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

Formation of convective cells in the scrape-off layer of the CASTOR tokamak

Understanding of the scrape-off layer (SOL) physics in tokamaks requires diagnostics with sufficient temporal and spatial resolution. This contribution describes results of experiments performed in the SOL of the CASTOR tokamak (R=40 cm, a = 6 cm) by means of a ring of 124 Langmuir probes surrounding the whole poloidal cross section. The individual probes measure either the ion saturation current of the floating potential with the spatial resolution up to 3 mm. Experiments are performed in a particular magnetic configuration, characterized by a long parallel connection length in the SOL, L_par ~q2piR. We report on measurements in discharges, where the edge electric field is modified by inserting a biased electrode into the edge plasma. In particular, a complex picture is observed, if the biased electrode is located inside the SOL. The poloidal distribution of the floating potential appears to be strongly non-uniform at biasing. The peaks of potential are observed at particular poloidal angles. This is interpreted as formation of a biased flux tube, which emanates from the electrode along the magnetic field lines and snakes q times around the torus. The resulting electric field in the SOL is 2-dimensional, having the radial as well as the poloidal component. It is demonstrated that the poloidal electric field E_pol convects the edge plasma radially due to the E_pol x B_T drift either inward or outward depending on its sign. The convective particle flux is by two orders of magnitude larger than the fluctuation-induced one and consequently dominates.Comment: 12th International Congress on Plasma Physics, 25-29 October 2004, Nice (France

Analysis of neural crest-derived clones reveals novel aspects of facial development

Cranial neural crest cells populate the future facial region and produce ectomesenchyme-derived tissues, such as cartilage, bone, dermis, smooth muscle, adipocytes, and many others. However, the contribution of individual neural crest cells to certain facial locations and the general spatial clonal organization of the ectomesenchyme have not been determined. We investigated how neural crest cells give rise to clonally organized ectomesenchyme and how this early ectomesenchyme behaves during the developmental processes that shape the face. Using a combination of mouse and zebrafish models, we analyzed individual migration, cell crowd movement, oriented cell division, clonal spatial overlapping, and multilineage differentiation. The early face appears to be built from multiple spatially defined overlapping ectomesenchymal clones. During early face development, these clones remain oligopotent and generate various tissues in a given location. By combining clonal analysis, computer simulations, mouse mutants, and live imaging, we show that facial shaping results from an array of local cellular activities in the ectomesenchyme. These activities mostly involve oriented divisions and crowd movements of cells during morphogenetic events. Cellular behavior that can be recognized as individual cell migration is very limited and short-ranged and likely results from cellular mixing due to the proliferation activity of the tissue. These cellular mechanisms resemble the strategy behind limb bud morphogenesis, suggesting the possibility of common principles and deep homology between facial and limb outgrowth

Altered developmental programs and oriented cell divisions lead to bulky bones during salamander limb regeneration

There are major differences in duration and scale at which limb development and regeneration proceed, raising the question to what extent regeneration is a recapitulation of development. We address this by analyzing skeletal elements using a combination of micro-CT imaging, molecular profiling and clonal cell tracing. We find that, in contrast to development, regenerative skeletal growth is accomplished based entirely on cartilage expansion prior to ossification, not limiting the transversal cartilage expansion and resulting in bulkier skeletal parts. The oriented extension of salamander cartilage and bone appear similar to the development of basicranial synchondroses in mammals, as we found no evidence for cartilage stem cell niches or growth plate-like structures during neither development nor regeneration. Both regenerative and developmental ossification in salamanders start from the cortical bone and proceeds inwards, showing the diversity of schemes for the synchrony of cortical and endochondral ossification among vertebrates

Markovian Dynamics on Complex Reaction Networks

Complex networks, comprised of individual elements that interact with each other through reaction channels, are ubiquitous across many scientific and engineering disciplines. Examples include biochemical, pharmacokinetic, epidemiological, ecological, social, neural, and multi-agent networks. A common approach to modeling such networks is by a master equation that governs the dynamic evolution of the joint probability mass function of the underling population process and naturally leads to Markovian dynamics for such process. Due however to the nonlinear nature of most reactions, the computation and analysis of the resulting stochastic population dynamics is a difficult task. This review article provides a coherent and comprehensive coverage of recently developed approaches and methods to tackle this problem. After reviewing a general framework for modeling Markovian reaction networks and giving specific examples, the authors present numerical and computational techniques capable of evaluating or approximating the solution of the master equation, discuss a recently developed approach for studying the stationary behavior of Markovian reaction networks using a potential energy landscape perspective, and provide an introduction to the emerging theory of thermodynamic analysis of such networks. Three representative problems of opinion formation, transcription regulation, and neural network dynamics are used as illustrative examples.Comment: 52 pages, 11 figures, for freely available MATLAB software, see http://www.cis.jhu.edu/~goutsias/CSS%20lab/software.htm

Solving the chemical master equation using sliding windows

<p>Abstract</p> <p>Background</p> <p>The chemical master equation (CME) is a system of ordinary differential equations that describes the evolution of a network of chemical reactions as a stochastic process. Its solution yields the probability density vector of the system at each point in time. Solving the CME numerically is in many cases computationally expensive or even infeasible as the number of reachable states can be very large or infinite. We introduce the sliding window method, which computes an approximate solution of the CME by performing a sequence of local analysis steps. In each step, only a manageable subset of states is considered, representing a "window" into the state space. In subsequent steps, the window follows the direction in which the probability mass moves, until the time period of interest has elapsed. We construct the window based on a deterministic approximation of the future behavior of the system by estimating upper and lower bounds on the populations of the chemical species.</p> <p>Results</p> <p>In order to show the effectiveness of our approach, we apply it to several examples previously described in the literature. The experimental results show that the proposed method speeds up the analysis considerably, compared to a global analysis, while still providing high accuracy.</p> <p>Conclusions</p> <p>The sliding window method is a novel approach to address the performance problems of numerical algorithms for the solution of the chemical master equation. The method efficiently approximates the probability distributions at the time points of interest for a variety of chemically reacting systems, including systems for which no upper bound on the population sizes of the chemical species is known a priori.</p

Oriented clonal cell dynamics enables accurate growth and shaping of vertebrate cartilage

Cartilaginous structures are at the core of embryo growth and shaping before the bone forms. Here we report a novel principle of vertebrate cartilage growth that is based on introducing transversally-oriented clones into pre-existing cartilage. This mechanism of growth uncouples the lateral expansion of curved cartilaginous sheets from the control of cartilage thickness, a process which might be the evolutionary mechanism underlying adaptations of facial shape. In rod-shaped cartilage structures (Meckel, ribs and skeletal elements in developing limbs), the transverse integration of clonal columns determines the well-defined diameter and resulting rod-like morphology. We were able to alter cartilage shape by experimentally manipulating clonal geometries. Using in silico modeling, we discovered that anisotropic proliferation might explain cartilage bending and groove formation at the macro-scale