**Project Heads**

*Maximilian Engel*

**Project Members**

Guillermo Olicon Mendez

**Project Duration**

15.02.2021 – 14.02.2023

**Located at**

FU Berlin

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
**

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