A diagram technique for cumulant equations in biomolecular reaction networks with mass-action kinetics

2016 ◽  
Author(s):  
Leo Bronstein ◽  
Heinz Koeppl
Keyword(s):  
Mass Action ◽  
Diagram Technique ◽  
Reaction Networks ◽  
Mass Action Kinetics ◽  
Cumulant Equations

scholarly journals A general technique for the detection of switch-like bistability in chemical reaction networks governed by mass action kinetics with conservation laws

2020 ◽  
Author(s):  
Brandon C Reyes ◽  
Irene Otero-Muras ◽  
Vladislav A Petyuk
Keyword(s):  
Conservation Laws ◽  
Signaling Pathways ◽  
Chemical Reaction ◽  
Reaction Network ◽  
Mass Action ◽  
Numerical Continuation ◽  
Reaction Networks ◽  
General Technique ◽  
Bistable Switch ◽  
Mass Action Kinetics

AbstractBackgroundTheoretical 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.ResultsIn 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.ConclusionsWe have developed a variation of an optimization-based method for discovery of bistability, which is not limited to the structure of the chemical reaction network. 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.

The Gompertz model revisited and modified using reaction networks: Mathematical analysis

BIOMATH ◽  
2021 ◽  
Vol 10 (2) ◽  
pp. 2110023
Author(s):  
Svetoslav Marinov Markov
Keyword(s):  
Exponential Decay ◽  
Mathematical Analysis ◽  
Reaction Network ◽  
Gompertz Model ◽  
Mass Action ◽  
Reaction Networks ◽  
Data Sets ◽  
Growth Data ◽  
Mass Action Kinetics ◽  
Additional Degree

In the present work we discuss the?usage of the framework of chemical reaction networks for the construction of dynamical models and their mathematical analysis. To this end, the process of construction of reaction-network-based models via mass action kinetics is introduced and illustrated on several familiar examples,?such as the exponential (radioactive) decay, the logistic and the Gompertz models. Our final goal is to modify the reaction network of the classic Gompertz model in a natural way using certain features of the exponential decay and the logistic models. The growth function of the obtained new Gompertz-type hybrid model possesses an additional degree of freedom (one more rate parameter) and is thus more flexible when applied to numerical simulation of measurement and experimental data sets. More specifically, the ordinate (height) of the inflection point of the new generalized Gompertz model can vary in the interval (0, 1/e], whereas the respective height of the classic Gompertz model is fixed at 1/e (assuming the height of the upper asymptote is one). It is shown that?the new model is a generalization of both the classic Gompertz model and the one-step exponential decay model.?Historically the Gompertz function has been first used for statistical/insurance purposes, much later this function has been applied to simulate biological growth data sets coming from various fields of science, the reaction network approach explains and unifies the two approaches.

Contributions to the Theory of Mass Action Kinetics. I. Enumeration of Second Order Mass Action Kinetics

1978 ◽  
Vol 33 (7) ◽  
pp. 827-833 ◽  
Author(s):  
K.-D. Willamowski ◽  
O. E. Rössler
Keyword(s):  
Second Order ◽  
Mass Action ◽  
Reversible Reaction ◽  
Reaction Networks ◽  
Mass Action Kinetics ◽  
Number Of Species ◽  
Different Types

By investigating the reaction diagram in its own right, it is possible to solve the problem of enumerating all the different types of mass action kinetics up to second order. The amount of non isomorphic complex sets for a given number of species and of non isomorphic reaction networks and reversible reaction networks which can be derived from a given number of complexes is given.

scholarly journals Deciphering noise amplification and reduction in open chemical reaction networks

2018 ◽  
Author(s):  
Fabrizio Pucci ◽  
Marianne Rooman
Keyword(s):  
Differential Equation ◽  
Molecular Species ◽  
Dynamical Behavior ◽  
Intrinsic Noise ◽  
Mass Action ◽  
Model Systems ◽  
Reaction Networks ◽  
Mass Action Kinetics ◽  
Noise Amplification ◽  
The Impact

AbstractThe impact of fluctuations on the dynamical behavior of complex biological systems is a longstanding issue, whose understanding would elucidate how evolutionary pressure tends to modulate intrinsic noise. Using the Itō stochastic differential equation formalism, we performed analytic and numerical analyses of model systems containing different molecular species in contact with the environment and interacting with each other through mass-action kinetics. For networks of zero deficiency, which admit a detailed- or complex-balanced steady state, all molecular species are uncorrelated and their Fano factors are Poissonian. Systems of higher deficiency have non-equilibrium steady states and non-zero reaction fluxes flowing between the complexes. When they model homooligomerization, the noise on each species is reduced when the flux flows from the oligomers of lowest to highest degree, and amplified otherwise. In the case of hetero-oligomerization systems, only the noise on the highest-degree species shows this behavior.

scholarly journals Inferring Reaction Networks using Perturbation Data

2018 ◽  
Author(s):  
Kiri Choi ◽  
Joesph Hellerstein ◽  
H. Steven Wiley ◽  
Herbert M. Sauro
Keyword(s):  
Linear Chain ◽  
Mass Action ◽  
Evolutionary Strategy ◽  
Reaction Networks ◽  
Second Phase ◽  
Mass Action Kinetics ◽  
Rate Laws ◽  
Perturbation Data ◽  
Parameter Values ◽  
The Impact

AbstractIn this paper we examine the use of perturbation data to infer the underlying mechanistic dynamic model. The approach uses an evolutionary strategy to evolve networks based on a fitness criterion that measures the difference between the experimentally determined set of perturbation data and proposed mechanistic models. At present we only deal with reaction networks that use mass-action kinetics employing uni-uni, bi-uni, uni-bi and bi-bi reactions. The key to our approach is to split the algorithm into two phases. The first phase focuses on evolving network topologies that are consistent with the perturbation data followed by a second phase that evolves the parameter values. This results in almost an exact match between the evolved network and the original network from which the perturbation data was generated from. We test the approach on four models that include linear chain, feed-forward loop, cyclic pathway and a branched pathway. Currently the algorithm is implemented using Python and libRoadRunner but could at a later date be rewritten in a compiled language to improve performance. Future studies will focus on the impact of noise in the perturbation data on convergence and variability in the evolved parameter values and topologies. In addition we will investigate the effect of nonlinear rate laws on generating unique solutions.

scholarly journals Noise-induced mixing and multimodality in reaction networks

2018 ◽  
Vol 30 (5) ◽  
pp. 887-911 ◽  
Author(s):  
TOMISLAV PLESA ◽  
RADEK ERBAN ◽  
HANS G. OTHMER
Keyword(s):  
Probability Distribution ◽  
Regulatory Networks ◽  
Probability Distributions ◽  
Mass Action ◽  
Reaction Networks ◽  
Multiple Time Scales ◽  
Multiple Time ◽  
Gene Products ◽  
Mass Action Kinetics

We analyse a class of chemical reaction networks under mass-action kinetics involving multiple time scales, whose deterministic and stochastic models display qualitative differences. The networks are inspired by gene-regulatory networks and consist of a slow subnetwork, describing conversions among the different gene states, and fast subnetworks, describing biochemical interactions involving the gene products. We show that the long-term dynamics of such networks can consist of a unique attractor at the deterministic level (unistability), while the long-term probability distribution at the stochastic level may display multiple maxima (multimodality). The dynamical differences stem from a phenomenon we call noise-induced mixing, whereby the probability distribution of the gene products is a linear combination of the probability distributions of the fast subnetworks which are ‘mixed’ by the slow subnetworks. The results are applied in the context of systems biology, where noise-induced mixing is shown to play a biochemically important role, producing phenomena such as stochastic multimodality and oscillations.

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