Generic algebraic conditions for the occurrence of switch-like behavior of a chemical kinetic system of the hypoxia pathway.

Weakly reversible chemical reaction networks with zero deficiency associated with mass-action kinetics admit, within each positive stoichiometric compatibility class, one positive steady state which is locally asymptotically stable and this irrespective of the values of the kinetics constants. Networks which do not enjoy these structural properties potentially exhibit more diverse dynamical behaviors. In this article, we consider a chemical reaction network associated with mass-action kinetics which is not weakly reversible and has a deficiency larger than one. The chemical reactions are at most bimolecular, but inflow and outflow reactions are present. Our results are as follows. We establish the existence of positive steady-state solutions and obtain their analytic expressions in the concentration space and in convex coordinates. We show that the system fulfills necessary conditions for a saddle-node and for a bifurcation into a saddle and a node. We apply a constructive approach to obtain a set of numerical values for the state variables and kinetic parameters, not reported previously, such that the reduced Jacobian is characterized by a zero eigenvalue with all other eigenvalues having negative real parts. The bifurcation diagram confirms the presence of the switch-like behavior.

Chemical Systems with Limit Cycles.

The dynamics of a chemical reaction network (CRN) is often modeled under the assumption of mass action kinetics by a system of ordinary differential equations (ODEs) with polynomial right-hand sides that describe the time evolution of concentrations of chemical species involved. Given an arbitrarily large integer [Formula: see text], we show that there exists a CRN such that its ODE model has at least K stable limit cycles. Such a CRN can be constructed with reactions of at most second-order provided that the number of chemical species grows linearly with K. Bounds on the minimal number of chemical species and the minimal number of chemical reactions are presented for CRNs with K stable limit cycles and at most second order or seventh-order kinetics. We also show that CRNs with only two chemical species can have K stable limit cycles, when the order of chemical reactions grows linearly with K.

A numerical approach for detecting switch-like bistability in mass action chemical reaction networks with conservation laws.

BACKGROUND: Theoretical analysis of signaling pathways can provide a substantial amount of insight into their function. One particular area of research considers signaling pathways capable of assuming two or more stable states given the same amount of signaling ligand. This phenomenon of bistability can give rise to switch-like behavior, a mechanism that governs cellular decision making. Investigation of whether or not a signaling pathway can confer bistability and switch-like behavior, without knowledge of specific kinetic rate constant values, is a mathematically challenging problem. Recently a technique based on optimization has been introduced, which is capable of finding example parameter values that confer switch-like behavior for a given pathway. Although this approach has made it possible to analyze moderately sized pathways, it is limited to reaction networks that presume a uniterminal structure. It is this limited structure we address by developing a general technique that applies to any mass action reaction network with conservation laws. RESULTS: In this paper we developed a generalized method for detecting switch-like bistable behavior in any mass action reaction network with conservation laws. The method involves (1) construction of a constrained optimization problem using the determinant of the Jacobian of the underlying rate equations, (2) minimization of the objective function to search for conditions resulting in a zero eigenvalue, (3) computation of a confidence level that describes if the global minimum has been found and (4) evaluation of optimization values, using either numerical continuation or directly simulating the ODE system, to verify that a bistability region exists. The generalized method has been tested on three motifs known to be capable of bistability. CONCLUSIONS: We have developed a variation of an optimization-based method for the discovery of bistability, which is not limited to uniterminal chemical reaction networks. Successful completion of the method provides an S-shaped bifurcation diagram, which indicates that the network acts as a bistable switch for the given optimization parameters.

A computational approach to investigate constitutive activation of NF-κB.

Nuclear factor kappa B (NF-κB) signaling is the master regulator of inflammatory pathways; therefore, its regulation has been the subject of investigation since last two decades. Multiple models have been published that describes the dynamics of NF-κB activity by stimulated activation and feedback loops. However, there is also paramount evidence of the critical role of posttranslational modifications (PTMs) in the regulation of NF-κB pathway. With the premise that PTMs present alternate routes for activation or repression of the NF-κB pathway, we have developed a model including all PTMs known so far describing the system behavior. We present a pathway network model consisting of 171 proteins forming 315 molecular species and consisting of 482 reactions that describe the NF-κB activity regulation in totality. The overexpression or knockdown of interacting molecular partners that regulate NF-κB transcriptional activity by PTMs is used to infer the dynamics of NF-κB activity and offers qualitative agreement between model predictions and the experimental results heuristically. Finally, we have demonstrated an instance of NF-κB constitutive activation through positive upregulation of cytokines (the stimuli) and IKK complex (NF-κB activator), the characteristic features in several cancer types and metabolic disorders, and its reversal by employing combinatorial activation of PPARG, PIAS3, and P50-homodimer. For the first time, we have presented a NF-κB model that includes transcriptional regulation by PTMs and presented a theoretical strategy for the reversal of NF-κB constitutive activation. The presented model would be important in understanding the NF-κB system, and the described method can be used for other pathways as well.

Multistationarity in Cyclic Sequestration-Transmutation Networks.

We consider a natural class of reaction networks which consist of reactions where either two species can inactivate each other (i.e., sequestration), or some species can be transformed into another (i.e., transmutation), in a way that gives rise to a feedback cycle. We completely characterize the capacity of multistationarity of these networks. This is especially interesting because such networks provide simple examples of "atoms of multistationarity", i.e., minimal networks that can give rise to multiple positive steady states.

Side Reactions Do Not Completely Disrupt Linear Self-Replicating Chemical Reaction Systems.

A crucial question within the fields of origins of life and metabolic networks is whether or not a self-replicating chemical reaction system is able to persist in the presence of side reactions. Due to the strong nonlinear effects involved in such systems, they are often difficult to study analytically. There are however certain conditions that allow for a wide range of these reaction systems to be well described by a set of linear ordinary differential equations. In this article, we elucidate these conditions and present a method to construct and solve such equations. For those linear self-replicating systems, we quantitatively find that the growth rate of the system is simply proportional to the sum of all the rate constants of the reactions that constitute the system (but is nontrivially determined by the relative values). We also give quantitative descriptions of how strongly side reactions need to be coupled with the system in order to completely disrupt the system.

Joining and decomposing reaction networks.

In systems and synthetic biology, much research has focused on the behavior and design of single pathways, while, more recently, experimental efforts have focused on how cross-talk (coupling two or more pathways) or inhibiting molecular function (isolating one part of the pathway) affects systems-level behavior. However, the theory for tackling these larger systems in general has lagged behind. Here, we analyze how joining networks (e.g., cross-talk) or decomposing networks (e.g., inhibition or knock-outs) affects three properties that reaction networks may possess-identifiability (recoverability of parameter values from data), steady-state invariants (relationships among species concentrations at steady state, used in model selection), and multistationarity (capacity for multiple steady states, which correspond to multiple cell decisions). Specifically, we prove results that clarify, for a network obtained by joining two smaller networks, how properties of the smaller networks can be inferred from or can imply similar properties of the original network. Our proofs use techniques from computational algebraic geometry, including elimination theory and differential algebra.

Identifiability from a Few Species for a Class of Biochemical Reaction Networks.

Under mass-action kinetics, biochemical reaction networks give rise to polynomial autonomous dynamical systems whose parameters are often difficult to estimate. We deal in this paper with the problem of identifying the kinetic parameters of a class of biochemical networks which are abundant, such as multisite phosphorylation systems and phosphorylation cascades (for example, MAPK cascades). For any system of this class, we explicitly exhibit a single species for each connected component of the associated digraph such that the successive total derivatives of its concentration allow us to identify all the parameters occurring in the component. The number of derivatives needed is bounded essentially by the length of the corresponding connected component of the digraph. Moreover, in the particular case of the cascades, we show that the parameters can be identified from a bounded number of successive derivatives of the last product of the last layer. This theoretical result induces also a heuristic interpolation-based identifiability procedure to recover the values of the rate constants from exact measurements.

Multistationarity in Structured Reaction Networks.

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 Gröbner bases.

Monostationarity and Multistationarity in Tree Networks of Goldbeter-Koshland Loops.

A major challenge in systems biology is to elicit general properties in the face of molecular complexity. Here, we introduce a class of enzyme-catalysed biochemical networks and examine how the existence of a single positive steady state (monostationarity) depends on the network structure, enzyme mechanisms, kinetic rate laws and parameter values. We consider Goldbeter-Koshland (GK) covalent modification loops arranged in a tree network, so that a substrate form in one loop can be an enzyme in another loop. GK loops are a canonical motif in cell signalling and trees offer a generalisation of linear cascades which accommodate network complexity while remaining mathematically tractable. In particular, they permit a modular, recursive proof strategy which may be more widely applicable. We show that if each enzyme follows its own complex reaction mechanism under mass action kinetics, then any network is monostationary for all appropriate parameter values. If the kinetics is non-mass action with a plausible monotonicity requirement, and each enzyme follows the Michaelis-Menten mechanism, then monostationarity is preserved. Surprisingly, a single GK loop with a complex enzyme mechanism under non-mass action monotone kinetics can have more than one positive steady state (multistationarity). The broader interplay between network structure, enzyme mechanism and kinetics remains an intriguing open problem.

Global stability of a class of futile cycles.

In this paper, we prove the global asymptotic stability of a class of mass action futile cycle networks which includes a model of processive multisite phosphorylation networks. The proof consists of two parts. In the first part, we prove that there is a unique equilibrium in every positive compatibility class. In the second part, we make use of a piecewise linear in rates Lyapunov function in order to prove the global asymptotic stability of the unique equilibrium corresponding to a given initial concentration vector. The main novelty of the paper is the use of a simple algebraic approach based on the intermediate value property of continuous functions in order to prove the uniqueness of equilibrium in every positive compatibility class.

Metabolic control of signalling pathways and metabolic auto-regulation.

Metabolic alterations have emerged as an important hallmark in the development of various diseases. Thus, understanding the complex interplay of metabolism with other cellular processes such as cell signalling is critical to rationally control and modulate cellular physiology. Here, we review in the context of mammalian target of rapamycin, AMP-activated protein kinase and p53, the orchestrated interplay between metabolism and cellular signalling as well as transcriptional regulation. Moreover, we discuss recent discoveries in auto-regulation of metabolism (i.e. how metabolic parameters such as metabolite levels activate or inhibit enzymes and thus metabolic pathways). Finally, we review functional consequences of post-translational modification on metabolic enzyme abundance and/or activities.

Minimal reaction schemes for pattern formation.

We link continuum models of reaction-diffusion systems that exhibit diffusion-driven instability to constraints on the particle-scale interactions underpinning this instability. While innumerable biological, chemical and physical patterns have been studied through the lens of Alan Turing's reaction-diffusion pattern-forming mechanism, the connections between models of pattern formation and the nature of the particle interactions generating them have been relatively understudied in comparison with the substantial efforts that have been focused on understanding proposed continuum systems. To derive the necessary reactant combinations for the most parsimonious reaction schemes, we analyse the emergent continuum models in terms of possible generating elementary reaction schemes. This analysis results in the complete list of such schemes containing the fewest reactions; these are the simplest possible hypothetical mass-action models for a pattern-forming system of two interacting species.

Oscillations in Planar Deficiency-One Mass-Action Systems.

Whereas the positive equilibrium of a planar mass-action system with deficiency zero is always globally stable, for deficiency-one networks there are many different scenarios, mainly involving oscillatory behaviour. We present several examples, with centers or multiple limit cycles.

Parameter Estimation for Kinetic Models of Chemical Reaction Networks from Partial Experimental Data of Species' Concentrations.

The current manuscript addresses the problem of parameter estimation for kinetic models of chemical reaction networks from observed time series partial experimental data of species concentrations. It is demonstrated how the Kron reduction method of kinetic models, in conjunction with the (weighted) least squares optimization technique, can be used as a tool to solve the above-mentioned ill-posed parameter estimation problem. First, a new trajectory-independent measure is introduced to quantify the dynamical difference between the original mathematical model and the corresponding Kron-reduced model. This measure is then crucially used to estimate the parameters contained in the kinetic model so that the corresponding values of the species' concentrations predicted by the model fit the available experimental data. The new parameter estimation method is tested on two real-life examples of chemical reaction networks: nicotinic acetylcholine receptors and Trypanosoma brucei trypanothione synthetase. Both weighted and unweighted least squares techniques, combined with Kron reduction, are used to find the best-fitting parameter values. The method of leave-one-out cross-validation is utilized to determine the preferred technique. For nicotinic receptors, the training errors due to the application of unweighted and weighted least squares are 3.22 and 3.61 respectively, while for Trypanosoma synthetase, the application of unweighted and weighted least squares result in training errors of 0.82 and 0.70 respectively. Furthermore, the problem of identifiability of dynamical systems, i.e., the possibility of uniquely determining the parameters from certain types of output, has also been addressed.

Unifying mass-action kinetics and Newtonian mechanics by means of Nambu brackets.

We demonstrate that elementary biochemical reactions defined by mass-action kinetics satisfy a particular Nambu structure. To this end, we express biochemical reaction equations in terms of Nambu brackets and certain ω-factors. The ω-factors account for the fact that mass-action kinetics exhibits in general flow fields with finite divergence. The proposed approach by means of Nambu brackets and ω-factors unifies divergence freeflow fields of Newtonian mechanics and flow fields with finite divergence of mass-action kinetics.

Complex-balanced equilibria of generalized mass-action systems: necessary conditions for linear stability.

It is well known that, for mass-action systems, complex-balanced equilibria are asymptotically stable. For generalized mass-action systems, even if there exists a unique complex-balanced equilibrium (in every stoichiometric class and for all rate constants), it need not be stable. We first discuss several notions of matrix stability (on a linear subspace) such as D-stability and diagonal stability, and then we apply abstract results on matrix stability to complex-balanced equilibria of generalized mass-action systems. In particular, we show that linear stability (on the stoichiometric subspace and for all rate constants) implies uniqueness. For cyclic networks, we characterize linear stability (in terms of D-stability of the Jacobian matrix); and for weakly reversible networks, we give necessary conditions for linear stability (in terms of D-semistability of the Jacobian matrices of all cycles in the network). Moreover, we show that, for classical mass-action systems, complex-balanced equilibria are not just asymptotically stable, but even diagonally stable (and hence linearly stable). Finally, we recall and extend characterizations of D-stability and diagonal stability for matrices of dimension up to three, and we illustrate our results by examples of irreversible cycles (of dimension up to three) and of reversible chains and S-systems (of arbitrary dimension).

Clumped-MCEM: Inference for multistep transcriptional processes.

Many biochemical events involve multistep reactions. Among them, an important biological process that involves multistep reaction is the transcriptional process. A widely used approach for simplifying multistep reactions is the delayed reaction method. In this work, we devise a model reduction strategy that represents several OFF states by a single state, accompanied by specifying a time delay for burst frequency. Using this model reduction, we develop Clumped-MCEM which enables simulation and parameter inference. We apply this method to time-series data of endogenous mouse glutaminase promoter, to validate the model assumptions and infer the kinetic parameters. Further, we compare efficiency of Clumped-MCEM with state-of-the-art methods - Bursty MCEM2 and delay Bursty MCEM. Simulation results show that Clumped-MCEM inference is more efficient for time-series data and is able to produce similar numerical accuracy as state-of-the-art methods - Bursty MCEM2 and delay Bursty MCEM in less time. Clumped-MCEM reduces computational cost by 57.58% when compared with Bursty MCEM2 and 32.19% when compared with delay Bursty MCEM.

Realizations of kinetic differential equations.

The induced kinetic differential equations of a reaction network endowed with mass action type kinetics is a system of polynomial differential equations. The problem studied here is: Given a system of polynomial differential equations, is it possible to find a network which induces these equations; in other words: is it possible to find a kinetic realization of this system of differential equations? If yes, can we find a network with some chemically relevant properties (implying also important dynamic consequences), such as reversibility, weak reversibility, zero deficiency, detailed balancing, complex balancing, mass conservation, etc.? The constructive answers presented to a series of questions of the above type are useful when fitting differential equations to datasets, or when trying to find out the dynamic behavior of the solutions of differential equations. It turns out that some of these results can be applied when trying to solve seemingly unrelated mathematical problems, like the existence of positive solutions to algebraic equations.

Results on stochastic reaction networks with non-mass action kinetics.

In 2010, Anderson, Craciun, and Kurtz showed that if a deterministically modeled reaction network is complex balanced, then the associated stochastic model admits a stationary distribution that is a product of Poissons [1]. That work spurred a number of followup analyses. In 2015, Anderson, Craciun, Gopalkrishnan, and Wiuf considered a particular scaling limit of the stationary distribution detailed in [1], and proved it is a well known Lyapunov function [2]. In 2016, Cappelletti and Wiuf showed the converse of the main result in [1]: if a reaction network with stochastic mass action kinetics admits a stationary distribution that is a product of Poissons, then the deterministic model is complex balanced [3]. In 2017, Anderson, Koyama, Cappelletti, and Kurtz showed that the mass action models considered in [1] are non-explosive (so the stationary distribution characterizes the limiting behavior). In this paper, we generalize each of the three followup results detailed above to the case when the stochastic model has a particular form of non-mass action kinetics.