# SURF 2015: Design of equilibrium and transient distributions of stochastic biochemical reaction networks

2015 SURF **project description**

- Mentor: Richard M. Murray

- Co-mentors: Ania Baetica and Vipul Singhal

Understanding how to design biocircuits subject to stochastic effects is one of the goals of the Murray lab. Experimental evidence has made it clear that stochastic fluctuations have a significant and even beneficial role in genetic circuits [1]. These fluctuations are particularly strong in the low molecule numbers regime that both our biocircuits and numerous cellular processes run in [2]. To better understand stochastic behavior, we model our biocircuits as stochastic biochemical reaction networks and perform design over time-evolving distributions of molecular species from our system. The goal of this project is to perform biologically-informed design of the equilibrium (stationary) and transient distributions in pertinent examples of stochastic biochemical reaction networks.

Stochastic behavior of biochemical reaction networks is described analytically by the Chemical Master Equation (CME). The time-evolution of molecular species distributions is described by an infinite-dimensional ODE. Approximations to the solution of the CME are used routinely for computer simulations (SSA algorithm), but analytical solutions are difficult to obtain due to the infinite-dimensionality of the equation. Nonetheless, the CME has been solved analytically for special case gene regulatory systems [3] and for monomolecular reaction systems [4]. Analytic solutions were determined for both the stationary and the transient distributions of molecular species in these examples. However, the problem of designing these probability distributions to reflect biologically relevant constraints remains largely unaddressed, with the exception of reference [5].

In this SURF project, we seek to understand how to choose parameters (rate constants, e.g.) such that the equilibrium and transient distributions of molecular species display desirable properties with direct biological significance (e.g. the distribution of switching times for the toggle switch [6] should be short-tailed so that fast/slow switching is rare). We will perform design on the distributions that are analytic solutions to the CME from reference papers [3] and [4]. Students will pick the biocircuit examples of their choice from the references provided and/or other sources.

The design process will take place in steps:

- Students will model the biocircuit in MATLAB to gain intuition of stationary and transient behavior under different inputs. We will use the analytic solution for the transient and equilibrium distributions of the CME.
- Students will identify appropriate biologically relevant constraints for the design problem. Then they will translate them into design optimization problems. Some of the problems will have analytical solutions and some will require the use of computational methods (MATLAB toolboxes such as SOSTOOLS or sampling techniques such as MCMC).
- We will perform sensitivity analysis and robustness analysis for the solutions of the design problems.

*Required Skills*: Students are required to be flexible about learning new analytical and computational techniques. Experience with a scientific programming language of your choice is required (MATLAB/C/Python). Background in probability (ACM/EE/CMS 116) and optimization (ACM/CMS 113) is recommended, but not required.

**References**

[1] *Functional roles for noise in genetic circuits*, A. Eldar, M. Elowitz. Nature Reviews, 2010.

[2] *Stochastic Gene Expression in a Single Cell*, M. Elowitz, A. Levine, E. Siggia, P. Swain. Science, 2002.

[3] *Equilibrium Distributions of Simple Biochemical Reaction Systems for Time-Scale Separation in Stochastic Reaction Networks*, B. Mélykúti, J.P. Hespanha, M. Khammash. J. R. Soc. Interface, 2014.

[4] *Solving the Chemical Master Equation for Monomolecular Reaction Systems Analytically*, T. Jahnke, W. Huisinga. J. Math. Biol., 2007.

[5] *A Linear Programming Approach to Parameter Fitting for the Master Equation*, N. Martins, J. Gonçalves. IEEE, 2009.

[6] *Construction of a Genetic Toggle Switch in Escherichia Coli*, T. Gardner, C. Cantor, J. Collins. Letters to Nature, 2000.