Modeling delayed processes in biological systems

TL;DR

This paper compares explicit intermediate-step models with traditional delay differential equations in biological systems, revealing significant differences in dynamics and highlighting limitations of DDEs in predicting system behavior.

Contribution

It introduces explicit models with intermediate states for delayed biological processes and compares their dynamics to DDEs, exposing key differences and ambiguities.

Findings

Explicit models often differ in equilibrium distributions and transition times from DDEs.

Different explicit models can produce similar DDEs despite having different dynamics.

DDE predictions of oscillations may not hold in explicit models.

Abstract

Delayed processes are ubiquitous in biological systems and are often characterized by delay differential equations (DDEs) and their extension to include stochastic effects. DDEs do not explicitly incorporate intermediate states associated with a delayed process but instead use an estimated average delay time. In an effort to examine the validity of this approach, we study systems with significant delays by explicitly incorporating intermediate steps. We show by that such explicit models often yield significantly different equilibrium distributions and transition times as compared to DDEs with deterministic delay values. Additionally, different explicit models with qualitatively different dynamics can give rise to the same DDEs revealing important ambiguities. We also show that DDE-based predictions of oscillatory behavior may fail for the corresponding explicit model.