Stochastic Modeling of Naïve T Cell Homeostasis for Competing Clonotypes via the Master Equation

Abstract

Stochastic models for competing clonotypes of T cells by multivariate, continuous-time, discrete state, Markov processes have recently been proposed in the literature. A stochastic modeling 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 projections 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. Both experimental and theoretical investigations have emphasized the need to take into account effects due to aging. Thus time-dependent propensities naturally arise in immunological processes. 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.