Stochastic Processes, Multiscale Modeling, and Numerical Methods for Computational Cellular Biology

Theoretical results regarding two-dimensional ordinary-differential equations (ODEs) with second-degree polynomial right-hand sides are summarized, with an emphasis on limit cycles, limit cycle bifurcations, and multistability. The results are then used for construction of two reaction systems, which are at the deterministic level described by two-dimensional third-degree kinetic ODEs. The first system displays a homoclinic bifurcation, and a coexistence of a stable critical point and a stable limit cycle in the phase plane. The second system displays a multiple limit cycle bifurcation, and a coexistence of two stable limit cycles. The deterministic solutions (obtained by solving the kinetic ODEs) and stochastic solutions [noisy time-series generating by the Gillespie algorithm, and the underlying probability distributions obtained by solving the chemical master equation (CME)] of the constructed systems are compared, and the observed differences highlighted. The constructed systems are proposed as test problems for statistical methods, which are designed to detect and classify properties of given noisy time-series arising from biological applications.

In this article, we address the issues that come up in the design of importance sampling schemes for rare events associated to stochastic dynamical systems. We focus on the issue of metastability and on the effect of multiple scales. We discuss why seemingly reasonable schemes that follow large deviations optimal paths may perform poorly in practice, even though they are asymptotically optimal. Pre-asymptotic optimality is important when one deals with metastable dynamics and we discuss possible ways as to how to address this issue. Moreover, we discuss how the effect of the multiple scales (either in periodic or random environments) on the efficient design of importance sampling should be addressed. We discuss the mathematical and practical issues that come up, how to overcome some of the issues and discuss future challenges.

The most commonly employed spatial stochastic simulation methods for biochemical systems in molecular systems biology are reviewed from a multiscale perspective. Three levels of approximation are distinguished: macroscopic, mesoscopic, and microscopic levels. The relation between the levels of approximation is discussed for both reactions between molecules and transport of the molecules through a solvent. Computational methods are described for each level separately and for hybrid methods involving two levels. Free software implementing these methods in space and time is surveyed.

In this paper we discuss a framework recently introduced to construct and analyze accurate stochastic integrators for the computation of expectation of functionals of a stochastic process for both finite time or long-time dynamics. Such accurate integrators are of interest for many applications in biology, chemistry or physics and are also often needed in multiscale stochastic simulations. We describe how ideas originating from geometric numerical integration or structure preserving methods for deterministic differential equations can help to design new integrators for weak approximation of stochastic differential equations or for long-time simulation of ergodic stochastic systems.

We review conditions for the well-posedness of models of stochastic jump kinetics. Our focus is on obtaining bounds in the sense of mean square, implying in particular so-called strong convergence. We look especially on problems posed in a spatial setting, formed by merging a local reaction process with a connecting transport mechanism. This type of network jump process occurs naturally in many applications and is an attractive modeling framework, yet is a challenge from the perspective of numerical analysis. Since the stochastic modeling itself is motivated by the presence of nonlinear feedback terms, by small number of participating agents, and by an overall noisy environment, a consistent analysis framework is clearly required. The review summarizes the required mathematical framework and techniques used for obtaining a priori bounds and stability estimates.

The multiscale problem that a modeller in biology is presented with, trying to provide a systematic description of many agents, their properties, their internal dynamics and interactions, is daunting. On the other hand, biology provides a natural scale, with individual cells as agents. In agent-based computation, variables representing cell population sizes may be evaluated by counting cells of various types, but the governing dynamical rules are laid down one event at a time (J Theor Biol 231(3):357–376, 2004; CPT: Pharmacometrics Syst Pharmacol 4(11):615–629, 2015). Every cell is an individual, with its own set of attributes (state of activation, surface molecule profile, spatial location, for example). Populations of cells decrease or increase because individual cells die or divide. Here, by way of a tutorial on agent-based immune system modelling, we implement a model of the behaviour of the set of T cells in a body—numbering more than 10¹¹ in an adult human, and more than 10⁷ in an adult mouse (Ann Rev Immunol 28:275–294, 2010).

It is often the case that one or more species in a system are much more abundant than the rest of the species. For such cases solving or simulating the master equation of the whole system is unnecessary since only the less abundant species will exhibit significant fluctuations number of fluctuations. Here we present a simple technique by which one can obtain, from the master equation of the full system, a reduced master equation for the less abundant species only. The method is illustrated on various examples drawn from chemistry, molecular biology and ecology.

We use an example to present in exhaustive detail the algorithmic steps of the zero-information (ZI) closure scheme (Smadbeck and Kaznessis, Proc Natl Acad Sci USA 110:14261–14265, 2013). ZI-closure is a method for solving the chemical master equation (CME) of stochastic chemical reaction networks.

Becker and Dörimy introduced their equations in 1935 to describes nucleation in supersaturated vapors. Since then, these equations have been popularized to a wide range of applications and Slemrod in 2000 said they “provide perhaps the simplest kinetic model to describe [...] phase transitions”. In this survey we attempt to give an overview of the results obtained on these equations in the parts decades until today. Particularly we gathered results on both deterministic and stochastic versions of the Becker–Dörimy equations.

Coagulation-fragmentation processes with a finite number of particles is a recent class of mathematical questions that serves modeling some cell biology dynamics. The analysis of the models offers new challenging questions in probability and analysis: the model is the clustering of particles after binding, the formation of local subclusters of arbitrary sizes and the dissociation into subclusters. We review here modeling and analytical approaches to compute the size and number of clusters with a finite size. Applications are clustering of chromosome ends (telomeres) in yeast nucleus and the formation of viral capsid assembly from molecular components. The methods to compute the probability distribution functions of clusters and to estimate the statistical properties of clustering are based on combinatorics and hybrid Gillespie-spatial simulations. Finally, we review models of capsid formation, the mean-field approximation, and jump processes used to compute first passage times to a finite size cluster. These models become even more relevant for extracting parameters from live cell imaging data.

Heterogeneity and variability is ubiquitous in biology and physiology and one of the great modelling challenges is how we cope with and quantify this variability. There are a wide variety of approaches. We can attempt to ignore spatial effects and represent the heterogeneity through stochastic models that evolve only in time, or we can attempt to capture some key spatial components. Alternatively we can perform very detailed spatial simulations or we can attempt to use other approaches that mimic stochasticity in some way, such as by the use of delay models or by using populations of deterministic models. The skill is knowing when a particular model is appropriate to the questions that are being addressed.

In this review, we give a brief introduction to modelling and simulation in Computational Biology and discuss the various different sources of heterogeneity, pointing out useful modelling and analysis approaches. The starting point is how we deal with intrinsic noise; that is, the uncertainty of knowing when a chemical reaction takes place and what that reaction is. These discrete stochastic methods do not follow individual molecules over time; rather they track only total molecular numbers. This leads, in the first instance, to the Stochastic Simulation Algorithm that describes the time evolution of a discrete nonlinear Markov process. From there we consider approaches that are more efficient and effective but still preserve the discreteness of the simulation, the so-called tau-leaping algorithms. We then move to approximations that are continuous in time based around the Chemical Langevin stochastic differential equation. In these contexts we will focus, later in this chapter, on a particular application, namely the behaviour of ion channels dynamics.

In the second part of the review, we address the question of spatial heterogeneity. This involves consideration on the nature of diffusion in crowded spaces and, in particular, anomalous diffusion, a relevant topic (for example) in the analysis and simulation of cell membrane dynamics. We discuss different approaches for capturing this spatial heterogeneity through generalisations of the Stochastic Simulation Algorithm and that eventually leads us to the concept of fractional differential equations. Finally we consider the use of delays in capturing stochastic effects. For each case we attempt to give a discussion of applicable methods and an indication of their advantages and disadvantages.

An axon is a long thin projection of a neuron that allows for rapid electrochemical communications with other cells over long distances. Axonal transport refers to the stochastic, bidirectional movement of organelles and proteins along cytoskeletal polymers inside an axon, powered by molecular motor proteins. The movement from the cell body to the axon terminal is called anterograde transport and the movement in the opposite direction is called retrograde transport. Axonal transport is a vital process for the axon to survive and maintain its regular shape. Mathematical models have been developed to help understand how cargoes are transported inside an axon and how impairment of axonal transport affects cargo distribution. In this chapter, we review recent mathematical models of axonal transport and discuss open problems in this area.

Mitochondria are essential cellular organelles whose dysfunction is associated with ageing, cancer, mitochondrial diseases, and many other disorders. They contain their own genomes (mtDNA), of which thousands can be present in a single cell. These genomes are repeatedly replicated and degraded over time, and are prone to mutations. If the fraction of mutated genomes (heteroplasmy) exceeds a certain threshold, cellular defects can arise. The dynamics of mtDNAs over time and the accumulation of mutant genomes form a rich and vital stochastic process, the understanding of which provides important insights into disease progression. Numerous mathematical models have been constructed to provide a better understanding of how mitochondrial dysfunctions arise and, importantly, how clinical interventions can alleviate disease symptoms. For a given mean heteroplasmy, an increased variance—and thus a wider cell-to-cell heteroplasmy distribution—implies a higher probability of exceeding a given threshold value, meaning that stochastic models are essential to describe mtDNA disease. Mitochondria can undergo fusion and fission events with each other making the mitochondrial population a dynamic network that continuously changes its morphology, and allowing for the possibility of exchange of mtDNA molecules: coupled stochastic physical and genetic dynamics thus govern cellular mtDNA populations. Here, an overview is given of the kinds of stochastic mathematical models constructed describing mitochondria, their implications, and currently existing open problems.

Rod photoreceptors have the remarkable ability to respond to a single photon. A photon absorption triggers the activation of a receptor which is subsequently amplified by the activation of only 5–10 molecules. Because of such low numbers, the activation process has to be proceed in a coordinated manner in order to generate a reproducible signal. In addition, this signal has to overcome the background noise generated by spontaneous activations and deactivation of millions of enzymatic molecules. We review here recent modeling and stochastic analysis of the molecular events underlying the single photon response and the background noise. The homogenization procedure of the rod geometry is the first step for reducing the three into one dimension, so that numerical simulations become possible and reveal the fundamental relation between proteins concentrations, biochemical rate constant, and rod geometry. The stochastic modeling is used to analyze electrophysiological recordings and to extract in vivo biochemical constants. Modeling phototransduction has evolved at the far front of cell transduction and system biology and thus the approach presented here can be applied to many transduction mechanisms.

Over the last few decades, ecologists have come to appreciate that key ecological patterns, which describe ecological communities at relatively large spatial scales, are not only scale dependent, but also intimately intertwined. The relative abundance of species—which informs us about the commonness and rarity of species—changes its shape from small to large spatial scales. The average number of species as a function of area has a steep initial increase, followed by decreasing slopes at large scales. Finally, if we find a species in a given location, it is more likely we find an individual of the same species close-by, rather than farther apart. Such spatial turnover depends on the geographical distribution of species, which often are spatially aggregated. This reverberates on the abundances as well as the richness of species within a region, but so far it has been difficult to quantify such relationships.

Within a neutral framework—which considers all individuals competitively equivalent—we introduce a spatial stochastic model, which phenomenologically accounts for birth, death, immigration and local dispersal of individuals. We calculate the pair correlation function—which encapsulates spatial turnover—and the conditional probability to find a species with a certain population within a given circular area. Also, we calculate the macro-ecological patterns, which we have referred to above, and compare the analytical formulæ with the numerical integration of the model. Finally, we contrast the model predictions with the empirical data for two lowland tropical forest inventories, showing always a good agreement.