Higher-order numerical methods for stochastic simulation of\ud chemical reaction systems

In this paper, using the framework of extrapolation, we present an approach for obtaining higher-order -leap methods for the Monte Carlo simulation of stochastic chemical kinetics. Specifically, Richardson extrapolation is applied to the expectations of functionals obtained by a fixed-step -leap algorithm. We prove that this procedure gives rise to second-order approximations for the first two moments obtained by the chemical master equation for zeroth- and first-order chemical systems. Numerical simulations verify that this is also the case for higher-order chemical systems of biological importance. This approach, as in the case of ordinary and stochastic differential equations, can be repeated to obtain even higher-order approximations. We illustrate the results of a second extrapolation on two systems. The biggest barrier for observing higher-order convergence is the Monte Carlo error; we discuss different strategies for reducing it

Analysis of a stochastic chemical system close to a sniper bifurcation of its mean field model

A framework for the analysis of stochastic models of chemical systems for which the deterministic mean-field description is undergoing a saddle-node infinite period (SNIPER) bifurcation is presented. Such a bifurcation occurs for example in the modelling of cell-cycle regulation. It is shown that the stochastic system possesses oscillatory solutions even for parameter values for which the mean-field model does not oscillate. The dependence of the mean period of these oscillations on the parameters of the model (kinetic rate constants) and the size of the system (number of molecules present) is studied. Our approach is based on the chemical Fokker Planck equation. To get some insights into advantages and disadvantages of the method, a simple one-dimensional chemical switch is first analyzed, before the chemical SNIPER problem is studied in detail. First, results obtained by solving the Fokker-Planck equation numerically are presented. Then an asymptotic analysis of the Fokker-Planck equation is used to derive explicit formulae for the period of oscillation as a function of the rate constants and as a function of the system size

Mathematical description of bacterial traveling pulses

The Keller-Segel system has been widely proposed as a model for bacterial waves driven by chemotactic processes. Current experiments on {\em E. coli} have shown precise structure of traveling pulses. We present here an alternative mathematical description of traveling pulses at a macroscopic scale. This modeling task is complemented with numerical simulations in accordance with the experimental observations. Our model is derived from an accurate kinetic description of the mesoscopic run-and-tumble process performed by bacteria. This model can account for recent experimental observations with {\em E. coli}. Qualitative agreements include the asymmetry of the pulse and transition in the collective behaviour (clustered motion versus dispersion). In addition we can capture quantitatively the main characteristics of the pulse such as the speed and the relative size of tails. This work opens several experimental and theoretical perspectives. Coefficients at the macroscopic level are derived from considerations at the cellular scale. For instance the stiffness of the signal integration process turns out to have a strong effect on collective motion. Furthermore the bottom-up scaling allows to perform preliminary mathematical analysis and write efficient numerical schemes. This model is intended as a predictive tool for the investigation of bacterial collective motion

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

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

Singular Cucker-Smale Dynamics

The existing state of the art for singular models of flocking is overviewed, starting from microscopic model of Cucker and Smale with singular communication weight, through its mesoscopic mean-filed limit, up to the corresponding macroscopic regime. For the microscopic Cucker-Smale (CS) model, the collision-avoidance phenomenon is discussed, also in the presence of bonding forces and the decentralized control. For the kinetic mean-field model, the existence of global-in-time measure-valued solutions, with a special emphasis on a weak atomic uniqueness of solutions is sketched. Ultimately, for the macroscopic singular model, the summary of the existence results for the Euler-type alignment system is provided, including existence of strong solutions on one-dimensional torus, and the extension of this result to higher dimensions upon restriction on the smallness of initial data. Additionally, the pressureless Navier-Stokes-type system corresponding to particular choice of alignment kernel is presented, and compared - analytically and numerically - to the porous medium equation

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

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

The Ciliate Paramecium Shows Higher Motility in Non-Uniform Chemical Landscapes

We study the motility behavior of the unicellular protozoan Paramecium tetraurelia in a microfluidic device that can be prepared with a landscape of attracting or repelling chemicals. We investigate the spatial distribution of the positions of the individuals at different time points with methods from spatial statistics and Poisson random point fields. This makes quantitative the informal notion of “uniform distribution” (or lack thereof). Our device is characterized by the absence of large systematic biases due to gravitation and fluid flow. It has the potential to be applied to the study of other aquatic chemosensitive organisms as well. This may result in better diagnostic devices for environmental pollutants.University of Wisconsin--Milwaukee (SURF (Salary for Undergraduate Research Fellows) Award)National Science Foundation (U.S.) (grant DMS-016214

From Microscopic to Macroscopic Descriptions of Cell Migration on Growing Domains

Cell migration and growth are essential components of the development of multicellular organisms. The role of various cues in directing cell migration is widespread, in particular, the role of signals in the environment in the control of cell motility and directional guidance. In many cases, especially in developmental biology, growth of the domain also plays a large role in the distribution of cells and, in some cases, cell or signal distribution may actually drive domain growth. There is an almost ubiquitous use of partial differential equations (PDEs) for modelling the time evolution of cellular density and environmental cues. In the last 20 years, a lot of attention has been devoted to connecting macroscopic PDEs with more detailed microscopic models of cellular motility, including models of directional sensing and signal transduction pathways. However, domain growth is largely omitted in the literature. In this paper, individual-based models describing cell movement and domain growth are studied, and correspondence with a macroscopic-level PDE describing the evolution of cell density is demonstrated. The individual-based models are formulated in terms of random walkers on a lattice. Domain growth provides an extra mathematical challenge by making the lattice size variable over time. A reaction–diffusion master equation formalism is generalised to the case of growing lattices and used in the derivation of the macroscopic PDEs