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Adaptive Fast-Slow Operator Splitting for Multiscale Biochemical Stochastic Dynamics

Yuming Zeng, Wei Xie, Keqi Wang

Abstract

Stochastic reaction networks governed by Chemical Langevin Equations (CLE) exhibit pronounced multiscale dynamics spanning fast molecular reactions, intermediate transport, and slow cellular regulation, posing significant challenges for efficient and accurate simulation. Although operator splitting naturally decouples fast and slow subsystems, a rigorous error characterization for CLE splitting schemes has been lacking. We propose a modular operator-splitting framework with adaptive discretization that enables reliable and efficient simulation across fast-slow dynamics with explicit control of discretization error. Using stochastic logarithmic representations, we present a complete error analysis of the fast-slow Lie-Trotter splitting method, decomposing the one-step error into stochastic flow truncation error, commutator errors due to subsystem noncommutativity, and numerical discretization errors from fast and slow integrations. Guided by this analysis, we develop a proportional-integral (PI) adaptive controller that jointly selects macro time steps and fast microsteps, achieving substantial efficiency gains while maintaining accuracy.

Adaptive Fast-Slow Operator Splitting for Multiscale Biochemical Stochastic Dynamics

Abstract

Stochastic reaction networks governed by Chemical Langevin Equations (CLE) exhibit pronounced multiscale dynamics spanning fast molecular reactions, intermediate transport, and slow cellular regulation, posing significant challenges for efficient and accurate simulation. Although operator splitting naturally decouples fast and slow subsystems, a rigorous error characterization for CLE splitting schemes has been lacking. We propose a modular operator-splitting framework with adaptive discretization that enables reliable and efficient simulation across fast-slow dynamics with explicit control of discretization error. Using stochastic logarithmic representations, we present a complete error analysis of the fast-slow Lie-Trotter splitting method, decomposing the one-step error into stochastic flow truncation error, commutator errors due to subsystem noncommutativity, and numerical discretization errors from fast and slow integrations. Guided by this analysis, we develop a proportional-integral (PI) adaptive controller that jointly selects macro time steps and fast microsteps, achieving substantial efficiency gains while maintaining accuracy.

Paper Structure

This paper contains 9 sections, 5 theorems, 17 equations, 1 figure, 1 table.

Table of Contents

  1. INTRODUCTION
  2. FAST--SLOW STOCHASTIC REACTION NETWORK MODEL
  3. MEAN‑SQUARE ERROR ANALYSIS OF FAST–SLOW OPERATOR SPLITTING
  4. KUNITA TRUNCATION AND FAST–SLOW OPERATOR‑SPLITTING ERRORS FOR SDES
  5. FAST SUBSTEPPING AND SLOW DISCRETIZATION ERRORS
  6. TOTAL ONE‑STEP MEAN‑SQUARE ERROR ANALYSIS
  7. MSE--DRIVEN ADAPTIVE CONTROL FOR FAST--SLOW SPLITTING
  8. EMPIRICAL STUDY
  9. CONCLUSION

Key Result

Theorem 1

Let $S_{n+1} = \exp(Y_{\Delta t})(S_n)$ and $\widehat{S}_{n+1} = \exp(\widehat{Y}_{\Delta t})(S_n),$ where $Y_{\Delta t}$ is the exact Stratonovich logarithm of the stochastic flow over a time step $\Delta t$ starting from state $S_n=\pmb{s}$, and $\widehat{Y}_{\Delta t}$ denotes its first-order tru where $[X_i,X_j]$ denotes the Lie bracket of the vector fields. $\blacktriangleleft$$\blacktriangle

Figures (1)

  • Figure 1: Empirical marginal distributions for species $X_0$--$X_2$ (rows) under increasing stiffness parameter $\kappa_5$ (columns). In each panel, the solid black curve denotes the SSA reference density, dashed colored curves correspond to fixed-step splitting, Euler--Maruyama, and the adaptive method, and the gray shaded bars indicate the empirical histogram of SSA samples.

Theorems & Definitions (9)

  • Theorem 1: Kunita Truncation Error
  • proof : Proof sketch
  • Theorem 2: Fast--Slow Noncommutativity Error
  • proof
  • Theorem 3: Accumulated Fast-Substepping MSE with Propagation
  • proof
  • Theorem 4: Euler Discretization Error of Slow Flow
  • Theorem 5: One-step MSE Decomposition
  • proof : Proof sketch