Switching and diffusion models for gene regulation networks

We analyze a hierarchy of three regimes for modeling gene regulation. The most complete model is a continuous time, discrete state space, Markov jump process. An intermediate 'switch plus diffusion' model takes the form of a stochastic differential equation driven by an independent continuous time Markov switch. In the third 'switch plus ODE' model the switch remains but the diffusion is removed. The latter two models allow for multi-scale simulation where, for the sake of computational efficiency, system components are treated differently according to their abundance. The 'switch plus ODE' regime was proposed by Paszek (Modeling stochasticity in gene regulation: characterization in the terms of the underlying distribution function, Bulletin of Mathematical Biology, 2007), who analyzed the steady state behavior, showing that the mean was preserved but the variance only approximated that of the full model. Here, we show that the tools of stochastic calculus can be used to analyze first and second moments for all time. A technical issue to be addressed is that the state space for the discrete-valued switch is infinite. We show that the new 'switch plus diffusion' regime preserves the biologically relevant measures of mean and variance, whereas the 'switch plus ODE' model uniformly underestimates the variance in the protein level. We also show that, for biologically relevant parameters, the transient behaviour can differ significantly from the steady state, justifying our time-dependent analysis. Extra computational results are also given for a protein dimerization model that is beyond the scope of the current analysis

Stochastic Representations of Ion Channel Kinetics and Exact Stochastic Simulation of Neuronal Dynamics

In this paper we provide two representations for stochastic ion channel kinetics, and compare the performance of exact simulation with a commonly used numerical approximation strategy. The first representation we present is a random time change representation, popularized by Thomas Kurtz, with the second being analogous to a "Gillespie" representation. Exact stochastic algorithms are provided for the different representations, which are preferable to either (a) fixed time step or (b) piecewise constant propensity algorithms, which still appear in the literature. As examples, we provide versions of the exact algorithms for the Morris-Lecar conductance based model, and detail the error induced, both in a weak and a strong sense, by the use of approximate algorithms on this model. We include ready-to-use implementations of the random time change algorithm in both XPP and Matlab. Finally, through the consideration of parametric sensitivity analysis, we show how the representations presented here are useful in the development of further computational methods. The general representations and simulation strategies provided here are known in other parts of the sciences, but less so in the present setting.Comment: 39 pages, 6 figures, appendix with XPP and Matlab cod

Jump-Diffusion Approximation of Stochastic Reaction Dynamics: Error bounds and Algorithms

Biochemical reactions can happen on different time scales and also the abundance of species in these reactions can be very different from each other. Classical approaches, such as deterministic or stochastic approach, fail to account for or to exploit this multi-scale nature, respectively. In this paper, we propose a jump-diffusion approximation for multi-scale Markov jump processes that couples the two modeling approaches. An error bound of the proposed approximation is derived and used to partition the reactions into fast and slow sets, where the fast set is simulated by a stochastic differential equation and the slow set is modeled by a discrete chain. The error bound leads to a very efficient dynamic partitioning algorithm which has been implemented for several multi-scale reaction systems. The gain in computational efficiency is illustrated by a realistically sized model of a signal transduction cascade coupled to a gene expression dynamics.Comment: 32 pages, 7 figure

Validity of the Cauchy-Born rule applied to discrete cellular-scale models of biological tissues

The development of new models of biological tissues that consider cells in a discrete manner is becoming increasingly popular as an alternative to PDE-based continuum methods, although formal relationships between the discrete and continuum frameworks remain to be established. For crystal mechanics, the discrete-to-continuum bridge is often made by assuming that local atom displacements can be mapped homogeneously from the mesoscale deformation gradient, an assumption known as the Cauchy-Born rule (CBR). Although the CBR does not hold exactly for non-crystalline materials, it may still be used as a first order approximation for analytic calculations of effective stresses or strain energies. In this work, our goal is to investigate numerically the applicability of the CBR to 2-D cellular-scale models by assessing the mechanical behaviour of model biological tissues, including crystalline (honeycomb) and non-crystalline reference states. The numerical procedure consists in precribing an affine deformation on the boundary cells and computing the position of internal cells. The position of internal cells is then compared with the prediction of the CBR and an average deviation is calculated in the strain domain. For centre-based models, we show that the CBR holds exactly when the deformation gradient is relatively small and the reference stress-free configuration is defined by a honeycomb lattice. We show further that the CBR may be used approximately when the reference state is perturbed from the honeycomb configuration. By contrast, for vertex-based models, a similar analysis reveals that the CBR does not provide a good representation of the tissue mechanics, even when the reference configuration is defined by a honeycomb lattice. The paper concludes with a discussion of the implications of these results for concurrent discrete/continuous modelling, adaptation of atom-to-continuum (AtC) techniques to biological tissues and model classification

Validity of the Cauchy-Born rule applied to discrete cellular-scale models of biological tissues.

The development of new models of biological tissues that consider cells in a discrete manner is becoming increasingly popular as an alternative to continuum methods based on partial differential equations, although formal relationships between the discrete and continuum frameworks remain to be established. For crystal mechanics, the discrete-to-continuum bridge is often made by assuming that local atom displacements can be mapped homogeneously from the mesoscale deformation gradient, an assumption known as the Cauchy-Born rule (CBR). Although the CBR does not hold exactly for noncrystalline materials, it may still be used as a first-order approximation for analytic calculations of effective stresses or strain energies. In this work, our goal is to investigate numerically the applicability of the CBR to two-dimensional cellular-scale models by assessing the mechanical behavior of model biological tissues, including crystalline (honeycomb) and noncrystalline reference states. The numerical procedure involves applying an affine deformation to the boundary cells and computing the quasistatic position of internal cells. The position of internal cells is then compared with the prediction of the CBR and an average deviation is calculated in the strain domain. For center-based cell models, we show that the CBR holds exactly when the deformation gradient is relatively small and the reference stress-free configuration is defined by a honeycomb lattice. We show further that the CBR may be used approximately when the reference state is perturbed from the honeycomb configuration. By contrast, for vertex-based cell models, a similar analysis reveals that the CBR does not provide a good representation of the tissue mechanics, even when the reference configuration is defined by a honeycomb lattice. The paper concludes with a discussion of the implications of these results for concurrent discrete and continuous modeling, adaptation of atom-to-continuum techniques to biological tissues, and model classification