We discuss a method of approximate model reduction for networks of biochemical reactions. This method can be applied to networks with polynomial or rational reaction rates and whose parameters are given by their orders of magnitude. In order to obtain reduced models we solve the problem of tropical equilibration that is a system of equations in max-plus algebra. In the case of networks with nonlinear fast cycles we have to solve the problem of tropical equilibration at least twice, once for the initial system and a second time for an extended system obtained by adding to the initial system the differential equations satisfied by the conservation laws of the fast subsystem. The two steps can be reiterated until the fast subsystem has no conservation laws different from the ones of the full model. Our method can be used for formal model reduction in computational systems biology.
We use the Litvinov-Maslov correspondence principle to reduce and hybridize networks of biochemical reactions. We apply this method to a cell cycle oscillator model. The reduced and hybridized model can be used as a hybrid model for the cell cycle. We also propose a practical recipe for detecting quasi-equilibrium QE reactions and quasi-steady state QSS species in biochemical models with rational rate functions and use this recipe for model reduction. Interestingly, the QE/QSS invariant manifold of the smooth model and the reduced dynamics along this manifold can be put into correspondence to …
We discuss the symbolic dynamics of biochemical networks with separate timescales. We show that symbolic dynamics of monomolecular reaction networks with separated rate constants can be described by deterministic, acyclic automata with a number of states that is inferior to the number of biochemical species. For nonlinear pathways, we propose a general approach to approximate their dynamics by finite state machines working on the metastable states of the network (long life states where the system has slow dynamics). For networks with polynomial rate functions we propose to compute metastable s…
Biochemical molecules interact through modification and binding reactions, giving raise to a combinatorial number of possible biochemical species. The time-dependent evolution of concentrations of the species is commonly described by a system of coupled ordinary differential equations (ODEs). However, the analysis of such high-dimensional, non-linear system of equations is often computationally expensive and even prohibitive in practice. The major challenge towards reducing such models is providing the guarantees as to how the solution of the reduced model relates to that of the original model…
Systems biology uses large networks of biochemical reactions to model the functioning of biological cells from the molecular to the cellular scale. The dynamics of dissipative reaction networks with many well separated time scales can be described as a sequence of successive equilibrations of different subsets of variables of the system. Polynomial systems with separation are equilibrated when at least two monomials, of opposite signs, have the same order of magnitude and dominate the others. These equilibrations and the corresponding truncated dynamics, obtained by eliminating the dominated t…
We have developed a coarse-grained formulation for modeling the dynamic behavior of cells quantitatively, based on stochasticity and heterogeneity, rather than on biochemical reactions. We treat each reaction as a continuous-time stochastic process, while reducing each biochemical quantity to a binary value at the level of individual cells. The system can be analytically represented by a finite set of ordinary linear differential equations, which provides a continuous time course prediction of each molecular state. In this letter, we introduce our formalism and demonstrate it with several exam…
Motivation: An essential step in developing computational tools for the inference, optimization, and simulation of biochemical reaction networks is gauging tool performance against earlier efforts using an appropriate set of benchmarks. General strategies for the assembly of benchmark models include collection from the literature, creation via subnetwork extraction and de novo generation. However, with respect to biochemical reaction networks, these approaches and their associated tools are either poorly suited to generate models that reflect the wide range of properties found in natural bioch…
We discuss a novel analysis method for reaction network systems with polynomial or rational rate functions. This method is based on computing tropical equilibrations defined by the equality of at least two dominant monomials of opposite signs in the differential equations of each dynamic variable. In algebraic geometry, the tropical equilibration problem is tantamount to finding tropical prevarieties, that are finite intersections of tropical hypersurfaces. Tropical equilibrations with the same set of dominant monomials define a branch or equivalence class. Minimal branches are particularly in…
Living cells encode and transmit information in the temporal dynamics of biochemical components. Gaining a detailed understanding of the input-output relationship in biological systems therefore requires quantitative measures that capture the interdependence between complete time trajectories of biochemical components. Mutual information provides such a measure but its calculation in the context of stochastic reaction networks is associated with mathematical challenges. Here we show how to estimate the mutual information between complete paths of two molecular species that interact with each o…
Stochasticity is a key characteristic of intracellular processes such as gene regulation and chemical signalling. Therefore, characterising stochastic effects in biochemical systems is essential to understand the complex dynamics of living things. Mathematical idealisations of biochemically reacting systems must be able to capture stochastic phenomena. While robust theory exists to describe such stochastic models, the computational challenges in exploring these models can be a significant burden in practice since realistic models are analytically intractable. Determining the expected behaviour…
Model reduction of biochemical networks relies on the knowledge of slow and fast variables. We provide a geometric method, based on the Newton polytope, to identify slow variables of a biochemical network with polynomial rate functions. The gist of the method is the notion of tropical equilibration that provides approximate descriptions of slow invariant manifolds. Compared to extant numerical algorithms such as the intrinsic low dimensional manifold method, our approach is symbolic and utilizes orders of magnitude instead of precise values of the model parameters. Application of this method t…
Sensitivity analysis of biochemical reactions aims at quantifying the dependence of the reaction dynamics on the reaction rates. The computation of the parameter sensitivities, however, poses many computational challenges when taking stochastic noise into account. This paper proposes a new finite difference method for efficiently computing sensitivities of biochemical reactions. We employ propensity bounds of reactions to couple the simulation of the nominal and perturbed processes. The exactness of the simulation is reserved by applying the rejection-based mechanism. For each simulation step,…
We present a simple method for determining the exact noise power spectra in linear chemical reaction networks. We apply the method to networks which are representative of biochemical processes such as gene expression and signal detection. Our results clarify how noise is transmitted by signal detection motifs, and indicate how to coarse-grain networks by the elimination of fast reactions.
The evolutionary origin of universal statistics in biochemical reaction network is studied, to explain the power-law distribution of reaction links and the power-law distributions of chemical abundances. Using cell models with catalytic reaction network, we find evidence that the power-law distribution in abundances of chemicals emerges by the selection of cells with higher growth speeds. Through the further evolution, this inhomogeneity in chemical abundances is shown to be embedded in the distribution of links, leading to the power-law distribution. These findings provide novel insights into…
The importance of stochasticity within biological systems has been shown repeatedly during the last years and has raised the need for efficient stochastic tools. We present SABRE, a tool for stochastic analysis of biochemical reaction networks. SABRE implements fast adaptive uniformization (FAU), a direct numerical approximation algorithm for computing transient solutions of biochemical reaction networks. Biochemical reactions networks represent biological systems studied at a molecular level and these reactions can be modeled as transitions of a Markov chain. SABRE accepts as input the formal…
We present a novel method for identifying a biochemical reaction network based on multiple sets of estimated reaction rates in the corresponding reaction rate equations arriving from various (possibly different) experiments. The current method, unlike some of the graphical approaches proposed in the literature, uses the values of the experimental measurements only relative to the geometry of the biochemical reactions under the assumption that the underlying reaction network is the same for all the experiments. The proposed approach utilizes algebraic statistical methods in order to parametrize…
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 identif…
We present a method for estimating parameters in stochastic models of biochemical reaction networks by fitting steady-state distributions using Wasserstein distances. We simulate a reaction network at different parameter settings and train a Gaussian process to learn the Wasserstein distance between observations and the simulator output for all parameters. We then use Bayesian optimization to find parameters minimizing this distance based on the trained Gaussian process. The effectiveness of our method is demonstrated on the three-stage model of gene expression and a genetic feedback loop for …
Thermodynamic constraints on reactions directions are inherent in the structure of a given biochemical network. However, concrete procedures for determining feasible reaction directions for large-scale metabolic networks are not well established. This work introduces a systematic approach to compute reaction directions, which are constrained by mass balance and thermodynamics, for genome-scale networks. In addition, it is shown that the nonconvex solution space constrained by physicochemical constraints can be approximated by a set of linearized subspaces in which mass and thermodynamic balanc…
We study the problem of computing outer bounds for the region of steady states of biochemical reaction networks modelled by ordinary differential equations, with respect to parameters that are allowed to vary within a predefined region. Using a relaxed version of the corresponding feasibility problem and its Lagrangian dual, we show how to compute certificates for regions in state space not containing any steady states. Based on these results, we develop an algorithm to compute outer bounds for the region of all feasible steady states. We apply our algorithm to the sensitivity analysis of a Go…
In many stochastic simulations of biochemical reaction networks, it is desirable to ``coarse-grain'' the reaction set, removing fast reactions while retaining the correct system dynamics. Various coarse-graining methods have been proposed, but it remains unclear which methods are reliable and which reactions can safely be eliminated. We address these issues for a model gene regulatory network that is particularly sensitive to dynamical fluctuations: a bistable genetic switch. We remove protein-DNA and/or protein-protein association-dissociation reactions from the reaction set, using various co…
Bazhin has analyzed ATP coupling in terms of quasiequilibrium states where fast reactions have reached an approximate steady state while slow reactions have not yet reached equilibrium. After an expository introduction to the relevant aspects of reaction network theory, we review his work and explain the role of emergent conserved quantities in coupling. These are quantities, left unchanged by fast reactions, whose conservation forces exergonic processes such as ATP hydrolysis to drive desired endergonic processes.
Model reduction is a central topic in systems biology and dynamical systems theory, for reducing the complexity of detailed models, finding important parameters, and developing multi-scale models for instance. While perturbation theory is a standard mathematical tool to analyze the different time scales of a dynamical system, and decompose the system accordingly, tropical methods provide a simple algebraic framework to perform these analyses systematically in polynomial systems. The crux of these tropicalization methods is in the computation of tropical equilibrations. In this paper we show th…
According to the chemical reaction network theory, the topology of a certain class of chemical reaction networks, regardless of the kinetic details, sets a limit on the dynamical properties that a particular network can potentially admit; the structure of a network predetermines the dynamic capacity of the network. We note that stochastic fluctuations can possibly confer a new dynamical capability to a network. Thus, it is of tremendous value to understand and be able to control the landscape of stochastic dynamical behaviors of a biochemical reaction network as a function of network architect…
Biochemical reaction networks in living cells usually involve reversible covalent modification of signaling molecules, such as protein phosphorylation. Under conditions of small molecule numbers, as is frequently the case in living cells, mass action theory fails to describe the dynamics of such systems. Instead, the biochemical reactions must be treated as stochastic processes that intrinsically generate concentration fluctuations of the chemicals. We investigate the stochastic reaction kinetics of covalent modification cycles (CMCs) by analytical modeling and numerically exact Monte-Carlo si…
Biochemical networks are the analog computers of life. They allow living cells to control a large number of biological processes, such as gene expression and cell signalling. In biochemical networks, the concentrations of the components are often low. This means that the discrete nature of the reactants and the stochastic character of their interactions have to be taken into account. Moreover, the spatial distribution of the components can be of crucial importance. However, the current numerical techniques for simulating biochemical networks either ignore the particulate nature of matter or tr…
We discuss a method to describe the qualitative dynamics of chemical reaction networks in terms of symbolic dynamics. The method, that can be applied to mass-action reaction networks with separated timescales, uses solutions of the partial tropical equilibration problem as proxies for symbolic states. The partial tropical equilibration solutions are found algorithmically. These solutions also provide the scaling needed for slow-fast decomposition and model reduction. Any trace of the model can thus be represented as a sequence of local approximations of the full model. We illustrate the method…
Metabolic networks have two properties that are generally regarded as unrelated: One, they have metabolic reactions whose single knockout is lethal for the organism, and two, they have correlated sets of reactions forming functional modules. In this review we argue that both essentiality and modularity seem to arise as a consequence of the same structural property: the existence of low degree metabolites. This observation allows a prediction of (a) essential metabolic reactions which are potential drug targets in pathogenic microorganisms and (b) regulatory modules within biological networks, …
A cell has the ability to convert an environmental change into the expression of genetic information through a chain of intracellular signal transduction reactions. Here, we aimed to develop a method for quantifying this signal transduction. We showed that the channel capacities of individual steps in a given general model cascade were equivalent in an independent manner, and were given by the entropy production rate. Signal transduction was transmitted by fluctuation of the entropy production rate and quantified transduction was estimated by the work done in individual steps. If the individua…
Statistical mechanics provides an effective framework to investigate information processing in biochemical reactions. Within such framework far-reaching analogies are established among (anti-) cooperative collective behaviors in chemical kinetics, (anti-)ferromagnetic spin models in statistical mechanics and operational amplifiers/flip-flops in cybernetics. The underlying modeling -- based on spin systems -- has been proved to be accurate for a wide class of systems matching classical (e.g. Michaelis--Menten, Hill, Adair) scenarios in the infinite-size approximation. However, the current resea…
Stochasticity is an indispensable aspect of biochemical processes at the cellular level. Studies on how the noise enters and propagates in biochemical systems provided us with nontrivial insights into the origins of stochasticity, in total however they constitute a patchwork of different theoretical analyses. Here we present a flexible and generally applicable noise decomposition tool, that allows us to calculate contributions of individual reactions to the total variability of a system's output. With the package it is therefore possible to quantify how the noise enters and propagates in bioch…
Understanding cell function requires an accurate description of how noise is transmitted through biochemical networks. We present an analytical result for the power spectrum of the output signal of a biochemical network that takes into account the correlations between the noise in the input signal (the extrinsic noise) and the noise in the reactions that constitute the network (the intrinsic noise). These correlations arise from the fact that the reactions by which biochemical signals are detected necessarily affect the signaling molecules and the detection components of the network simultaneo…
Intra-cellular biochemical reactions exhibit a rich dynamical phenomenology which cannot be explained within the framework of mean-field rate equations and additive noise. Here, we show that the presence of metastable states and radically different timescales are general features of a broad class of autocatalyic reaction networks, and moreover, that this fact may be exploited to gain analytical results. The latter point is demonstrated by a treatment of the paradigmatic Togashi-Kaneko reaction, which has resisted theoretical analysis for the last decade.
How do single cell fate decisions induced by activation of key signaling proteins above threshold concentrations within a time interval are affected by stochastic fluctuations in biochemical reactions? We address this question using minimal models of stochastic chemical reactions commonly found in cell signaling and gene regulatory systems. Employing exact solutions and semi-analytical methods we calculate distributions of the maximum value ($N$) of activated species concentrations ($P_{max}(N)$) and the time ($t$) taken to reach the maximum value ($P_{max}(t)$) within a time window in the min…
We present herein an extension of an algebraic statistical method for inferring biochemical reaction networks from experimental data, proposed recently in [3]. This extension allows us to analyze reaction networks that are not necessarily full-dimensional, i.e., the dimension of their stoichiometric space is smaller than the number of species. Specifically, we propose to augment the original algebraic-statistical algorithm for network inference with a preprocessing step that identifies the subspace spanned by the correct reaction vectors, within the space spanned by the species. This dimension…
Biochemical networks play a crucial role in biological systems, implementing a broad range of vital functions. They normally operate at low copy numbers and in spatial settings, but this is often ignored and well-stirred conditions are assumed. Yet, it is increasingly becoming clear that even microscopic spatial inhomogeneities oftentimes can induce significant differences on the macroscopic level. Since experimental observation of single-molecule behavior is extremely challenging, theoretical modeling of biochemical reactions on the single-particle level is an important tool for understanding…
We consider the general problem of describing the dynamics of subnetworks of larger biochemical reaction networks, e.g. protein interaction networks involving complex formation and dissociation reactions. We propose the use of model reduction strategies to understand the 'extrinsic' sources of stochasticity arising from the rest of the network. Our approaches are based on subnetwork dynamical equations derived by projection methods and by path integrals. The results provide a principled derivation of the different components of the extrinsic noise that is observed experimentally in cellular bi…
Mathematical and computational modelling of biochemical networks is often done in terms of either the concentrations of molecular species or the fluxes of biochemical reactions. When is mathematical modelling from either perspective equivalent to the other? Mathematical duality translates concepts, theorems or mathematical structures into other concepts, theorems or structures, in a one-to-one manner. We present a novel stoichiometric condition that is necessary and sufficient for duality between unidirectional fluxes and concentrations. Our numerical experiments, with computational models der…
Biochemical reaction networks frequently consist of species evolving on multiple timescales. Stochastic simulations of such networks are often computationally challenging and therefore various methods have been developed to obtain sensible stochastic approximations on the timescale of interest. One of the rigorous and popular approaches is the multiscale approximation method for continuous time Markov processes. In this approach, by scaling species abundances and reaction rates, a family of processes parameterized by a scaling parameter is defined. The limiting process of this family is then u…
Genetically identical cells under the same environmental conditions can show strong variations in protein copy numbers due to inherently stochastic events in individual cells. We here develop a theoretical framework to address how variations in enzyme abundance affect the collective kinetics of metabolic reactions observed within a population of cells. Kinetic parameters measured at the cell population level are shown to be systematically deviated from those of single cells, even within populations of homogeneous parameters. Because of these considerations, Michaelis-Menten kinetics can even b…
Cells must continuously sense and respond to time-varying environmental stimuli. These signals are transmitted and processed by biochemical signalling networks. However, the biochemical reactions making up these networks are intrinsically noisy, which limits the reliability of intracellular signalling. Here we use information theory to characterise the reliability of transmission of time-varying signals through elementary biochemical reactions in the presence of noise. We calculate the mutual information for both instantaneous measurements and trajectories of biochemical systems for a Gaussian…
Many processes in biology, from the regulation of gene expression in bacteria to memory in the brain, involve switches constructed from networks of biochemical reactions. Crucial molecules are present in small numbers, raising questions about noise and stability. Analysis of noise in simple reaction schemes indicates that switches stable for years and switchable in milliseconds can be built from fewer than one hundred molecules. Prospects for direct tests of this prediction, as well as implications, are discussed.
Heterogeneity in gene expression across isogenic cell populations can give rise to phenotypic diversity, even when cells are in homogenous environments. This diversity arises from the discrete, stochastic nature of biochemical reactions, which naturally arise due to the very small numbers of genes, RNA, or protein molecules in single cells. Modern measurements of single biomolecules have created a vast wealth of information about the fluctuations of these molecules, but a quantitative understanding of these complex, stochastic systems requires precise computational tools. In this article, we p…
Synthetic biology is an interdisciplinary field aiming to design biochemical systems with desired behaviors. To this end, molecular controllers have been developed which, when embedded into a pre-existing ambient biochemical network, control the dynamics of the underlying target molecular species. When integrated into smaller compartments, such as biological cells in vivo, or vesicles in vitro, controllers have to be calibrated to factor in the intrinsic noise. In this context, molecular controllers put forward in the literature have focused on manipulating the mean (first moment), and reducin…
State- versus Reaction-Based Information Processing in Biochemical Networks
Max Planck Institute of Molecular Cell Biology and Genetics（马克斯·普朗克分子细胞生物学与遗传学研究所）；Center for Systems Biology Dresden（德累斯顿系统生物学中心）；AMOLF（阿莫夫基金会）；Scuola Internazionale Superiore di Studi Avanzati（高等高级研究国际学院）
We present a kinetic Monte Carlo method for simulating chemical transformations specified by reaction rules, which can be viewed as generators of chemical reactions, or equivalently, definitions of reaction classes. A rule identifies the molecular components involved in a transformation, how these components change, conditions that affect whether a transformation occurs, and a rate law. The computational cost of the method, unlike conventional simulation approaches, is independent of the number of possible reactions, which need not be specified in advance or explicitly generated in a simulatio…
Nature presents multiple intriguing examples of processes which proceed at high precision and regularity. This remarkable stability is frequently counter to modelers' experience with the inherent stochasticity of chemical reactions in the regime of low copy numbers. Moreover, the effects of noise and nonlinearities can lead to "counter-intuitive" behavior, as demonstrated for a basic enzymatic reaction scheme that can display stochastic focusing (SF). Under the assumption of rapid signal fluctuations, SF has been shown to convert a graded response into a threshold mechanism, thus attenuating t…
The application of the quasi-steady-state approximation to the Michaelis-Menten reaction embedded in large open chemical reaction networks is a popular model reduction technique in deterministic and stochastic simulations of biochemical reactions inside cells. It is frequently assumed that the predictions of the reduced master equations obtained using the stochastic quasi-steady-state approach are in very good agreement with the predictions of the full master equations, provided the conditions for the validity of the deterministic quasi-steady-state approximation are fulfilled. We here use the…
Long-term behaviors of biochemical systems are described by steady states in deterministic models and stationary distributions in stochastic models. Obtaining their analytic solutions can be done for limited cases, such as linear or finite-state systems, as it generally requires solving many coupled equations. Interestingly, analytic solutions can be easily obtained when underlying networks have special topologies, called weak reversibility (WR) and zero deficiency (ZD), and the kinetic law follows a generalized form of mass-action kinetics. However, such desired topological conditions do not …
Processes involving multi-input multi-step reaction cascades are used in developing novel biosensing, biocomputing, and decision making systems. In various applications different changes in responses of the constituent processing steps (reactions) in a cascade are desirable in order to allow control of the system's response. Here we consider conversion of convex response to sigmoid by "intensity filtering," as well as "threshold filtering," and we offer a general overview of this field of research. Specifically, we survey rate equation modelling that has been used for enzymatic reactions. This…
Biochemical networks are used in computational biology, to model the static and dynamical details of systems involved in cell signaling, metabolism, and regulation of gene expression. Parametric and structural uncertainty, as well as combinatorial explosion are strong obstacles against analyzing the dynamics of large models of this type. Multi-scaleness is another property of these networks, that can be used to get past some of these obstacles. Networks with many well separated time scales, can be reduced to simpler networks, in a way that depends only on the orders of magnitude and not on the…
In an experimental study of single enzyme reactions, it has been proposed that the rate constants of the enzymatic reactions fluctuate randomly, according to a given distribution. To quantify the uncertainty arising from random rate constants, it is necessary to investigate how one can simulate such a biochemical system. To do this, we will take the Gillespie's stochastic simulation algorithm for simulating the evolution of the state of a chemical system, and study a modification of the algorithm that incorporates the random rate constants. In addition to simulating the waiting time of each re…
Exact first passage time distribution for nonlinear chemical reaction networks II: monomolecular reactions and a A + B - C type of second-order reaction with arbitrary initial conditions
Many biological, physical, and social interactions have a particular dependence on where they take place. In living cells, protein movement between the nucleus and cytoplasm affects cellular response (i.e., proteins must be present in the nucleus to regulate their target genes). Here we use recent developments from dynamical systems and chemical reaction network theory to identify and characterize the key-role of the spatial organization of eukaryotic cells in cellular information processing. In particular the existence of distinct compartments plays a pivotal role in whether a system is capab…
