AA1 – Life Sciences

Project

AA1-8

Random bifurcations in chemical reaction networks

Project Heads

Maximilian Engel

Project Members

Guillermo Olicon Mendez

Project Duration

01.02.2020 – 01.02.2022

Located at

FU Berlin

Description

In this project, we develop a random bifurcation theory for chemical reaction networks whose kinetics are properly modeled by Markov jump processes. For large reactants, model reduction to reaction rate equations (ODEs) and Langevin equations (SDEs) is possible and computationally alleviating. We investigate which model reductions succeed in preserving pathwise properties of the jump process in terms of dynamical stability and bifurcation behavior stemming from varying reaction rates.

In more details, we consider general models of large sets of molecules that can be allocated to a certain number of different species. Reactions, changing the number of molecules of a subset of species, occur at certain rates  which are defined as propensity funtions involving reaction rate constants. Assuming that these functions depend only on the state of the whole reaction system, one can describe the changes of the system over time, i.e. the dynamics, as a Markov jump process driven by independent Poisson processes. For high accumulations of reaction events, the Poisson variables can be approximated by normal random variables, yielding the chemical Langevin equation (CLE) which is a stochastic differential equation (SDE). In the thermodynamic limit where volume and volume-dependent molecule numbers scale in a fixed proportion approaching infinity, we may average over the noise terms to obtain the purely deterministic version of the CLE, i.e. an ODE also called reaction rate equation (RRE).

For all three approximation levels, there are corresponding differential equations, describing the evolution of the probability distribution of states ; in the stochastic case, this evolution is described by the forward Kolmogorov equation, which takes the form of the chemical master equation for the Markov jump process and the form of a Fokker-Planck equation for the CLE. The corresponding PDE for the RRE is called Liouville equation. Many investigations focus on these equations comparing distributions of the reaction process and its approximation models, and by that only capture statistical properties of the processes.

We employ the perspective of random dynamical systems (RDS)  in order to understand trajectory-wise behavior of the Markov jump process in comparison to the SDE solution, and to demonstrate what can be missed, also in the thermodynamic limit, by only focusing on expectations given by solutions of the RRE. The RDS approach compares two trajectories with the same noise realizations and analyzes whether they synchronize or diverge asymptotically, determining the form of a random attractor. By applying techniques of linearization and multiplicative ergodic theory, we are the first to investigate and compare thoroughly the stability behavior of all three models; this becomes highly relevant when changes of rate constants  are considered such that bifurcations can be detected. A crucial role is played by the Hopf bifurcation, where an attracting equilibrium loses stability and an attracting periodic orbit occurs, which exhibits complicated behavior under noise perturbations (see Figures below).

Project Webpages

Selected Publications

M. Engel, J.S.W. Lamb and M. Rasmussen. Conditioned Lyapunov exponents for random dynamical systems. Transactions of the American Mathematical Society, 372(9): 6343-6370, 2019. https://doi.org/10.1090/tran/7803

M. Engel, J.S.W. Lamb and M. Rasmussen. Bifurcation analysis of a stochastically driven limit cycle. Communications in Mathematical Physics, 365: 935-942, 2019.
https://doi.org/10.1007/s00220-019-03298-7

Selected Pictures

Chaotic random attractor from a stochastic Hopf bifurcation
Such a fractal-like attractor with chaotic properties can occur in stochastically perturbed systems (CLE) that experience a Hopf bifurcation in the deterministic case (RRE), for example in a typical biological modle like the so-called Brusselator.

Synchronization of stochastic trajectories driven by the same noise
Synchonization of trajectories, i.e. the random attractor shrinking to a point, is the other typical behavior of random dynamical systems, also occurring in Hopf bifurcations under the influence of noise.

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