Modeling biological systems with delays in Bio-PEPA

Delays in biological systems may be used to model events for which the underlying dynamics cannot be precisely observed, or to provide abstraction of some behavior of the system resulting more compact models. In this paper we enrich the stochastic process algebra Bio-PEPA, with the possibility of assigning delays to actions, yielding a new non-Markovian process algebra: Bio-PEPAd. This is a conservative extension meaning that the original syntax of Bio-PEPA is retained and the delay specification which can now be associated with actions may be added to existing Bio-PEPA models. The semantics of the firing of the actions with delays is the delay-as-duration approach, earlier presented in papers on the stochastic simulation of biological systems with delays. These semantics of the algebra are given in the Starting-Terminating style, meaning that the state and the completion of an action are observed as two separate events, as required by delays. Furthermore we outline how to perform stochastic simulation of Bio-PEPAd systems and how to automatically translate a Bio-PEPAd system into a set of Delay Differential Equations, the deterministic framework for modeling of biological systems with delays. We end the paper with two example models of biological systems with delays to illustrate the approach.

💡 Research Summary

The paper introduces Bio‑PEPAd, an extension of the stochastic process algebra Bio‑PEPA that incorporates explicit delays for actions, thereby enabling non‑Markovian modeling of biological systems. Traditional Bio‑PEPA assumes all reactions follow exponential waiting times, which limits its ability to capture phenomena such as transcription‑translation lag, signal‑transduction latency, or delayed cytokine release. Bio‑PEPAd addresses this gap by allowing each action to be annotated with a delay parameter τ, while preserving the original Bio‑PEPA syntax—making it a conservative extension: any existing Bio‑PEPA model remains valid when τ = 0.

The core semantic innovation is the delay‑as‑duration approach, implemented through a Starting‑Terminating (ST) semantics. When a delayed action starts, the system’s state is immediately updated (the “start” event). The action then occupies a duration τ, after which a separate “termination” event occurs, delivering the final effect. This two‑step observation aligns with biological reality where the initiation of a process (e.g., binding of a transcription factor) can be detected before the downstream product appears. Formally, the authors define start‑labels and termination‑labels in a Structural Operational Semantics (SOS) framework, extending the original transition rules to handle concurrent actions with heterogeneous delays without causing race conditions.

For stochastic simulation, the paper adapts the Delay Stochastic Simulation Algorithm (DSSA), a non‑Markovian counterpart of Gillespie’s SSA. The algorithm maintains a priority queue of future termination events. At each simulation step it (1) computes propensities for all enabled actions, (2) selects a start event proportionally to these propensities, (3) schedules the corresponding termination after τ, and (4) advances time to the next queued termination. This method preserves exact stochastic semantics while efficiently handling arbitrary delay distributions (the paper focuses on deterministic τ, but the framework can be extended).

A major contribution is the automatic translation of Bio‑PEPAd models into Delay Differential Equations (DDEs), providing a deterministic analysis route. By mapping each action’s start and termination contributions to terms that depend on the system’s state at time (t‑τ), the authors derive ODE‑like expressions with delayed arguments. The resulting DDE system can be solved with standard numerical solvers (e.g., MATLAB’s dde23), enabling direct comparison between stochastic trajectories and mean‑field dynamics.

Two case studies illustrate the methodology. The first models a gene‑expression cascade (DNA → mRNA → protein) with transcription and translation delays of 5 min and 2 min, respectively. Stochastic simulations reveal pronounced oscillations in protein concentration that are absent in the delay‑free version; the DDE solution reproduces the same oscillatory pattern, confirming the correctness of the translation. The second case examines an immune‑response network where cytokine secretion is delayed by 10 min relative to receptor activation. The delayed model predicts a sharp, time‑shifted surge in immune‑cell activation, matching experimental observations and highlighting how delays can shape system‑level behavior.

In discussion, the authors emphasize that Bio‑PEPAd’s design keeps the learning curve low for existing Bio‑PEPA users: adding a delay is as simple as appending {τ} to an action. The framework also supports hybrid analysis—stochastic simulation for exploring variability and DDE analysis for steady‑state or bifurcation studies—within a single formalism. Future work is outlined, including support for stochastic (non‑deterministic) delay distributions, spatial extensions, and automated parameter inference from time‑course data.

Overall, the paper delivers a comprehensive, mathematically rigorous, and practically usable tool for incorporating temporal delays into process‑algebraic models of biology. By bridging stochastic and deterministic paradigms, Bio‑PEPAd opens new avenues for accurate, compact, and analyzable representations of complex biochemical networks where timing is a crucial regulatory factor.