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Posters (SDBW03)

     Deficiency and Entropy Production in Chemical Reaction Networks

Multilevel Drift-Implicit Tau-Leap

Equilibrium distributions of simple biochemical reaction systems for time-scale separation in stochastic reaction networks

Living organisms transform food into waste, living on the energetic and entropic difference - thus fighting the Second Law of Thermodynamics which states that the entropy production is always positive. In this perspective organisms are reminiscent of thermodynamic engines, rendering cyclic transformations the key concept for the thermodynamic analysis of life. Biochemical reaction networks play a crucial role in understanding how cells operate on a molecular level modeling metabolic and signalling pathways. At high particle numbers noise can be neglected but for low particle numbers stochasticity can play a crucial role in the dynamics.

Here, we show how the cycle structure of a chemical reaction network is linked to its thermodynamics on small scales. One of the key quantities is the deficiency of a chemical network - a purely topological number restricting the possible dynamics and thermodynamics.


The dynamics of biochemical reactive systems with small copy numbers of one or more reactant molecules is dominated by stochastic effects. For those systems, discrete state-space and stochastic simulation approaches were proved to be more relevant than continuous state-space and deterministic ones. In systems characterized by having simultaneously fast and slow timescales, the existing discrete space-state stochastic path simulation methods such as the stochastic simulation algorithm (SSA) and the explicit tauleap method can be very slow. Implicit approximations were developed in the literature to improve numerical stability and provide efficient simulation algorithms for those systems. In this work, we propose an efficient Multilevel Monte Carlo method in the spirit of the work by Anderson and Higham (2012) that uses drift-implicit tau-leap approximations at levels where the explicit tauleap method is not applicable due to numerical stability issues. We present numerical examples that illustrate the performance of the proposed method.


Our goal is to catalogue all reaction systems where the equilibrium distribution can be computed analytically, and also to collect all such computation strategies. These can be used in averaging (quasi-steady-state analysis) but also provide insight into the stimulus—response of enzymatic reactions and gene regulatory models. We derive the equilibrium distribution for the binding of dimer transcription factors (TFs) to a gene that first have to form from monomers. This allows a comparison of four different gene regulatory mechanisms. The new technique may be applicable to other cases where for marginalisation, a summing of a product-form stationary distribution with a conservation law is required. [1]

We also study how to compute the equilibrium of a continuous-time Markov chain from those of two smaller parts. We glue the two small parts together at 1 or 2 states. [2]

  1. Mélykúti, Hespanha, Khammash; Journal of the Royal Society Interface, 11(97), 20140054, 2014. doi:10.1098/rsif.2014.0054
  2. Mélykúti, Pfaffelhuber; Stochastic Models, 31(4), 525—553, 2015. doi:10.1080/15326349.2015.1055769; arXiv:1401.6400

University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons