The developed method has been motivated by laboratory tracer experiments. To investigate the kinetic properties and behavior of a substance of interest, we computationally label this species in order to track it throughout its life cycle. The corresponding mathematical model is extended by an additional set of reactions for the labeled species, avoiding any double-counting within closed circuits, correcting for the influences of upstream fluxes, and taking into account combinatorial multiplicity for complexes or reactions with several reactants or products. A profile likelihood approach is used to estimate confidence intervals on the label half-life and transit-time.Mathematical models of dynamical systems facilitate the computation of characteristic properties that are not accessible experimentally. In cell biology, two main properties of interest are (1) the time-period a protein is accessible to other molecules in a certain state - its half-life - and (2) the time it spends when passing through a subsystem - its transit-time. We discuss two approaches to quantify the half-life, present the novel method of Application to the JAK-STAT signaling pathway in Epo-stimulated BaF3-EpoR cells enabled the calculation of the time-dependent label half-life and transit-time of STAT species. The results were robust against parameter uncertainties.http://www.potterswheel.de).Our approach renders possible the estimation of species and label half-lives and transit-times. It is applicable to large non-linear systems and an implementation is provided within the PottersWheel modeling framework ( The second one, the label transit-time, is the time-period it takes for a fraction of labeled entities to pass through a subsystem of the network. Both quantities are calculated using a novel approach called in silico labeling, which is also introduced in the present work.An increasing number of biological phenomena are described by mathematical models, specifically on the basis of biochemical reaction networks ,2. The din silico labeling approach, where an additional set of reactions is added to an existing mathematical model describing the kinetic behavior of a tracer, called the label. In contrast to real tracer experiments, the in silico method offers the opportunity to define dead-ends, avoid double-counting of cycling label, and to restrict the label to a sub-network of reactions. This allows asking specific questions about the original system, like how long it takes for 50% of the molecules of a substance to travel along a certain path, while in reality an alternative path may exist. In addition, predominant paths can be identified in deterministic models as has been done previously for stochastic systems  ≐ ix. The time-profile of each species can then be calculated for given initial values ku(t). The flux j vof reaction j may be a non-linear function of one or more species concentrations i xand externally defined ku. To improve readability, we omit explicitly denoting the time-dependency, i.e. ix(t) is rather written as ix.Here, i xof interest is determined by extending the differential equation network (4) by one equation for an artificial quantity y depending only on the outfluxes of ix,The half-life of a species i xis given by T1/2 for whichwith initial value t0. In a numerical integration, it is important to limit the maximum integrator step size for an accurate approximation of the y0/2 threshold crossing.Note that a half-life characterizes the decay of a quantity, independent of any production rates. Therefore, all influx contributions are neglected in equation (5). In general, only linear processes possess a constant half-life. Otherwise, the half-life depends on the initial concentration i xis only partially related to the time it takes for 50% of an experimental tracer to leave the source pool. The two values coincide if i xhas either no influx or when the outflux from i xis described by a linear process, which will be proved in the next two subsections. Therefore, we suggest the in silico labeling half-life as a means to determine a time-characteristic which is motivated by laboratory tracer experiment with the additional property to avoid tracer-double counting in kinetic cycles.The half-life of a species x ∈ {x1, . . . , mx} is consumed only in one reaction. In in silico labeling, the flux of the corresponding label z depends on the outflux of x byFor simplicity, we assume that the species of interest in silico labeling half-life of x is defined as the time when z drops to z0/2. We will show that this time equals the species half-life of x if its influx in vis zero. This property is independent from the amount of initially labeled entities, i.e. it holds for any z0/x0 ∈ ℝ+:The Proof:x be determined by the processing with an unknown, potentially non-linear outflux out vand no influx in v= 0, i.e. v = outv,Let z(t) is given byThen, the kinetics of the label species It can be shown that the factor t = 0, the proportionality constant is given by Since this relation holds also true for x(t) and z(t) share the same half-life T1/2, sinceBoth processes in v≠ 0, because the fraction z/x becomes time-dependent as the labeling gets diluted, except for linear outfluxes as shown in the next section.This relation does not hold for processes with x which is produced by an unknown, potentially non-linear influx in vand is consumed by a linear process.In this section, we prove that the label half-life coincides with the half-life of a species Proof:in vand a linear outflux, kx,Let x can be determined viaThen, the analytical half-life of z it holds thatFor the labeled system i xbelong to the group of tracked, i.e. potentially labeled entities. Let us assume that they are given by x1, . . . , α xand untracked ones by xα+1, . . . , mx. Further, it can be assumed without loss of generality that (1) x1, . . . , xγ ≤ α are not complexes consisting of two or more tracked single entities, and (2) that the tracked single entities within each complex xγ+1, . . . , α xbelong to the set x1, . . . , γx. In the JAK-STAT example, S, pR_S, and pS belong to x1, . . . , α xand pS_pS to xα+1,. . . , m xas it contains two labeled single entities pS.Some entities xLF is created based on the original x, by applying the following rules:A new set of labeled or free entities xLF = {}• Start with an empty set, i x∈{x1, . . . , γx}, xLF is enlarged by a labeled • Single entities: For each i x∈ {xγ+1, . . . , xα} is decomposed into • Complex entities: Each complex ix, and are added to xLF. The complex pS_pS for instance leads to the four new complexes F_pSF, pSL_pSF, pSF_pSLpS, and L_pSLpS.possible combinations using labeled rLF , the combinatorial multiplicity has to be applied not only to complexes but also to the ordered lists of reactants and products. Suppose an ordered list I of entities from the set {ix}i≤α 1 ≤ with possible repetition, as for example the reactants of the reaction A + A + pA_pA → A_A_pA_pA corresponding to I = . Summing up all single reactants and elements of the complexes leads to p single entities, in this case p = 4. Taking into account all combinations of labeled and free entities, 2p different lists can be derived, in the exampleIn order to create a new set of reactions δ reactions of the original system are assumed to affect a tracked entity. In these reactions, at least one reactant or product is a tracked entity. Then, a new set of reactions rLF can be established. Starting with the empty set rLF = {}, for each reaction i r∈ {r1, . . . , δr} with one or more reactants of tracked entities,Without loss of generality, only the first 1. all reactants and products not belonging to the group of tracked entities are removed,I of the remaining reactants leading to 2. the combinatorial multiplicity approach is applied to the ordered list p reactions are added to rLF with reactants 3. 24. the fluxes Note that again the same symbol has been used for the entity name and its concentration. The sum over all weighting factors is 1.i r∈ {r1, . . . , δr}without reactants produce only free entities, which simplifies the conversion of i rbefore adding to rLF: All untracked entities are removed, all i xare replaced by Reactions When calculating the label half-life, products that coincide with the initially labeled entity are replaced by the corresponding free entity. This corresponds to removing the label and is necessary to avoid double-counting and to exclude upstream fluxes.rLF by their free counterparts. At the same time, a new product is added to those reactions where the target entity is a product to accumulate the returned label, RL.In order to calculate the label transit-time, entities entering the target pool must be released from their labeling, again, to avoid double-counting. Therefore, all labeled target entities are replaced in the reaction network injected at time-point 0, but the procedure is repeated for a series of time-points t (let i xbe the source species):Since the label half-life and transit-time characteristics are time-dependent, the label is not only 1. Set all initial values for labeled entities and RL, if available, to 0. Set the initial value of free entities to the value of their counterpart in the original network.r, rLF} from 0 to t.2. Numerically integrate the ordinary differential equations corresponding to the extended reaction network {injection.3. Apply a complete labeling of the source species: Set 4. Continue the numerical integration.t" of the time-profiles t' > t) with t"-t, respectively. The threshold crossing is determined by linear interpolation of the discrete samples given by the numerical integration.Threshold crossing at We recently suggested a profile likelihood-based approach to determine simultaneous and separate confidence intervals for calibrated unknown model parameters . In ordeT/21(t) of simple and isolated biochemical reactions can be calculated analytically. Except for first-order processes, it usually depends on the concentration x0 = x(t0) at the time-point of interest t0 and is therefore time-dependent:The half-life n >1 with The half-life calculation for a process of order x(t) with known x0 at t = t0:In order to calculate the half-life for Michaelis-Menten kinetics, rived in , for x:pR(t) triggering the phosphorylation of STAT (S → pS). After dimerization (pS + pS → -pSpS), the complexes enter the nucleus (pS_pS → npS_npS). Then they dissociate and are dephosphorylated (npS_npS → nS + nS). Finally, single STAT molecules leave the nucleus again (nS → S). Model parameters were estimated using a Levenberg-Marquardt approach and the PottersWheel modeling software. The pools of total and phosphorylated cytoplasmic STAT have been used as observation functions. The kinetic parameters were estimated as k1 = 1.37, k2 = 0.22, k3 = 0.63, k4 = 0.59, and k5 = 0.59. The initial value of S was calibrated to 0.96 and the scaling factors for the observables to 1.45 for -obs pSand 0.98 for -obsS.A smoothing spline approximation of the phosphorylated receptor served as the input function S is both, the initially labeled entity and the target pool. The flux of the label is illustrated in Figure in silico labeling approach allowed for discrimination between these two dynamics. In order to determine the transit-time for t >0, the label is injected at a series of time-points, which is visualized in Figure In order to determine the label half-life and transit-time of STAT, LS. It reaches a minimum of 0.6 ± 0.1 minutes after ten minutes compared to a half-life of approximately 3 to 4 minutes at the starting point of the time-course analysis. For later time-points, the stimulus decreases (not shown) and STAT is no longer phosphorylated, resulting in an increased label half-life of STAT molecules. The minimum label transit-time for a complete cycle of STAT molecules was estimated as 12 ± 2 minutes.Figure k1), dimerization (k2), nuclear import (k3) and export (k5) were systematically varied consecutively within four orders of magnitude between 0.01fit kand 100fitk, with fit kbeing the parameter value for the best fit. For each variation, the other free parameters were calibrated resulting in a profile likelihood estimation (see Fig. S2 in additional file X2-threshold of the separate 95% confidence interval are used to recalculate the label half-life and transit-time. Figure t = 0 and t = 20 minutes. The label transit-time has a wider confidence interval reflecting the larger number of reactions involved in a complete cycle of shuttling STAT.In order to investigate how uncertainties in calibrated model parameter values propagate to the estimated time-characteristics, we applied the profile likelihood approach on an identifiable version of the model. The kinetic parameters involved in phosphorylation k, dimerizin silico labeling method, which relies on an extended reaction network taking into account constraints concerning double-counting, upstream fluxes and combinatorial multiplicity. Our model-based in silico approach allows for insights into reaction networks that cannot be determined experimentally.In this paper, the half-life of a species has been compared conceptually, analytically, and numerically to the half-life of a label in a hypothetical tracer experiment. Two time-characteristics, the label half-life and label transit time have been introduced, which capture the kinetics of a dynamical system on a higher level than e.g. single rate constants. Calculation of the time-characteristics and their profile likelihood-based confidence intervals for an identifiable pathway model showed that the approach is robust against parameter uncertainties. The quantities are calculated based on the novel The proposed method provides important information for a wide spectrum of biological applications ranging from cell biology and pharmacokinetics to population dynamics. We applied it to a non-linear model of the cellular JAK-STAT signaling pathway, which allowed for calculating the time-dependent label half-life and transit-time of cytoplasmic STAT.In summary our approach enables to calculate the amount of time a molecule spends in a certain state or compartment and therefore provides novel insights into the temporal scale of networks. This knowledge will have profound impact on drug design, as it offers the possibility to predict the life-time of a specific molecule and provides a basis to improve drug targeting.TM and JB: Definition of the biological question, initiation, development, and implementation of the proposed method, writing of the manuscript. AR: Development of the proposed method, writing of the manuscript. SH, MS, SS, VB, UK, JT: Definition of the biological question, initiation of the method, critical discussion and contribution to manuscript. All authors read and approved the final manuscript.Application within PottersWheel. This additional file contains MATLAB scripts to run various tasks related to the in silico labeling approach. http://www.biomedcentral.com/imedia/4654854926777309/supp1.pdf..Click here for file


Networks are ubiquitous in modern cell biology and physiology. A large literature exists for inferring/proposing biological pathways/networks using statistical or machine learning algorithms. Despite these advances a formal testing procedure for analyzing network-level observations is in need of further development. Comparing the behaviour of a pharmacologically altered pathway to its canonical form is an example of a salient one-sample comparison. Locating which pathways differentiate disease from no-disease phenotype may be recast as a two-sample network inference problem.We outline an inferential method for performing one- and two-sample hypothesis tests where the sampling unit is a network and the hypotheses are stated via network model(s). We propose a dissimilarity measure that incorporates nearby neighbour information to contrast one or more networks in a statistical test. We demonstrate and explore the utility of our approach with both simulated and microarray data; random graphs and weighted  correlation networks are used to form network models. Using both a well-known diabetes dataset and an ovarian cancer dataset, the methods outlined here could better elucidate co-regulation changes for one or more pathways between two clinically relevant phenotypes.Formal hypothesis tests for gene- or protein-based networks are a logical progression from existing gene-based and gene-set tests for differential expression. Commensurate with the growing appreciation and development of systems biology, the dissimilarity-based testing methods presented here may allow us to improve our understanding of pathways and other complex regulatory systems. The benefit of our method was illustrated under select scenarios. Networks, a construct emphasizing the interrelations between objects, have application in studying human behaviour, mathematics, physics, econometrics, etc. With the introduction of microarrays and other high-throughput systems, networks increasingly provide a means to organize and study the interdependencies of genes, proteins, metabolites, etc. Gene transcription/regulatory networks, metabolic pathways, protein-protein interaction systems (PPIs), signal transduction pathways, and phylogenetic trees are established tools in biology. Prone-to-noise experiments are routinely coupled with computational algorithms to infer relationships. Given the empirical reliance on uncertain data it seems natural to ask, “Do these networks differ from one another?” This paper outlines and demonstrates an inferential process for performing one- and two-sample hypothesis tests when the sample data are biological networks.et al.[et al. [et al. [et al. [Numerous books are devoted to networks. Bornholdt et al. and Junk.[et al.  provide  [et al.  is a col [et al.  appears  [et al.  provide p, play an important conceptual role in our understanding of graphs [et al. [p* models in social network analysis, offer a theoretical model for stochastic networks and have been used in biological applications [et al. [C. elegans protein interaction growth model. Cardelli [et al. [Testing graphs is not trivial; comparing two static graphs is conceptually different from comparing a sample of stochastic graphs under one or more treatments. Defining a suitable null probability model for a network is a difficult consideration . Erdős-Rf graphs . The usef graphs . Chung e [et al.  explore  [et al. , in discications . ERGM paications . Motif fications . Wiuf et [et al.  outline Cardelli  suggests [et al.  argue anet al. [p* models and correspondence analysis to compare networks. Banks et al. [et al. [et al. [et al. [et al. [Approaches for both two- and one-sample comparisons occur in the literature. Faust et al.  use a cos et al. , using a [et al.  extend [ [et al.  to netwo [et al. , in a re [et al.  propose  [et al.  use an aet al. [et al. [et al. [et al. [While our focus here is on forming statistical comparisons of network model parameters, computer scientists can divide the graph comparison problem into two areas – exact graph matching and graph similarity . Graph set al.  propose  [et al.  combine  [et al.  extend a [et al.  capture et al. [et al. [The use and analysis of weighted (correlation-based) networks is increasingly relevant in biological applications. Zhang et al. , in a cl [et al. , as part [et al.  contains [et al.  uses a det al. [et al.[Resampling methods are common in network analysis, e.g., see , and wilet al.  use bootet al.  combines. [et al. use a pe0 state of a signal transduction network. Algorithms that derive networks using in silico or in vitro/vivo data could be tested for the comparability of their network model estimates in either a one- or two-sample context. For obvious reasons, two-sample comparisons have a broad application range. Two-sample tests are conducted by drug developers to compare pathways between an investigative compound at various doses or to a competitor’s compound. We conduct one- and two-sample tests using both simulated and microarray-based network data. To evaluate both Type I error control and the power of our procedure, we examine both null and non-null cases under a small set of network models. We assume that each sampling unit is an independent realization of a network. Our testing approach follows the traditional hypothesis testing route: define the null and alternate hypotheses  and the risk associated with a decision; define a ‘nearby neighbour’ dissimilarity-based test statistic for testing the hypothesis; compute the test statistic; compute the distribution of the test statistic under the null hypothesis; and, make a decision using the sampling distribution and the calculated test statistic. Our approach applies to edge/no-edge graphs, weighted graphs, and can be extended to directional networks. In the one-sample hypothesis testing context a null network model is assumed or based on resamples from an a priori null sample to generate the sampling distribution of our test statistic under the null hypothesis. To implement a two-sample comparison we rely on the permutation testing principle as the formal basis of our inferential approach [Hypothesis tests for networks have a variety of obvious applications. A one-sample comparison can occur when comparing a network estimate to a known ‘gold standard’ model. The standard could reside in an online ontology, be defined via a data processing algorithm , or reflect the starting tapproach .xi, i = 1, …, n} of independent observations is therefore a set of networks. In contrast to social networks, where the sampling unit is often the node, we assume that the biological network is inherent to each sample observation. We propose a dissimilarity measure as a test statistic to capture the separation between two networks. We assume that we can align the nodes across the networks to be compared. Since each node in the network represents a gene or other molecular entity, the ability to align nodes across a set of networks implies that we know a unique identifier for each node to allow for a molecule-by-molecule comparison across a set of networks. We determine the sampling distribution of the test statistic under the null hypothesis via resampling techniques.We consider an observed network as a realization of a stochastic process. A sample {G is an ordered pair of disjoint sets  where both V and E are finite sets. V = V(G) is the set of vertices and E = E(G) is the set of edges. An edge {x,y} is said to join, or tie, the vertices x and y and is denoted xy. If xy ∈ E(G) then x and y are adjacent, or neighbouring, vertices of G and the vertices x and y are incident with the edge xy. Two edges are adjacent if they have exactly one common endvertex. G’ =  is a subgraph of G =  if V’ ⊂ V and E’ ⊂ E. If x is a vertex of G we write x ∈ G in place of x ∈ V(G). The order of G is the number of vertices in G; the size of G is the number of edges of G. G denotes an arbitrary graph of order n and size m. A graph of order n and size n-graph. A covariance matrix of nonzero elements with dimension n, Σn, is a complete n-graph. The set of vertices adjacent to x ∈ G, the neighbourhood of x, is Γ(x). For adjacent vertices x and y we have x ∈ Г(y) and y ∈ Г(x). The degree of x is |Γ(x)|. A vertex of degree 0 is an isolated vertex (or isolate). A path is a graph P where V(P) = {x0, x1, …, xl} and E(P) = {x0x1, x1x2, …, xl-1xl}. The length of P is the size of P. A graph without any cycles, a path with length greater than or equal to three and comprised of distinct vertices, is an acylic graph. Unlike trees, we allow for cyclic graphs with isolates. Paths have an obvious tie to motifs and other regulatory functions. By definition, a loop (xx ∈ E(G)) is not allowed; multiple edges joining the same two vertices are not allowed. A graph G can contain a subgraph G’ that is not connected to the remainder of G. Isolated nodes and subgraphs occur in biological networks. Accommodating isolates is necessary to align nodes between two graphs.First, we begin with some definitions. Our definitions are consistent with Bollobás . A graphG in matrix form. The adjacency matrix A = A(G) = (aij) of a graph G is the nxn matrix given by wij, where wij is the strength, covariance, etc., between vertices vi and vj. Given nxn network matrices G = (gij) and H = (hij) we define G-H in the standard algebraic sense, i.e., gij-hij. We do not require a matrix be square; some directed network forms are nxm matrices. But, our use of element-wise subtraction is key; we are not suggesting a definition based on subspaces/subgraphs or set complements. We need to map an nxn network onto the real line in order to define a measure of separation. Under this definition of matrix subtraction, G-H = 0, where 0 is an appropriately dimensioned matrix of zeros, implies no separation between two networks.It is common to represent a graph drs between r and s satisfies the following: drs≥ 0 for every r, s, drr= 0 for every r, and drs= dsr for every r, s. Refer to Gan et al. [x and y are generally based on a simple matching distance, The concept of dissimilarity (or similarity) is standard fare, especially in cluster analysis and pattern recognition. The dissimilarity measure n et al.  for an edistance  is a symWO= (wijO) be a (weighted) adjacency (or directed incidence) matrix for the observed network estimate and WT= (wijT) the target network. In a one-sample comparison WT represents the true network model. For a two-sample comparison the distinction between the two labels is arbitrary. Both WO and WT represent graphs of order n; the nodes are labeled, common to, and aligned between both graphs. For node i define the dissimilarity at that node to be a combination of that node’s dissimilarity, i’s neighbours, j ≠ i, j ∈ Г(i). For the overall network, the dissimilarity D is defined asLet I is defined as the standard mathematical indicator function. For a graph of order n a set/neighbourhood is formed at each node wii, i = 1,…,n. We measure the dissimilarity between a node and its adjacent neighbours between the two graphs. To account for the intrinsic network structure the neighbourhood is then extended to those neighbouring nodes that are incident to nodes in Γ(wii) for both the target and observed networks. I.e., we now measure the dissimilarity for the two subgraphs induced by Γ(wjj), where i ≠ j and wjj is an element of Γ(wii). This ‘extended’ neighbourhood dissimilarity is added to the dissimilarity measured at wii. The contribution of the second nearest neighbours is weighted by cij in the definition of D. In a weighted network, e.g., a correlation network, this weight is easily motivated. In an unweighted network cij is set by the researcher. Assuming a weight of cij= 0 for an unweighted 0–1 graph reduces D to an unscaled version of the familiar Hamming distance and is comparable to the edit distance approach listed in [isted in .D using separate edge and weight L1-norms. We elaborate on this choice later. The need to align the two graphs is critical to calculating D. Our approach does result in additional computational overhead since edge xy will be counted for nodes x and y. But, the counting is consistent and avoids the need for complex single-count network partitioning schemes. Only those nodes with a path length of 2 or less from wii are included; D can easily be extended to include path lengths greater than 2.We form n, G, the obvious parameter to test is p. Apart from ERGMs and  correlation networks, explicit network parametric models may not be readily apparent. The basic form of a one-sample network hypothesis test here is H0: η = η0 versus H1: η ≠ η0, where η is anticipated to be a vector-valued parameter for most realistic (weighted) biological networks. For a G graph we have η = p and one could test H0:G = G versus H1:G ≠ G. We intentionally provide no explicit guidance for how to determine p. And, we make explicit the probability model for the graph rather than state the hypothesis in a more compact manner, i.e., H0:p = p0. For a network defined using a correlation matrix Ω we construct hypotheses of the form H0:Ω = Ω0 versus H1:Ω ≠ Ω0. One needs to make explicit the procedure used to establish the edges in the network and the probability model for the observation data. In an example we test H0:G = G versus H1:G > G. Here, one needs to recognize that the amount of randomness/entropy for a G graph is less when p is close to 0 or 1 relative to p close to 0.5 to successfully perform a one-sided test. Primarily, we are interested in the question, “Does this differ from the target?”Defining an appropriate hypothesis in the context of networks can be nontrivial. For an Erdős-Rényi graph of order D may be suitable in its stock form or require customization for the problem at hand. Third, compute the test statistic. Fourth, determine the distribution of the test statistic under the null hypothesis via a suitable resampling procedure. Finally, make a decision using the sampling distribution of the test statistics as a guide.We employ a resampling approach to perform one- and two-sample network hypothesis tests. Following the five-step procedure outlined in Good  we firstD in the one-sample case we assume a parametric model or explicit generative algorithm is available in order to draw samples from the null network model. In most customary testing situations a test statistic is an estimate for a parameter of interest. However, in some cases the sample itself is the statistic – concise reductions of the data may be limited or not obvious/possible. Biological networks, where each edge or weight may be associated with transcription activity or a regulatory cascade, are inherently high-dimensional objects and may therefore lack parsimonious model parameterizations. In many applications a network algorithm F is required to produce an observed network. In these instances we may need to resample from F(x) instead of resampling from the observed data {xi}. In other cases, e.g., the Erdős-Rényi G graph, the role of the xi may be suppressed or not apparent since we observe F(x). For networks based on explicit probability models, e.g.,  correlation networks, parametric bootstraps or Monte Carlo procedures may be possible under suitable assumptions [To generate a null distribution for umptions .H0: η1= η2 versus H1: η1≠ η2 two-sample problem allows one to draw upon established parametric and nonparametric comparisons. A null hypothesis of network equality versus an alternate hypothesis of network inequality is expected to be commonplace. Alternate hypotheses such as H1: η1> η2 are possible but not explored here. We make the standard assumption of two independent and identically distributed samples {x1, …, xn} and {y1, …, ym}, where xi and yj are network-valued. Following the notation of [P be a family of distributions for {X1, …, Xn} that are symmetric in the sense that for a permutation π of the subscripts {1, …, n} we have P{ ∈ B} = P{( Xπ(1), …, Xπ(n)) ∈ B} for all Borel sets B. The random variables Xi are said to be exchangeable – a condition established under the assumption of independent and identically distributed samples or via the principle of randomization/random allocation in experimental design. As noted in Good, permutation tests rely on the assumption of exchangeability under the null hypothesis.Although a one-sample comparison has application for biomedical researchers, relative comparisons are of broad practical relevance. Research clinicians and pharmacologists are interested in exploring standard-of-care and new treatment comparisons for therapeutics. Transitioning to the ation of , let P bn<p’ wide data and the lack of closed-form sampling distributions for various test statistics proposed. Pesarin [Permutation testing, a procedure which relies on samples drawn from the combined pool of experimental units and the random assignment of a treatment label to each unit, is common in the bioinformatics literature due to the prevalence of ‘ Pesarin  lists a  Pesarin , to the D using an Erdős-Rényi G graph. We test H0:G = G versus H1:G > G. The order of G is 25 and p0 is 0.20. We simulate four cases with 100 hypothesis tests (or experiments) in each case. For the null case we assume that the observed network is a p = p0 = 0.20 model. For the remaining two cases we assume that p = 0.25. We set cij = 0 for both a null and an alternate case and cij = exp(−2) for the remaining two cases. These four cases illustrate the Type I and II error rates of our procedure both with and without the inclusion of the neighbouring information in calculating D. 1,000 resamples were used to estimate the null distribution for D. The execution time for the set of 100 experiments using 1,000 resamples was approximately 1 hour on a standard personal computer. The R package Statnet [G graphs. All of our computations were conducted using R (http://www.r-project.org/). For testing large networks or large numbers of networks a more computationally efficient language such as Fortran or C++ is recommended. The representation diversity and size of networks, combined with a need or interest to tailor D for a particular application, poses a challenge for software developers.We first demonstrate  Statnet  was usedD for a correlation-based network model we assume that the p-dimensional observations follow a multivariate normal distribution, Np, with mean vector μ and positive definite covariance matrix Σ. Transforming Σ into correlation form Ω allows us to form a partial correlation matrix. Applying a threshold to each ρ estimate or a testing procedure to the entries of Ω can be used to define a correlation network. Given Ω-1= (ωij) we compute the partial correlation matrix Π = (πij) via (πij) = −ωij/√(ωiiωjj). Under multivariate normality two variables are conditionally independent given the remaining variables if and only if the partial correlation vanishes. The zeros in Ω-1 determine the conditional independence graph. As before, a threshold or testing procedure is used to define the partial correlation network. We evaluate both the Type I error rate and the power of D under an alternate hypothesis.To evaluate xi~ N. To mimic a sparse biological network Ω consists of 6 nonzero 5x5 blocks along the diagonal. A sample rejection scheme guaranteed that the magnitude of each block entry exceeded a predefined threshold. The number of nodes (30) and the threshold (ρ = 0.2) was common to all simulations. The same threshold ρ was used to define the data generation model and to estimate the observed correlation network in the one-sample case. For both comparisons we evaluated D using sample sizes of n1= n2 = 200. For the one-sample comparison the target correlation network Ω0 is estimated from the sample data and the resamples are drawn from the observed samples to determine the null distribution for D. Admittedly, this approach violates the true spirit of a one-sample test under an assumed null model. But, if historical sample observations are available for analysis and reflect the null hypothesis then these samples might provide the most scientifically defensible null distribution for Ω0. The same approach is used for the derived microarray-based correlation network comparison; a case where the true null model is unknown. To simulate the alternate hypothesis of network inequality at least 10% of the 5 × 5 blocks for a 30 × 30 correlation matrix, with a minimum of one block per experiment, were varied using a random number generator. A total of 100 experiments were performed and 1,000 resamples used in calculating each p-value. For both the correlation and partial correlation networks we calculate D using only the weight portion of the index since the existence of an edge was defined via the  correlation estimate. The execution time for the 100 experiments using 1,000 resamples was approximately 1–2 hours on a standard personal computer for the one-sample comparison. The two-sample setting execution time was on the order of 4–6 hours. For select simulation data the algorithm used to estimate the partial correlation network, see below, would abruptly terminate. All of the resample p-values shown here were obtained upon a successful completion of the estimation process. The computation time for the actual biological data was negligible. This is likely due to the small networks involved; network comparisons involving a large number of nodes and/or edges present a non-trivial computational burden. Refer to Additional file Apart from the Erdős-Rényi example, the one- and two-sample simulation comparisons assume that the observations are in their correlation form, i.e., et al. [et al. [et al. [et al. [http://cran.r-project.org). The algorithm broadly consists of 3 steps. First, the correlation (or covariance) matrix Ω is converted to a partial correlation matrix Π; pseudoinverses may be involved. The second step tests for the presence of ‘significant’ edges. The final step extracts the significant edges based on a user-defined criterion. We used the magnitude of the estimated (shrunken) partial coefficient to establish this cut-off – the cutoff.ggm parameter for πij was set to 0.5. Default settings were used for the remainder of the GeneNet settings. We do not claim that GGMs are superior for modeling gene/protein networks. [We selected partial correlation networks as presented in   is a serious metabolic disorder affecting a large number of people worldwide. Mootha et al. , using m website .ρ. To facilitate a one-sample comparison the normal tissue samples are used to define a target network. The resampling procedure was outlined in the previous section. Rather than analyze severely ill-conditioned correlation matrices (numerous gene sets contained over 100 genes), we formed correlation networks only for those gene sets with fewer than 18 probes in the pathway. A significant one-sample finding for one or more gene sets may suggest differences in co-regulation network behaviour of a person with DM2 relative to a normal subject.The original study examined average expression levels across the two groups. Here, we explore the differences in covariance or correlation structures between the two phenotypes as expressed in a correlation network formed via a threshold for et al. [et al. [E2F/Rb pathway in SBTs and SCAs. Chien et al. [p53 and E2F pathways in serous carcinomas and reinforced the role of E2Fs listed in [Ovarian cancer is the foremost lethal neoplasm of the female genital tract. We examine here the gene expression signatures of ovarian serous carcinomas (SCAs) relative to serous borderline tumors (SBTs) based on three recent studies. Sieben et al.  confirme [et al.  investign et al.  demonstristed in . These tet al. [E2F target genes with the studies of Bracken et al. [et al. [BRCA2), 1 development gene, 5 DNA damage and repair genes, and 13 DNA synthesis/replication genes. A table of the specific genes included is in Additional file The microarray data analyzed here was obtained from the NCBI GEO database  via acceet al.  screenedn et al.  and Bied [et al. . 43 of t [et al. : 5 for tG random graph comparison refer to Figure cij = 0 and cij = exp(−2) cases the p-values are approximately uniformly distributed under the null hypothesis. In panels (a) and (c) we see that the nominal Type I α level is approximately 0.05. Comparing panel (b) to (d) we see the improvement in the power of D when p = 0.25 and cij = exp(−2). When the neighbouring information was not used in calculating D 34% of the resample p-values were below a nominal α level of 0.05. When the neighbouring information was included in D and scaled by cij = exp(−2) 55% of the p-values were below the nominal 0.05 level. The likelihood of D detecting the true alternative hypothesis increased from 34% to 55% just by including the neighbouring information here.For the Erdős-Rényi D to reject H0: Ω = Ω0 versus H1: Ω ≠ Ω0. The tests were performed using cij = 0, i.e., excluding the neighbouring information, and cij= rij, the thresholded correlation estimate. Under the assumed null model the distribution of p-values, for both cij = 0 and cij= rij, had less mass at the extremes of the p-value range. Additional simulation work, see [n1= n2 = 100 the Type I error rate was conservative, produced an inflated error rate when n1= n2 = 2,000, and the test achieved a proper size when the a priori sample size was a factor of 10 larger than the observed network based on a sample of n = 200. These results suggest caution when trying to determine the null distribution for D using a finite set of a priori samples. Figure D via cij= rij penalized our ability to reject the null hypothesis when compared to the use of cij = 0.For the one-sample correlation network comparison we examined both the Type I error rate and the power of ork, see , suggestH0: Π1= Π2. In Figure D  suggests that the edge portion of D is redundant to the weight component and panels (b-c) suggest that including the neighbouring information detracts from the power of D. Table ρ = 0.35 and ρ = 0.5  in Table For the two-sample comparisons we examined a limited ordered comparison of the SBT, SCA1, and SCA3 phenotypes for the 5 biological processes. Our subjective rationale was that a comparison of the SBT and SCA1 phenotypes might suggest a biomarker candidate; comparing the SCA1 and SCA3 phenotypes might better characterize disease progression or provide insight regarding resistance to chemotherapy agents. Table et al. . Apart fD was inspired by the question, “How long is the coast of Britain?” This question motivates the definition of fractal dimension, a concept suited for complex objects. Cutler’s [et al. [D to internodal path lengths strictly less than three since this is the minimum distance necessary to capture a feedforward/feedback loop, it limits the range of topological comparisons, and D is prevented from revisiting the ‘centre’ of a neighbourhood in a cyclic graph. D can be extended to include a larger neighbourhood. Apart from obvious computational implications, defining a second set of cij’s is necessary and the efficiency of D may deteriorate as D integrates a larger set of imprecise or variable estimates. In contrast to fractals, viewing a network as an inhomogeneous mixture of subgraphs suggests that a local distance may be preferable to large distances that span a large set of nodes.The definition of Cutler’s  definiti [et al.  suggest D is or will be optimal under a broad range of network models. Apart from the literature related to ERGMs, weighted biological network models with well-developed inferential theory analogous to maximum likelihood, uniformly minimum-variance unbiased, or minimax estimators is largely unavailable. The ability to apply D to a broad range of edge/no-edge and weighted/unweighted networks is sure to involve tradeoffs. Banks et al. [1-based edit distance for unweighted binary networks using protein signaling logic. While not constituting a definitive argument, we speculate that the value of including the neighbouring information in the G simulation example is due to the uniformity of p across the set of nodes. The loss in performance using the neighbouring information for a pairwise correlation network model is likely due to D unnecessarily integrating across a larger set of imprecise estimates. For the two-sample partial correlation network comparison, the gain in performance via the use of neighbours may be influenced by the fact that each edge is formed using a plurality of variables. For researchers interested in alternate network models, e.g., preferential attachment or small-world models, evaluating D (or comparing it to a competing procedure) under various scenarios is recommended.We do not claim that s et al.  illustras et al.  state ths et al.  modify aD on a more general form of a Hamming distance parallels efforts in the area of global network alignment scoring. Others have proposed alignment-based operations for use with biological networks, e.g., [D incorporate neighbouring information might be obvious to a systems biologist; see [D and may interact with network estimation procedures, data collection requirements, etc.Basing s, e.g.,  for comps, e.g., ,21,56. Hist; see  for an eist; see  for a paist; see , using pist; see  use leveist; see  outline ist; see  present D’s simplicity and potential for use across a broad set of network models, the importance (and weighting) of D’s components should receive serious consideration by the investigator. Gower [D to gage or improve its performance for a specific network model is advised. The selection of the weight constant cij for weighted graphs was motivated by the idea of conditional probabilities. If A and B represent two adjacent edges and information flows through their common vertex then it is reasonable to assume that some form regarding the state of B is meaningful to A. It is plausible to assume that the force two objects exert on one another is proportional to their proximity. Preferential attachment networks assume that new edges are formed at a node conditional on the number of existing edges at that node. Wei et al. [et al. [D is complicated. Gao et al. [et al. [D and the total sums of squares in regression modeling we allow for nodes with a high degree  to contribute more to the calculation of D. If one chooses to normalize portions of D at each node by some topological property then one has to justify the scaling factor. Does one scale by the node’s (weighted) degree, a clustering coefficient, in- or out-degree for a directed network, etc.? If a network model has an efficient estimator whose sampling distribution is well characterized, the use of D may be contraindicated. At present, we do not expect this to be the case for experiment-based weighted biological networks. As a conservative measure, D could be reduced to its Hamming-like form and only the inferential procedures outlined here applied. Evaluating D using an in silico model under plausible scientific scenarios of interest to the investigator could also reduce concerns about aspects of the definition of D.Despite r. Gower  concedesr. Gower . For boti et al.  employ g [et al.  found quo et al.  allow hu [et al.  normalizD was limited. Computational expediency, an ability to explore and contrast common parametric models and their associated weighted network forms, and the use of a canonical structure  graphs) motivated our network model selection. Both Marbach et al. [et al. [et al. [et al. [D may be restrictive in the one-sample case; but, the complexities surrounding network probability models and the intractability of network-based test statistics does not suggest easy alternatives. Large (weighted) networks are likely to be costly in terms of the data needed to estimate a family of network parameters. The literature on the analysis of large networks, with its emphasis on select topological comparisons, typically imposes a vast reduction in network complexity. Thorne et al. [Our choice of network architectures to evaluate with h et al.  and Alta [et al.  discuss  [et al.  found th [et al.  suggest e et al. , in propD for observation studies raises the topic of partial exchangeability. It is unreasonable to assume that either of the biological datasets presented here met the assumption of exchangeability. However, unconditional procedures also struggle when confronted with observational data and/or missing or hidden covariates. Our one- and two-sample comparison approaches require or assume a hypothesized network model for the purpose of sampling; we do not employ fixed-network random rewiring schemes or variations that seek to align a set of topological measures across a set of stochastic networks.General criticisms leveled against the use of resampling methods are applicable here; see Berger . The praD is a sum of node-based dissimilarities, a test for dissimilarity between two aligned nodes or subgraphs can be formed using the portion of D attributable to that node or subgraph. Such an approach may help isolate molecular targets or subnetworks for closer study. Determining the sampling distribution for the modified test statistic is identical to the procedure developed here. See Yates [A significant one- or two-sample finding naturally suggests the question, “Where do the networks differ?” Similar to individual effect tests in regression, the answer to this question will involve one or more nodes or subgraphs. Answering targeted questions routinely appear in the literature, see ,72. Dongee Yates  for moreIn this paper we have demonstrated an inferential framework for performing one- and two-sample hypothesis tests for biological networks. We proposed a suitable test statistic and evaluated it with both simulated and microarray data under a variety of situations. The dissimilarity test statistic proposed is a logical extension of existing measures used in sequence alignment/nominal data comparisons. In generating the null distribution of the test statistic we used an assumed parametric model, resamples from a collection of null samples, and made use of permutation testing principles. Resampling methods suggest that counterintuitive behaviours may occur when dealing with a  network model. While not explicitly demonstrated here, our procedure easily facilitates node-level or subgraph post hoc tests.The authors declare that they have no competing interests.This work is based on the PhD dissertation of PDY. NDM suggested the problem and supervised the research. All authors read and approved the final manuscript.R code.Click here for fileOvarian cancer genes analyzed. Subset of analyzed genes as categorized by Bracken et al. [n et al. .Click here for fileTwo-sample comparison for partial correlation networks under H0. A uniform qq-plot of the 100 resample p-values for a test of H0 Π1= Π2 versus H1 Π1≠ Π2 under the null hypothesis. The y-axis is the observed p-value; the x-axis the expected p-value.Click here for file


A/GABAB receptors. All metabotropic neurotransmitter receptors are GPCRs. There are, however, a large number of non-GPCRs that also fulfill the original definition for a metabotropic receptor, namely “that the transmitter acts indirectly, by triggering a chemical reaction or a series of reactions”  in the postsynaptic cell from other glutamate receptors that pass an ion current (ionotropic) -spanning proteins as typical for GPCRs but do not couple to a heterotrimeric G-protein. With the given 7-TM topology it was as yet not clear how the channelrhodopsin proteins have to arrange to form an ion channel. Recently, the non-selective cation channel, channelrhodopsin-2 from


Reading habits are thought to play an important role in the emergence of cultural differences in visuo-spatial and numerical tasks. Left-to-right readers show a slight visuo-spatial bias to the left side of space, and automatically associate small numbers to the left and larger numbers to the right side of space, respectively. A paradigm that demonstrated an automatic spatial-numerical association involved the generation of random numbers while participants performed lateral head turns. That is, Westerners have been shown to produce more small numbers when the head was turned to the left compared to the right side. We here employed the head turning/random number generation (RNG) paradigm and a line bisection (LB) task with a group of 34 Iranians in their home country. In the participants’ native language (Farsi) text is read from right-to-left, but numbers are read from left-to-right. If the reading direction for text determines the layout of spatial-numerical mappings we expected to find more small numbers after right than left head turns. Yet, the generation of small or large numbers was not modulated by lateral head turns and the Iranians showed therefore no association of numbers with space. There was, however, a significant rightward shift in the LB task. Thus, while the current results are congruent with the idea that text reading habits play an important role in the cultural differences observed in visuo-spatial tasks, our data also imply that these habits on their own are not strong enough to induce significant horizontal spatial-numerical associations. In agreement with previous suggestions, we assume that for the emergence of horizontal numerical mappings a congruency between reading habits for words and numbers is required. Our thoughts, perception and actions are shaped by the culture in which we live. Our way of thinking, for example, depends on the social systems we grew up with. That is, East Asians tend to reason in a holistic way, while Westerners exhibit a more analytical thinking style . Based on the previous literature we here expected to find a slight bias to the right of the line’s true center  participated in this study. The 25 right and 9 left-handed participants  as a within-subjects factor and Handedness  as a between-subjects factor. The analysis revealed neither a main effect for Condition  = 0.17, p = 0.85) nor for Handedness  = 0.32, p = 0.57). The interaction between Condition and Handedness was not significant either  = 0.73, p = 0.49). Due to its theoretical importance for the current study we also directly compared the number of generated “small” numbers during left and right head turns. Paired t-tests revealed no significant differences in the number of “small” numbers between lateral head turns  = 0.65, p = 0.52; only right-handers: t(24) = 0.91, p = 0.37; only left-handers: t(8) = −0.21, p = 0.84).The number of “small” numbers was submitted to a repeated-measure ANOVA with t-tests were conducted to investigate whether there was a bias for naming too many “small” numbers in any of the three conditions. As there was no handedness effect in the ANOVA, data were collapsed across this factor for this analysis. The number of “small” numbers generated did not differ from the expected value of 20.0 in any of the three conditions (t(33) < 1.61, p > 0.11, see Figure t(33) = 1.21, p = 0.23).One sample t-test comparing the participants’ mean deviation to 0 indicated that there was a significant rightward bias (t(33) = 2.67, p < 0.02) for the 34 participants. The performance in the LB task differed between left  and right-handers  = 2.08, p < 0.05). One sample t-tests showed that right-handers deviated significantly to the right of the true center (t(24) = 3.02, p < 0.01) and that there was no LB bias for left-handers (t(8) = −1.79, p > 0.11).The subjective midpoint in the LB task was shifted 0.93 mm (SEM = 0.35) to the right side of the lines’ true center. A one sample r = 0.21, p = 0.23).The bias in the LB task was not related to the average magnitude of generated “small” numbers across all three conditions . A paradigm that revealed an automatic mapping of small and large numbers to left and right head turns respectively in Westerners  and numbers (left-to-right) differ in Farsi. Our working hypothesis is that these opposite reading habits lead to the disappearance of any preferred lateral association of numbers along the horizontal mental number line. Hebrew readers also use opposite reading directions for words and numbers, and these readers also lacked reliable spatial-numerical associations in a SNARC paradigm  exhibit a radial spatial-numerical mapping when response buttons in a SNARC paradigm were placed in a radial instead of the conventional horizontal arrangement (Shaki and Fischer, Participants’ handedness only affected performance in the visuo-motor LB task, but not in the RNG task. This finding is consistent with previous research. Differences between left and right-handers in LB tasks are commonly observed (Sampaio and Chokron, The observed rightward shift in the LB task is analogous to that described in previous studies assessing visuo-spatial biases in right-to-left reading cultures (Chokron and Imbert, The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.


Finally, a polynomial-time approximation algorithm for the problem of sorting unsigned genomes by double-cut-and-join operations is devised, which achieves the approximation ratio In this paper we study the problem of sorting unsigned genomes by double-cut-and-join operations, where genomes allow a mix of linear and circular chromosomes to be present. First, we formulate an equivalent optimization problem, called maximum cycle/path decomposition, which is aimed at finding a largest collection of edge-disjoint cycles/AA-paths/AB-paths in a breakpoint graph. Then, we show that the problem of finding a largest collection of edge-disjoint cycles/AA-paths/AB-paths of length no more than  A fundamental problem in the study of genome rearrangements is to compute the genomic distance between two genomes based on their gene orders, where the genomic distance is generally defined as the minimum number of evolutionary operations necessary to transform one genome to another. This problem has been extensively studied in the last two decades -4.double-cut-and-join (DCJ). It basically cuts a genome in two places and then joins the resulting four ends in a new way. Noticeably, computing the genomic distance based on DCJ operations can be applied between two genomes that allow a mix of linear and circular chromosomes to be present.The choices of genome evolutionary operations include reversals , translocations, fissions, fusions, transpositions and block-interchanges. To unify all these classical operations, Yancopoulos et al  introducsigned genomes in [unsigned genomes in [The complexity of computing the genomic distance between two genomes seems to largely depend on the availability of gene strand information rather than on the choice of evolutionary operations. For instance, the problem of sorting by reversals is tractable when gene strand information is available ,6, but bnomes in ,8, wherenomes in ,10; in tuni-chromosome genomes, Chen presented in [linear genomes, Jiang et al [To tackle an NP-hard optimization problem, it is of highly practical interest to develop a polynomial-time approximation algorithm with provable performance guarantee. For the problem of sorting by double-cut-and-join operations, in the case of unsigned ented in  an approng et al  recentlylinear and circular chromosomes to be present. The main goal is to devise a polynomial-time approximation algorithm for this NP-hard problem. To this end, we first formulate a new and equivalent combinatorial optimization problem, called maximum cycle/path decomposition, which is aimed at finding a largest collection of edge-disjoint cycles/AA-paths/AB-paths in the breakpoint graph constructed from two input genomes. Then, we show that the problem of finding a largest collection of edge-disjoint cycles/AA-paths/AB-paths of length no more than l in the breakpoint graph can be reduced to the well-known degree-bounded k-set packing problem, where k = 2l. Finally, we present a polynomial-time approximation algorithm for the problem of sorting unsigned genomes by DCJ operations and then obtain its approximation ratio In this paper we study the problem of sorting unsigned genomes by double-cut-and-join operations in the more general case where genomes allow a mix of gene is a stretch of DNA with two ends: the 3′ end and the 5′ end. Both are called the extremities of a gene. A chromosome is a single double-stranded DNA molecule that contains a sequence of genes. It can be either a linear molecule or a circular molecule. For a linear chromosome, there is a telomere marker located at each of its two ends. A genome is the whole collection of chromosomes in a cell. For example, the following gives a genome containing two linear chromosomes and one circular chromosome.A a, c, d, o), , }Genome A = {, then the corresponding genome is called signed. For a signed genome, Bergeron et al [adjacencies between extremities from different genes or between telomeres and extremities. For example, we may represent a signed genomeHere we use the symbol ‘o’ to represent a telomere marker, further indicating that the corresponding chromosome is linear. An on et al  introducasha and ta represents the 5′ end and 3′ ends of gene a, respectively.where double-cut-and-join (DCJ) is an operation that cuts a genome in two places and joins the resulting four ends in a new way. Specifically, the cuts are applied between two adjacent extremities from different genes, between a telomere and its adjacent gene extremity, or between the two extremities of a null chromosome if necessary. The DCJ operation was first introduced by Yancopoulos et al [The os et al  and lateos et al  to unifyA and B on the same set of genes, the problem of sorting unsigned genomes by DCJ operations (UDCJ) is defined to find a shortest sequence of DCJ operations that transform one genome into the other. The length of such a sequence is called the double-cut-and-join distance between two genomes A and B, and denoted by DdCJ.Given two unsigned genomes Example 1. Let two unsigned genomes beGenome A has two linear chromosomes and one circular chromosome, while genome B has two linear chromosomes and two circular chromosomes. Sorting A into B can, for example, be done in the following four DCJ operations, where the places to be cut are underlined:We will see later that at least four DCJ operations are needed to transform one genome into the other. Therefore, DCJthe DCJ distance between A and B is d = 4.S and a collection S of subsets of S, the set packing problem asks for the maximum number of pairwise disjoint subsets in S. The k-set packing problem is a restricted variant of the set packing problem where every subset in S has size at most k. If in addition the number of occurrences in S of any element is upper bounded by a constant Δ, then it reduces to the degree-bounded k-set packing problem. The following theorem states the best-to-date approximability and its detailed proof can be found in [Given a base set found in .Theorem 2  and will be used in our approximation for computing the DCJ distance between two unsigned genomes. Its running time complexity is This algorithm is denoted by APPROX-SP =  as follows. Each vertex in |V| corresponds to a distinct gene or a telomere, so |V| = n + 2Al + 2Bl. Every adjacency in A forms a black edge belonging to bE and every adjacency in B forms a gray edge belonging to g.E It is easy to see that |bE| = n + Al, g||E = n + BI, and |E| = 2n + Al + B.l Moreover, every telomere of genome A (resp. genome B) has degree one and is incident to a black (resp. gray) edge, whereas every gene has degree four and is incident to two black and two gray edges. For short, a telomere of genome A is referred to as an A-telomere, and a telomere of genome B as a B-telomere. Note that the number of A-telomeres is not necessarily equal to the number of B-telomeres in a breakpoint graph.The breakpoint graph  is first introduced in  and wideExample 3. Consider the two unsigned genomes A and B given in Example 1, Awhere l = 2 Band l = 2. The breakpoint graph G =  is depicted in Figurein which |V| = 15, |bE| = 9, g||E = 9, and |E| = 18.G is called alternating if its edges are alternatively black and gray. From now on, whenever we mention cycles/paths in a breakpoint graph, they are alternating and edge-disjoint. A path is called an AA-path (resp. BB-path) if it connects two A-telomeres (resp. two B-telomeres) or an AB-path if it connects an A-telomere and a B-telomere. The length of a cycle/path is referred to as the number of black edges that it contains.A cycle/path in the breakpoint graph cycle/path decomposition of G into edge-disjoint cycles, AA-paths, AB-paths and BB-paths. A maximum cycle/path decomposition refers to a cycle/path decomposition that contains the maximum number of edge-disjoint cycles, AA-paths and AB-paths. Note that this maximum number does not take into account any BB-paths. The maximum cycle/path decomposition problem is hence defined as the problem of finding such a maximum cycle/path decomposition of a breakpoint graph. See Figure Since every telomere vertex has degree one and every gene vertex has two incident black edges and two incident gray edges, there always exists a Before proceeding to study the problem of sorting unsigned genomes by DCJ operations, we take a first look at the case of signed genomes. Note that the above concept of breakpoint graph extends naturally to two signed genomes , et al.  further Given a breakpoint graph for two signed genomes Theorem 4 .Lemma 5. For every cycle/path decomposition of G, there exists a signed versionandof genomes A and B such thatwhere b is the number of black edges in the breakpoint graph G and c (AA and IABresp. I) is the number of cycles (resp. AA-paths and AB-paths) in the given cycle/path decomposition of G.Proof. Note that every gene vertex would be visited twice if we traverse all the cycles/paths in a fixed cycle/path decomposition of G. When a gene vertex  is visited for the first time, we may assume that we are visiting the 5′ end of the gene (denoted as ha). When it is visited for the second time, we may assume that we are visiting the 3′ end of the gene (denoted as ta). To obtain a signed genome A and B, respectively.G—that is, there are still b black edges, AAI AA-paths, ABI AB-paths and BBI BB-paths. After closing paths into cycles, the breakpoint graph Moreover, the breakpoint graph Lemma 6. For every signed versionandof genomes A and B, there exists a cycle/path decomposition of G such thatwhere b is the number of black edges in the breakpoint graph G and c (AA and IABresp. I) is the number of cycles (resp. AA-paths and AB-paths) in this cycle/path decomposition of G.Proof. Observe that we would obtain the breakpoint graph G if we combine two extremity vertices of a same gene into a single vertex in the breakpoint graph G which preserves the same numbers of black edges/cycles/AA-paths/AB-paths/BB-paths . After closing paths into cycles, the breakpoint graph Therefore, the trivial cycle/path decomposition of Theorem 7. Let A and B be two unsigned genomes defined on the same set of genes. Then, we haveDCJd = b – (c + AAI + ABI)where b is the number of black edges in G and c  is the number of cycles  in a maximum cycle/path decomposition of G.Proof. Let us consider a maximum cycle/path decomposition of G. By Lemma 5, we would obtain a signed version A and B such that A into genome B. It hence follows that DCJd DCJ operations can be applied to transform genome A into genome B. Let A in which all genes are positive. We then apply the same sequence of DCJd DCJ operations to the signed genome B if we disregard all the gene signs; in other words, B. Thus, G such that DCJd ≥ b – (c + AAI + ABI).Assume now that a sequence of Note that, for a given pair of genomes, the number of black edges is fixed without regard to any cycle/path decomposition. Theorem 7 hence suggests a way to approximate the DCJ distance between two unsigned genomes via maximizing the number of cycles/AA-paths/AB-paths in a cycle/decomposition of the breakpoint graph. To do so, our proposed approximation algorithm performs three subroutines to find edge-disjoint cycles/paths of length one, of length two, and of length no more than three, respectively.Lemma 8. There exists a maximum cycle/path decomposition of G which contains a largest collection of edge-disjoint cycles/AA-paths/AB-paths of length one.Proof. Let C be a maximum cycle/path decomposition of G and C1 a largest collection of edge-disjoint cycles/AA-paths/AB-paths of length one in G. If C3 is a cycle/AA-path/AB-path contained in C1 but not in C . In this case, we would modify the maximum cycle/path decomposition C as follows: take one gene vertex of the only black edge of C3 and re-connect its incident black edges and gray edges in a new way. Consequently, C1 and C2 would be replaced by the two distinct cycles/paths C3 and C4 in C. Suppose by contradiction that the above modification decreases the number of cycles/AA-paths/AB-paths so that the new C is no longer a maximum cycle/path decomposition. Note first that this would happen only when C1 and C2 are two distinct cycles/AA-paths/AB-paths but C4 is a BB-path. Since no AA-path in G could be of length one, C3 is either a cycle or an AB-path. Hence, C3 and C4 together use at least two B-telomeres and at most one A-telomere, and so do C1 and C2 together. Since neither C1 nor C2 is a BB-path, they together shall use A-telomeres no less than B-telomeres, contrasting the previously established fact that they together use at least two B-telomeres and at most one A-telomere. Continue this process with the remaining cycles/AA-paths/AB-paths that are contained in C1 but not in C. It would necessarily end up with a maximum cycle/path decomposition that contains a largest collection of edge-disjoint cycles/AA-paths/AB-paths of length one.G which could contain all the cycles/AA-paths/AB-paths of length one.It is worth noticing that there might be no maximum cycle/path decomposition of Lemma 9. The problem of finding a largest collection of edge-disjoint cycles, AA-paths and AB-paths of length one in the breakpoint graph G is solvable in polynomial time.Proof. We can transform a problem instance of finding a largest collection of edge-disjoint cycles/AA-paths/AB-paths of length one into an instance of the 2-set packing problem, where the base set S contains all the edges of G and each subset of the collection S is comprised of edges of a cycle/AA-path/AB-path of length one in G. The 2-set packing problem can be reduced to the maximal matching problem which is well-known to be solvable in polynomial time by a simple greedy algorithm [lgorithm .G.Following Lemma 9, we assume from now on that there does not exist any cycle/AA-path/AB-path of length one in the breakpoint graph l that traverse a common edge. Although this upper bound is no way the tight one, it is already enough for our purpose to devise an approximation algorithm.The following lemma gives an upper bound on the number of distinct cycles/paths of length less than or equal to Lemma 10. Every edge in the breakpoint graph G belongs to at most (22l+1 × 2l) distinct cycles/AA-paths/AB-paths of length less than or equal to l.Proof. Let us consider a breadth-first traversal of edges in the breakpoint graph G that starts at a given edge e, and then count all the cycles/AA-paths/AB-paths of length less than or equal to l that use the edge e. Note that every vertex is incident to at most two black edges and at most two gray edges and also that every edge is at distance at most (2l – 1) from another edge of the same cyle/path of length less than or equal to l. The traversal of every such cycle/path will end at two edges at distance i and j from the root edge e, respectively, such that i ,j ≥ 0 and i + j ≤ 2l – 1. If we fix values for i and j, there are at most 2i × 2j = 2i+j ≤ 22l–1 cycles/paths whose traversals end at two edges in distance i and j from the root edge e, respectively. For all the combinations of values i and j such that i ≤ 2l – 1 and j ≤ 2l – 1, there are at most 2l × 2l × 22l–1 = 22l+1 × l2 cycles/paths to be reached; in other words, there are at most 22l+1 × l2 cycles/paths of length less than or equal to l that use the edge e.l, we may construct a collection S of subsets of the base set S, where S is comprised of all the edges in G and each subset of S is comprised of edges of a distinct cycle/AA-path/AB-path of length no more than l in G. Then, we obtain an instance  of the k-set packing problem where k = 2l. Further by the above lemma, we obtain the following observations.To find a largest collection of edge-disjoint cycles/paths of length no more than Corollary 11. Finding a largest collection of edge-disjoint cycles/AA-paths/AB-paths of length two can he transformed into an instance of the degree-bounded 4-set packing problem.Corollary 12. Finding a largest collection of edge-disjoint cycles/AA-paths/AB-paths of length no more than three can he transformed into an instance of the degree-bounded 6-set packing problem.It is worth noting that, if, as previously done in  on the bLemma 13. The problem of finding a largest collection of edge-disjoint cycles/AA-paths/AB-paths of length two in the breakpoint graph G can be approximated with ratioin polynomial time, for any positive ε.Lemma 14. The problem of finding a largest collection of edge-disjoint cycles/AA-paths/AB-paths of length no more than three in the breakpoint graph G can be approximated with ratioin polynomial time, for any positive ε.A and B be two unsigned genomes defined on the same set of genes. Given a breakpoint graph G, our proposed algorithm for the cycle/path decomposition is summarized below:Let C1 of cycles/AA-paths/AB-paths of length one by a greedy algorithm;1. Find a largest collection C1;2. Remove from the breakpoint graph all the edges used by the cycles/AA-paths/AB-paths of C2 of cycles/AA-paths/AB-paths of length two by Algorithm APPROX-SP;3. Find a collection C3 of cycles/AA-paths/AB-paths of length no more than three by Algorithm APPROX-SP;4. Find a collection C4 of cycles/AA-paths/AB-paths/BB-paths;5. Decompose the remaining edges arbitrarily into a collection C1 ∪ C2 ∪ C4 or C1 ∪ C3 ∪ C4, depending on which one has the larger size.6. Output either r2 denote the number of cycles/AA-paths/AB-paths of length two, r3 the number of cycles/AA-paths/AB-paths of length three, and r′ the total number of cycles/AA-paths/AB-paths. Let r be the total number of cycles/AA-paths/AB-paths in the cycle/path decomposition returned by our proposed algorithm. The cycles/AA-paths/AB-paths of length one are all excluded from the above counts since it would not affect the worst-case algorithmic performance. We find the worst-case approximation ratio of our algorithm by solving the following optimization problem:For a maximum cycle/path decomposition, let fractional linear programming problem  can be approximated with ratioin polynomial time, for any positive ε.emma 16 . Let A In view of this lemma, the third constraint in the above fractional linear programming problem can be replaced by the inequality Theorem 17. The problem of sorting unsigned permutations by DCJ operations can be approximated within ratioin polynomial time, for any positive ε.Since the introduction of the NP-hard problem of sorting unsigned genomes by double-cut-and-join operations in , the polsitive ε . The secsitive ε . In thisdegree-bounded k-set packing problem . In this paper we established this reduction, which then leads to the improved approximation ratios.Conceptually, our proposed algorithm operates in the same spirit as many previous algorithms for approximating the genomic distance via genome rearrangement operations ,10,14,15The authors declare that they have no competing interests.XC conceived the study. All authors contributed to the algorithm analysis, read and approved the final manuscript.


Flux coupling analysis (FCA) is a useful method for finding dependencies between fluxes of a metabolic network at steady-state. FCA classifies reactions into subsets  in which activity of one reaction implies activity of another reaction. Several approaches for FCA have been proposed in the literature.We introduce a new FCA algorithm, FFCA , which is based on checking the feasibility of a system of linear inequalities. We show on a set of benchmarks that for genome-scale networks FFCA is faster than other existing FCA methods.http://www.bioinformatics.org/ffca/.We present FFCA as a new method for flux coupling analysis and prove it to be faster than existing approaches. A corresponding software tool is freely available for non-commercial use at  Constraint-based analysis of metabolic networks has become increasingly important for describing and predicting the behavior of living organisms ,2. WhileFlux coupling analysis (FCA)  is a useIn the rest of this section, we first recall some basic definitions. Then we briefly review the previously proposed FCA methods.m  metabolites and n reactions, the stoichiometric matrix S is an m × n matrix, where element ij Sis the stoichiometric coefficient of metabolite i in reaction j. We denote by R = {1, ..., n} the set of reactions and by M = {1, ..., m} the set of metabolites.For a metabolic network with Irr is the set of irreversible reactions which are assumed to always have non-negative flux values. Rev is the set of reversible reactions which are allowed to have positive, negative or zero flux values. If a reversible reaction has a positive (resp. negative) flux, we say that it is working in forward (resp. backward) direction. Splitting a reversible reaction i means making reaction i irreversible and adding one more irreversible reaction i + n to the network, which works in the backward direction. Without loss of generality, we assume that in the numbering of reactions, the first |Rev| reactions are the reversible ones.There are two types of reactions in a metabolic network. v is the vector of metabolic fluxes through all the reactions, the steady-state condition holds for v if S · v = 0. The steady-state flux cone is defined as C = {v ∈ ℝn | S · v = 0, i ≥v0, for all i ∈ Irr}. The lineality space of C is given by lin.space(C) = {v ∈ ℝn | S · v = 0, i v= 0, for all i ∈ Irr}.If r ∈ R, r v= 0 for all v ∈ C, then r is called blocked, otherwise r is unblocked. The set of blocked reactions will be denoted by Blk. The set of unblocked reactions can be further partitioned based on the reversibility type of reactions [Irev as the set of all reactions that can work only in one direction, i.e., those reactions that take only non-negative or only non-positive flux values at steady-state. If i ∈ Irev and i v≤ 0 for all v ∈ C, one can multiply by -1 the i-th column of the stoichiometric matrix, such that in the modified network, we have i v≥ 0 for all v ∈ C. Without loss of generality, we assume that all reactions in Irev are operating in the forward direction.If for some reaction eactions . We defia) Prev, the set of pseudo-irreversible reactions, and (b) Frev, the set of fully reversible reactions. A reaction i is in Prev if it can work in both directions, and additionally, for all flux vectors v in the lineality space of the flux cone, we have i v= 0. Accordingly, we define Frev as the set of those reactions, which can work in both directions, and additionally, can have a non-zero flux value if fluxes through all irreversible reactions are set to zero.The set of reactions that can work in both directions at steady-state is further partitioned into two subsets:  = {i ∈ R | i v≠ 0} of reactions active in v is called the support of v. Suppose v ∈ C is a flux vector and A ⊆ R is a subnetwork. The flux pattern of v for A is defined as A ∩ supp(v), which is the set of those reactions in A which have non-zero values in v[elementary flux pattern (EFP) if it cannot be written as the union of other flux patterns. For studying EFPs, Kaleta et al. [The set lues in v. A flux a et al.  assume tIn this subsection, we briefly recall existing approaches to flux coupling analysis. For additional information and the technical details on the implementation of these algorithms, we refer to Additional file i and j, two LPs are solved. Depending on the optimal values obtained, the coupling relation between i and j is determined. There is a post-processing step in FCF. Since the reversible reactions have been split, flux coupling relations for these reactions have to be obtained from the coupling relations of the corresponding irreversible forward and backward reactions.The most widely used method for FCA is the Flux Coupling Finder (FCF) algorithm . This apThe FCF algorithm has been successfully used for finding coupling relations in a number of metabolic networks -13,16. HLarhlimi and Bockmayr ,17 have i and j. If each of these reactions can have a non-zero flux independently of the other  that i is directionally coupled to j, then the EFPs of this subnetwork are {i, j} and {j}. Finally, if the reactions are partially coupled, we will have only one EFP, which is {i, j}. With this method, it is not possible to distinguish between partial and full coupling, since flux patterns only contain the information about the activity or inactivity of the fluxes, but not the flux values.Recently, Kaleta et al.  introducIn FCF, and some other FCA approaches, every reversible reaction is split into a forward and a backward reaction. This splitting procedure results in an increase in the number of LPs and also in the size of each LP to be solved. Moreover, a complicated postprocessing step is required to infer the flux coupling relations of the original reversible reactions. For all these reasons, it might be better to perform flux coupling analysis without splitting the reversible reactions. An implementation of FCF without splitting is presented in Additional file i and j can be coupled only if one of the following 4 cases holds  would be possible):Additionally, Larhlimi and Bockmayr  show thai, j ∈ Irev: In this case, i and j can be directionally, partially or fully coupled.1. i ∈ Irev and j ∈ Prev: The only possibility is j → i.2. i, j ∈ Prev: In this case, we can only have i ⇔ j.3. i, j ∈ Frev: In this case, we can only have i ⇔ j.4. i and j, and then check if the corresponding coupling relation holds. This will be referred to as "Reversibility-Type prunings" . RT-prunings were originally used in MMB-FCA as presented in [Therefore, it is enough to determine the reversibility types of ented in .S, the columns corresponding to the blocked reactions have been removed. According to Proposition 6.13 in [i and j such that i, j ∈ Prev or i, j ∈ Frev, the following two statements are equivalent:After RT-prunings, a further improvement can be applied to LP-based FCA methods. Assume that in the stoichiomtric matrix  6.13 in , for two• i ⇔ j• For all v such that Sv = 0, i v= 0 implies j v= 0.This means that it is sufficient to solve only one LP:i and j will be fully coupled if and only if in the above LP Prev/Frev-based improvement .The two reactions An improved version of FCF, which takes into account all of the above-mentioned improvements, has been suggested in . Here weFCA is a promising tool for metabolic network analysis. However, to perform FCA most authors seem to use their personal implementation of the FCF algorithm ,13. Onlymaximize(T x c| Ax ≤ b) can be solved by enumerating a sequence of feasible solutions, i.e., solutions of the system of linear inequalities Ax ≤ b such that at each solution x the objective function value T x cimproves. Accordingly, finding one feasible solution could be faster than computing an optimal solution . If (P1) and (P2) are both infeasible, then i and j are partially (and maybe fully) coupled. To check whether they are fully coupled, one has to use other methods, e.g. computing enzyme subsets [If (P1) and (P2) are both feasible, then  subsets  or solvi subsets .i ∈ Irev and j ∈ Prev: The only possible coupling relation is j → i (but not i → j). Hence, (P1) will always be feasible, because feasibility of (P1) means that i → j does not hold. However, we need to check the feasibility of (P2). Additionally, since j can take negative values, one more system should be checked for feasibility:2. j → i. Otherwise, i and j are uncoupled.It can be easily shown that if (P2) and (P3) are both infeasible, then i, j ∈ Prev or i, j ∈ Frev: In this case, the following system of linear inequalities should be checked for feasibility:3. i ⇔ j (because feasibility of (P4) implies j → i, which in turn implies i ⇔ j based on Proposition 2 in [i and j are uncoupled.If (P4) is infeasible, then ion 2 in ). If , one can solve the following LP:i vis constant, the optimal value exists if and only if this problem is feasible. Similar LPs can be solved to check the feasibility of (P2) and (P3).Since In Table H. pylori network and especially for the large networks of S. cerevisiae and E. coli, FFCA proves to be faster than MMB-FCA. The computation time required for MMB-FCA rapidly grows when the number of reactions increases. This is due to the possibly exponential size of the set of generating vectors which has to be computed before finding the coupled reactions . EFP-FCBoth the FCF algorithm and the current implementation of FFCA solve LPs for flux coupling analysis. One might ask why FFCA is faster than the classical FCF method. There are at least five major differences between FCF and FFCA:• When an LP is solved in FFCA, finding the first feasible solution is sufficient, while the LPs should be solved to optimality in case of the FCF algorithm.• In the FCF method, in contrast to FFCA, every reversible reaction is split into two (forward and backward) irreversible reactions. This step slows down the procedure and increases the size of the LPs to be solved.• For computing the flux coupling relation between any pair of reactions, we always need two LPs in FFCA, while in FCF sometimes more LPs have to be solved. For example, for computing the coupling relation between an irreversible and a reversible reaction (after splitting), four LPs are solved  FCF was re-implemented without splitting reactions (W-FCF); (ii) FCF was re-implemented without splitting reactions and with the RT-prunings (WR-FCF); and (iii) FCF was re-implemented without splitting reactions and with the RT-prunings and the PF-improvement (WRP-FCF), as suggested in [One can think of implementing FCF without splitting reversible reactions and/or with the RT-prunings and/or PF-improvement. In order to assess the importance of these issues, three improved versions of FCF were implemented  FCA hasE. coli [E. coli [Since FFCA includes the RT-prunings and PF-improvement, it might be interesting to see how the number of coupling relations  depends on the number of reversible reactions in a network. This problem is studied in Additional file E. coli , for the[E. coli   genome-scale metabolic network [E. coli (iJR904) [E. coli (iAF1260) [Nine metabolic networks were used in this study: (i) ork from ; (ii) RBood cell ; (iii) E E. coli ; (iv) H. network ; (v) yea network ; (vi) th(iJR904) ; (vii) tiAF1260) ; (viii) iAF1260) ; and (ixiAF1260) .For FCA, all uptake reactions were assumed to be able to carry non-zero fluxes.Unless indicated otherwise, all tools were implemented in Matlab v7.4. Details about the implementation of the different FCA approaches, together with the corresponding pseudo-codes are presented in Additional file All computations were performed on a 64-bit Debian Linux system with Intel Xeon 3.0 GHz processor. The running times include the CPU time for preprocessing, computation of flux coupling relations, and post-processing (where necessary).Prev/Frev-based improvement.FCA: Flux Coupling Analysis; FFCA: Feasibility-based Flux Coupling Analysis; FCF: Flux Coupling Finder; MMB-FCA: Flux Coupling Analysis based on Minimal Metabolic Behaviors; EFP-FCA: Flux Coupling Analysis based on Elementary Flux Patterns; LP: Linear Program; MILP: Mixed-Integer Linear Program; RT-prunings: Reversibility-Type prunings; PF-improvement: The authors declare that they have no competing interests.SAM, LD and AB presented the original idea of the work and designed the study. AL and LD introduced the improved versions of FCF. BM and LD implemented the tools and carried out the experiments. The manuscript was first drafted by SAM. The final manuscript was approved by all authors.Different approaches to flux coupling analysis and implementation details. In this file, pseudocodes and implementation details of different FCA methods are presented.Click here for fileDependence of flux coupling analysis on the number of reversible reactions. In this file, we show how flux coupling relations depend on the number of reversible reactions in the E. coli metabolic network.Click here for file


In the sensing applications of optical fiber grating, it is necessary to reduce the transmission-type polarization dependence to isolate the sensing parameter. It is experimentally shown that the polarization-dependent spectrum of acousto-optic long-period fiber grating sensors can be suppressed in the transmission port of a π-shifted Sagnac loop interferometer. General expressions for the transmittance and reflectance are derived for transmission-type, reflection-type, and partially reflecting/transmitting-type polarization-dependent optical devices. The compensation of polarization dependence through the counter propagation in the Sagnac loop interferometer is quantitatively measured for a commercial in-line polarizer and an acousto-optic long-period fiber grating sensor. In adIn this work, three types of polarization-dependent devices, a transmission-type, reflection-type, and mixed (both transmitting and reflecting)-type, are analyzed by use of Jones matrix method. It is shown that the polarization-dependent spectra of transmission-type optical devices can be suppressed in the transmission port of π-shifted Sagnac loop interferometer ,6, and tWe also characterize experimentally PDL suppression using Sagnac loop interferometer for various devices exhibiting various degrees of PDL such as an in-line polarizer (ILP) and an acousto-optic LPFG sensor. The cancellation of polarization dependence through the counter propagation in the Sagnac loop interferometer is quantitatively measured using a commericial PDL measurement control system. This simple PDL suppression method can be used to minimize the various PDL effect.2.deviceT, between the input and transmitted field and the reflection, deviceR, between the input field and reflected field are described by the Jones matrices in α, β, δ, and ε are all real numbers and birefringence of the device is considered to be negligible. These transmission and reflection relations hold for bi-directional propagation.A schematic diagram of the Sagnac loop interferometer with a polarization-dependent device, similar to the one demonstrated in , is showSagnacT, and reflectance, SagnacR, of a Sagnac interferometer can be solved analytically using Jones matrix analysis. In the waveplate in θ, with respect to the laboratory coordinates.  The two cases when θ = 0, Γ = 0 and θ = π/4, Γ = π are expressed in θ = 0, Γ = 0 (no waveplate), the Sagnac loop interferometer works as mirror; that is, it exhibits maximum reflection and minimum transmission. Contrarily, for a waveplate with θ = π/4, Γ = π (π-shifted waveplate at angle π/4), maximum transmission and minimum reflection can be achieved in Ports 2 and 1, respectively. We consider Port 1 as the reflection port and Port 2 as the transmission port of a Sagnac interferometer with a nested optical device. With 1l set equal to the length between Ports 3 and 5 and 2l set equal to the arm length between between Ports 4 and 6, Δl is defined as 1l − 2l. When Δl = 0 or 1l = 2l, as the Sagnac interferometer is referred to as a length-symmetric Sagnac loop interferometer. The longitudinal phase constant for a traveling wave in the fiber is represented as k.Using a procedure similar to the one shown in , the traSagnacR, contains polarization-dependent terms in each of the three types of devices. The transmittance, SagnacT, also includes polarization-dependent terms, except for the case of a transmission-type device, in which the intensity is null at the transmission port. In comparison, the π-shifted Sagnac loop interferometer shows us an interesting theoretical result. Both the reflectance, SagnacR, and the transmittance, SagnacT, do not have polarization-dependent terms in both transmission-type and reflection-type devices. The case of In this research, more general cases are considered. For a length-symmetric Sagnac loop interferometer, the expression for the reflectance in 3.As a representative polarization-dependent device having the highest polarization dependence to the input polarization state, a commercial in-line polarizer (ILP-155-09-09-1) is selected in this experiment  because Without a Sagnac loop, the average PDL of the ILP is measured to be ∼55 dB in each transmission direction. However, the average PDL value is dramatically reduced to less than 0.45 dB when it is placed within a Sagnac loop. The polarization dependence, quantified as PDL, can be measured from a control system consisting of a tunable laser , polarizer, and polarimeter . The intrinsic polarization dependence of the transmission-type polarization-dependent optical device can be measured by placing it between polarizer and polarimeter in any direction when the device has similar transmission characteristics in both transmission directions. Thus, the measured PDL values are similarly measured in both directions. As shown in These can in principle be minimized by optimizing all components used to construct the Sagnac loop. A large number of the packaged optical devices have been treated as failure products at their final test stage just due to the high PDL value only. It is expected that the intrinsic and fabrication-induced PDL of these products can be easily compensated with the incorporation of a Sagnac loop in the packaging process.4.01, TM01, and HE21 modes is clearly observed for an acousto-optic LPFG sensor with an interaction region consisting of a 28-cm-long dispersion compensating fiber (OFS EHS100), [The polarization dependent transmission and reflection of an acousto-optic LPFG sensor are also experimentally examined by placing it inside the Sagnac loop interferometer with waveplate and are compared with expressions in EHS100), . The outθ = 0). The spectral reflectance is easily changed by tuning the input polarization as shown in θ = π/4 by maximizing the intensity in the transmission Port 2 as shown in For the case of 4.We have generally analyzed and measured the polarization dependence cancellation of optical fiber grating sensor using a π-shifted Sagnac loop interferometer. For the feasible intrinsic PDL-free integrated optical devices, a commercial-grade in-line polarizer device was tested to verify the proposed PDL compensating method. A representative type of polarization-dependent device, an acousto-optic LPFG sensor, is also experimentally proved for the polarization-dependence suppression. It is expected that the use of the Sagnac loop can save many optical sensor products, such as FBG sensors and micro-bending LPFG sensors which have been limited in their applications solely due to their high PDL.