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Fixed and Distributed Gene Expression Time Delays in Reaction-Diffusion Systems

Sargood, Alec and Gaffney, Eamonn A. and Krause, Andrew L. (2022) 'Fixed and Distributed Gene Expression Time Delays in Reaction-Diffusion Systems.', Bulletin of Mathematical Biology, 84 (9). p. 98.


Time delays, modelling the process of intracellular gene expression, have been shown to have important impacts on the dynamics of pattern formation in reaction-diffusion systems. In particular, past work has shown that such time delays can shrink the Turing space, thereby inhibiting patterns from forming across large ranges of parameters. Such delays can also increase the time taken for pattern formation even when Turing instabilities occur. Here we consider reaction-diffusion models incorporating fixed or distributed time delays, modelling the underlying stochastic nature of gene expression dynamics, and analyze these through a systematic linear instability analysis and numerical simulations for several sets of different reaction kinetics. We find that even complicated distribution kernels (skewed Gaussian probability density functions) have little impact on the reaction-diffusion dynamics compared to fixed delays with the same mean delay. We show that the location of the delay terms in the model can lead to changes in the size of the Turing space (increasing or decreasing) as the mean time delay, t, is increased. We show that the time to pattern formation from a perturbation of the homogeneous steady state scales linearly with t, and conjecture that this is a general impact of time delay on reactiondiffusion dynamics, independent of the form of the kinetics or location of the delayed terms. Finally we show that while initial and boundary conditions can influence these dynamics, particularly the timeto- pattern, the effects of delay appear robust under variations of initial and boundary data. Overall our results help clarify the role of gene expression time delays in reaction-diffusion patterning, and suggest clear directions for further work in studying more realistic models of pattern formation.

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Publisher statement:This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit
Date accepted:30 June 2022
Date deposited:01 August 2022
Date of first online publication:07 August 2022
Date first made open access:17 August 2022

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