[Paper Review] On Robustness Analysis of Stochastic Biochemical Systems by Probabilistic Model Checking

Luboš Brim, Milan Češka+2 more|arXiv (Cornell University)|Oct 17, 2013

Gene Regulatory Network Analysis49 references3 citations

TL;DR

This paper proposes a probabilistic model checking framework for rigorous robustness analysis of stochastic biochemical systems modeled as continuous-time Markov chains (CTMCs). By adapting Kitano's robustness definition to stochastic systems and using bounded CSL with rewards, the method quantifies how parameter uncertainty and intrinsic noise affect system functionality, revealing that stochasticity significantly alters predictions compared to deterministic models, especially in gene regulation and two-component signaling pathways.

ABSTRACT

This report proposes a novel framework for a rigorous robustness analysis of stochastic biochemical systems. The technique is based on probabilistic model checking. We adapt the general definition of robustness introduced by Kitano to the class of stochastic systems modelled as continuous time Markov Chains in order to extensively analyse and compare robustness of biological models with uncertain parameters. The framework utilises novel computational methods that enable to effectively evaluate the robustness of models with respect to quantitative temporal properties and parameters such as reaction rate constants and initial conditions. The framework is applied to gene regulation as an example of a central biological mechanism where intrinsic and extrinsic stochasticity plays crucial role due to low numbers of DNA and RNA molecules. Using our methods we have obtained a comprehensive and precise analysis of stochastic dynamics under parameter uncertainty. Furthermore, we apply our framework to compare several variants of two-component signalling networks from the perspective of robustness with respect to intrinsic noise caused by low populations of signalling components. We succeeded to extend previous studies performed on deterministic models (ODE) and show that stochasticity may significantly affect obtained predictions. Our case studies demonstrate that the framework can provide deeper insight into the role of key parameters in maintaining the system functionality and thus it significantly contributes to formal methods in computational systems biology.

Motivation & Objective

• To address the lack of rigorous methods for robustness analysis in stochastic biochemical systems with uncertain parameters.
• To extend Kitano's general robustness definition to continuous-time Markov chains (CTMCs) modeling stochastic biochemical kinetics.
• To enable quantitative analysis of system functionality under perturbations in rate constants and initial conditions using formal verification techniques.
• To compare robustness of different two-component signaling network topologies under intrinsic noise due to low molecule counts.
• To provide a formal, precise, and scalable method for assessing how likely a system is to maintain functionality under parameter uncertainty.

Proposed method

• Adapts Kitano's robustness integral to stochastic systems by modeling perturbations as changes in kinetic parameters or initial conditions.
• Uses Continuous Stochastic Logic (CSL) with bounded time and reward extensions to formally express temporal and quantitative properties of interest.
• Employs post-processing functions over probability density vectors to evaluate system functionality beyond standard CSL properties.
• Applies computational techniques such as adaptive uniformisation and piecewise linear approximation to improve efficiency and accuracy of robustness evaluation.
• Performs parallel analysis of multiple perturbation subsets on high-performance hardware to scale to complex models.
• Uses the chemical master equation (CME) to define the semantics of CTMCs governing stochastic system dynamics.

Experimental results

Research questions

• RQ1 How can Kitano's general robustness definition be formally adapted to stochastic biochemical systems modeled as CTMCs?
• RQ2 To what extent does intrinsic stochasticity affect the robustness of gene regulatory networks under parameter uncertainty?
• RQ3 How do different two-component signaling pathway topologies compare in robustness to intrinsic noise and varying input signals?
• RQ4 Can probabilistic model checking provide more precise and comprehensive insights into system robustness than deterministic ODE-based approaches?
• RQ5 What role do key parameters—such as degradation rates—play in maintaining functional states under uncertainty?

Key findings

• The framework enables exact, global robustness analysis of bistable gene regulatory systems, showing that the cancer-inducing high state of the retinoblastoma-binding transcription factor is almost always reachable regardless of initial conditions.
• Robustness to avoid malfunction is positively influenced by increasing the degradation rate of the retinoblastoma-binding transcription factor.
• For low input signals, the synthetic two-component pathway topology exhibits lower output variance and is therefore more robust than the basic topology.
• For high input signals, the output variance of the synthetic pathway increases rapidly, making the basic topology more suitable for strong signals.
• Both pathway topologies are robust to scaling of noise in signaling component dynamics, but their relative performance depends on input signal level.
• The study reveals that stochasticity can significantly alter predictions compared to deterministic models, highlighting limitations of ODE-based robustness analysis.