Exact Probability Distributions of Selected Species in Stochastic Chemical Reaction Networks

Chemical reactions are discrete, stochastic events. As such, the species’ molecular numbers can be described by an associated master equation. However, handling such an equation may become difficult due to the large size of reaction networks. A commonly used approach to forecast the behaviour of reaction networks is to perform computational simulations of such systems and analyse their outcome statistically. This approach, however, might require high computational costs to provide accurate results. In this paper we opt for an analytical approach to obtain the time-dependent solution of the Chemical Master Equation for selected species in a general reaction network. When the reaction networks are composed exclusively of zeroth and first-order reactions, this analytical approach significantly alleviates the computational burden required by simulation-based methods. By building upon these analytical solutions, we analyse a general monomolecular reaction network with an arbitrary number of species to obtain the exact marginal probability distribution for selected species. Additionally, we study two particular topologies of monomolecular reaction networks, namely (i) an unbranched chain of monomolecular reactions with and without synthesis and degradation reactions and (ii) a circular chain of monomolecular reactions. We illustrate our methodology and alternative ways to use it for non-linear systems by analysing a protein autoactivation mechanism. Later, we compare the computational load required for the implementation of our results and a pure computational approach to analyse an unbranched chain of monomolecular reactions. Finally, we study calcium ions gates in the sarco/endoplasmic reticulum mediated by ryanodine receptors.     

Translated Chemical Reaction Networks

Many biochemical and industrial applications involve complicated networks of simultaneously occurring chemical reactions. Under the assumption of mass action kinetics, the dynamics of these chemical reaction networks are governed by systems of polynomial ordinary differential equations. The steady states of these mass action systems have been analyzed via a variety of techniques, including stoichiometric network analysis, deficiency theory, and algebraic techniques (e.g., Gröbner bases). In this paper, we present a novel method for characterizing the steady states of mass action systems. Our method explicitly links a network’s capacity to permit a particular class of steady states, called toric steady states, to topological properties of a generalized network called a translated chemical reaction network. These networks share their reaction vectors with their source network but are permitted to have different complex stoichiometries and different network topologies. We apply the results to examples drawn from the biochemical literature.     

Siphons in Chemical Reaction Networks

Siphons in a chemical reaction system are subsets of the species that have the potential of being absent in a steady state. We present a characterization of minimal siphons in terms of primary decomposition of binomial ideals, we explore the underlying geometry, and we demonstrate the effective computation of siphons using computer algebra software. This leads to a new method for determining whether given initial concentrations allow for various boundary steady states.     

A Stronger Necessary Condition for the Multistationarity of Chemical Reaction Networks

Biochemical reaction networks grow bigger and bigger, fed by the high-throughput data provided by biologists and bred in open repositories of models allowing merging and evolution. Nevertheless, since the available data is still very far from permitting the identification of the increasing number of kinetic parameters of such models, the necessity of structural analyses for describing the dynamics of chemical networks appears stronger every day. Using the structural information, notably from the stoichiometric matrix, of a biochemical reaction system, we state a more strict version of the famous Thomas’ necessary condition for multistationarity. In particular, the obvious cases where Thomas’ condition was trivially satisfied, mutual inhibition due to a multimolecular reaction and mutual activation due to a reversible reaction, can now easily be ruled out. This more strict condition shall not be seen as some version of Thomas’ circuit functionality for the continuous case but rather as related and complementary to the whole domain of the structural analysis of (bio)chemical reaction systems, as pioneered by the chemical reaction network theory.     

Evolution of Autocatalytic Sets in Computational Models of Chemical Reaction Networks

Several computational models of chemical reaction networks have been presented in the literature in the past, showing the appearance and (potential) evolution of autocatalytic sets. However, the notion of autocatalytic sets has been defined differently in different modeling contexts, each one having some shortcoming or limitation. Here, we review four such models and definitions, and then formally describe and analyze them in the context of a mathematical framework for studying autocatalytic sets known as RAF theory. The main results are that: (1) RAF theory can capture the various previous definitions of autocatalytic sets and is therefore more complete and general, (2) the formal framework can be used to efficiently detect and analyze autocatalytic sets in all of these different computational models, (3) autocatalytic (RAF) sets are indeed likely to appear and evolve in such models, and (4) this could have important implications for a possible metabolism-first scenario for the origin of life.     

Microtubule Patterning in the Presence of Stationary Motor Distributions

In this paper, we construct a novel nonlocal transport model that describes the evolution of microtubules (MTs) as they interact with stationary distributions of motor proteins. An advection term accounts for directed MT transport (sliding due to motor protein action), and an integral term accounts for reorientation of MTs due to their interactions with cross-linking motor proteins. Simulations of our model show how MT patterns depend on boundary constraints, as well as model parameters that represent motor speed, cross-linking capability (motor activity), and directionality. In large domains, and using motor parameter values consistent with experimentally-derived values, we find that patterns such as asters, vortices, and bundles are able to persist. In vivo, MTs take on aster patterns during interphase and they form bundles in neurons and polarized epithelial cells. Vortex patterns have not been observed in vivo, however, are found in in vitro experiments. In constrained domains, we find that similar patterns form (asters, bundles, and vortices). However, we also find that when two opposing motors are present, anti-parallel bundles are able to form, resembling the mitotic spindle during cell division. This model demonstrates how MT sliding and MT reorientation are sufficient to produce experimentally observed patterns.     

Steady State Detection of Chemical Reaction Networks Using a Simplified Analytical Method

Chemical reaction networks (CRNs) are susceptible to mathematical modelling. The dynamic behavior of CRNs can be investigated by solving the polynomial equations derived from its structure. However, simple CRN give rise to non-linear polynomials that are difficult to resolve. Here we propose a procedure to locate the steady states of CRNs from a formula derived through algebraic geometry methods. We have applied this procedure to define the steady states of a classic CRN that exhibits instability, and to a model of programmed cell death.     

Catalysis in Reaction Networks

We define catalytic networks as chemical reaction networks with an essentially catalytic reaction pathway: one which is “on” in the presence of certain catalysts and “off” in their absence. We show that examples of catalytic networks include synthetic DNA molecular circuits that have been shown to perform signal amplification and molecular logic. Recall that a critical siphon is a subset of the species in a chemical reaction network whose absence is forward invariant and stoichiometrically compatible with a positive point. Our main theorem is that all weakly-reversible networks with critical siphons are catalytic. Consequently, we obtain new proofs for the persistence of atomic event-systems of Adleman et al., and normal networks of Gnacadja. We define autocatalytic networks, and conjecture that a weakly-reversible reaction network has critical siphons if and only if it is autocatalytic.     

Autocatalysis in Reaction Networks

The persistence conjecture is a long-standing open problem in chemical reaction network theory. It concerns the behavior of solutions to coupled ODE systems that arise from applying mass-action kinetics to a network of chemical reactions. The idea is that if all reactions are reversible in a weak sense, then no species can go extinct. A notion that has been found useful in thinking about persistence is that of “critical siphon.” We explore the combinatorics of critical siphons, with a view toward the persistence conjecture. We introduce the notions of “drainable” and “self-replicable” (or autocatalytic) siphons. We show that: Every minimal critical siphon is either drainable or self-replicable; reaction networks without drainable siphons are persistent; and nonautocatalytic weakly reversible networks are persistent. Our results clarify that the difficulties in proving the persistence conjecture are essentially due to competition between drainable and self-replicable siphons.     

Optimal Constrained Stationary Intervention in Gene Regulatory Networks

A key objective of gene network modeling is to develop intervention strategies to alter regulatory dynamics in such a way as to reduce the likelihood of undesirable phenotypes. Optimal stationary intervention policies have been developed for gene regulation in the framework of probabilistic Boolean networks in a number of settings. To mitigate the possibility of detrimental side effects, for instance, in the treatment of cancer, it may be desirable to limit the expected number of treatments beneath some bound. This paper formulates a general constraint approach for optimal therapeutic intervention by suitably adapting the reward function and then applies this formulation to bound the expected number of treatments. A mutated mammalian cell cycle is considered as a case study.     

Thermodynamics of Random Reaction Networks

Reaction networks are useful for analyzing reaction systems occurring in chemistry, systems biology, or Earth system science. Despite the importance of thermodynamic disequilibrium for many of those systems, the general thermodynamic properties of reaction networks are poorly understood. To circumvent the problem of sparse thermodynamic data, we generate artificial reaction networks and investigate their non-equilibrium steady state for various boundary fluxes. We generate linear and nonlinear networks using four different complex network models (Erdős-Rényi, Barabási-Albert, Watts-Strogatz, Pan-Sinha) and compare their topological properties with real reaction networks. For similar boundary conditions the steady state flow through the linear networks is about one order of magnitude higher than the flow through comparable nonlinear networks. In all networks, the flow decreases with the distance between the inflow and outflow boundary species, with Watts-Strogatz networks showing a significantly smaller slope compared to the three other network types. The distribution of entropy production of the individual reactions inside the network follows a power law in the intermediate region with an exponent of circa −1.5 for linear and −1.66 for nonlinear networks. An elevated entropy production rate is found in reactions associated with weakly connected species. This effect is stronger in nonlinear networks than in the linear ones. Increasing the flow through the nonlinear networks also increases the number of cycles and leads to a narrower distribution of chemical potentials. We conclude that the relation between distribution of dissipation, network topology and strength of disequilibrium is nontrivial and can be studied systematically by artificial reaction networks.     

Dynamical properties of Discrete Reaction Networks

Reaction networks are commonly used to model the dynamics of populations subject to transformations that follow an imposed stoichiometry. This paper focuses on the efficient characterisation of dynamical properties of Discrete Reaction Networks (DRNs). DRNs can be seen as modeling the underlying discrete nondeterministic transitions of stochastic models of reaction networks. In that sense, a proof of non-reachability in a given DRN has immediate implications for any concrete stochastic model based on that DRN, independent of the choice of kinetic laws and constants. Moreover, if we assume that stochastic kinetic rates are given by the mass-action law (or any other kinetic law that gives non-vanishing probability to each reaction if the required number of interacting substrates is present), then reachability properties are equivalent in the two settings. The analysis of two types of global dynamical properties of DRNs is addressed: irreducibility, i.e., the ability to reach any discrete state from any other state; and recurrence, i.e., the ability to return to any initial state. Our results consider both the verification of such properties when species are present in a large copy number, and in the general case. The necessary and sufficient conditions obtained involve algebraic conditions on the network reactions which in most cases can be verified using linear programming. Finally, the relationship of DRN irreducibility and recurrence with dynamical properties of stochastic and continuous models of reaction networks is discussed.     

Chemical Reaction Networks’ Programming for Solving Equations

The computational ability of the chemical reaction networks (CRNs) using DNA as the substrate has been verified previously. To solve more complex computational problems and perform the computational steps as expected, the practical design of the basic modules of calculation and the steps in the reactions have become the basic requirements for biomolecular computing. This paper presents a method for solving nonlinear equations in the CRNs with DNA as the substrate. We used the basic calculation module of the CRNs with a gateless structure to design discrete and analog algorithms and realized the nonlinear equations that could not be solved in the previous work, such as exponential, logarithmic, and simple triangle equations. The solution of the equation uses the transformation method, Taylor expansion, and Newton iteration method, and the simulation verified this through examples. We used and improved the basic calculation module of the CRN++ programming language, optimized the error in the basic module, and analyzed the error’s variation over time.     

Large-Scale Chemical Similarity Networks for Target Profiling of Compounds Identified in Cell-Based Chemical Screens

Target identification is one of the most critical steps following cell-based phenotypic chemical screens aimed at identifying compounds with potential uses in cell biology and for developing novel disease therapies. Current in silico target identification methods, including chemical similarity database searches, are limited to single or sequential ligand analysis that have limited capabilities for accurate deconvolution of a large number of compounds with diverse chemical structures. Here, we present CSNAP (Chemical Similarity Network Analysis Pulldown), a new computational target identification method that utilizes chemical similarity networks for large-scale chemotype (consensus chemical pattern) recognition and drug target profiling. Our benchmark study showed that CSNAP can achieve an overall higher accuracy (>80%) of target prediction with respect to representative chemotypes in large (>200) compound sets, in comparison to the SEA approach (60–70%). Additionally, CSNAP is capable of integrating with biological knowledge-based databases (Uniprot, GO) and high-throughput biology platforms (proteomic, genetic, etc) for system-wise drug target validation. To demonstrate the utility of the CSNAP approach, we combined CSNAP''s target prediction with experimental ligand evaluation to identify the major mitotic targets of hit compounds from a cell-based chemical screen and we highlight novel compounds targeting microtubules, an important cancer therapeutic target. The CSNAP method is freely available and can be accessed from the CSNAP web server (http://services.mbi.ucla.edu/CSNAP/).