On the Interpretation of Delays in Delay Stochastic Simulation of Biological Systems

TL;DR

This paper examines the limitations of the delay as duration approach in stochastic simulation of biological systems with delays and proposes a new algorithm based on a purely delayed interpretation, improving accuracy.

Contribution

It introduces a novel stochastic simulation algorithm that interprets delays as purely delayed events, addressing shortcomings of existing methods.

Findings

Delay as duration approach can be inaccurate for certain biological interactions.

The proposed purely delayed interpretation yields more precise simulation results.

Application to tumor growth model demonstrates improved simulation accuracy.

Abstract

Delays in biological systems may be used to model events for which the underlying dynamics cannot be precisely observed. Mathematical modeling of biological systems with delays is usually based on Delay Differential Equations (DDEs), a kind of differential equations in which the derivative of the unknown function at a certain time is given in terms of the values of the function at previous times. In the literature, delay stochastic simulation algorithms have been proposed. These algorithms follow a "delay as duration" approach, namely they are based on an interpretation of a delay as the elapsing time between the start and the termination of a chemical reaction. This interpretation is not suitable for some classes of biological systems in which species involved in a delayed interaction can be involved at the same time in other interactions. We show on a DDE model of tumor growth that the delay as duration approach for stochastic simulation is not precise, and we propose a simulation algorithm based on a ``purely delayed'' interpretation of delays which provides better results on the considered model.