Realistic boundary conditions for stochastic simulations of reaction-diffusion processes

Many cellular and subcellular biological processes can be described in terms of diffusing and chemically reacting species (e.g. enzymes). Such reaction-diffusion processes can be mathematically modelled using either deterministic partial-differential equations or stochastic simulation algorithms. The latter provide a more detailed and precise picture, and several stochastic simulation algorithms have been proposed in recent years. Such models typically give the same description of the reaction-diffusion processes far from the boundary of the simulated domain, but the behaviour close to a reactive boundary (e.g. a membrane with receptors) is unfortunately model-dependent. In this paper, we study four different approaches to stochastic modelling of reaction-diffusion problems and show the correct choice of the boundary condition for each model. The reactive boundary is treated as partially reflective, which means that some molecules hitting the boundary are adsorbed (e.g. bound to the receptor) and some molecules are reflected. The probability that the molecule is adsorbed rather than reflected depends on the reactivity of the boundary (e.g. on the rate constant of the adsorbing chemical reaction and on the number of available receptors), and on the stochastic model used. This dependence is derived for each model.Comment: 24 pages, submitted to Physical Biolog

Stochastic modelling of reaction-diffusion processes: algorithms for bimolecular reactions

Several stochastic simulation algorithms (SSAs) have been recently proposed for modelling reaction-diffusion processes in cellular and molecular biology. In this paper, two commonly used SSAs are studied. The first SSA is an on-lattice model described by the reaction-diffusion master equation. The second SSA is an off-lattice model based on the simulation of Brownian motion of individual molecules and their reactive collisions. In both cases, it is shown that the commonly used implementation of bimolecular reactions (i.e. the reactions of the form A + B -> C, or A + A -> C) might lead to incorrect results. Improvements of both SSAs are suggested which overcome the difficulties highlighted. In particular, a formula is presented for the smallest possible compartment size (lattice spacing) which can be correctly implemented in the first model. This implementation uses a new formula for the rate of bimolecular reactions per compartment (lattice site).Comment: 33 pages, submitted to Physical Biolog

Multiscale Computations on Neural Networks: From the Individual Neuron Interactions to the Macroscopic-Level Analysis

We show how the Equation-Free approach for multi-scale computations can be exploited to systematically study the dynamics of neural interactions on a random regular connected graph under a pairwise representation perspective. Using an individual-based microscopic simulator as a black box coarse-grained timestepper and with the aid of simulated annealing we compute the coarse-grained equilibrium bifurcation diagram and analyze the stability of the stationary states sidestepping the necessity of obtaining explicit closures at the macroscopic level. We also exploit the scheme to perform a rare-events analysis by estimating an effective Fokker-Planck describing the evolving probability density function of the corresponding coarse-grained observables

Equation-Free Analysis of Two-Component System Signalling Model Reveals the Emergence of Co-Existing Phenotypes in the Absence of Multistationarity

Phenotypic differences of genetically identical cells under the same environmental conditions have been attributed to the inherent stochasticity of biochemical processes. Various mechanisms have been suggested, including the existence of alternative steady states in regulatory networks that are reached by means of stochastic fluctuations, long transient excursions from a stable state to an unstable excited state, and the switching on and off of a reaction network according to the availability of a constituent chemical species. Here we analyse a detailed stochastic kinetic model of two-component system signalling in bacteria, and show that alternative phenotypes emerge in the absence of these features. We perform a bifurcation analysis of deterministic reaction rate equations derived from the model, and find that they cannot reproduce the whole range of qualitative responses to external signals demonstrated by direct stochastic simulations. In particular, the mixed mode, where stochastic switching and a graded response are seen simultaneously, is absent. However, probabilistic and equation-free analyses of the stochastic model that calculate stationary states for the mean of an ensemble of stochastic trajectories reveal that slow transcription of either response regulator or histidine kinase leads to the coexistence of an approximate basal solution and a graded response that combine to produce the mixed mode, thus establishing its essential stochastic nature. The same techniques also show that stochasticity results in the observation of an all-or-none bistable response over a much wider range of external signals than would be expected on deterministic grounds. Thus we demonstrate the application of numerical equation-free methods to a detailed biochemical reaction network model, and show that it can provide new insight into the role of stochasticity in the emergence of phenotypic diversity

A Proposal for Integrated Efficacy-to-Effectiveness (E2E) Clinical Trials

We propose an “efficacy-to-effectiveness” (E2E) clinical trial design, in which an effectiveness trial would commence seamlessly upon completion of the efficacy trial. Efficacy trials use inclusion/exclusion criteria to produce relatively homogeneous samples of participants with the target condition, conducted in settings that foster adherence to rigorous clinical protocols. Effectiveness trials use inclusion/exclusion criteria that generate heterogeneous samples that are more similar to the general patient spectrum, conducted in more varied settings, with protocols that approximate typical clinical care. In E2E trials, results from the efficacy trial component would be used to design the effectiveness trial component, to confirm and/or discern associations between clinical characteristics and treatment effects in typical care, and potentially to test new hypotheses. An E2E approach may improve the evidentiary basis for selecting treatments, expand understanding of the effectiveness of treatments in subgroups with particular clinical features, and foster incorporation of effectiveness information into regulatory processes.National Center for Research Resources (U.S.) (Grant UL1 RR025752)National Center for Advancing Translational Sciences (U.S.) (Grant UL1 TR000073

Partial differential equations for self-organization in cellular and developmental biology

Understanding the mechanisms governing and regulating the emergence of structure and heterogeneity within cellular systems, such as the developing embryo, represents a multiscale challenge typifying current integrative biology research, namely, explaining the macroscale behaviour of a system from microscale dynamics. This review will focus upon modelling how cell-based dynamics orchestrate the emergence of higher level structure. After surveying representative biological examples and the models used to describe them, we will assess how developments at the scale of molecular biology have impacted on current theoretical frameworks, and the new modelling opportunities that are emerging as a result. We shall restrict our survey of mathematical approaches to partial differential equations and the tools required for their analysis. We will discuss the gap between the modelling abstraction and biological reality, the challenges this presents and highlight some open problems in the field

Overview of mathematical approaches used to model bacterial chemotaxis II: bacterial populations

We review the application of mathematical modeling to understanding the behavior of populations of chemotactic bacteria. The application of continuum mathematical models, in particular generalized Keller–Segel models, is discussed along with attempts to incorporate the microscale (individual) behavior on the macroscale, modeling the interaction between different species of bacteria, the interaction of bacteria with their environment, and methods used to obtain experimentally verified parameter values. We allude briefly to the role of modeling pattern formation in understanding collective behavior within bacterial populations. Various aspects of each model are discussed and areas for possible future research are postulated

Localized bioconvection of Euglena caused by phototaxis in the lateral direction

Euglena, a swimming micro-organism, exhibited a characteristic bioconvection that was localized at the center of a sealed chamber under bright illumination to induce negative phototaxis. This localized pattern consisted of high-density spots, in which convection was found. These observations were reproduced by a mathematical model that was based on the phototaxis of individual cells in both the vertical and lateral directions. Our results indicate that this convection is maintained by upward swimming, as with general bioconvection, and the localization originates from lateral phototaxis

Comparison of bacterial microbiota of the predatory mite Neoseiulus cucumeris (Acari: Phytoseiidae) and its factitious prey Tyrophagus putrescentiae (Acari: Acaridae)

Neoseiulus cucumeris is a predatory mite used for biological control of arthropod pests. Mass-reared predators are fed with factitious prey mites such as Tyrophagus putrescentiae. Although some information on certain endosymbionts of N. cucumeris and T. putrescentiae exists, it is unclear whether both species share bacterial communities. The bacterial communities in populations of predator and prey mites, as well as the occurence of potential acaropathogenic bacteria were analyzed. The comparisons were based on the following groups: (i) N. cucumeris mass-production; (ii) N. cucumeris laboratory population with disease symptoms; (iii) T. putrescentiae pure populations and; (iv) T. putrescentiae from rearing units of N. cucumeris. Only 15% of OTUs were present in all samples from predatory and prey mite populations (core OTUs): the intracellular symbionts Wolbachia, Cardinium, plus other Blattabacterium-like, Solitalea-like, and Bartonella-like symbionts. Environmental bacteria were more abundant in predatory mites, while symbiotic bacteria prevailed in prey mites. Relative numbers of certain bacterial taxa were significantly different between the microbiota of prey mites reared with and without N. cucumeris. No significant differences were found in the bacterial communities of healthy N. cucumeris compared to N. cucumeris showing disease symptoms. We did not identify any confirmed acaropathogenic bacteria among microbiota

The Role of Regulated mRNA Stability in Establishing Bicoid Morphogen Gradient in Drosophila Embryonic Development

The Bicoid morphogen is amongst the earliest triggers of differential spatial pattern of gene expression and subsequent cell fate determination in the embryonic development of Drosophila. This maternally deposited morphogen is thought to diffuse in the embryo, establishing a concentration gradient which is sensed by downstream genes. In most model based analyses of this process, the translation of the bicoid mRNA is thought to take place at a fixed rate from the anterior pole of the embryo and a supply of the resulting protein at a constant rate is assumed. Is this process of morphogen generation a passive one as assumed in the modelling literature so far, or would available data support an alternate hypothesis that the stability of the mRNA is regulated by active processes? We introduce a model in which the stability of the maternal mRNA is regulated by being held constant for a length of time, followed by rapid degradation. With this more realistic model of the source, we have analysed three computational models of spatial morphogen propagation along the anterior-posterior axis: (a) passive diffusion modelled as a deterministic differential equation, (b) diffusion enhanced by a cytoplasmic flow term; and (c) diffusion modelled by stochastic simulation of the corresponding chemical reactions. Parameter estimation on these models by matching to publicly available data on spatio-temporal Bicoid profiles suggests strong support for regulated stability over either a constant supply rate or one where the maternal mRNA is permitted to degrade in a passive manner