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Open Access Research article

Probability distributed time delays: integrating spatial effects into temporal models

Tatiana T Marquez-Lago12*, André Leier12* and Kevin Burrage34

Author affiliations

1 Department of Biosystems Science and Engineering, Swiss Federal Institute of Technology (ETH) Zurich, Mattenstrasse 26, CH-4058 Basel, Switzerland

2 Swiss Institute of Bioinformatics, ETH Zurich, CH-8092 Zurich, Switzerland

3 COMLAB, University of Oxford, OX1 3QD, UK

4 Institute for Molecular Biology, The University of Queensland, Brisbane, QLD 4072, Australia

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Citation and License

BMC Systems Biology 2010, 4:19  doi:10.1186/1752-0509-4-19

Published: 4 March 2010



In order to provide insights into the complex biochemical processes inside a cell, modelling approaches must find a balance between achieving an adequate representation of the physical phenomena and keeping the associated computational cost within reasonable limits. This issue is particularly stressed when spatial inhomogeneities have a significant effect on system's behaviour. In such cases, a spatially-resolved stochastic method can better portray the biological reality, but the corresponding computer simulations can in turn be prohibitively expensive.


We present a method that incorporates spatial information by means of tailored, probability distributed time-delays. These distributions can be directly obtained by single in silico or a suitable set of in vitro experiments and are subsequently fed into a delay stochastic simulation algorithm (DSSA), achieving a good compromise between computational costs and a much more accurate representation of spatial processes such as molecular diffusion and translocation between cell compartments. Additionally, we present a novel alternative approach based on delay differential equations (DDE) that can be used in scenarios of high molecular concentrations and low noise propagation.


Our proposed methodologies accurately capture and incorporate certain spatial processes into temporal stochastic and deterministic simulations, increasing their accuracy at low computational costs. This is of particular importance given that time spans of cellular processes are generally larger (possibly by several orders of magnitude) than those achievable by current spatially-resolved stochastic simulators. Hence, our methodology allows users to explore cellular scenarios under the effects of diffusion and stochasticity in time spans that were, until now, simply unfeasible. Our methodologies are supported by theoretical considerations on the different modelling regimes, i.e. spatial vs. delay-temporal, as indicated by the corresponding Master Equations and presented elsewhere.