Stochastic simulation of chemical reactions in spatially complex media

Discrete stochastic simulations, via techniques such as the Stochastic Simulation Algorithm (SSA) are a powerful tool for understanding the dynamics of chemical kinetics when there are low numbers of certain molecular species. However, an important constraint is the assumption of well-mixedness and homogeneity. In this paper, we show how to use Monte Carlo simulations to estimate an anomalous diffusion parameter that encapsulates the crowdedness of the spatial environment. We then use this parameter to replace the rate constants of bimolecular reactions by a time-dependent power law to produce an SSA valid in cases where anomalous diffusion occurs or the system is not well-mixed (ASSA). Simulations then show that ASSA can successfully predict the temporal dynamics of chemical kinetics in a spatially constrained environment.

Oscillatory regulation of Hes1: Discrete stochastic delay modelling and simulation

Discrete stochastic simulations are a powerful tool for understanding the dynamics of chemical kinetics when there are small-to-moderate numbers of certain molecular species. In this paper we introduce delays into the stochastic simulation algorithm, thus mimicking delays associated with transcription and translation. We then show that this process may well explain more faithfully than continuous deterministic models the observed sustained oscillations in expression levels of hes1 mRNA and Hes1 protein.

High-probability regret bounds for bandit online linear optimization

We present a modification of the algorithm of Dani et al. [8] for the online linear optimization problem in the bandit setting, which with high probability has regret at most O ∗ ( √ T) against an adaptive adversary. This improves on the previous algorithm [8] whose regret is bounded in expectation against an oblivious adversary. We obtain the same dependence on the dimension (n 3/2) as that exhibited by Dani et al. The results of this paper rest firmly on those of [8] and the remarkable technique of Auer et al. [2] for obtaining high probability bounds via optimistic estimates. This paper answers an open question: it eliminates the gap between the high-probability bounds obtained in the full-information vs bandit settings.

Stochastic models and simulation of ion channel dynamics

The behaviour of ion channels within cardiac and neuronal cells is intrinsically stochastic in nature. When the number of channels is small this stochastic noise is large and can have an impact on the dynamics of the system which is potentially an issue when modelling small neurons and drug block in cardiac cells. While exact methods correctly capture the stochastic dynamics of a system they are computationally expensive, restricting their inclusion into tissue level models and so approximations to exact methods are often used instead. The other issue in modelling ion channel dynamics is that the transition rates are voltage dependent, adding a level of complexity as the channel dynamics are coupled to the membrane potential. By assuming that such transition rates are constant over each time step, it is possible to derive a stochastic differential equation (SDE), in the same manner as for biochemical reaction networks, that describes the stochastic dynamics of ion channels. While such a model is more computationally efficient than exact methods we show that there are analytical problems with the resulting SDE as well as issues in using current numerical schemes to solve such an equation. We therefore make two contributions: develop a different model to describe the stochastic ion channel dynamics that analytically behaves in the correct manner and also discuss numerical methods that preserve the analytical properties of the model.

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

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.

Stochastic simulation for spatial modelling of dynamic process in a living cell

One of the fundamental motivations underlying computational cell biology is to gain insight into the complicated dynamical processes taking place, for example, on the plasma membrane or in the cytosol of a cell. These processes are often so complicated that purely temporal mathematical models cannot adequately capture the complex chemical kinetics and transport processes of, for example, proteins or vesicles. On the other hand, spatial models such as Monte Carlo approaches can have very large computational overheads. This chapter gives an overview of the state of the art in the development of stochastic simulation techniques for the spatial modelling of dynamic processes in a living cell.

Linear models for endocytic transformations from live cell imaging

Endocytosis is the process by which cells internalise molecules including nutrient proteins from the extracellular media. In one form, macropinocytosis, the membrane at the cell surface ruffles and folds over to give rise to an internalised vesicle. Negatively charged phospholipids within the membrane called phosphoinositides then undergo a series of transformations that are critical for the correct trafficking of the vesicle within the cell, and which are often pirated by pathogens such as Salmonella. Advanced fluorescent video microscopy imaging now allows the detailed observation and quantification of these events in live cells over time. Here we use these observations as a basis for building differential equation models of the transformations. An initial investigation of these interactions was modelled with reaction rates proportional to the sum of the concentrations of the individual constituents. A first order linear system for the concentrations results. The structure of the system enables analytical expressions to be obtained and the problem becomes one of determining the reaction rates which generate the observed data plots. We present results with reaction rates which capture the general behaviour of the reactions so that we now have a complete mathematical model of phosphoinositide transformations that fits the experimental observations. Some excellent fits are obtained with modulated exponential functions; however, these are not solutions of the linear system. The question arises as to how the model may be modified to obtain a system whose solution provides a more accurate fit.

Stochastic regularization method for backward Cauchy problem in Banach spaces

We consider a stochastic regularization method for solving the backward Cauchy problem in Banach spaces. An order of convergence is obtained on sourcewise representative elements.