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Zero-one reaction networks play key roles in cell signaling such as signalling pathways regulated by protein phosphorylation. Multistability of zero-one networks is a key dynamics feature enabling decision-making in cells. Since multistability (or, nondegenerate multistationarity) can be lifted from a smaller subnetwork (low-dimensional networks with less species and fewer reactions) to large networks, we aim to explore the multistability problem of small zero-one networks. We prove that any zero-one network with a one-dimensional stoichiometric subspace admits at most one (stable) positive steady state (this steady state is also called a structural attractor), and we completely classify all the one-dimensional zero-one networks according to if they indeed admits a (stable) positive steady state or not. Also, we prove that any two-dimensional zero-one network with up to three species either admits only degenerate positive steady states, or admits at most one (stable) positive steady state. In these proofs, we apply the theorem based on the Brouwer degree theory and the theory of real algebraic geometry. Moreover, using the tools of computational algebraic geometry, we provide a systematical way for detecting the smallest zero-one networks that admit nondegenerate multistationarity/multistability. We show that the smallest zero-one networks that admit nondegenerate multistationarity contain three species and five reactions, and the smallest zero-one networks that admit multistability contain three species and six reactions., Comment: 45 pages, 6 figures

We provide a sufficient and necessary condition in terms of the stoichiometric coefficients for a bi-reaction network to admit multistability. Also, this result completely characterizes the bi-reaction networks according to if they admit multistability., Comment: 39 pages

For the reaction networks with zero-one stoichiometric coefficients (or simply zero-one networks), we prove that if a network admits a Hopf bifurcation, then the rank of the stoichiometric matrix is at least four. As a corollary, we show that if a zero-one network admits a Hopf bifurcation, then it contains at least four species and five reactions. As applications, we show that there exist rank-four subnetworks, which have the capacity for Hopf bifurcations/oscillations, in two biologically significant networks: the MAPK cascades and the ERK network. We provide a computational tool for computing all four-species, five-reaction, zero-one networks that have the capacity for Hopf bifurcations., Comment: 27 pages, 5 figures

In the tropical projective torus, it is not guaranteed that the projection of a Fermat-Weber point of a given data set is a Fermat-Weber point of the projection of the data set. In this paper, we focus on the projection on the tropical triangle (the three-point tropical convex hull), and we develop one algorithm (Algorithm 1) and its improved version (Algorithm 4), such that for a given data set in the tropical projective torus, these algorithms output a tropical triangle, on which the projection of a Fermat-Weber point of the data set is a Fermat-Weber point of the projection of the data set. We implement these algorithms in R and test how it works with random data sets. The experimental results show that, these algorithms can succeed with a much higher probability than choosing the tropical triangle randomly, the succeed rate of these two algorithms is stable while data sets are changing randomly, and Algorithm 4 can output the results much faster than Algorithm 1 averagely., Comment: 21 pages, 5 figures, 4 tables

We study the multistationarity for the reaction networks with one-dimensional stoichiometric subspaces, and we focus on the networks admitting finitely many positive steady states. We prove that if a network admits multistationarity, then network has an embedded one-species network with arrow diagram (->,<-) and another with arrow diagram (<-,->). The inverse is also true if there exist two reactions in the network such that the subnetwork consisting of the two reactions admits at least one and finitely many positive steady states. We also prove that if a network admits at least three positive steady states, then it contains at least three bi-arrow diagrams. More than that, we completely characterize the bi-reaction networks that admit at least three positive steady states., Comment: 31 pages

For the reaction networks with one-dimensional stoichiometric subspaces, we show the following results. (1) If the maximum number of positive steady states is an even number N, then the maximum number of stable positive steady states is N/2. (2) If the maximum number of positive steady states is an odd number N, then we provide a condition on the network such that the maximum number of stable positive steady states is (N-1)/2 if this condition is satisfied, and this maximum number is (N+1)/2 otherwise., Comment: 26 pages, 1 figure

For three typical sets of small reaction networks (networks with two reactions, one irreversible and one reversible reaction, or two reversible-reaction pairs), we completely answer the challenging question: what is the smallest subset of all multistable networks such that any multistable network outside of the subset contains either more species or more reactants than any network in this subset?, Comment: 23 pages, 5 tables

Most data in genome-wide phylogenetic analysis (phylogenomics) is essentially multidimensional, posing a major challenge to human comprehension and computational analysis. Also, we can not directly apply statistical learning models in data science to a set of phylogenetic trees since the space of phylogenetic trees is not Euclidean. In fact, the space of phylogenetic trees is a tropical Grassmannian in terms of max-plus algebra. Therefore, to classify multi-locus data sets for phylogenetic analysis, we propose tropical support vector machines (SVMs). Like classical SVMs, a tropical SVM is a discriminative classifier defined by the tropical hyperplane which maximizes the minimum tropical distance from data points to itself in order to separate these data points into sectors (half-spaces) in the tropical projective torus. Both hard margin tropical SVMs and soft margin tropical SVMs can be formulated as linear programming problems. We focus on classifying two categories of data, and we study a simpler case by assuming the data points from the same category ideally stay in the same sector of a tropical separating hyperplane. For hard margin tropical SVMs, we prove the necessary and sufficient conditions for two categories of data points to be separated, and we show an explicit formula for the optimal value of the feasible linear programming problem. For soft margin tropical SVMs, we develop novel methods to compute an optimal tropical separating hyperplane. Computational experiments show our methods work well. We end this paper with open problems., Comment: 27 pages, 6 figures, 2 tables

We consider a model of extracellular signal-regulated kinase (ERK) regulation by dual-site phosphorylation and dephosphorylation, which exhibits bistability and oscillations, but loses these properties in the limit in which the mechanisms underlying phosphorylation and dephosphorylation become processive. Our results suggest that anywhere along the way to becoming processive, the model remains bistable and oscillatory. More precisely, in simplified versions of the model, precursors to bistability and oscillations (specifically, multistationarity and Hopf bifurcations, respectively) exist at all "processivity levels". Finally, we investigate whether bistability and oscillations can exist together., Comment: 22 pages, 2 figures, 3 tables, builds on arXiv:1903.02617

We solve a conjecture on multiple nondegenerate steady states, and prove bistability for sequestration networks. More specifically, we prove that for any odd number of species, and for any production factor, the fully open extension of a sequestration network admits three nondegenerate positive steady states, two of which are locally asymptotically stable. In addition, we provide a non-empty open set in the parameter space where a sequestration network admits bistability., Comment: 20 pages

This work concerns the question of how two important dynamical properties, oscillations and bistability, emerge in an important biological signaling network. Specifically, we consider a model for dual-site phosphorylation and dephosphorylation of extracellular signal-regulated kinase (ERK). We prove that oscillations persist even as the model is greatly simplified (reactions are made irreversible and intermediates are removed). Bistability, however, is much less robust -- this property is lost when intermediates are removed or even when all reactions are made irreversible. Moreover, bistability is characterized by the presence of two reversible, catalytic reactions: as other reactions are made irreversible, bistability persists as long as one or both of the specified reactions is preserved. Finally, we investigate the maximum number of steady states, aided by a network's "mixed volume" (a concept from convex geometry). Taken together, our results shed light on the question of how oscillations and bistability emerge from a limiting network of the ERK network -- namely, the fully processive dual-site network -- which is known to be globally stable and therefore lack both oscillations and bistability. Our proofs are enabled by a Hopf bifurcation criterion due to Yang, analyses of Newton polytopes arising from Hurwitz determinants, and recent characterizations of multistationarity for networks having a steady-state parametrization., Comment: 33 pages, 4 figures, 4 tables, 3 appendices

We develop a probabilistic algorithm for computing elimination ideals of likelihood equations, which is for larger models by far more efficient than directly computing Groebner bases or the interpolation method proposed in the first author's previous work. The efficiency is improved by a theoretical result showing that the sum of data variables appears in most coefficients of the generator polynomial of elimination ideal. Furthermore, applying the known structures of Newton polytopes of discriminants, we can also efficiently deduce discriminants of the elimination ideals. For instance, the discriminants of 3 by 3 matrix model and one Jukes-Cantor model in phylogenetics (with sizes over 30 GB and 8 GB text files, respectively) can be computed by our methods., Comment: 31 pages, 1 figure, 3 tables

Many dynamical systems arising in biology and other areas exhibit multistationarity (two or more positive steady states with the same conserved quantities). Although deciding multistationarity for a polynomial dynamical system is an effective question in real algebraic geometry, it is in general difficult to determine whether a given network can give rise to a multistationary system, and if so, to identify witnesses to multistationarity, that is, specific parameter values for which the system exhibits multiple steady states. Here we investigate both problems. First, we build on work of Conradi, Feliu, Mincheva, and Wiuf, who showed that for certain reaction networks whose steady states admit a positive parametrization, multistationarity is characterized by whether a certain "critical function" changes sign. Here, we allow for more general parametrizations, which make it much easier to determine the existence of a sign change. This is particularly simple when the steady-state equations are linearly equivalent to binomials; we give necessary conditions for this to happen, which hold for many networks studied in the literature. We also give a sufficient condition for multistationarity of networks whose steady-state equations can be replaced by equivalent triangular-form equations. Finally, we present methods for finding witnesses to multistationarity, which we show work well for certain structured reaction networks, including those common to biological signaling pathways. Our work relies on results from degree theory, on the existence of explicit rational parametrizations of the steady states, and on the specialization of Groebner bases., Comment: 44 pages, 2 figures, 1 table

Most biological data are multidimensional, posing a major challenge to human comprehension and computational analysis. Principal component analysis is the most popular approach to rendering two- or three-dimensional representations of the major trends in such multidimensional data. The problem of multidimensionality is acute in the rapidly growing area of phylogenomics. Evolutionary relationships are represented by phylogenetic trees, and very typically a phylogenomic analysis results in a collection of such trees, one for each gene in the analysis. Principal component analysis offers a means of quantifying variation and summarizing a collection of phylogenies by dimensional reduction. However, the space of all possible phylogenies on a fixed set of species does not form a Euclidean vector space, so principal component analysis must be reformulated in the geometry of tree-space, which is a CAT(0) geodesic metric space. Previous work has focused on construction of the first principal component, or principal geodesic. Here we propose a geometric object which represents a $k$-th order principal component: the locus of the weighted Fr\'echet mean of $k+1$ points in tree-space, where the weights vary over the standard $k$-dimensional simplex. We establish basic properties of these objects, in particular that locally they generically have dimension $k$, and we propose an efficient algorithm for projection onto these surfaces. Combined with a stochastic optimization algorithm, this projection algorithm gives a procedure for constructing a principal component of arbitrary order in tree-space. Simulation studies confirm these algorithms perform well, and they are applied to data sets of Apicomplexa gene trees and the African coelacanth genome. The results enable visualizations of slices of tree-space, revealing structure within these complex data sets., Comment: 26 pages, 5 figures

We derive a procedure for computing an upper bound on the number of equiangular lines in various Euclidean vector spaces by generalizing the classical pillar decomposition developed by (Lemmens and Seidel, 1973); namely, we use linear algebra and combinatorial arguments to bound the number of vectors within an equiangular set which have inner products of certain signs with a negative clique. After projection and rescaling, such sets are also certain spherical two-distance sets, and semidefinite programming techniques may be used to bound the size. Applying our method, we prove new relative bounds for the angle arccos(1/5). Experiments show that our relative bounds for all possible angles are considerably less than the known SDP bounds for a range of larger dimension r. Our computational results also show an explicit bound on the size of a set of equiangular lines regardless of angle, which is strictly less than the well-known Gerzon's bound if r+2 is not a square of an odd number., Comment: 32 pages, 2 figures, 2 tables, 4 appendixes with 1 webpage for supplementary materials

An algebraic approach to the maximum likelihood estimation problem is to solve a very structured parameterized polynomial system called likelihood equations that have finitely many complex (real or non-real) solutions. The only solutions that are statistically meaningful are the real solutions with positive coordinates. In order to classify the parameters (data) according to the number of real/positive solutions, we study how to efficiently compute the discriminants, say data-discriminants (DD), of the likelihood equations. We develop a probabilistic algorithm with three different strategies for computing DDs. Our implemented probabilistic algorithm based on Maple and FGb is more efficient than our previous version presented in ISSAC2015, and is also more efficient than the standard elimination for larger benchmarks. By applying RAGlib to a DD we compute, we give the real root classification of 3 by 3 symmetric matrix model., Comment: 4 tables. arXiv admin note: substantial text overlap with arXiv:1501.00334. authors' note: The paper the extended and improved version of our paper presented in ISSAC2015 (arXiv:1501.00334) and the paper is accepted by Journal of Symbolic Computation Special Issue of ISSAC2015

We study the geometry of metrics and convexity structures on the space of phylogenetic trees, which is here realized as the tropical linear space of all \ ultrametrics. The ${\rm CAT}(0)$-metric of Billera-Holmes-Vogtman arises from the theory of orthant spaces. While its geodesics can be computed by the Owen-Provan algorithm, geodesic triangles are complicated. We show that the dimension of such a triangle can be arbitrarily high. Tropical convexity and the tropical metric behave better. They exhibit properties desirable for geometric statistics, such as geodesics of small depth., Comment: 21 pages, 5 figures; Theorem 13 is now proved in all dimensions

Maximum likelihood estimation (MLE) is a fundamental computational problem in statistics. The problem is to maximize the likelihood function with respect to given data on a statistical model. An algebraic approach to this problem is to solve a very structured parameterized polynomial system called likelihood equations. For general choices of data, the number of complex solutions to the likelihood equations is finite and called the ML-degree of the model. The only solutions to the likelihood equations that are statistically meaningful are the real/positive solutions. However, the number of real/positive solutions is not characterized by the ML-degree. We use discriminants to classify data according to the number of real/positive solutions of the likelihood equations. We call these discriminants data-discriminants (DD). We develop a probabilistic algorithm for computing DDs. Experimental results show that, for the benchmarks we have tried, the probabilistic algorithm is more efficient than the standard elimination algorithm. Based on the computational results, we discuss the real root classification problem for the 3 by 3 symmetric matrix~model., Comment: 2 tables

The concept of comprehensive triangular decomposition (CTD) was first introduced by Chen et al. in their CASC'2007 paper and could be viewed as an analogue of comprehensive Grobner systems for parametric polynomial systems. The first complete algorithm for computing CTD was also proposed in that paper and implemented in the RegularChains library in Maple. Following our previous work on generic regular decomposition for parametric polynomial systems, we introduce in this paper a so-called hierarchical strategy for computing CTDs. Roughly speaking, for a given parametric system, the parametric space is divided into several sub-spaces of different dimensions and we compute CTDs over those sub-spaces one by one. So, it is possible that, for some benchmarks, it is difficult to compute CTDs in reasonable time while this strategy can obtain some "partial" solutions over some parametric sub-spaces. The program based on this strategy has been tested on a number of benchmarks from the literature. Experimental results on these benchmarks with comparison to RegularChains are reported and may be valuable for developing more efficient triangularization tools.

We consider the problem of counting (stable) equilibriums of an important family of algebraic differential equations modeling multistable biological regulatory systems. The problem can be solved, in principle, using real quantifier elimination algorithms, in particular real root classification algorithms. However, it is well known that they can handle only very small cases due to the enormous computing time requirements. In this paper, we present a special algorithm which is much more efficient than the general methods. Its efficiency comes from the exploitation of certain interesting structures of the family of differential equations., Comment: 24 pages, 5 algorithms, 10 figures

This paper presents a generalization of our earlier work in [19]. In this paper, the two concepts, generic regular decomposition (GRD) and regular-decomposition-unstable (RDU) variety introduced in [19] for generic zero-dimensional systems, are extended to the case where the parametric systems are not necessarily zero-dimensional. An algorithm is provided to compute GRDs and the associated RDU varieties of parametric systems simultaneously on the basis of the algorithm for generic zero-dimensional systems proposed in [19]. Then the solutions of any parametric system can be represented by the solutions of finitely many regular systems and the decomposition is stable at any parameter value in the complement of the associated RDU variety of the parameter space. The related definitions and the results presented in [19] are also generalized and a further discussion on RDU varieties is given from an experimental point of view. The new algorithm has been implemented on the basis of DISCOVERER with Maple 16 and experimented with a number of benchmarks from the literature., Comment: It is the latest version. arXiv admin note: text overlap with arXiv:1208.6112

Two new concepts, generic regular decomposition and regular-decomposition-unstable (RDU) variety for generic zero-dimensional systems, are introduced in this paper and an algorithm is proposed for computing a generic regular decomposition and the associated RDU variety of a given generic zero-dimensional system simultaneously. The solutions of the given system can be expressed by finitely many zero-dimensional regular chains if the parameter value is not on the RDU variety. The so called weakly relatively simplicial decomposition plays a crucial role in the algorithm, which is based on the theories of subresultant chains. Furthermore, the algorithm can be naturally adopted to compute a non-redundant Wu's decomposition and the decomposition is stable at any parameter value that is not on the RDU variety. The algorithm has been implemented with Maple 15 and experimented with a number of benchmarks from the literature. Empirical results are also presented to show the good performance of the algorithm., Comment: Accepted by Science China: Information Science

A new concept, decomposition-unstable (DU) variety of a parametric polynomial system, is introduced in this paper and the stabilities of several triangular decomposition methods, such as characteristic set decomposition, relatively simplicial decomposition and regular chain decomposition, for parametric polynomial systems are discussed in detail. The concept leads to a definition of weakly comprehensive triangular decomposition (WCTD) and a new algorithm for computing comprehensive triangular decomposition (CTD) which was first introduced in [4] for computing an analogue of comprehensive Groebner systems for parametric polynomial systems. Our algorithm takes advantage of a hierarchical solving strategy and a self-adaptive order of parameters. The algorithm has been implemented with Maple 15 and experimented with a number of benchmarks from the literature. Comparison with the Maple package RegularChains, which contains an implementation of the algorithm in [4], is provided and the results illustrate that the time costs by our program for computing CTDs of most examples are no more than those by RegularChains., Comment: This paper has been withdrawn by the author due to a crucial error in a proof