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Stochastic modelling of T cell homeostasis for two competing clonotypes via the master equation

MacNamara, Shev & Burrage, Kevin (2011) Stochastic modelling of T cell homeostasis for two competing clonotypes via the master equation. In Molina-Paris, Carmen & Lythe, Grant (Eds.) Mathematical Models and Immune Cell Biology. Springer, New York, pp. 207-225.

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    Abstract

    Stochastic models for competing clonotypes of T cells by multivariate, continuous-time, discrete state, Markov processes have been proposed in the literature by Stirk, Molina-París and van den Berg (2008). A stochastic modelling framework is important because of rare events associated with small populations of some critical cell types. Usually, computational methods for these problems employ a trajectory-based approach, based on Monte Carlo simulation. This is partly because the complementary, probability density function (PDF) approaches can be expensive but here we describe some efficient PDF approaches by directly solving the governing equations, known as the Master Equation. These computations are made very efficient through an approximation of the state space by the Finite State Projection and through the use of Krylov subspace methods when evolving the matrix exponential. These computational methods allow us to explore the evolution of the PDFs associated with these stochastic models, and bimodal distributions arise in some parameter regimes. Time-dependent propensities naturally arise in immunological processes due to, for example, age-dependent effects. Incorporating time-dependent propensities into the framework of the Master Equation significantly complicates the corresponding computational methods but here we describe an efficient approach via Magnus formulas. Although this contribution focuses on the example of competing clonotypes, the general principles are relevant to multivariate Markov processes and provide fundamental techniques for computational immunology.

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    ID Code: 45866
    Item Type: Book Chapter
    Keywords: Markov processes, Monte Carlo simulation, probability density function (PDF)
    DOI: 10.1007/978-1-4419-7725-0_10
    ISBN: 9781441977250
    Subjects: Australian and New Zealand Standard Research Classification > MATHEMATICAL SCIENCES (010000) > OTHER MATHEMATICAL SCIENCES (019900)
    Divisions: Past > QUT Faculties & Divisions > Faculty of Science and Technology
    Past > Schools > Mathematical Sciences
    Copyright Owner: Copyright 2011 Springer
    Copyright Statement: The original publication is available at SpringerLink http://www.springerlink.com
    Deposited On: 12 Sep 2011 11:39
    Last Modified: 14 Sep 2011 03:06

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