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A hybrid tau-leap for simulating chemical kinetics with applications to parameter estimation

Thomas Trigo Trindade, Konstantinos C. Zygalakis

TL;DR

The paper tackles the computational bottleneck of simulating stochastic chemical kinetics, especially for Bayesian parameter estimation where the likelihood is intractable. It introduces a Hybrid $τ$-leap that blends SSA and $τ$-leap on a per-reaction basis through Poisson-process splitting, enabling accurate CME simulations across multiscale regimes. The authors formalize region-based blending, develop a practical PP-MMH inference framework using a bootstrap particle filter, and demonstrate substantial speed-ups with preserved accuracy across diverse models (Lotka–Volterra, Birth–Death, Schlögl, Autoregressive). This approach lowers the computational cost of Bayesian inference in stochastic kinetics, making parameter estimation more scalable for complex biochemical networks.

Abstract

We consider the problem of efficiently simulating stochastic models of chemical kinetics. The Gillespie Stochastic Simulation algorithm (SSA) is often used to simulate these models, however, in many scenarios of interest, the computational cost quickly becomes prohibitive. This is further exasperated in the Bayesian inference context when estimating parameters of chemical models, as the intractability of the likelihood requires multiple simulations of the underlying system. To deal with issues of computational complexity in this paper, we propose a novel hybrid $τ$-leap algorithm for simulating well-mixed chemical systems. In particular, the algorithm uses $τ$-leap when appropriate (high population densities), and SSA when necessary (low population densities, when discrete effects become non-negligible). In the intermediate regime, a combination of the two methods, which leverages the properties of the underlying Poisson formulation, is employed. As illustrated through a number of numerical experiments the hybrid $τ$ offers significant computational savings when compared to SSA without however sacrificing the overall accuracy. This feature is particularly welcomed in the Bayesian inference context, as it allows for parameter estimation of stochastic chemical kinetics at reduced computational cost.

A hybrid tau-leap for simulating chemical kinetics with applications to parameter estimation

TL;DR

The paper tackles the computational bottleneck of simulating stochastic chemical kinetics, especially for Bayesian parameter estimation where the likelihood is intractable. It introduces a Hybrid -leap that blends SSA and -leap on a per-reaction basis through Poisson-process splitting, enabling accurate CME simulations across multiscale regimes. The authors formalize region-based blending, develop a practical PP-MMH inference framework using a bootstrap particle filter, and demonstrate substantial speed-ups with preserved accuracy across diverse models (Lotka–Volterra, Birth–Death, Schlögl, Autoregressive). This approach lowers the computational cost of Bayesian inference in stochastic kinetics, making parameter estimation more scalable for complex biochemical networks.

Abstract

We consider the problem of efficiently simulating stochastic models of chemical kinetics. The Gillespie Stochastic Simulation algorithm (SSA) is often used to simulate these models, however, in many scenarios of interest, the computational cost quickly becomes prohibitive. This is further exasperated in the Bayesian inference context when estimating parameters of chemical models, as the intractability of the likelihood requires multiple simulations of the underlying system. To deal with issues of computational complexity in this paper, we propose a novel hybrid -leap algorithm for simulating well-mixed chemical systems. In particular, the algorithm uses -leap when appropriate (high population densities), and SSA when necessary (low population densities, when discrete effects become non-negligible). In the intermediate regime, a combination of the two methods, which leverages the properties of the underlying Poisson formulation, is employed. As illustrated through a number of numerical experiments the hybrid offers significant computational savings when compared to SSA without however sacrificing the overall accuracy. This feature is particularly welcomed in the Bayesian inference context, as it allows for parameter estimation of stochastic chemical kinetics at reduced computational cost.
Paper Structure (18 sections, 26 equations, 10 figures, 6 tables, 5 algorithms)

This paper contains 18 sections, 26 equations, 10 figures, 6 tables, 5 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Algorithms for simulating chemical kinetics
  4. Parameter estimation for chemical kinetics
  5. Particle Pseudo-Marginal Metropolis–Hastings algorithm
  6. Hybrid $\tau$-leap
  7. Numerical Investigations
  8. Comparison with other numerical methods
  9. Lotka-Volterra System
  10. Birth-death system
  11. Schlögl System
  12. Autorepressive System
  13. Parameter estimation
  14. Experiment setting
  15. Methodology
  16. ...and 3 more sections

Figures (10)

  • Figure 1: (a)-(c): Illustration of $C^{\textcolor{black}{j}}_{\mathrm{SSA}}$, $C^{j}_{\mathrm{hybrid}}$ and $C^j_{\tau\mathrm{-leap}}$ associated with each of the reactions of the chemical system in Example \ref{['ex:simple']}; (d): Partition of state space as per applicable simulation regime
  • Figure 2: Histogram of $10^3$ SSA simulations up to time $T=5$ of the Lotka-Volterra system with $A(0) = 50 , B(0) = 60$.
  • Figure 3: Chemical System \ref{['chem-sys:lv']} (Lotka-Volterra). Simulators : SSA, $\tau$-leap (with reflective boundary conditions), CLE (also with reflective BC), Hybrid CLE, Hybrid $\tau$. All averages were performed over $10^4$ simulations.
  • Figure 4: Birth-death system. Statistical comparisons of extinction times obtained via SSA, $\tau$-leap, CLE, Hybrid $\tau$ and Hybrid CLE, taken over $10^4$ simulations.
  • Figure 5: Trajectories of the Schlögl chemical system
  • ...and 5 more figures

Theorems & Definitions (6)

  • Example 2.1
  • Remark 2.2
  • Example 3.1
  • Remark 3.2
  • Remark 3.3
  • Remark 3.4