Guided simulation of conditioned chemical reaction networks

TL;DR

This work develops a principled method to sample paths of chemical reaction networks conditioned on future linear observations by introducing guiding functions that replace the intractable Doob $h$-transform with a tractable change of measure. By constructing a tractable $g$ and a path-space likelihood ratio, the conditioned process is represented through ${\mathbb P}^g$, enabling weighted sampling of conditioned trajectories. The authors provide sufficient conditions for absolute continuity and, under a greedy-type condition, equivalence between the true conditioned law and the guided law, extending previous single-observation results to multiple partial observations. Several guiding strategies are explored, including CLE-based, LNA-based, scaled Brownian, and Poisson-guiding terms, with detailed simulation algorithms such as Next Reaction Thinning tailored for time-dependent rates. Numerical studies on death processes, gene transcription/translation, and enzyme kinetics demonstrate the practical utility and trade-offs of different guiding choices, highlighting how monotone components can be effectively handled via Poisson guidance. The framework offers a flexible, theoretically grounded toolkit for Bayesian inference and smoothing in CRNs, with potential to improve data augmentation and parameter estimation in systems biology and chemistry.

Abstract

Let $X$ be a chemical reaction process, modeled as a multi-dimensional continuous-time jump process. Assume that at given times $0< t_1 < \cdots <t_n$, linear combinations $v_i = L_i X(t_i),\, i=1,\dots ,n$ are observed for given matrices $L_i$. We show how the process that is conditioned on hitting the states $v_1,\dots, v_n$ is obtained by a change of measure on the law of the unconditioned process. This results in an algorithm for obtaining weighted samples from the conditioned process. Our results are illustrated by numerical simulations.

Figures (10)

• Figure 1: Realization of \ref{['eq:GTT-formofX']} using $\kappa_1 = 200$, $\kappa_2 = 10$, $d_M = 25$, $d_P = 1$ and initial position $x_0 = (1, 50, 10)$. Note that the gene count is constant in this process. Therefore it was omitted from the figure.
• Figure 2: Realization of the forward process of \ref{['ex:Enzyme-kinetics']} using $\kappa_1 = 5$, $\kappa_2 = 5$, $\kappa_3=3$ and initial position $x_0 = (12, 10, 10, 10)$.
• Figure 3: Estimates of the probability mass function of $X(T)\mid X(0)=x_0$ using the guiding functions $f$ from the LNA method, a scaled diffusion and a Poisson process. The upper barplot is the true density. For each $v$, we estimated using \ref{['eq:deathprocess-estimators']} with $N = 15000$.
• Figure 4: The percentage of paths ending in the point of conditioning ($v$) versus $v$ for the four methods considered (with the same settings as in \ref{['fig:deathprocess-density_estimates']}).
• Figure 5: Realization of a guided process starting from $x_0=(1,50,10)$ conditioned to be at $(1,10,50)$ at time $T=1$ of the GTT-model from \ref{['ex:GTT']} with $\kappa_1 = 100$, $\kappa_2 = 10$, $d_M = 25$ and $d_P=1$. The thick line is the original (unconditioned) process.

Theorems & Definitions (53)

• Example 2.2: Pure death process
• Example 2.3: Gene Transcription and Translation (GTT)
• Example 2.4: Enzyme kinetics
• Definition 3.1
• Definition 3.2
• Lemma 3.3
• proof
• Theorem 3.4
• proof
• Proposition 3.5