA logarithmic plot of the eigenvalue data in Figs.~\\ref{F11} and \\ref{F12} is \nshown in Fig.~\\ref{F13}. Once again we see a transition, in this case near\n$9.5$, at which the continuum eigenvalues abruptly drop in magnitude by about\nten orders of magnitude. The location of the transition is again insensitive to\nthe value of $L$.\n\n\\begin{figure}[h!]\n\\begin{center}\n\\includegraphics[scale=0.50]{Fig13.png}\n\\end{center}\n\\caption{[Color online] Logarithmic plot of the eigenvalue data in\nFigs.~\\ref{F11} and \\ref{F12}. The data in this figure, like the data in\nFigs.~\\ref{F3}, \\ref{F7}, and \\ref{F10}, indicates that the sharp transition,\nwhich in this case is close to $9.5$ is not sensitive to the value of $L$.}\n\\label{F13}\n\\end{figure}\n\n\\section{Eigenvalue behavior of $V_5(x)$}\n\\label{s6}\nThe most interesting and surprising results that we have obtained concern the \neigenspectrum associated with $V_5(x)$. For this potential we take the strength\nparameter $A_5=10$. Because this is a long-range potential, it is not easy to\nobtain accurate and trustworthy numerical results, and we have had to do the\n$L=1000$ calculation in {\\it quadruple} precision (for all other results in\nthis paper double precision is sufficient). In Fig.~\\ref{F14} we plot the\neigenvalues for $L=10$ in panel (a) and $L=100$ in panel (b). There is one pair\nof bound-state eigenvalues in panel (a). When we increase the size of the\ninterval, we see in panel (b) that the continuum spectrum has dropped much\ncloser to the real axis and has uncovered three new pairs of bound-state\neigenvalues.\n\nFigure~\\ref{F14} reveals two new effects that we have not observed in our\nstudies of short-range potentials. First, the sequence of bound-state\neigenvalues has {\\it turned around} and is heading backward towards the origin.\nIn Figs.~\\ref{F1}, \\ref{F5}, \\ref{F8}, and \\ref{F11} the real parts of the\neigenvalues are increasing, not decreasing. Second, the transition in the\ncontinuum part of the spectrum at which the eigenvalues become real is no\nlonger a fixed point on the real axis; rather, the transition point is moving\nup the real axis as $L$ increases. In panel (a) the transition is near\n$16$ but in panel (b) it is near $28$.\n\n\\begin{figure}[h!]\n\\begin{center}\n\\includegraphics[scale=0.50]{Fig14a.png}\\hspace{-.1cm}\n\\includegraphics[scale=0.50]{Fig14b.png}\n\\end{center}\n\\caption{[Color online] Energy eigenvalues associated with $V_5(x)$ with\nstrength parameter $A_5=10$. In panel (a) we take $L=10$ and we observe one\ncomplex-conjugate pair of bound-state eigenvalues; in panel (b) we take $L=100$\nand observe three new pairs of complex bound-state eigenvalues. Unlike the\nresults for short-range potentials, this figure shows that the sequence of\nbound-state eigenvalues is turning around and heading back towards the\norigin. Also, the transition points in the continuum part of the spectrum\nare not fixed but are moving up the real axis from about $16$ in panel (a)\nto about $28$ in panel (b).}\n\\label{F14}\n\\end{figure}\n\nIf we increase $L$ to $1000$, Fig.~\\ref{F15} shows that there are now {\\it\nnine} complex-conjugate pairs of bound-state eigenvalues (which are not easy\nto see clearly). This sequence of bound-state eigenvalues is tending towards the\norigin. To observe the bound-state eigenvalues more clearly we have replotted in\nFig.~\\ref{F16} the data in Fig.~\\ref{F15} on a log-log plot. We can see on\nthis plot that the sequence of bound-state eigenvalues is becoming linear. \n\n\\begin{figure}[h!]\n\\begin{center}\n\\includegraphics[scale=0.50]{Fig15.png}\n\\end{center}\n\\caption{[Color online] Energy eigenvalues associated with $V_5(x)$ with\nstrength parameter $A_5=10$. In this figure we have have increased $L$ from \n$100$ in Fig.~\\ref{F14} (right panel) to $1000$ and there are now nine complex\npairs of bound-state energies (not easy to distinguish). The numerical\ncalculations needed to produce this figure required quadruple precision.}\n\\label{F15}\n\\end{figure}\n\n\\begin{figure}[h!]\n\\begin{center}\n\\includegraphics[scale=0.50]{Fig16.png}\n\\end{center}\n\\caption{[Color online] Plot of the eigenvalue data in Fig.~\\ref{F15} on a\nlog-log graph. In this plot we can now easily see nine bound-state eigenvalues.}\n\\label{F16}\n\\end{figure}\n\nTo observe the transition points in the continuum part of the eigenspectrum,\nwe have plotted the data in Figs.~\\ref{F14} and \\ref{F15} on a logarithmic graph\nin Fig.~\\ref{F17}. Note that the transition point in the continuum eigenvalues\nfrom complex to real is moving up the real axis; it is no longer fixed as it was\nfor finite-range potentials. Observe that at the transition near $40$ there is a\ndrop of nearly $20$ (and not $10$) orders of magnitude. To see this effect\nrequires that we use quadruple and not double precision in our numerical\ncalculations.\n\n\\begin{figure}[h!]\n\\begin{center}\n\\includegraphics[scale=0.50]{Fig17.png}\n\\end{center}\n\\caption{[Color online] Logarithmic plot of the eigenvalue data in\nFigs.~\\ref{F14} and \\ref{F15}. Observe that the transition point in the\ncontinuum part of the spectrum is moving up the real axis as $L$ increases\nfrom $L=10$ to $L=100$ to $L=1000$.}\n\\label{F17}\n\\end{figure}\n\nThe most interesting aspect of the long-range potential $V_5(x)$ is that it\nappears to confine an infinite number of bound states and the complex\nbound-state energies appear to be approaching $0$. To verify this we use\nRichardson extrapolation \\cite{R14} to study the behavior of the sequence of\nbound-state energies.\n\nRichardson extrapolation enables one to find the limit of the sequence $\\{a_k\\}$\nas $k\\to\\infty$ if the limit is a finite number. Given such a sequence we can\ncalculate more and more accurate Richardson extrapolants, which converge faster\nto the limiting value. The formulas for the first five Richardson extrapolants\nare given by\n\\begin{eqnarray}\nR_k^{(1)}&=&(k+1)a_{k+1}-ka_k,\\nonumber\\\\\nR_k^{(2)}&=&\\left[(k+2)^2a_{k+2}-2(k+1)^2a_{k+1}+k^2a_k\\right]\/2!,\\nonumber\\\\\nR_k^{(3)}&=&\\left[(k+3)^3a_{k+3}-3(k+2)^3a_{k+2}+3(k+1)^3a_{k+1}-k^3a_k\\right]\/\n3!,\\nonumber\\\\\nR_k^{(4)}&=&\\left[(k+4)^4a_{k+4}-4(k+3)^4a_{k+3}+6(k+2)^4a_{k+2}\\right.\n\\nonumber\\\\\n&&\\left.\\qquad\\qquad-4(k+1)^4a_{k+1}+k^4a_k\\right]\/4!,\\nonumber\\\\\nR_k^{(5)}&=&\\left[(k+5)^5a_{k+5}-5(k+4)^5a_{k+4}+10(k+3)^5a_{k+3}\\right.\n\\nonumber\\\\\n&&\\left.\\qquad\\qquad-10(k+2)^5 a_{k+2}+5(k+1)^5a_{k+1}+k^5a_k\\right]\/5!.\n\\label{e2}\n\\end{eqnarray}\n\nFrom our numerical analysis, we have determined that the $k$th bound-state\nenergy $E_k$ has the asymptotic form \n\\begin{equation}\nE_k\\sim\\frac{\\alpha}{k^2}\\pm i\\frac{\\beta}{k^3}\\qquad(k\\gg1),\n\\label{e3}\n\\end{equation}\nwhere $\\alpha$ and $\\beta$ are real numbers. This shows that the bound-state\neigenvalues associated with $V_5(x)$ share many of the quantitative features of\nthe Balmer series for the hydrogen atom. As indicated in the tables \\ref{t1} and\n\\ref{t2}, we have determined that the numerical value of $\\alpha$ is about $25$\nand the numerical value of $\\beta$ is about $61$. To obtain these results we\nhave multiplied the real part of $E_k$ by $k^2$ and the imaginary part of\n$E_k$ by $k^3$ and computed the first five Richardson extrapolations.\n\n\\begin{table}[h!]\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|c|c|}\n\\hline\n$k$ & ${\\rm Re}\\,E_k$ & $k^2\\,{\\rm Re}\\,E_k$ & $R_k^{(1)}$ & $R_k^{(2)}$ &\n$R_k^{(3)}$ & $R_k^{(4)}$ & $R_k^{(5)}$ \\\\\n\\hline\n1& 0.83298288 & 0.83298 & 10.2355 & 26.6628 & 29.8431 & 23.8084 & 24.2927\\\\\n2& 1.38356468 & 5.53426 & 21.1871 & 29.0481 & 25.0154 & 24.2120 & 25.5106\\\\\n3& 1.19465086 & 10.7519 & 25.1176 & 26.6284 & 24.4798 & 25.1396 & 25.2280\\\\\n4& 0.89645517 & 14.3433 & 25.7219 & 25.5541 & 24.8568 & 25.1949 & 25.0599\\\\\n5& 0.66476032 & 16.6190 & 25.6660 & 25.2553 & 25.0258 & 25.1199 &    --- \\\\\n6& 0.50352322 & 18.1268 & 25.5486 & 25.1692 & 25.0676 &    ---  &    --- \\\\\n7& 0.39157331 & 19.1871 & 25.4538 & 25.1354 &    ---  &    ---  &    --- \\\\\n8& 0.31203794 & 19.9704 & 25.3830 &    ---  &    ---  &    ---  &    --- \\\\\n9& 0.25397318 & 20.5718 &    ---  &    ---  &    ---  &    ---  &    --- \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{\\label{t1} Real parts of the first nine eigenvalues $E_k$ $(1\\leq k\\leq\n9)$ associated with $V_5(x)$ and the first five Richardson extrapolants\nconstructed from the sequence $\\{k^2\\,{\\rm Re}\\,E_k\\}$. Evidently, the real\nparts of the eigenvalues vanish like $\\alpha k^{-2}$, where $\\alpha$ is roughly\n$25$.}\n\\end{table}\n\n\\begin{table}[h!]\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|c|c|}\n\\hline\n$k$ & ${\\rm Im}\\,E_k$ & $k^3\\,{\\rm Im}\\,E_k$ & $R_k^{(1)}$ & $R_k^{(2)}$ &\n$R_k^{(3)}$ & $R_k^{(4)}$ & $R_k^{(5)}$ \\\\\n\\hline\n1& 3.90859038 & 3.90859 & 29.8484 & 63.7004 & 62.5280 & 53.5920 & 61.0166\\\\\n2& 2.10981263 & 16.8785 & 52.4164 & 62.8211 & 55.3792 & 59.7791 & 62.5903\\\\\n3& 1.06386938 & 28.7245 & 57.6188 & 58.3560 & 58.3125 & 61.7871 & 61.3005\\\\\n4& 0.56168819 & 35.9480 & 57.9136 & 58.3342 & 60.2980 & 61.4830 & 60.9267\\\\\n5& 0.32272930 & 40.3412 & 58.0538 & 59.1758 & 60.8905 & 61.1739 &    --- \\\\\n6& 0.20043182 & 43.2933 & 58.3744 & 59.8188 & 61.0165 &    ---  &    --- \\\\\n7& 0.13250064 & 45.4477 & 58.7355 & 60.2180 &    ---  &    ---  &    --- \\\\\n8& 0.09200917 & 47.1087 & 59.0650 &    ---  &    ---  &    ---  &    --- \\\\\n9& 0.06644330 & 48.4372 &    ---  &    ---  &    ---  &    ---  &    --- \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{\\label{t2} Imaginary parts of the first nine eigenvalues $E_k$ $(1\\leq\nk\\leq9)$ associated with $V_5(x)$ and the first five Richardson extrapolants\nconstructed from the sequence $\\{k^3\\,{\\rm Im}\\,E_k\\}$. The imaginary parts of\nthe eigenvalues vanish like $\\beta k^{-3}$, where $\\beta$ is roughly $61$.}\n\\end{table}\n\n\\section{Conclusions}\n\\label{s7}\n\nIn this paper we have studied numerically the five $\\ensuremath{\\mathcal{PT}}$-symmetric potentials\nin (\\ref{e1}) that have continuous spectra. Each of these potentials is pure\nimaginary and vanishes as $|x| \\to\\infty$. The interesting feature of these\npotentials is that even though they vanish at $\\pm\\infty$, they still confine\nbound states. Of course, an imaginary potential can confine bound states. For\nexample, the $ix^3$ potential has an infinite number of bound states \\cite{R1}.\nHowever, this potential becomes {\\it stronger} as $|x|\\to\\infty$. We emphasize\nthat the five potentials that we have studied here {\\it decay} and become {\\it\nweaker} as $|x|\\to\\infty$. Even more remarkable is the fact that the potential\n$V_5(x)$, which decays very slowly as $|x|=\\infty$, binds an {\\it infinite}\nnumber of bound states and that the sequence of bound-state energies\nasymptotically approaches the Balmer series for the hydrogen atom.\n\n\\vspace{0.5cm}\n\\footnotesize\n\\noindent\nCMB thanks the Alexander von Humboldt Foundation for partial financial support.\n\\normalsize\n\n\\vspace{0.5cm}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{Sec:Introduction}\n\nThe propagation of nonlinear waves in inhomogeneous media has captured much recent interest in physics and biophysics \\cite{Ivancevic2013}. Nonlinear\npulse excitations regulate the functioning in many physical models of biological systems, such as the Purkinje's fibres of the cardiac muscles\n\\cite{Aslanidi1999, Mornev2000}, neural fibres \\cite{Scott1975}, DNA chains \\cite{Polozov1988, Yakushevich2004} and muscular networks\n\\cite{Gojkovic2016}. The study of the effects that can be produced on these regulatory signals by the presence of spatial inhomogeneities, such as\nlocalised injuries and necrotic sites, has gained major interest in recent works \\cite{Mornev2000, Grinevich2015, Grinevich2016}. For instance,\ninjuries in the heart muscular tissue may produce annihilation, transmission or reflection of incident waves, depending on the geometrical size and\nshape of the defects. These disturbances may produce a disorder in the rhythmic contractions of the muscular walls,\nleading to cardiac arrhythmias and different pathologies \\cite{Mornev2000}. A similar situation occurs in the propagation of topological\none-dimensional solitons in DNA chains, which has been considered as a suitable model to explain the formation of open states or {\\it bubbles} in the\ndouble helix \\cite{Grinevich2015}. This bubble formation is an essential mechanism for the regulation and control of gene transcription\n\\cite{Polozov1988, Yakushevich2004}. Structure heterogeneities in the DNA chain and the presence of boundary interphase walls may affect the\npropagation regime of these regulatory signals during replication processes.\n\nIn the scattering of waves by localised defects, a crucial point to consider is that nonlinear waves exhibit a highly complex behaviour. Transmission,\nreflection, and annihilation are far from being the only possible phenomena that may occur. Indeed, solitons with an internal structure, such\nas sine-Gordon (sG) kinks and breathers, may produce many remarkable phenomena when colliding with localised inhomogeneities. The main phenomenon is\nthe internal mode instability \\cite{Gonzalez2002, Gonzalez2003}, which may cause the breaking of solitons, the formation of multikinks, the creation of\nkink-antikink pairs and the formation of localised structures \\cite{GarciaNustes2012, GarciaNustes2017}. The formation of such structures in sG\nsystems may have important implications not only in biological systems \\cite{Ivancevic2013}, but also in physical systems such as\nJosephson Junctions (JJs) \\cite{Barone1982} and ferromagnetic materials \\cite{Mikeska1978}.\n\nIn this article, we investigate the complex phenomena exhibited by a two-dimensional sG system with localised inhomogeneities. Formation of localised structures with bubble-like and drop-like shape are produced in two situations: (a) for line solitons trapped by a single localised\ninhomogeneity, and (b) for travelling line solitons colliding with an array of localised inhomogeneities. The structure formation is understood using\nthe one-dimensional theory of activation of internal modes in sG solitons. The outline of the article is as follows: Section \\ref{Sec:Theory} gives an overview of the one-dimensional theory of activation of internal modes of sG kinks under the action of inhomogeneous external forces. In Sec.\n\\ref{Sec:StationarySolitons} the two-dimensional model is introduced, and the internal mode instabilities of line solitons are studied. To avoid any scattering effect, we\nconsider steady line solitons. In Sec. \\ref{Sec:Arrays}, the scattering of line solitons with a spatial\narray of localised inhomogeneities is investigated. Finally, Sec. \\ref{Sec:Conclusions} gives some concluding remarks.\n\n\n\n\\section{Preliminaries}\n\\label{Sec:Theory}\n\nIn this section, an overview of well-known preliminary results of sG kinks under the action of external forces is given. These results provide\nthe theoretical framework for the understanding of the phenomena reported in the subsequent sections.\n\n\\subsection{Kinks driven by homogeneous external forces}\n\nThe driven and damped two-dimensional sG system is a particular case of the more general nonlinear Klein-Gordon (KG) system\n\\begin{equation}\n \\label{Eq01}\n  \\partial_{tt}\\phi(\\mathbf{r},t)-\\nabla^2\\phi(\\mathbf{r},t)+\\gamma\\partial_t\\phi(\\mathbf{r},t)-G(\\phi)=f,\n\\end{equation}\nwhere $\\mathbf{r}=(x,y)$, $\\gamma$ is a linear-damping coefficient, $f$ is a constant external force and $G(\\phi)=-\\partial U(\\phi)\/\\partial\\phi$,\nwith the nonlinear potential $U(\\phi)$ possessing at least two minima separated by a barrier.\nIt is well known that these two minima are fixed points of the\nassociated dynamical system, and that kink solutions are heteroclinic trajectories joining such fixed points \\cite{vanSaarloos1990}. For sG systems,\nthe potential function is given by $U(\\phi)=1-\\cos\\phi$ \\cite{Peyrard2004}, and the equation (\\ref{Eq01}) for $f=0$ has the well known stationary\nkink-antikink solutions (line solitons), given by $ \\phi_{\\pm}(\\mathbf{r})=4\\arctan\\exp\\left(\\pm x\\right)$. Here, $\\phi_+$ and $\\phi_-$ reads\nfor the kink and antikink, respectively.\n\nA space independent force with a constant value $f$ leads to an effective potential of the form $U_{eff}(\\phi)=U(\\phi)-f\\phi$. In this case, kink\nsolutions exist for equation (\\ref{Eq01}) only if this effective potential still have at least two minima separated by a barrier. That condition ceases to\nbe fulfilled for a certain critical value $f_c$ of the external force. When $f>|f_c|$ kink solutions do not exist, which means that an initially at\nrest kink will be destroyed by the external force \\cite{Gonzalez1992, Gonzalez2007}. The exact value of $f_c$ depends indeed on the particular\npotential function of the system. For instance, using the sG potential, minima $\\phi_j$ $(j=0,1\\ldots)$ of the effective potential satisfy the\nequation $\\sin\\phi_j=f$, which has no real solutions for $\\phi_j$ if $|f|>1$. Thus, the critical value is $f_c=1$ for sG systems. \n \nIf $f<|f_c|$, kink solutions in perturbed sG systems exists, and its dynamics in the neighbourhood of fixed points and separatrices can be\ninvestigated using the so-called qualitative theory of dynamical systems \\cite{Guckenheimer1986, McLaughlin1978, Kivshar1989, Sanchez1998,\nChacon2008, Gonzalez2007-II}. Based on this, it is possible to generalise the results to other equations that are topologically equivalent to those with the exact\nsolutions \\cite{Gonzalez1996, Gonzalez2003}.\n\n\\subsection{Soliton breaking by inhomogeneous forces}\n\nIn the sense of the qualitative theory of dynamical systems, for a general $x$-dependent force $F(\\mathbf{r})=f(x)$, it is known that a zero of\n$f(x)$ at $x=x_*$ is an equilibrium point for the centre-of-mass position of the kink \\cite{Gonzalez1992, Gonzalez1996, Gonzalez2003, Gonzalez2008}. The\nequilibrium is stable if $\\left.df(x)\/dx\\right|_{x=x_*}>0$, and unstable if $\\left.df(x)\/dx\\right|_{x=x_*}<0$. If the kink is on a stable\nequilibrium point, the shape and position of the kink will be recovered for any initial perturbation \\cite{Gonzalez2003}. On the contrary, if the\nkink is in an unstable equilibrium position, it is stretched by the inhomogeneous force that is acting on its body in opposite directions.\nThe value $f_c$ is the limit of $|f(x)|$ that the kink can resist the stretching without being destroyed \\cite{Gonzalez2007}. If $|f(x)|>f_c\\forall x$, the force will destroy the line soliton.\n\nNotwithstanding, there is a more subtle mechanism for the kink destruction when $|f(x)|>f_c$ only in a localised region in space. This mechanism\nis the internal mode instability \\cite{Gonzalez2007-II}. In one-dimensional systems,\nthe stability analysis of internal modes of sG kinks has been solved exactly \\cite{Gonzalez2002, Gonzalez2003}. The procedure starts by introducing\nthe ansatz\n\\begin{equation}\n \\label{Eq02}\n\\phi(x)=4\\arctan\\exp\\left(\\pm Bx\\right), \n\\end{equation}\nwhere $B$ is a parameter that controls the width of the kink. Solving an inverse problem, one obtains that equation (\\ref{Eq02}) is a solution of the \none-dimensional sG equation if the external force is given by\n\\begin{equation}\n \\label{Eq02Force}\nF(x) = 2(B^2 -1)\\mbox{sinh}(Bx)\\mbox{sech}^2(Bx),\n\\end{equation}\nwhich is an antisymmetric spatial function that vanishes exponentially for $x\\to\\pm\\infty$. For increasing (decreasing) values\nof $B$, the force is less (more) localised in space and has decreasing (increasing) extreme values. Thus, the parameter $B$ also controls the extreme\nvalues and the extension of $F(x)$ in space.\nInteractions of DNA special sites with ligands or perturbation of fluxons by dipole currents in JJs are typical examples of strong and local\nexternal influences with the same topological form as equation (\\ref{Eq02Force}) \\cite{Polozov1988, Malomed2004}.\nFrom the stability analysis follows that the first stable internal shape mode arises for $1\/6<B^2<1\/3$, while for $B^2<1\/6$ many other internal modes can\nappear. For $B^2<2\/[\\Lambda_*(\\Lambda_*+1)]$, where $\\Lambda_*=(5+\\sqrt{17})\/2$, the first internal mode becomes unstable, leading to soliton\nbreakups and shape instabilities \\cite{Gonzalez1992,Gonzalez2002, Gonzalez2003}. These instabilities play a central role in the formation of\nstructures reported in the following sections.\n\n\n\\section{Stationary line solitons in the presence of a single localised inhomogeneity}\n\\label{Sec:StationarySolitons}\n\n\\begin{figure}[t]\n \\begin{center}\n  \\scalebox{0.27}{\\includegraphics{fig01}}\\scalebox{0.27}{\\includegraphics{fig02}}\\\\\n  \\scalebox{0.27}{\\includegraphics{fig03}}\\scalebox{0.27}{\\includegraphics{fig04}}\n  \\caption{(Colour online) (a) Inhomogeneous force of equation (\\ref{Eq04}) for $B=0.1$ and $\\sigma=15$. The inset shows the static line soliton\n  used as initial condition in the numerical simulations of section \\ref{Sec:StationarySolitons}. (b) A drop-like structure after $10\\,000$ time\n  iterations for $B=0.28$ and $\\sigma=8.9$. The inset shows the elliptic-like structure that precedes the drop formation. The value of the field\n  $\\phi$ is indicated with the same colour scale. (c) A bubble-like structure after $10\\,000$ time iterations for $B=0.27$ and $\\sigma=10$.\n  The inset shows the elliptic-like structure that precedes the bubble formation. (d) A multi-structure formed after $8\\,300$ time iterations for\n  $B=0.1$ and $\\sigma=6$.}\n  \\label{fig01}\n \\end{center}\n\\end{figure}\n\n\nFollowing the one-dimensional system, consider the two-dimensional perturbed sG equation\n\\begin{equation}\n \\label{Eq03}\n \\partial_{tt}\\phi(\\mathbf{r},t)=\\nabla^2\\phi(\\mathbf{r},t)-\\sin\\phi(\\mathbf{r},t)-\\gamma\\partial_t\\phi(\\mathbf{r}, t)+F(\\mathbf{r}),\n\\end{equation}\nwhere the inhomogeneous force $F(x,y)$ is given by \\cite{GarciaNustes2017}\n\\begin{equation}\n\\label{Eq04}\nF(x, y)=2(B^2 -1)\\mbox{sech}(Bx)\\mbox{tanh}(Bx)e^{-y^2\/\\sigma^2},\n\\end{equation}\nand a real parameter $\\sigma$ has been introduced to control the spread of the force in the $y$-direction.\nFigure \\ref{fig01}(a) shows the force of equation (\\ref{Eq04}), which represents a single localised inhomogeneity centred at the origin. The point\n$x_*=0$ will be an equilibrium point for two-dimensional kinks. The external force of equation (\\ref{Eq04}) is physically realisable in real\nexperiments. Indeed, the sG equation (\\ref{Eq03}) with this kind of external perturbation models the propagation of fluxons in two-dimensional JJs\nwith a current dipole device inserted into one of the superconducting electrodes \\cite{Ustinov2002, Malomed2004, GarciaNustes2017}.\n\nThis section shows how the inhomogeneity of equation (\\ref{Eq04}) affects the dynamics of a line soliton, having a special focus on the dynamics of\nthe soliton internal structure. To avoid any scattering effect, consider an initially at rest kink with its centre-of-mass located at the equilibrium point. This initial condition is depicted in the inset of figure \\ref{fig01}(a) and is given by\n$\\phi(x,y,0)=4\\arctan\\,\\exp(x\/L)$ with $\\partial_t{\\phi}(x,y,0)=0$, where $L$ is an arbitrary parameter.\nWe have used an explicit finite difference scheme of the second order of\naccuracy for the numerical solution of equation (\\ref{Eq03}), with $\\gamma=0.01$ and homogeneous Neumann boundary conditions. The used regular mesh size is $1\\,200\\times 1\\,200$. The space intervals are $\\Delta x=\\Delta y=0.1$, and the time increment is \\hbox{$\\Delta t=0.02$}.\n\nFigure \\ref{fig01} shows the main results obtained from numerical simulations.\nFigure \\ref{fig01}(b) shows a drop-like structure created for $B=0.28$ and $\\sigma=8.9$. The local instability of the shape mode leads to a\nlocal break-up of the initially at rest line soliton. This initial breaking consists in a local creation of a pair kink-antikink, giving an\nelliptic-like structure that grows in time [see the inset of figure \\ref{fig01}(b)]. This kink zone propagates until the localised force is\nweak enough, and the contribution of curvature effects and kink interactions are important. The elliptic form eventually collapses due to the attractive interaction forces between the kink and\nantikink, developing a final stable drop-like structure in the negative region of $x$. Notice that the drop is sustained by the positive part of the external force. A kink is still present in the system due to the conservation of the topological charge and propagates\ntowards the negative $x$-direction with a velocity $\\mathbf{v}$, as showed in figure \\ref{fig01}(b).\n\nIf we proceed to diminish the value of $B$, the preferred structure is one with the reverse topological charge: a bubble-like\nstructure. Figure \\ref{fig01}(c) shows a bubble-like structure created for $B=0.27$ and $\\sigma=10$. In this case, the force is extended enough in\nthe $x$-direction to prevent the collapse and sustain the walls of the elliptic form, forming a bubble sustained by the negative part of the force. The mechanism of bubble formation\nis similar to the drop formation, giving that both are preceded by an elliptic-like form that grows in time [see inset of figure \\ref{fig01}(c)].\nOver a critical value of the elliptical area, the kink-antikink interaction is too weak to produce the collapse of the bubble. In this case, the external force supports the structure. Otherwise, the external force cannot sustain the walls of the bubble and the structure collapses by kink-antikink interaction, triggering the formation of a drop-like structure. Therefore, the interplay of both parameters $B$ and $\\sigma$ will determine which kind of structure will be formed.\nIndeed, further numerical simulations have show that these instabilities in sG kinks can be activated varying any of both parameters $B$ or $\\sigma$\n\\cite{GarciaNustes2017}.\n\nIf we continue further diminishing the values of parameters $B$ and $\\sigma$, it is expected that the line soliton will exhibit more complex\nphenomena due to the local destabilisation of more internal modes. Indeed, there is a coexistence region in the parameter space where a bubble\nand a drop merge in a stable structure, like the bubble-drop bound state depicted in figure \\ref{fig01}(d) for $B=0.1$ and $\\sigma=6$. From the\none-dimensional theory, it is known that for $B\\lesssim0.2564$ the formation of two-kink solitons occurs \\cite{GarciaNustes2012}. The later explains the\ntwo-kink profile of the bubble showed in figure \\ref{fig01}(d). More complex multi-structures can be observed for smaller values of $B$ and\n$\\sigma$, such as multi-kink bubbles and drops.\n\nNotice that the localised structures showed in Fig.~\\ref{fig01} are called \\emph{bubbles} and \\emph{drops} giving that the equation (\\ref{Eq03}) also appears in the description of structural phase transitions \\cite{GarciaNustes2017, Gonzalez1999, Gonzalez2006}. The results of this article can\nbe interpreted as the driving of a system to a new phase by the activation of a local instability.\n\n\n\\section{Scattering of solitons in an array of localised inhomogeneities}\n\\label{Sec:Arrays}\n\nIn order to simulate some real situations in physical and biological systems, this section deals with travelling line solitons that collide with a\ndistribution of localised inhomogeneities. Consider in equation (\\ref{Eq03}) the following spatial distribution of inhomogeneous forces\n\\begin{equation}\n \\label{Eq05}\nF(x,y)= F_o+2\\sum_{i=1}^4(B_i^2-1)\\mbox{sech}\\left[B_i(x-x_i)\\right]\\mbox{tanh}\\left[B_i(x-x_i)\\right]e^{-(y-y_i)^2\/\\sigma^2},\n\\end{equation}\nwhere $F_o=0.1$ is a constant homogeneous contribution to the total external force, $(x_i,y_i)$ is the coordinate-position of the core of the $i$-th inhomogeneity, $B_i$ is its\nrespective control parameter, $\\sigma=10$, and $i\\in\\left\\{1,\\,2,\\,3,\\,4\\right\\}$. Notice that the inhomogeneous contribution of force (\\ref{Eq05})\nis given by the superposition of forces of the same form as equation (\\ref{Eq04}).\n\nThe position of the inhomogeneities are fixed for all simulations in this section, and are given by $\\mathbf{r}_1=(-15, 25)$, $\\mathbf{r}_2=(15, 25)$, $\\mathbf{r}_3=(-15, -25)$ and\n$\\mathbf{r}_4=(15, -25)$. With these coordinates, the inhomogeneities are placed at the vertices of a rectangle centred at the origin.\nThe initial condition is given by $\\phi(x,y,0)=4\\arctan\\,\\exp[(x-xo)\/L]$ with $\\partial_t\\phi(x,y,0)=0$, which is an initially at rest line soliton\nplaced some distance away from the array of inhomogeneities. The initial $x$-coordinate of the centre-of-mass of the kink is given by $x_o=50$. Due to the homogeneous part of the force (\\ref{Eq05}) and the damping term of equation (\\ref{Eq03}), the line soliton accelerates towards the negative $x$-direction before reaching an asymptotically constant velocity. When the soliton collides with the inhomogeneities, many remarkable phenomena may occur due to possible internal mode\ninstabilities.\n\n\\begin{figure}[t]\n \\begin{center}\n  \\scalebox{0.32}{\\includegraphics{fig05}}\\scalebox{0.32}{\\includegraphics{fig06}}\\\\\n  \\scalebox{0.32}{\\includegraphics{fig07}}\n  \\caption{(Colour online) Scattering of travelling line solitons after collisions with a distribution of localised inhomogeneities. (a) Generation\n  of a bubble induced by the collapse of an echo-wave, for $B_1=0.27$, $B_2=3.0$, $B_3=B_4=0.3$, and $\\gamma=0.5$. (b) Kink transmission for\n  $\\gamma=0.1$, $B_1=B_4=0.27$ and $B_2=B_3=1.1$. A bubble and a drop remains trapped at the inhomogeneities after the transmission of the soliton.\n  (c) Kink transmission and formation of complex interacting structures (bubbles and drops) for $\\sigma=10$, $\\gamma=0.01$, and\n  $B_i=0.27$ $(i=1,2,3,4)$.}\n  \\label{fig02}\n \\end{center}\n\\end{figure}\n\nFigure \\ref{fig02}(a) shows the results from numerical simulations for $B_1=0.27$, $B_2=3.0$, $B_3=B_4=0.3$, and $\\gamma=0.5$. For this combination of\nparameters, the $1$st inhomogeneity activates the internal modes of kinks, while the $2$nd inhomogeneity tends to trap them at position\n$\\mathbf{r}_2$. When the line soliton reaches the first line of inhomogeneities, part of the kink is deformed due to the collision with the $1$st\ninhomogeneity, while another part is trapped by the $2$nd inhomogeneity. Notice that this initial bent of the soliton is the so-called \\emph{echo effect} reported in\nbiophysical systems and is considered as responsible for certain types of disorders and cardiac arrhythmias \\cite{Mornev2000}. This echo-wave is eventually stabilised in size, and forms a stable bubble-like structure sustained by the $1$st inhomogeneity. Notice that the echo-waves observed\nin this section have some resemblance with the elliptic-like structures observed in section \\ref{Sec:StationarySolitons} [see insets of figure\n\\ref{fig01}(b) and (c)].\n\nFor smaller values of $\\gamma$ the soliton reaches the inhomogeneities with higher terminal speeds, and we would expect the\nenhancing of the transmission of the soliton across the inhomogeneities. Figure \\ref{fig02}(b) shows the results from numerical simulations for $\\gamma=0.1$, $B_1=B_4=0.27$ and\n$B_2=B_3=1.1$. In this case, the $1$st and $4$th inhomogeneities activate the internal modes of the soliton. The $2$nd and $3$rd\ninhomogeneities tend weakly to trap the soliton. Indeed, when the kink collides with the first pair of inhomogeneities, the $2$nd cannot capture the\nkink, while the $1$st inhomogeneity produces an echo-effect. This echo-wave eventually collapse and forms a stable drop, as showed in figure\n\\ref{fig02}(b). A similar mechanism occurs when the line soliton reaches the second pair of inhomogeneities. However, the speed of the line\nsoliton has diminished due to previous collisions. The echo-wave generated at the $4$th inhomogeneity partially collapse, as indicated in figure\n\\ref{fig02}(b), but is eventually stabilised and forms a bubble. Thus, the relative speed of collision also plays an important role in determining\nwhich kind of structure will be formed. \n\nA highly complex dynamics is observed diminishing the parameters $B_i$ in the array of inhomogeneities. Figure \\ref{fig02}(c) shows the\nresults from numerical simulations when $B_i=0.27$ for all $i$. After the collision, there is an echo effect followed by the formation of bubbles.\nThe inhomogeneities cannot sustain such bubbles, and they collapse to form drops eventually. However, during this drop formation, another pair of\nbubbles are created in the neighbour inhomogeneities. In figure \\ref{fig02}(c) is clearly seen repulsive interactions between these new bubbles and\nthe previously formed drops. Notice that these drops also collide, enhancing the formation of new structures with the topological charge of three around\nthe origin. More complex dynamics can be observed diminishing the values of $B_i$ or $\\sigma$.\n\n\\section{Conclusion}\n\\label{Sec:Conclusions}\n\nIn this article, the formation of localised structures in a two-dimensional sG system has been reported considering two different scenarios: (a) the\npresence of a single localised inhomogeneity, and (b) the presence of an array of localised inhomogeneities. These inhomogeneities are modelled by a family of topologically equivalent external forces. The localised structures are formed with a bubble-like and drop-like shape and are\nsustained by the same inhomogeneities that create them. The structure creation is entirely due to the dynamics of the internal modes of the soliton,\nand not to any scattering effects. However, the internal mode instabilities can also occur during scattering processes, and exhibit a highly complex\ndynamics.\n\nAs a final remark, notice that the reported phenomena may have important implications in biological systems. The formation of bubbles, and the \ntrapping and breaking of such structures by localised inhomogeneities may be related to different functional disorders in similar systems.\nMoreover, the study of the stability and transport of such bubbles may give some light on how to introduce shock perturbations, and produce the\ndesired effects in a controlled way. Gene transcription in DNA chains can also be regulated by the insertion of this kind of external perturbations.\n\n\n\\ack\n\nThis article was written after a conference at the XX Symposium of the Physical Society of Chile (SOCHIFI), held in Santiago de Chile on November 2016.\nThe author thanks the organising committee for this occasion. The author also thanks Prof. M\\'onica A. Garc\\'ia-\\~Nustes and Prof. Jorge A.\nGonz\\'alez for many fruitful discussions and the active collaboration on the subject of this article. This work was supported by CONICYT doctorado nacional No.\n21150292.\n\n\n\\section*{References}\n\n\\providecommand{\\noopsort}[1]{}\\providecommand{\\singleletter}[1]{#1}%\n\\providecommand{\\newblock}{}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{sec:one}\n\n\nThe microlensing phenomenon occurs by the gravity of lensing objects regardless of their \nluminosity. Due to this property, microlensing provides an important tool to detect \nvery faint and even dark objects that cannot be observed by other methods. However, \nit is difficult to conclude the faint\/dark nature of the lens just based on the event \ntimescale $t_{\\rm E}$, which is the only observable related to the lens mass for general \nlensing events,  because the timescale is related to not only the lens mass $M$ but \nalso the relative lens-source proper motion $\\mu$ and distance to the lens $D_{\\rm L}$ \nand source $D_{\\rm S}$ by\n\\begin{equation}\nt_{\\rm E} = {\\theta_{\\rm E}\\over \\mu};\\qquad\n\\theta_{\\rm E} = \\sqrt{\\kappa M \\pi_{\\rm rel}};\\qquad\n\\pi_{\\rm rel}={\\rm au} \\left( {1\\over D_{\\rm L}}-{1\\over D_{\\rm S}}\\right),\n\\label{eq1}\n\\end{equation}\nwhere $\\kappa=4G\/(c^2{\\rm au})$ and $\\theta_{\\rm E}$ is the angular Einstein radius.\nIn order to reveal the nature of lenses, their masses should be determined.\n\n\nFor the unique determination of the lens mass, it is required to measure two additional \nobservables. These observables are the angular Einstein radius $\\theta_{\\rm E}$ and the \nmicrolens parallax $\\pi_{\\rm E}$. They are related to the lens mass by \\citep{Gould2000}\n\\begin{equation}\nM={\\theta_{\\rm E}\\over \\kappa\\pi_{\\rm E}}.\n\\label{eq2}\n\\end{equation}\nWith $\\pi_{\\rm E}$ and $\\theta_{\\rm E}$, the distance to the lens is also determined by\n\\begin{equation}\nD_{\\rm L} = {{\\rm au}\\over \\pi_{\\rm E}\\theta_{\\rm E} + \\pi_{\\rm S}},\n\\label{eq3}\n\\end{equation}\nwhere $\\pi_{\\rm S}={\\rm au}\/D_{\\rm S}$.\n\n\nThe angular Einstein radius is determined by detecting deviations in lensing light \ncurves caused by finite-source effects.  Deviations induced by finite-source effects \narise due to the differential magnification in which different parts of the source \nsurface are magnified by different amount.  For events produced by single mass \nobjects, finite-source effects can be detected in the special case in which the \nlens passes over the surface of the source \\citep{Witt1994, Gould1994a, Nemiroff1994}.  \nHowever, the ratio of the angular source radius $\\theta_*$ to the angular Einstein \nradius $\\theta_{\\rm E}$ is very small, $\\rho~\\sim 1\/1000$ for events associated with \nmain-sequence source stars and $\\sim~1\/100$ even for events involved with giant \nsource stars, and thus the rate of source-crossing events is accordingly very low. \nThe chance to detect finite-source effects is relatively much higher for events \nproduced by binary objects. This is because a binary lens forms caustics that can \nextend over an important portion of the Einstein ring, and the event with a source \ncrossing the caustic exhibits a light curve affected by the finite-source effect \nduring the caustic crossing.\n\n\nIn the early-generation lensing surveys that were conducted with a $>1$ day cadence, \nmeasuring the angular Einstein radius by detecting finite-source effects was \nobservationally a challenging task. This is because the duration of the deviation \ninduced by finite-source effects is, in most cases, $< 1$ day and thus it was difficult \nto detect the deviation.  The unpredictable nature of caustic crossings also made it \ndifficult to cover crossings from high-cadence follow-up observations \\citep{Jaroszynski2001}.  \nHowever, with the inauguration of lensing surveys using globally distributed multiple \ntelescopes equipped with wide-field cameras, the observational cadence has been dramatically \nincreased to $< 1$ hour, making it possible to measure $\\theta_{\\rm E}$ for a greatly increased \nnumber of lensing events.  \n\n\nOne channel to measure the microlens parallax is simultaneously observing lensing \nevents from the ground and in space: `space-based microlens parallax' \n\\citep{Refsdal1966, Gould1994b}. The physical size of the Einstein radius for a \ntypical lensing event is of the order of au. Then, if space observations are \nconducted using a satellite in a heliocentric orbit, e.g., {\\it Deep Impact} \n\\citep{Muraki2011} spacecraft or {\\it Spitzer Space Telescope} \\citep{Dong2007, \nCalchi2015, Udalski2015b}, \nthe lensing light curve obtained from the satellite observation will be substantially \ndifferent from that obtained from the ground-based observation.  For events with well \ncovered light curves from both the ground and in space, then, the microlens parallax \ncan be precisely measured by comparing the two light curves. \n\n\n\nAnother channel to measure $\\pi_{\\rm E}$ is analyzing deviations induced by \nmicrolens-parallax effects in lensing light curves obtained from ground-based \nobservations: `annual microlens parallax' \\citep{Gould1992}. In the single frame \nof Earth, such deviations occur due to the positional change of an observer caused \nby the orbital motion of Earth around the Sun.  For typical lensing events produced \nby low-mass stars, however, the event timescale is several dozen days, which comprises \na small fraction of the orbital period of Earth, i.e., year, and thus deviations \ninduced by the annual microlens-parallax effects are usually very minor.  As a \nresult, it is difficult to detect the parallax-induced deviations for general events, \nand even for events with detected deviations, the uncertainties of the measured $\\pi_{\\rm E}$ \nand the resulting lens mass can be considerable.  \n\n\nIn this work, we analyze the binary-lensing event OGLE-2016-BLG-0156.  The light \ncurve of the event, which is characterized by three distinctive widely-separated \npeaks, exhibits pronounced parallax-induced deviations, from which we precisely \nmeasure the microlens parallax.  All the peaks are densely covered from continuous \nand high-cadence survey observations and the analysis of the peaks leads to the \nprecise measurement of the angular Einstein radius.  We characterize the lens by \nmeasuring the masses of the lens components from $\\pi_{\\rm E}$ and $\\theta_{\\rm E}$.\n\n\n\n\n\\section{Observation and Data}\n\nThe source star of the lensing event OGLE-2016-BLG-0156 is located in the bulge \nfield with equatorial coordinates \n$({\\rm R.A.},{\\rm decl.})_{\\rm J2000}=(17:56:36.63, -31:04:40.7)$.  The corresponding \ngalactic coordinates are $(l,b)=(359.4^\\circ, -3.14^\\circ)$. The event was found in \nthe very early part of the 2016 bulge season by the Optical Gravitational Lensing \nExperiment \\citep[OGLE:][]{Udalski2015a} survey.  The OGLE lensing survey was conducted \nusing the 1.3 m telescope of the Las Campanas Observatory, Chile.  The source brightness \nhad remained constant, with a baseline magnitude of $I_{\\rm base}\\sim 18.74$,   until \nthe end of the 2015 season since it was monitored by the survey in 2009.  When the \nevent was found, the source brightness was already $\\sim 0.5$ magnitude brighter than \nthe baseline magnitude, indicating that the event started during the $\\sim 4$ month \ntime gap between the 2015 and 2016 seasons when the bulge field could not be observed \nduring the passage of the field behind the Sun.  The OGLE observations were conducted \nat a $\\sim 1$--2 day cadence and data were acquired mainly in $I$ band with some $V$ \nband data obtained for the source color measurement.\n\n\nTwo other microlensing survey groups of the Microlensing Observations in Astrophysics \n\\citep[MOA:][]{Bond2001, Sumi2003} and the Korea Microlensing Telescope Network \n\\citep[KMTNet:][]{Kim2016} independently detected the event. The MOA survey observed \nthe event with a $\\sim 0.5$ hour cadence in a customized broad $R$ band using the 1.8 m \ntelescope of the Mt.~John University Observatory, New Zealand. The event was entitled \nMOA-2016-BLG-069 in the list of MOA transient \nevents.\\footnote{http:\/\/www.massey.ac.nz\/~iabond\/moa\/alert2016\/alert.php} \nThe KMTNet observations were \nconducted using 3 identical 1.6 m telescopes that are located at the Siding Spring \nObservatory, Australia, Cerro Tololo Interamerican Observatory, Chile, and the South \nAfrican Astronomical Observatory, South Africa. We refer to the individual KMTNet \ntelescopes as KMTA, KMTC, and KMTS, respectively. The event, dubbed as KMT-2016-BLG-1709\nin the 2016 KMTNet event list\\footnote{http:\/\/kmtnet.kasi.re.kr\/ulens\/2016\/}, \nwas located in the KMTNet BLG01 field toward which observations were conducted with a \n$\\sim 0.5$ hour cadence. The field almost overlaps the BLG41 field that was additionally \ncovered to fill the gaps between the CCD chips of the BLG01 field. The source happens to \nbe located in the gap between the chips of the BLG41 field and thus no data was obtained \nfrom the field. KMTNet observations were conducted mainly in $I$ band and occasional $V$ \nband observations were carried out to measure the source color.\n\n\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{f1.eps}\n\\caption{\nLight curve of OGLE-2016-BLG-0156. The colors of data points correspond to those of \ntelescopes, marked in the legend, used for observations. The times marked by arrows \nindicate the centers of the peaks at ${\\rm HJD}^\\prime={\\rm HJD}-2450000\\sim 7462.4$ \n($t_1$), 7493.7 ($t_2$), and 7512.1 ($t_3$).\n\\vskip0.2cm\n}\n\\label{fig:one}\n\\end{figure}\n\n\n\nPhotometry of the individual data sets are processed using the codes of the individual \nsurvey groups. All the photometry codes utilize the difference imaging technique \ndeveloped by \\citet{Alard1998}. For the KMTC data set, we additionally conduct \npyDIA photometry\\footnote{The pyDIA code is a python package for performing \ndifference imaging and photometry developed by \\citet{Albrow2017}. The difference-imaging \npart of this software implements the algorithm of \\citet{Bramich2013} with extended delta \nbasis functions, enabling independent control of the degrees of spatial variation for \nthe differential photometric scaling and differential PSF variations between images.} \nfor the source color measurement.  For the use of the multiple data \nsets reduced by different codes, we normalize the error bars of the individual data \nsets using the method described in \\citet{Yee2012}.\n\n\nFigure~\\ref{fig:one} shows the light curve of OGLE-2016-BLG-0156. The light \ncurve is characterized by three distinct peaks centered at \n${\\rm HJD}^\\prime={\\rm HJD}-2450000 \\sim 7462.4$ ($t_1$), 7493.7 ($t_2$), \nand 7512.1 ($t_3$). The peaks are widely separated with time gaps \n$\\Delta t_{1-2}\\sim 31.3$ days between the first and second peaks and \n$\\Delta t_{2-3}\\sim 18.4$ days between the second and the third peaks. \n\n\nIn Figure~\\ref{fig:two}, we present the enlarged views of the individual peaks. \nIt is found that the source became brighter by $\\gtrsim 3$ magnitudes during very \nshort periods of time, indicating that the peaks were produced by the source \ncrossings over the caustic. Caustics produced by a binary lens form closed curves \nand thus caustic crossings usually occur in multiples of two.  The region between \nthe second and third peaks shows a U-shape pattern, which is the characteristic \npattern appearing when the source passes inside a caustic, suggesting that the \npair of the peaks centered at $t_2$ and $t_3$ were produced when the source star \nentered and exited the caustic, respectively. On the other hand, the first peak \nhas no counterpart peak. Such a single-peak feature can be produced when the \nsource crosses a caustic tip in which the gap between the caustic entrance and \nexit is smaller than the source size.\n\n\nWe note that all the peaks were densely covered. The first peak was covered by the \ncombined data sets obtained using the three KMTNet telescopes, the second peak was \nresolved by the OGLE+KMTA data sets, and the last peak was covered by the MOA+KMTS \ndata sets. The dense and continuous coverage of all the caustic crossings were \npossible thanks to the coordination of the high-cadence survey experiments using \nglobally distributed telescopes.\n\n\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{f2.eps}\n\\caption{\nEnlarged view of the three peaks in the lensing light curve.\nThe locations of the individual peaks in the whole light curve \nare marked by $t_1$, $t_2$, and $t_3$ in Fig.~\\ref{fig:one}.\n\\vskip0.3cm\n}\n\\label{fig:two}\n\\end{figure}\n\n\n\n\\begin{figure*}\n\\epsscale{0.82}\n\\plotone{f3.eps}\n\\caption{\nThe top two panels show the best-fit model light curve, the curve superposed on the data \npoints, obtained by considering both the microlens-parallax and lens-orbital effects. \nThe lower four panels compare the residuals of the 4 tested models. The `parallax' \nand `orbit' models are obtained by separately considering the microlens-parallax and \nlens-orbital effects. The `standard' model consider neither of these higher-order \neffects.\n\\vskip0.3cm\n}\n\\label{fig:three}\n\\end{figure*}\n\n\n\\section{Modeling Light Curve}\n\n\\subsection{Model under Rectilinear Relative Lens-Source Motion}\n\nThe caustic-crossing features in the observed light curve indicates that the event is \nlikely to be produced by a binary lens and thus we conduct binary-lens modeling of \nthe light curve.  We begin by searching for the sets of the lensing parameters that \nbest explain the observed light curve under the assumption of the rectilinear \nlens-source motion, wherein the lensing light curve is described by 7 principal parameters. \nThe first three parameters $(t_0, u_0, t_{\\rm E})$ are identical to those of a \nsingle-lens events, describing the time of the closest lens-source approach, the \nseparation at that time, and the event timescale, respectively. Because a binary lens \nis composed of two masses, one needs a reference position for the lens. We set the \nbarycenter of the binary lens as the reference position.  Due to the binary nature \nof the lens, one needs another three parameters $(s, q, \\alpha)$, indicating the \nprojected binary separation (normalized to $\\theta_{\\rm E}$), the mass ratio between the \nlens components, and the angle between the binary axis and the source trajectory, \nrespectively. The last parameter $\\rho$, which represents the ratio of the angular \nsource radius $\\theta_*$ to the angular Einstein radius, i.e., $\\rho=\\theta_*\/\\theta_{\\rm E}$ \n(normalized source radius), is needed to describe the deviation of lensing \nmagnifications caused by finite-source effects during caustic crossings.\n\n\nModeling the light curve is done through a multiple-step process. In the first \nstep, we conduct a grid search for the parameters $s$ and $q$ and, for a given \nset of $s$ and $q$, the other parameters are searched for using a downhill approach \nbased on the Markov Chain Monte Carlo (MCMC) method. We identify local solutions \nfrom the $\\Delta\\chi^2$ maps obtained from this preliminary \nsearch. In the second step, we refine the individual local solutions first by \ngradually narrowing down the parameter space and then allowing all parameters \n(including the grid parameters $s$ and $q$) to vary. If a satisfactory solution is \nnot found from these searches, we repeat the process by changing the initial values \nof the parameters. We refer to the model based on these principal parameters as \n`standard model'.\n\n\nFrom these searches, we find that it is difficult to find a lensing model that \nadequately describes the observed light curve.\nIn the bottom panel of Figure~\\ref{fig:three} labeled as `standard', we \npresent the residual of the standard model. It shows that the model fit to the \nfirst peak is very poor although the model relatively well describes the second \nand third peaks.\n\n\n\\subsection{Model with Higher-order Effects}\n\nThe difficulty in finding a lensing model that fully explains all the features \nin the observed light curve under the assumption of the rectilinear lens-source \nmotion suggests that the motion may not be rectilinear. This possibility is \nfurther supported by the long duration of the event.\n\n\nTwo major effects cause accelerations in the relative lens-source motion. One is \nthe microlens-parallax effect. The other is the orbital motion of the lens: \nlens-orbital effect. We therefore conduct additional modeling considering \nthese higher-order effects. \n\n\nIncorporating the microlens-parallax effect into lensing modeling requires \ntwo additional parameters of $\\pi_{{\\rm E},N}$ and $\\pi_{{\\rm E},E}$. They represent the two components \nof the microlens-parallax vector $\\mbox{\\boldmath $\\pi$}_{\\rm E}$ directed to the north and east, \nrespectively. The microlens-parallax vector is related to \n$\\pi_{\\rm rel}$, $\\theta_{\\rm E}$, and the relative lens-source proper motion vector \n$\\mbox{\\boldmath $\\mu$}$ by\n\\begin{equation}\n\\mbox{\\boldmath $\\pi$}_{\\rm E}={\\pi_{\\rm rel}\\over \\theta_{\\rm E}}{\\mbox{\\boldmath $\\mu$} \\over \\mu}.\n\\label{eq4}\n\\end{equation}\n\n\nConsidering the lens-orbital effect also requires additional parameters. Under \nthe approximation that the positional changes of the lens components induced by \nthe lens-orbital effect during the event is small, the effect is described by \ntwo parameters of $ds\/dt$ and $d\\alpha\/dt$. They represent the change rates of \nthe binary separation and the source trajectory angle, respectively.\n\n\nWe conduct a series of additional modeling runs considering the higher-order effects. \nIn the `parallax' and `orbit' modeling runs, we separately consider the microlens-parallax \nand lens-orbital effects, respectively. In the `orbit+parallax' modeling run, we \nsimultaneously consider both the higher-order effects. For solutions considering \nmicrolens-parallax effects, it is known that there may exist a pair of degenerate \nsolutions with $u_0>0$ and $u_0<0$ due to the mirror symmetry of the source trajectory \nwith respect to the binary axis \\citep{Smith2003, Skowron2011}.  \nWe inspect this `ecliptic degeneracy' \nwhen microlens-parallax effects are considered in modeling. \n\n\n\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{f4.eps}\n\\caption{\nCumulative distribution of $\\chi^2$ difference between the `parallax' and \n`orbit+parallax' models. The light curve in the upper panel is presented to show \nthe region of the fit improvement.\n}\n\\label{fig:four}\n\\end{figure}\n\n\n\\begin{deluxetable}{lcc}\n\\tablecaption{Comparison of models\\label{table:one}}\n\\tablewidth{240pt}\n\\tablehead{\n\\multicolumn{2}{c}{Model}       &\n\\multicolumn{1}{c}{$\\chi^2$}  \n}\n\\startdata                                              \nStatic            &            &  34436.4   \\\\\nOrbit             &            &  20019.4   \\\\\nParallax          & $u_0>0$    &  5749.4   \\\\\n--                & $u_0<0$    &  6026.4   \\\\\nOrbit + parallax  & $u_0>0$    &  5215.4   \\\\\n--                & $u_0<0$    &  5221.2     \n\\enddata                            \n\\end{deluxetable}\n\n\n\nIn Table~\\ref{table:one}, we list the results of the individual modeling runs in terms \nof $\\chi^2$ values of the fits. In order to visualize the goodness of the fits, we also \npresent the residuals of the individual models in the lower panels of Figure~\\ref{fig:three}. \nFor the pair of solutions with $u_0>0$ and $u_0<0$ obtained considering microlens-parallax \neffects, we present the residuals of the solution yielding a better fit.\n\n\nWe compare the fits to judge the importance of the individual higher-order effects. \nFrom this, it is found that the major features of the light curve, i.e., the three \npeaks, still cannot be adequately explained by the orbital effect alone, although \nthe effect improves fit by \n$\\Delta\\chi^2\\sim 14407.0$ \nwith respect to the standard \nmodel. See the residual labeled as `orbit' in Figure~\\ref{fig:three}. For the parallax \nmodel, on the other hand, the fit greatly improves, by \n$\\Delta\\chi^2\\sim 28677.0$, \nand \nall the three peak features are approximately described.  See the residual labeled as \n`parallax' in Figure~\\ref{fig:three}. We also find that the fit further improves, by \n$\\Delta\\chi^2\\sim 534.0$ \nwith respect to the parallax model, by additionally considering \nthe lens-orbital effects. This indicates that although the lens-orbital effect is not the \nprime higher-order effects, it is important to precisely describe the light curve.  Due \nto the relatively minor improvement, it is not easy to see the additional fit improvement \nby the lens-orbital effect from the comparison of the residuals of the `parallax' and \n`orbit+parallax' models.  We, therefore, present the cumulative distribution of $\\Delta\\chi^2$ \nbetween the two models as a function of time in Figure~\\ref{fig:four}. \nIt is found that the fit improves throughout the event and major improvement occurs at the \nfirst and the second peaks and after the third peak. \nThis indicates that the widely separated \nmultiple peak features in the lensing light curve help to constrain the subtle higher-order \neffects. \nTo check the consistency of the fit improvement, we \nalso plot the distributions for the individual data sets.\nFrom the distributions, one finds that the $\\chi^2$ improvement shows up in all data sets \n(OGLE, MOA, KMTC, and KMTS) except for the KMTA data set. \nWe judge that the marginal orbital signal in the KMTA data set is \ncaused by the relatively lower photometry quality than the other \nKMTNet data sets and the resulting smaller number of data points (626 points \ncompared to 975 and 1214 points of the KMTS and KMTC data sets, respectively).\nWe find the ecliptic degeneracy is quite severe although the model with $u_0>0$ is \npreferred over the model with $u_0<0$ by \n$\\Delta\\chi^2\\sim 5.8$.\n\n\n\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{f5.eps}\n\\caption{\nTriangular diagram showing the \n$\\Delta\\chi^2$ distributions of MCMC points in the planes of the pair of the \nhigher-order lensing lensing parameters $\\pi_{{\\rm E},N}$, $\\pi_{{\\rm E},E}$, $ds\/dt$, and $d\\alpha\/dt$.\nPoints marked in red, yellow, green, cyan, and blue \nrepresent those with $1\\sigma$, $2\\sigma$, $3\\sigma$, $4\\sigma$, and $5\\sigma$, respectively.\n}\n\\label{fig:five}\n\\end{figure}\n\n\n\n\\begin{deluxetable}{lcc}\n\\tablecaption{Best-fit lensing parameters\\label{table:two}}\n\\tablewidth{240pt}\n\\tablehead{\n\\multicolumn{1}{c}{parameter}       &\n\\multicolumn{1}{c}{$u_0>0$}      & \n\\multicolumn{1}{c}{$u_0<0$}  \n}\n\\startdata                                              \n$t_0$ (${\\rm HJD}^\\prime$) &     7504.730 $\\pm$ 0.012       &          7504.644 $\\pm$ 0.015     \\\\\n$u_0$                      &     0.083 $\\pm$ 0.001          &          -0.085 $\\pm$ 0.001       \\\\\n$t_{\\rm E}$ (days)         &     68.19 $\\pm$ 0.18           &          67.56 $\\pm$ 0.39         \\\\\n$s$                        &     0.727 $\\pm$ 0.001          &          0.731 $\\pm$ 0.002        \\\\\n$q$                        &     0.869 $\\pm$ 0.006          &          0.841 $\\pm$ 0.006        \\\\\n$\\alpha$ (rad)             &     1.408 $\\pm$ 0.003          &          -1.408 $\\pm$ 0.002       \\\\\n$\\rho$ ($10^{-3}$)         &     0.615 $\\pm$ 0.013          &          0.620 $\\pm$ 0.012        \\\\\n$\\pi_{{\\rm E},N}$          &     0.334 $\\pm$ 0.008          &          -0.347 $\\pm$ 0.003       \\\\\n$\\pi_{{\\rm E},E}$          &     -0.335 $\\pm$ 0.011         &          -0.406 $\\pm$ 0.012       \\\\\n$ds\/dt$ (yr$^{-1}$)        &     0.168 $\\pm$ 0.010          &          0.198 $\\pm$ 0.013        \\\\\n$d\\alpha\/dt$ (yr$^{-1}$)   &     -0.958 $\\pm$ 0.039         &          1.059 $\\pm$ 0.011        \\\\\n$I_{s,{\\rm OGLE}}$         &      19.37 $\\pm$ 0.01          &          19.37 $\\pm$ 0.01         \\\\   \n$I_{b,{\\rm OGLE}}$         &      19.77 $\\pm$ 0.01          &          19.77 $\\pm$ 0.01    \n\\enddata                            \n\\tablecomments{${\\rm HJD}^\\prime={\\rm HJD-2450000}$. }\n\\end{deluxetable}\n\n\n\nIn Table~\\ref{table:two}, we present the lensing parameters of the best-fit models. Because \nthe ecliptic degeneracy is severe, we present both the $u_0>0$ and $u_0<0$ \nsolutions.  Also presented are the $I$-band magnitudes of the source, $I_{s,{\\rm OGLE}}$, \nand the blend, $I_{b,{\\rm OGLE}}$, estimated based on the OGLE data.  We note that the \nlensing parameters of the two solutions are roughly in the relation \n$(u_0, \\alpha, \\pi_{{\\rm E},N}, d\\alpha\/dt)\\leftrightarrow -(u_0, \\alpha, \\pi_{{\\rm E},N}, d\\alpha\/dt)$\n\\citep{Skowron2011}. \nSeveral facts should noted for the obtained lensing parameters. First, the event timescale, \n$t_{\\rm E}\\sim 68$ days, is substantially longer than typical lensing events with \n$t_{\\rm E} \\sim 20$ days. Second, the binary parameters $(s,q)\\sim (0.73, 0.87)$ indicate \nthat the lens is comprised of two similar masses with a projected separation slightly smaller \nthan $\\theta_{\\rm E}$. Third, the normalized source radius $\\rho\\sim 0.62\\times 10^{-3}$ is smaller \nby about a factor $\\sim 2.5$ than the value of an event typically occurring on a star with a \nsimilar stellar type to the source of OGLE-2016-BLG-0156.  \nSince $\\rho=\\theta_*\/\\theta_{\\rm E}$, the \nsmall $\\rho$ value suggests that the angular Einstein radius is likely to be big. Finally, \nthe parameters describing the higher-order effects, i.e., $\\pi_{{\\rm E},N}$, $\\pi_{{\\rm E},E}$, $ds\/dt$, and \n$d\\alpha\/dt$, are precisely determined with fractional uncertainties $\\sim 2.4\\%$, $3.3\\%$, \n$6.3\\%$, and $4.1\\%$, respectively. In Figure~\\ref{fig:five}, we present the $\\Delta\\chi^2$ \ndistributions of MCMC points in the planes of the pair of the higher-order lensing parameters.\n\n\n\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{f6.eps}\n\\caption{\nLens system configuration. The curve with an arrows represents the source trajectory with \nrespect to the caustic (closed curve composed of concave segments).  To show the variation \nof the caustic caused by the lens orbital motion, we present caustics at 3 moments \ncorresponding to the times of the three peaks in the lensing light curve. We mark the \npositions of source crossings for the individual peaks occurring at $t_1$, $t_2$, and $t_3$. \nThe small dots marked by $M_1$ and $M_2$ present the positions of the binary lens components. \nLengths are scaled to the angular Einstein radius corresponding to the total mass of the lens.\n}\n\\label{fig:six}\n\\end{figure}\n\n\n\nIn Figure~\\ref{fig:three}, we present the model light curve, which is plotted over the \ndata points, of the best-fit solution, i.e., the orbit+parallax model with $u_0>0$. \nIn Figure~\\ref{fig:six}, we present the corresponding lens-system configuration, showing \nthe trajectory of the source with respect to the caustic. When the binary separation $s$ \nis close to unity, caustics form a single closed curve, `resonant caustic', and as the \nseparation becomes smaller, the caustic becomes elongated along the direction perpendicular \nto the binary axis and eventually splits into 3 segments, in which one 4-cusp central \ncaustic is located around the center of mass and the other two triangular caustics are \nlocated away from the center of mass.   \\citep{Erdl1993, Dominik1999}. For OGLE-2016-BLG-0156, \nthe caustic topology corresponds to the boundary between the single closed-curve \n(`resonant') and triple closed-curve (`close') topologies.  The source moved approximately \nin parallel with the elongated caustic, crossing the caustic 3 times at the positions marked \nby $t_1$, $t_2$, and $t_3$. The first peak was produced by the source crossing over the slim \nbridge part of the caustic connecting the 4-cusp central caustic and one of the triangular \nperipheral caustics.  The peak could in principle have been produced by the source star's \napproach to the the right cusp of the upper triangular caustic.  We check this possibility \nand find that it cannot explain the light curve in the region around the first peak.  The \nsecond and third peaks were produced when the source passed the upper and lower right \nparts of the central caustic, respectively.\n\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{f7.eps}\n\\caption{\nModel light curve (curve superposed on data points) obtained under the assumption \nof the rectilinear relative lens-source motion. The lower panel shows the \nresidual from the model. The inset in the upper panel shows the lens system \nconfiguration corresponding to the model.\n\\vskip0.2cm\n}\n\\label{fig:seven}\n\\end{figure}\n\n\n\nWe note that the well-covered 3-peak feature in the lensing light curve provides a very \ntight constraint on the source trajectory, and thus on the higher-order effects.  To \ndemonstrate the high sensitivity of the light curve to the slight change of the source \ntrajectory induced by the higher-order effects, in Figure~\\ref{fig:seven}, we present \nthe model fit of the standard solution and the corresponding lens system configuration. \nOne finds that the straight source trajectory without higher-order effects can describe \nthe second and third peaks by crossing similar parts of the central caustic to those of \nthe solution obtained considering the higher-order effects. However, the extension of \nthe trajectory crosses the upper triangular caustic, resulting in a light curve that \ndiffers greatly from the observed one. The importance of well-covered multiple peaks \nin determining $\\mbox{\\boldmath $\\pi$}_{\\rm E}$ was first pointed out by \\citet{An2001} and a good \nexample was presented by \\citet{Udalski2018} for the quintuple-peak lensing event \nOGLE-2014-BLG-0289.\n\n\n\n\n\n\n\\section{Physical Lens Parameters}\n\n\\subsection{Angular Einstein Radius}\n\nFor the unique determinations of the mass and distance to the lens, one needs to \ndetermine $\\theta_{\\rm E}$ as well as $\\pi_{\\rm E}$. See Equations (\\ref{eq2}) and (\\ref{eq3}). \nThe angular Einstein radius is estimated from the combination of the normalized \nsource $\\rho$ and the angular source radius $\\theta_*$ by $\\theta_{\\rm E}=\\theta_*\/\\rho$. \nThe $\\rho$ value is determined from the light curve modeling. Then, one needs to \nestimate $\\theta_*$ for the determination of $\\theta_{\\rm E}$.  \n\n\n\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{f8.eps}\n\\caption{\nKMTC $I$ and $V$-band data sets processed using the pyDIA photometry code.\nThe data sets are used for the source color measurement.\n}\n\\label{fig:eight}\n\\end{figure}\n\n\nWe estimate the angular source radius from the de-reddened color, $(V-I)_0$, \nand brightness, $I_0$.  For this, we first measure the instrumental (uncalibrated) \nsource color and brightness from the KMTC $V$ and $I$-band data sets processed using the pyDIA \nphotometry. \nWe estimate the source color using the regression of the $V$ and $I$-band data sets.\nThe color can also be estimated by using the model and we find that the source \ncolor estimated by both ways are consistent.\nFigure~\\ref{fig:eight} shows the KMTC $I$ and $V$-band data.  In the \nsecond step, following the method of \\citet{Yoo2004}, we calibrate the color and \nbrightness of the source using the centroid of the red giant clump (RGC) in the \ncolor-magnitude diagram as a reference.  In Figure~\\ref{fig:nine}, we mark the \npositions of the source, with \n$(V-I, I)=(1.99\\pm 0.01, 19.31\\pm 0.01)$, \nand the RGC centroid, \n$(V-I, I)_{\\rm RGC}=(2.43,15.98)$, in the instrumental color-magnitude diagram.  \nWith the known de-reddened color and brightness of the RGC centroid, \n$(V-I,I)_{{\\rm RGC},0}=(1.06,14.46)$ \\citep{Bensby2011, Nataf2013}, combined with \nthe measured offsets in color, $\\Delta(V-I)=-0.44$, and brightness, $\\Delta I=3.33$, \nbetween the source and the RGC centroid, we estimate that the de-reddened color and \nbrightness of the source star are \n$(V-I,I)_{0}=(V-I,I)_{0,{\\rm RGC}}+\\Delta (V-I,I)=(0.62\\pm 0.01, 17.78\\pm 0.01)$, \nindicating that \nthe source is a turn-off star.  In the last step, we convert the measured $V-I$ \ninto $V-K$ using the color-color relation of \\citet{Bessell1988} and then estimate \nthe angular source radius using the relation between the color and surface \nbrightness of \\citet{Kervella2004}.  It is estimated that the angular source \nradius is\n\\begin{equation}\n\\theta_*=0.79 \\pm 0.06\\ \\mu{\\rm as}.\n\\label{eq5}\n\\end{equation} \nIn addition to the measurement error, the source color estimation is further \naffected by the uncertainty in determining RGC centroid and the differential \nreddening of the field.\n\\citet{Bensby2013} showed that for lensing events in the fields with \nwell defined RGCs, the typical error in the source color estimation is \nabout 0.07 mag. \nWe, therefore, estimate the errorbar of $\\theta_*$ by considering this \nadditional error.\n\n\nThe estimated angular Einstein radius is \n\\begin{equation}\n\\theta_{\\rm E}= 1.30 \\pm 0.09\\ {\\rm mas}.\n\\label{eq6}\n\\end{equation}\nFor a typical lensing event produced by a low-mass star ($\\sim 0.3~M_\\odot$) \nlocated halfway between the source and observer ($D_{\\rm L}\\sim 4~{\\rm kpc}$), \nthe angular Einstein radius is \n$\\theta_{\\rm E}=\\sqrt{\\kappa M \\pi_{\\rm rel}}\n\\sim 0.55~{\\rm mas}~(M\/0.3~M_\\odot)^{1\/2}$.\nThen the estimated angular Einstein radius is $\\gtrsim 2$ times \nbigger than the \nvalue of a typical lensing event.  This is expected from the small value of the \nnormalized source radius. Combined with the event timescale, the relative \nlens-source proper motion in the geocentric frame is estimated by\n\\begin{equation}\n\\mu_{\\rm geo} = {\\theta_{\\rm E}\\overt_{\\rm E}} =\n6.94 \\pm 0.69\\ {\\rm mas}\\ {\\rm yr}^{-1}.\n\\label{eq7}\n\\end{equation}\nThe corresponding proper motion in the heliocentric frame is estimated by\n\\begin{equation}\n\\mu_{\\rm helio} =\n\\left\\vert\n\\mu_{\\rm geo} {\\mbox{\\boldmath $\\pi$}_{\\rm E}\\over \\pi_{\\rm E}} +\n{\\bf v}_{\\oplus,\\perp}{\\pi_{\\rm rel} \\over {\\rm au}}\\right\\vert =\n5.94 \\pm 0.43\\ {\\rm mas}\\ {\\rm yr}^{-1}.\n\\label{eq8}\n\\end{equation}\nHere ${\\bf v}_{\\oplus,\\perp}=(v_{\\oplus,\\perp,N}, v_{\\oplus,\\perp,E})=\n(3.1,17.5)\\ {\\rm km}\\ {\\rm s}^{-1}$ denotes the \nprojected velocity of Earth at $t_0$. \n\n\nIn Table~\\ref{table:three}, we summarize the estimated values of the angular \nEinstein radius, relative lens-source proper motion (in both geocentric and \nheliocentric frames), and the direction of the relative motion, i.e., \n$\\phi=\\tan^{-1} (\\mu_{{\\rm helio},E}\/\\mu_{{\\rm helio},N})$.  We also present \nthe quantities resulting from the $u_0<0$ solution.  The obtained quantities are \nslightly different from those of the $u_0>0$ solution due to the slight differences \nin $\\rho$, $\\pi_{{\\rm E},N}$, and $\\pi_{{\\rm E},E}$.\n\n\n\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{f9.eps}\n\\caption{\nLocations of the source and the centroid of red giant clump (RGC) in the instrumental \ncolor-magnitude diagram of stars around the source.\nThe diagram is constructed using the pyDIA photometry of \nKMTC $I$ and $V$-band data.\n\\vskip0.5cm\n}\n\\label{fig:nine}\n\\end{figure}\n\n\n\n\\begin{deluxetable}{lcc}\n\\tablecaption{Einstein radius and Proper Motion\\label{table:three}}\n\\tablewidth{240pt}\n\\tablehead{\n\\multicolumn{1}{c}{Quantity}       &\n\\multicolumn{1}{c}{$u_0>0$}        &\n\\multicolumn{1}{c}{$u_0<0$}         \n}\n\\startdata                                              \n$\\theta_{\\rm E}$                            &   $1.30 \\pm 0.09$  &   $1.28 \\pm 0.09$    \\\\\n$\\mu_{\\rm geo}$ (mas yr$^{-1}$)      &   $6.94 \\pm 0.50$  &   $6.91 \\pm 0.50$    \\\\\n$\\mu_{\\rm helio}$ (mas yr$^{-1}$)    &   $5.94 \\pm 0.43$  &   $4.88 \\pm 0.35$    \\\\\n$\\phi$                               &   $334^\\circ$      &   $215^\\circ$        \n\\enddata                            \n\\end{deluxetable}\n\\smallskip\n\n\n\n\n\\subsection{Mass and Distance}\n\n\nWith the measured $\\pi_{\\rm E}$ and $\\theta_{\\rm E}$, the masses of the individual lens components are \ndetermined as\n\\begin{equation}\nM_1=\n0.18 \\pm 0.01\\ M_\\odot,\n\\label{eq9}\n\\end{equation}\nand\n\\begin{equation}\nM_2=qM_1=\n0.16 \\pm 0.01\\ M_\\odot.\n\\label{eq10}\n\\end{equation}\nIt is estimated that the lens is located at a distance\n\\begin{equation}\nD_{\\rm L} = \n1.35\\pm 0.09\\ {\\rm kpc}.\n\\label{eq11}\n\\end{equation}\nThe determined masses and distance indicates that the lens is a binary \ncomposed of two M dwarfs located in the disk.  The projected separation \nbetween the lens components is\n\\begin{equation}\na_\\perp = s\\theta_{\\rm E}D_{\\rm L} =  \n1.28 \\pm 0.09\\ {\\rm au}.\n\\label{eq12}\n\\end{equation}\nWe note that the $u_0<0$ solution yields similar lens parameters.\nIn Table~\\ref{table:four}, we list the physical lens parameters for both \nthe $u_0>0$ and $u_0<0$ solutions.\n\n\n\nWe check the validity of the solution by estimating the projected kinetic-to-potential \nenergy ratio. We compute the ratio from the physical lens parameters of $M=M_1+M_2$ \nand $a_\\perp$ and the measured lensing parameters of $s$, $\\alpha$, $ds\/dt$, \nand $d\\alpha\/dt$ by\n\\begin{equation}\n\\left( {{\\rm KE}\\over {\\rm PE}}\\right)_\\perp =\n{ (a_\\perp\/{\\rm au})^3\\over 8\\pi^2(M\/M_\\odot) }\n\\left[\n\\left( {1\\over s}{ds\/dt\\over {\\rm yr}^{-1}} \\right)^2 +\n\\left( {d\\alpha\/dt \\over {\\rm yr}^{-1}}\\right)^2 \n\\right].\n\\label{eq13}\n\\end{equation}\nIn order for the lens system to be a gravitationally bound system, \nthe solution should satisfy the condition of\n$({\\rm KE}\/{\\rm PE})_\\perp \\leq {\\rm KE}\/{\\rm PE} \\leq 1.0$,\nwhere ${\\rm KE}\/{\\rm PE}$ denotes the intrinsic energy ratio.\nThe estimated ratio $({\\rm KE}\/{\\rm PE})_\\perp \\sim 0.08$ satisfies\nthis condition. The low value of the ratio suggests that the binary\ncomponents are aligned along the line of sight.\n\n\n\n\\subsection{Lens Brightness}\n\n\nAlthough the lens components are M dwarfs, they are located at a close distance, \nand the flux from the lens can comprise a significant portion of the blended flux, \ne.g., OGLE-2017-BLG-0039 \\citep{Han2018}.  To check this possibility, we estimate \nthe expected brightness of the lens.  The stellar types of the lens components are \nabout M4.5V and M5.0V with absolute $I$-band magnitudes of $M_{I,1}\\sim 10.5$ and \n$M_{I,2}\\sim 11.0$ for the primary and companion, respectively, resulting in the \ncombined magnitude $M_I\\sim 10.0$.  With the known distance to the lens, the \nde-reddened $I$-band magnitude is then $I_{L,0}=M_I + 5\\logD_{\\rm L} -5 \\sim 20.6$.  \nFrom the OGLE extinction map \\citep{Nataf2013}, the total $I$-band extinction \ntoward the source is $A_{I,{\\rm tot}}\\sim 1.48$.  Assuming that about half of \nthe total extinction is caused by the dust and gas located in front of the lens, \ni.e., $A_I\\sim 0.7$, the expected brightness of the lens is\n\\begin{equation}\nI_{\\rm L} = I_{{\\rm L},0}+A_I \\sim \n21.3.\n\\label{eq14}\n\\end{equation}  \nCompared to the brightness of the blend, $I_{b,{\\rm OGLE}}\\sim 19.8$,\nit is found that the the flux from the lens comprises an important fraction,\n$\\sim 25\\%$, of the blended light. \nWe note that the color constraint of the blended light cannot be used because the \nuncertainty of the $V$-band blend flux measurement is bigger than the flux itself.\n\n\n\n\\begin{deluxetable}{lcc}\n\\tablecaption{Physical lens parameters\\label{table:four}}\n\\tablewidth{240pt}\n\\tablehead{\n\\multicolumn{1}{c}{Parameter}      &\n\\multicolumn{1}{c}{$u_0>0$}        &\n\\multicolumn{1}{c}{$u_0<0$}  \n}\n\\startdata                                              \n$M_1$ ($M_\\odot$)             &   $0.18 \\pm 0.01$  &  $0.16 \\pm 0.01$    \\\\ \n$M_2$ ($M_\\odot$)             &   $0.16 \\pm 0.01$  &  $0.13 \\pm 0.01$    \\\\  \n$D_{\\rm L}$ (kpc)                   &   $1.35 \\pm 0.09$  &  $1.24 \\pm 0.08$    \\\\ \n$a_\\perp$    (au)             &   $1.28 \\pm 0.09$  &  $1.16 \\pm 0.08$    \\\\   \n$({\\rm KE}\/{\\rm PE})_\\perp$   &   0.08             &  0.08\n\\enddata                            \n\\end{deluxetable}\n\\smallskip\n\n\nThe bright nature of the lens combined with the high relative lens-source proper \nmotion suggests that the lens can be directly observed from high-resolution \nfollow-up observations.  \nFor the case of the lensing event OGLE-2005-BLG-169, \nthe lens was resolved from the source on the Keck AO images when they were separated \nby $\\sim 50$ mas after $\\sim 8$ years after the event \\citep{Batista2015}.  By applying \nthe same criterion, the lens and source of OGLE-2016-BLG-0156 can be resolved if \nsimilar follow-up observations are conducted $\\sim 6$ years after the event, i.e., \nafter 2022. \nFor the case of the another lensing event OGLE-2012-BLG-0950, \n\\citet{Bhattacharya2018} resolved the source and lens using Keck and the \n{\\it Hubble Space Telescope} when they were separated by $\\sim 34$ mas. \nAccording to this criterion, then, \nthe source and lens of this event would be resolved in 2022.\n\n\nBecause follow-up observations are likely to to be conducted in near infrared bands, we estimate \nthe expected $H$-band brightness of the lens.  The absolute $H$-band magnitudes of \nthe individual lens components are \n$M_H\\sim 8.2$ and 8.7 resulting in the combined brightness of $M_{H,0}\\sim 7.7$.  \nWith $A_I=A_{I,{\\rm tot}}\/2\\sim 0.7$ and \n$E(V-I)=E_{\\rm tot}(V-I)\/2\\sim 0.6$ from \\citet{Nataf2013} and adopting the relation \n$A_H\\sim 0.108 A_V\\sim 0.14$ of \\citet{Nishiyama2008}, we estimate that $H$-band \nbrightness of the lens is \n\\begin{equation}\nH_{\\rm L} = M_{H,0} + A_H + 5\\log D_{\\rm L} -5 \n\\sim 17.9.\n\\label{eq15}\n\\end{equation}\nThe $H$-band brightness of the source is \n\\begin{equation}\nH_{\\rm S}=I_{\\rm S}-E(I-H)-(I-H)_0 \\sim 17.5,\n\\label{eq16}\n\\end{equation}\nwhich is similar to that of the the lens.\nWhen the lens brightness is similar to the brightness of the source,\nthe lens and source can be better resolved as demonstrated for events \nOGLE-2005-BLG-169 \\citep{Bennett2015}, \nMOA-2008-BLG-310 \\citep{Bhattacharya2017}, and\nOGLE-2012-BLG-0950 \\citep{Bhattacharya2018}.\n\n\n\n\n\n\\section{Conclusion}\n\nWe analyzed a binary microlensing event OGLE-2016-BLG-0156.  We found that \nthe light curve of the event exhibited pronounced deviations induced by \nhigher-order effects, especially the microlens effect. It is found that the \nmultiple-peak feature provided a very tight constraint on the microlens-parallax \nmeasurement.  In addition, the good coverage of all the peaks from the combined \nsurvey observations allowed us to precisely measure the angular Einstein radius.  \nWe uniquely determined the physical lens parameters from the measured values of \n$\\pi_{\\rm E}$ and $\\theta_{\\rm E}$ and found that the lens was a binary composed of two M dwarfs \nlocated in the disk.  \nWe also found that the flux from the lens comprises an important fraction \nof the blended flux.\nThe bright nature \nof the lens combined with the high relative lens-source motion suggested that the lens \ncould be directly observed from high-resolution follow-up observations. \n\n\n\\acknowledgments\nWork by CH was supported by the grant (2017R1A4A1015178) of National Research Foundation of Korea.\nWork by AG was supported by US NSF grant AST-1516842.\nWork by IGS and AG were supported by JPL grant 1500811.\nAG received support from the\nEuropean Research Council under the European Union's\nSeventh Framework Programme (FP 7) ERC Grant Agreement n.~[321035].\nThe MOA project is supported by JSPS KAKENHI Grant Number JSPS24253004,\nJSPS26247023, JSPS23340064, JSPS15H00781, and JP16H06287.\nYM acknowledges the support by the grant JP14002006.\nDPB, AB, and CR were supported by NASA through grant NASA-80NSSC18K0274. \nThe work by CR was supported by an appointment to the NASA Postdoctoral Program at the Goddard \nSpace Flight Center, administered by USRA through a contract with NASA. NJR is a Royal Society \nof New Zealand Rutherford Discovery Fellow.\nThe OGLE project has received funding from the National Science Centre, Poland, grant\nMAESTRO 2014\/14\/A\/ST9\/00121 to AU.\nThis research has made use of the KMTNet system operated by the Korea\nAstronomy and Space Science Institute (KASI) and the data were obtained at\nthree host sites of CTIO in Chile, SAAO in South Africa, and SSO in\nAustralia.\nWe acknowledge the high-speed internet service (KREONET)\nprovided by Korea Institute of Science and Technology Information (KISTI).\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\nQuantum theory (QT) is in conflict with noncontextual realism \\cite{Specker60,Bell66,KS67}, that is, with the assumption that the results of measurements reveal pre-existing properties that are not affected by compatible measurements. Noncontextual realism is a legitimate assumption in so far as the statistics of the measurement outcomes are not perturbed by other measurements, or, in other words, whenever measurements cannot be used to communicate information. No signaling holds when the measurements are spacelike separated. If this is the case, noncontextual realism is called local realism. It is well known that QT is also in conflict with local realism \\cite{Bell64}.\n\nHowever, most quantum violations of noncontextual realism do not occur in scenarios in which parties perform spacelike separated measurements. Still, they can be reveled in experiments \\cite{Cabello08}. This shows that contextuality, besides being ``a class of `paradoxes' which result from counterfactual logic'' \\cite{Peres02}, is a phenomenon that can be experimentally tested \\cite{Szangolies15}.\n\nAmong the violations of noncontextual realism that cannot be observed in Bell-like tests there are many cases of interest: Qutrit contextuality \\cite{KCBS08}, quantum-state-independent contextuality \\cite{Cabello08,BBCP09,YO12}, simple fully contextual correlations \\cite{Cabello13b}, contextuality associated to universal quantum computation via magic states \\cite{HWVE14,DGBR15,RBDOB15}, and absolute maximal contextuality \\cite{ATC15}. This variety of examples motivates the challenge of whether there is a unified universal way to test any form of contextuality.\n\nCertifying contextuality requires observing the violation of a noncontextuality (NC) inequality \\cite{KCBS08,Cabello08}, which is an inequality for correlations satisfied by any noncontextual hidden-variable theory (NCHVT), i.e., for any theory assuming noncontextual realism. Testing a NC inequality entails considerable difficulty and requires solving some problems.\n\n\n\\begin{figure}[tb]\n\\vspace{-2.4cm}\n \\includegraphics[width=0.48 \\textwidth]{Fig1.pdf}\n \\vspace{-2.6cm}\n \\caption{(color online). (a) A typical contextuality experiment requires preparing individual quantum systems in a quantum state $|\\psi\\rangle$ and performing a sequence of projective measurements on each of them. (b) Contextuality experiments can be substantially simplified by exploiting that any form of contextuality can be tested by measuring sequences of only two compatible binary observables. The problem is that in most quantum systems sequential projective measurements are unfeasible. In these cases, a possible test of contextuality (not without drawbacks) is one based on the scheme in panel (c). It requires replacing the projective measurement of $i$ in panel (b) by a demolition measurement of $i$ followed, in those trials with outcome $i=1$, by a preparation of the eigenstate of $i$ with eigenvalue $1$ or, in those trials with outcome $i=0$, by a preparation of the quantum state predicted by L\\\"uders's rule. \\label{Fig1}}\n\\end{figure}\n\n\nThe first difficulty is related to the fact that in a test of a NC inequality every observable must be measured using the same device in any context. An alternative would be, before conducting the experiment, certifying that different devices measure the same observable since they produce the same distributions of probability for any possible state. However, this is unfeasible using a finite sample of states. Without such certification it seems unreasonable to assume that measurement outcomes do not depend on the context when measurements themselves do \\cite{ABBCGKLW13}. This problem can be avoided by designing devices that exactly measure each of the required observables in any possible context.\n\nThe second difficulty comes from the fact that, except for the case of Bell inequalities (which are NC inequalities), testing other NC inequalities requires performing sequential measurements of quantum observables \\cite{LKGC11}, as illustrated in Fig.\\ \\ref{Fig1} (a). This is challenging. Recall that the measurement of a quantum observable, as defined by von Neumann \\cite{vonNeumann32} (i.e., a projective measurement), should not only produce an outcome (that should be ``indelibly recorded'' \\cite{MW84} before any further measurement) but also transform the quantum state according to L\\\"uders's rule \\cite{Luders51}. The problem is that only a few experiments have successfully implemented sequential projective measurements (e.g., Ref.\\ \\cite{KZGKGCBR09}). The typical measurements on quantum systems are actually demolition measurements which absorb the system, preventing subsequent measurements, or fail to transform the state of the system according to L\\\"uders's rule.\n\nNevertheless, the main difficulty is achieving the no-signaling condition with sequential measurements. Recall that in a contextuality test each measurement should not affect the probabilities of the outcomes of any subsequent measurement. That is, if the correlations between $A$ and $B$ and between $A'$ and $B$ both appear in the NC inequality and $B$ is measured in the second place, then $\\varepsilon \\equiv |\\sum_a P(a,b|A,B)-\\sum_{a'} P(a',b|A',B)|$ must be zero for any $b$. The problem is that it is impossible to have $\\varepsilon=0$ in actual experiments since the number of trials is finite, long-lasting experiments suffer drifts, and there are inevitable imperfections \\cite{Szangolies15,GKCLKZGR10}. One could argue that the condition of perfect no signaling is not even achieved in experiments with spacelike separated measurements (e.g., $\\varepsilon = 0.07 \\pm 0.10$ in Ref.\\ \\cite{Hensen15}). However, in these experiments no signaling can be assumed on physical grounds and a nonzero $\\varepsilon$ can be attributed to statistical fluctuations. In contrast, in experiments with sequential measurements, a nonzero $\\varepsilon$ should be taken into account in the analysis \\cite{Szangolies15,Winter14,DKL15}. A practical issue, however, is that these analysis are in general difficult or are not well defined beyond the case of two-point correlations.\n\nDespite these problems, little effort has been made for designing contextuality tests with small $\\varepsilon$ that are easy to analyze. From this perspective, a main challenge is to have tests in which the influence of past measurements on future measurements is experimentally indistinguishable from the influence of future measurements on past measurements. More precisely, experiments in which $\\varepsilon$ is zero within the same experimental error that $\\varepsilon' \\equiv |\\sum_b P(a,b|A,B)-\\sum_{b'} P(a,b'|A,B')|$. Assuming causality, $\\varepsilon'$ is, by definition, zero. However, when calculated from experimental data, this zero will be inevitably affected by an error. Only if the experimental value of $\\varepsilon$ is compatible with zero and is affected by a similar error can one reasonably assume that no signaling is satisfied within the experimental precision. In this case, the violation of the NC inequality can be taken as a compelling evidence of contextuality.\n\nAnother difficulty comes from the infeasibility of implementing projective sequential measurements on most quantum systems, particularly on high-dimensional quantum systems \\cite{MTT07,PB11,DBSBLC14,CEGSXLC14,CAEGCXL14} which are necessary for some forms of quantum contextuality \\cite{Cabello08,Cabello13b,ATC15,CDLP13}. One solution is implementing sequentially the unitary transformations corresponding to the desired measurements and detect the systems afterwards. This is the approach adopted in some experiments with photons \\cite{MWZ00,LLSNRWZ11}, neutrons \\cite{HLBBR03,BKSSCRH09}, and molecular nuclear spins \\cite{MRCL09}. Other solution, closer to the ideal contextuality experiment in Fig.\\ \\ref{Fig1} (a), consists of encoding the outcome of each measurement in an extra degree of freedom (e.g., extra spatial modes \\cite{ARBC09}, time bin \\cite{AACB13}, or polarization \\cite{MANCB14}) before performing the next one. However, this solution has one disadvantage: (I) The complexity of the setup grows exponentially with the number of sequential measurements. This does not only entail practical problems but also introduces a conceptual issue since these extra degrees of freedom could provide the memory classical systems need to simulate quantum contextuality \\cite{KGPLC11,FBVFLLFC15}. Moreover, both solutions have other disadvantage: (II) The observers cannot delay the choice of the next measurement to the moment in which the result of the previous measurement is indelibly recorded. As a result, the system is not really forced to answer before the context is fixed \\cite{PR88}. This indicates that other interesting challenge is designing contextuality tests for high dimensional quantum systems and free of disadvantages (I) and (II).\n\nIn this paper we show how to reveal any form of quantum contextuality in an experiment requiring sequences of only two measurements and allowing a better control on the experimental imperfections and a simple analysis. In addition, we show how to design contextuality tests for quantum systems in which sequential projective measurements are unfeasible and free of disadvantages (I) and (II).\n\n\n\\section{Main result}\n\n\nIn this section we show that two-point correlation experiments between binary compatible observables are sufficient to test any form of quantum contextuality. In the next section we will explain why this is important.\n\nTo understand what we mean by ``any form of quantum contextuality,'' we start by recalling that any given NC inequality can be associated to graph \\cite{CSW14}. This association requires converting the linear combination of probabilities in the initial NC inequality into a positive linear combination $S$ of probabilities of yes-no tests (represented in QT by rank 1 projectors). $S$ can then be associated to a graph $G$ in which each probability is represented by a vertex while edges connect vertices corresponding to probabilities of exclusive events (i.e., corresponding to alternative outcomes of a sharp measurement \\cite{CY14}). The maximum of $S$ for NCHVTs and QT are two characteristic numbers of $G$: The independence number, $\\alpha(G)$, and the Lov\\'asz number, $\\vartheta(G)$, respectively \\cite{CSW14}. Reciprocally, any graph $G$ can be converted into a NC inequality such that the bound for NCHVTs is $\\alpha(G)$ and the maximum in QT is $\\vartheta(G)$ \\cite{CSW14}. This establishes a one-to-one connection between any possible quantum violation of any possible NC inequality and a graph. It is in this sense that any form of quantum contextuality can be represented by a graph.\n\nHere we show that any NC inequality represented by a graph $G$ can be further transformed into a NC inequality involving only two-point correlations, while still having $\\alpha(G)$ as upper bound for NCHVTs and $\\vartheta(G)$ as maximum in QT.\n\n{\\em Theorem.} For any graph $G$ with vertex set $V(G)$ and edge set $E(G)$ the following inequalities are tight:\n\\begin{equation}\n\\begin{split}\n\\label{main}\n{\\cal S} \\equiv\n\\sum_{i \\in V(G)} P(1|i) - \\sum_{(i,j) \\in E(G)} P(1,1|i,j)\n& \\stackrel{\\mbox{\\tiny{NCHVTs}}}{\\leq} \\alpha(G) \\\\\n& \\stackrel{\\mbox{\\tiny{QT}}}{\\leq} \\vartheta(G),\n\\end{split}\n\\end{equation}\nwhere $P(1|i)$ is the probability of obtaining result $1$ when observable $i$ is measured and $P(1,1|i,j)$ is the joint probability of obtaining result $1$ for $i$ and result $1$ for $j$.\n\nThe quantum maximum $\\vartheta(G)$ can be attained by preparing a quantum state $|\\psi\\rangle$ and measuring a set of rank 1 projectors $\\{i=|i\\rangle \\langle i|\\}$ such that each projector is associated to a vertex of $G$, adjacent vertices are associated orthogonal projectors, and $\\sum_i |\\langle i | \\psi \\rangle |^2 = \\vartheta(G)$. The set $|\\psi \\rangle \\cup \\{|i\\rangle\\}$ is called a Lov\\'asz-optimum-orthogonal representation of $\\overline{G}$ \\cite{CDLP13}. Its existence is guaranteed by the definition of $\\vartheta(G)$.\n\n\n\\begin{figure}[tb!]\n\\includegraphics[trim = 3.6cm 5.2cm 3.6cm 5.4cm,clip,width=8cm]{Fig2.pdf}\n \\caption{(color online). Simplest nontrivial example of how the graph $G$ is related to the graph $G'$. (a) For any graph $G$ with vertex set $V(G)$, there is a NC inequality of the type $S = \\sum_{i \\in V} P(1,0,\\ldots,0|i,i_1,\\ldots,i_{n(i)}) \\stackrel{\\mbox{\\tiny{NCHVTs}}}{\\leq} \\alpha(G)$, which is violated by QT up to $\\vartheta(G)$; the set $\\{i_j\\}_{j=1}^{n(i)}$ contains the observables corresponding to vertices adjacent to vertex $i$. The probabilities in this inequality are well defined both in NCHVTs and in QT but, in general, involve incompatible tests. This is a problem for designing experiments. (b) By replacing the probabilities $P(1,0,\\ldots,0|i,i_1,\\ldots,i_{n(i)})$ in $G$ by $P(1|i)$ and every edge $(i,j) \\in G$ by the vertices corresponding to $P(0,0|i,j)$, $P(0,1|i,j)$, and $P(1,0|i,j)$ and adding edges between vertices corresponding to exclusive events, we construct a new graph $G'$ which is associated to a new NC inequality, given in Eq.\\ (\\ref{main}), which has the same bounds for NCHVTs and QT that the NC inequality associated to $G$, but involves only compatible observables and can be tested by measuring only two-point correlations. \\label{Fig2}}\n\\end{figure}\n\n\n{\\em Proof.} We can rewrite ${\\cal S}$ as\n\\begin{equation}\n\\begin{split}\n\\label{S}\n\\sum_{i \\in V(G)} P(1|i) - |E(G)| + \\sum_{(i,j) \\in E(G)} \\left[P(0,0|i,j) \\right.\\\\\n\\left. +P(0,1|i,j)+P(1,0|i,j)\\right] \\equiv {\\cal S'} - |E(G)|,\n\\end{split}\n\\end{equation}\nwhere $|E(G)|$ is the cardinal of $E(G)$. Then,\n\\begin{equation}\n{\\cal S'} \\stackrel{\\mbox{\\tiny{NCHVTs}}}{\\leq} \\alpha(G') \\stackrel{\\mbox{\\tiny{QT}}}{\\leq} \\vartheta(G'),\n\\end{equation}\nwhere $G'$ is the graph of the events in ${\\cal S'}$. Therefore,\n\\begin{equation}\n{\\cal S} \\stackrel{\\mbox{\\tiny{NCHVTs}}}{\\leq} \\alpha(G') - |E(G)| \\stackrel{\\mbox{\\tiny{QT}}}{\\leq} \\vartheta(G') - |E(G)|.\n\\end{equation}\nNow we have to prove that $\\alpha(G')=\\alpha(G)+|E(G)|$ and $\\vartheta(G')=\\vartheta(G)+|E(G)|$.\n\nThe maximum of ${\\cal S'}$ for any NCHVT is always attained by assigning probability $1$ to some events and $0$ to the others. For every $(i,j) \\in E(G)$, there are five vertices in $G'$: those corresponding to the probabilities of the events $1|i$, $1|j$, $0,0|i,j$, $0,1|i,j$, and $1,0|i,j$. See Fig.\\ \\ref{Fig2} (b). If $P(1|i)=1$ and $P(1|j)=1$, then $P(0,0|i,j)=0$, $P(0,1|i,j)=0$, and $P(1,0|i,j)=0$. Thus if we do this assignment for all edges in $G$ we obtain ${\\cal S'}=|V(G)|$ (i.e., the number of vertices of $G$), which is less than or equal to $\\alpha(G)+|E(G)|$. On the other hand, if $P(1|i)=0$ and $P(1|j)=0$, then $P(0,0|i,j)=1$, $P(0,1|i,j)=0$, and $P(1,0|i,j)=0$. Thus if we do this assignment for all edges in $G$, we obtain ${\\cal S'}=|E(G)|$, which is strictly smaller than $\\alpha(G)+|E(G)|$. If $P(1|i)=1$ and $P(1|j)=0$, then $P(0,0|i,j)=0$, $P(0,1|i,j)=0$, and $P(1,0|i,j)=1$. However, in general, we cannot do this assignment for all edges in $G$. An assignment maximizing ${\\cal S'}$ for NCHVTs is one maximizing ${\\cal S}$ (recall that ${\\cal S'} = {\\cal S} + |E(G)|$) and assigning probability $1$ to at most one of the vertices that are adjacent in $G$. This is exactly accomplished by any assignment leading to ${\\cal S}=\\alpha(G)$. Then, for every edge $(i,j)$ in $G$ with probabilities $1$ and $0$, the sum of the probabilities of the corresponding five-vertex subgraph of $G'$ is $2$, while for every $(i,j) \\in E(G)$ with probabilities both $0$, the sum of the probabilities of the corresponding five-vertex subgraph of $G'$ is $1$. Therefore, the sum of the probabilities of the events in $G'$ is $\\alpha(G)+|E(G)|$. This proves the upper bound of ${\\cal S}$ for NCHVTs.\n\nFor the maximum of ${\\cal S'}$ in QT consider the assignment of probabilities to the vertices of $G'$ corresponding, via Born's rule, to a Lov\\'asz-optimum-orthogonal representation of $\\overline{G}$. Since, for every edge $(i,j)$ of $G$, $P(1|i, 1|j)=0$, then $P(0,0|i,j)+P(0,1|i,j)+P(1,0|i,j)=1$. To prove that this is actually the quantum maximum notice that ${\\cal S}$ (and ${\\cal S'}$) contains only probabilities which are well defined within QT (irrespective of the order in which measurements are performed) only if all pairs $(i,j) \\in E(G)$ correspond to compatible observables. Then, maximizing $\\sum_{i\\in V} |\\langle i | \\psi \\rangle|^2-\\sum_{(i,j) \\in E(G)} | \\langle j | i \\rangle \\langle i | \\psi \\rangle |^2$ is equivalent to maximizing $\\sum_{i\\in V(G)} |\\langle i | \\psi \\rangle|^2$ under the restriction that $\\langle j | i \\rangle=0$ for $(i,j) \\in E(G)$. Consequently, $\\vartheta(G')=\\vartheta(G)+|E(G)|$. \\hfill \\vrule height6pt width6pt depth0pt\n\nFor clarity's sake we have not addressed the case of vertex-weighted graphs. However, the result presented in this section can be extended to vertex-weighted graphs with weights that are natural numbers (i.e., to the graphs corresponding to inequalities with rational coefficients) by noticing that any of these graphs can be converted into a standard graph by copying $n$ times each vertex with weight $n$.\n\n\n\\section{Discussion}\n\n\nThe observation that any NC inequality can be converted into a NC inequality of the form (\\ref{main}) may change the way contextuality is tested in the laboratory. This is so because testing (\\ref{main}) only requires measuring sequences of two observables, which allows us to simultaneously overcome many obstacles that, so far, had prevented experiments on most forms of contextuality. More specifically:\n\n(i) On one hand, the fact that only two sequential measurements are required will help to improve the quality of the experimental results by reducing the imperfections and making easier to identify and correct them.\n\n(ii) On the other hand, the fact that only two-point correlations are required allows us to apply existing tools for analyzing contextuality experiments with imperfections which are only defined in this case \\cite{DKL15}.\n\n(iii) Furthermore, it allows us to design tests free of problems (I) and (II) using systems in which sequential nondemolition measurements are not feasible. If only sequences of two measurements are needed, then the complexity of setup for testing each sequence is constant and thus problem (I) is solved. On the other hand, the first measurement can be simulated by a demolition measurement followed by a preparation that depends on the outcome. For example, the projective measurement of $i$ in Fig.\\ \\ref{Fig1} (b) can be simulated by a demolition measurement of $i$ followed by a preparation of a new system in state $|i\\rangle$ if the outcome of $i$ is $1$, or in the state predicted by L\\\"uders's rule for the state $|\\psi\\rangle$ if the outcome of $i$ is $0$; see Fig.\\ \\ref{Fig1} (c). Actually, only the first part is needed for obtaining the probabilities in EQ.\\ (\\ref{main}), which can then be expressed as $P_{|\\psi\\rangle}(1,1|i,j)=P_{|\\psi\\rangle}(1|i) P_{|i\\rangle}(1|j)$, where $P_{|\\phi\\rangle}(\\ldots)$ denotes the probability $P_{|\\phi\\rangle}(\\ldots)$ for the quantum state $|\\phi\\rangle$. Then, the choice of the second measurement is made after the result of the first measurement is indelibly recorded, thus solving problem (II).\n\nNevertheless, it is important to point out that the experiments using the scheme in Fig.\\ \\ref{Fig1} (c) present some conceptual disadvantages with respect to those using the schemes in Figs.\\ \\ref{Fig1} (a) and \\ref{Fig1} (b). For example, the scheme in Fig.\\ \\ref{Fig1} (c) provides an operational justification for the assumption of outcome noncontextuality, but only in experiments with sequences of two measurements on $|\\psi\\rangle$. There, it is possible to check that, when $|\\psi\\rangle$ is prepared, $i$ does not affect the outcome probabilities for any $j$ that is measured afterwards. However, this is not guaranteed for other possible preparations because the way the measurement of $i$ in Fig.\\ \\ref{Fig1} (c) transforms the state depends on $|\\psi\\rangle$ (and, in particular, fails to imitate a projective measurement when the outcome is $i=0$). In contrast, experiments using the schemes in Figs.\\ \\ref{Fig1} (a) and \\ref{Fig1} (b) provide a much more compelling operational justification of the assumption of outcome noncontextuality, since they allow us to check repeatability and no signaling for any preparation and any sequence of compatible measurements.\n\nNotice also that the experiments in Figs.\\ \\ref{Fig1} (b) and \\ref{Fig1} (c) are not thermodynamically equivalent. The measurement of $i$ in Fig.\\ \\ref{Fig1} (c), a demolition measurement followed by a preparation, dissipates more heat than the projective measurement of $i$ in Fig.\\ \\ref{Fig1} (b). Since classical simulations of quantum contextuality can be distinguished from truly quantum contextuality by the fact that the former produces more heat \\cite{CGGLW15}, then to certify quantum contextuality it is more convenient to use tests like those in Fig.\\ \\ref{Fig1} (b) than those in Fig.\\ \\ref{Fig1} (c).\n\nIn addition to the advantages for designing better experiments, the fact that any form of contextuality can be revealed in experiments measuring only two compatible observables also has some conceptual advantages:\n\n(iv) It allows for a novel black box approach to contextuality, extending the black box approach to nonlocality \\cite{NGHA15}. The basic elements would be pairs of boxes with respective inputs $x,y \\in V(G)$ such that $(x,y) \\in E(G)$ and respective outputs $a,b\\{0,1\\}$. Unlike the case of nonlocality, here we are not limited by the condition that each box should be attributable to a party spacelike separated from the others. Still, we can use tools like postselection, composition, and wiring of boxes. This approach may help to understand why certain sets of nonlocal correlations \\cite{NGHA15} are not realizable.\n\n(v) It suggests a way for developing a resource theory of contextuality in analogy to the resource theory of nonlocality \\cite{deVicente14}.\n\n(vi) It permits translating any NC inequality into a communication complexity scenario involving only two-point communication. This may help in elucidating the connection between the principle that constraints quantum contextuality \\cite{Cabello13} and information-theoretic principles that are essentially bipartite \\cite{PPKSWZ09}.\n\n(vii) It allows for building theory-independent sets of events with exclusivity relations leading to noncontextual and quantum bounds on demand. For any graph $G$, one can assign unit vectors to the vertices of $G$ such that adjacent vectors are orthogonal. This means that, according to QT, adjacent vertices correspond to alternative outcomes of a sharp measurement and, therefore, to exclusive events. However, these exclusivities cannot be justified in a theory-independent way unless we identify each of the vertices in $G$ with an operational procedure that explains why adjacent vertices correspond to exclusive events. The construction used in inequality (\\ref{main}) avoids the problem of finding such procedures and allows us to construct theory-independent sets of events with noncontextual and quantum bounds equal to those of $G$. Notice that the only assumption in the transition from $G$ to $G'$ is that edges in $G$ correspond to jointly measurable observables.\n\n(viii) The recent observation that contextuality is a necessary resource for fault-tolerant universal quantum computation via magic states \\cite{HWVE14,DGBR15,RBDOB15} depends on a definition of contextuality based on the violation of inequalities involving probabilities of noncompatible observables (following Ref.\\ \\cite{CSW14}). The fact that any of these inequalities can be converted into a NC inequality of the type (\\ref{main}) allows us to reformulate the contextuality-computation connection in terms of the traditional definition of contextuality as the violation of inequalities involving probabilities of compatible observables \\cite{Peres02}.\n\n\n\\begin{acknowledgments}\nWe thank L.\\ Aolita, M.\\ Ara\\'ujo, G.\\ Ca\\~{n}as, J.\\ Chen, V.\\ D'Ambrosio, E.\\ S.\\ G\\'omez, J.-\\AA.\\ Larsson, G.\\ Lima, A.\\ J.\\ L\\'opez-Tarrida, S.\\ P\\'adua, J.\\ R.\\ Portillo, F.\\ Sciarrino, S.\\ van Dam, and A.\\ Winter for useful conversations. This work was supported by the FQXi large grant project ``The Nature of Information in Sequential Quantum Measurements'' and Project No.\\ FIS2014-60843-P, ``Advanced Quantum Information'' (MINECO, Spain), with FEDER funds.\n\\end{acknowledgments}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}