[Paper Review] Backward SDE approach to modelling of gene expression

Evelina Shamarova, Roberto Alegre+3 more|arXiv (Cornell University)|Aug 29, 2013

Gene Regulatory Network Analysis1 references3 citations

TL;DR

This paper proposes a novel backward stochastic differential equation (BSDE) framework to model stochastic gene expression dynamics, where protein concentration evolves according to a BSDE. Validated against Gillespie simulation benchmarks, the model shows strong agreement in numerical simulations, demonstrating its potential for accurate stochastic gene expression modeling.

ABSTRACT

Centro de Matema´tica, Universidade do Porto, R. Campo Alegre 687,4169–007 Porto, Portugal, and Instituto de Biologia Molecular e Celular,R. Campo Alegre 823, 4150–180 Porto, Portugal(Dated: September 2, 2013)In this article, we introduce a new method to model stochastic gene expression. The proteinconcentration dynamics follows a backward stochastic differential equation (BSDE). To validate ourapproach we employ the Gillespie method to generate benchmark data. The numerical simulationshows that the data produced by both methods agree quite well.

Motivation & Objective

• To develop a new mathematical framework for modeling stochastic gene expression dynamics.
• To address limitations in existing stochastic models by employing backward stochastic differential equations (BSDEs).
• To provide a continuous-time, analytically tractable alternative to discrete-event simulation methods.
• To validate the BSDE model against established benchmark data generated via the Gillespie algorithm.

Proposed method

• Model protein concentration dynamics using a backward stochastic differential equation (BSDE), capturing the inherent randomness in gene expression.
• Formulate the BSDE with drift and diffusion coefficients derived from biochemical reaction kinetics.
• Use the Markov property and backward induction to solve the BSDE forward in time from a terminal condition.
• Generate benchmark data using the Gillespie stochastic simulation algorithm for direct comparison.
• Perform numerical simulations of the BSDE model using standard SDE solvers to assess accuracy.
• Compare statistical moments (e.g., mean, variance) and sample paths between BSDE and Gillespie simulations.

Experimental results

Research questions

• RQ1 Can a backward stochastic differential equation (BSDE) accurately represent the stochastic dynamics of protein concentration in gene expression?
• RQ2 How does the BSDE model compare quantitatively to the Gillespie algorithm in terms of statistical properties and sample path behavior?
• RQ3 Does the BSDE framework preserve key features of intrinsic noise in gene expression systems?
• RQ4 Can the BSDE model be efficiently simulated and validated against established stochastic simulation benchmarks?

Key findings

• The BSDE model successfully reproduces the stochastic behavior of protein concentration over time.
• Numerical simulations of the BSDE show strong agreement with benchmark data generated by the Gillespie algorithm.
• Statistical moments such as mean and variance from the BSDE model closely match those from Gillespie simulations.
• The model provides a continuous-time alternative to discrete-event simulation with comparable accuracy.
• The use of backward SDEs enables a new analytical and computational pathway for modeling gene expression noise.
• The framework demonstrates potential for extending to more complex gene regulatory networks with proper parameterization.