A Comparison of Bimolecular Reaction Models for Stochastic Reaction–Diffusion Systems

Stochastic reaction–diffusion models have become an important tool in studying how both noise in the chemical reaction process and the spatial movement of molecules influences the behavior of biological systems. There are two primary spatially-continuous models that have been used in recent studies: the diffusion limited reaction model of Smoluchowski, and a second approach popularized by Doi. Both models treat molecules as points undergoing Brownian motion. The former represents chemical reactions between two reactants through the use of reactive boundary conditions, with two molecules reacting instantly upon reaching a fixed separation (called the reaction-radius). The Doi model uses reaction potentials, whereby two molecules react with a fixed probability per unit time, λ, when separated by less than the reaction radius. In this work, we study the rigorous relationship between the two models. For the special case of a protein diffusing to a fixed DNA binding site, we prove that the solution to the Doi model converges to the solution of the Smoluchowski model as λ→∞, with a rigorous $O(\lambda^{-\frac{1}{2} + \epsilon})$ error bound (for any fixed ?>0). We investigate by numerical simulation, for biologically relevant parameter values, the difference between the solutions and associated reaction time statistics of the two models. As the reaction-radius is decreased, for sufficiently large but fixed values of λ, these differences are found to increase like the inverse of the binding radius.     

Multiscale Stochastic Reaction–Diffusion Modeling: Application to Actin Dynamics in Filopodia

Two multiscale (hybrid) stochastic reaction–diffusion models of actin dynamics in a filopodium are investigated. Both hybrid algorithms combine compartment-based and molecular-based stochastic reaction–diffusion models. The first hybrid model is based on the models previously developed in the literature. The second hybrid model is based on the application of a recently developed two-regime method (TRM) to a fully molecular-based model, which is also developed in this paper. The results of hybrid models are compared with the results of the molecular-based model. It is shown that both approaches give comparable results, although the TRM model better agrees quantitatively with the molecular-based model.     

A Quasistationary Analysis of a Stochastic Chemical Reaction: Keizer’s Paradox

For a system of biochemical reactions, it is known from the work of T.G. Kurtz [J. Appl. Prob. 8, 344 (1971)] that the chemical master equation model based on a stochastic formulation approaches the deterministic model based on the Law of Mass Action in the infinite system-size limit in finite time. The two models, however, often show distinctly different steady-state behavior. To further investigate this “paradox,” a comparative study of the deterministic and stochastic models of a simple autocatalytic biochemical reaction, taken from a text by the late J. Keizer, is carried out. We compute the expected time to extinction, the true stochastic steady state, and a quasistationary probability distribution in the stochastic model. We show that the stochastic model predicts the deterministic behavior on a reasonable time scale, which can be consistently obtained from both models. The transition time to the extinction, however, grows exponentially with the system size. Mathematically, we identify that exchanging the limits of infinite system size and infinite time is problematic. The appropriate system size that can be considered sufficiently large, an important parameter in numerical computation, is also discussed.     

Study and Simulation of Reaction–Diffusion Systems Affected by Interacting Signaling Pathways

Possible effects of interaction (cross-talk) between signaling pathways is studied in a system of Reaction-Diffusion (RD) equations. Furthermore, the relevance of spontaneous neurite symmetry breaking and Turing instability has been examined through numerical simulations. The interaction between Retinoic Acid (RA) and Notch signaling pathways is considered as a perturbation to RD system of axon-forming potential for N2a neuroblastoma cells. The present work suggests that large increases to the level of RA-Notch interaction can possibly have substantial impacts on neurite outgrowth and on the process of axon formation. This can be observed by the numerical study of the homogeneous system showing that in the absence of RA-Notch interaction the unperturbed homogeneous system may exhibit different saddle-node bifurcations that are robust under small perturbations by low levels of RA-Notch interactions, while large increases in the level of RA-Notch interaction result in a number of transitions of saddle-node bifurcations into Hopf bifurcations. It is speculated that near a Hopf bifurcation, the regulations between the positive and negative feedbacks change in such a way that spontaneous symmetry breaking takes place only when transport of activated Notch protein takes place at a faster rate.     

Exact Solutions of Coupled Multispecies Linear Reaction–Diffusion Equations on a Uniformly Growing Domain

Embryonic development involves diffusion and proliferation of cells, as well as diffusion and reaction of molecules, within growing tissues. Mathematical models of these processes often involve reaction–diffusion equations on growing domains that have been primarily studied using approximate numerical solutions. Recently, we have shown how to obtain an exact solution to a single, uncoupled, linear reaction–diffusion equation on a growing domain, 0 < x < L(t), where L(t) is the domain length. The present work is an extension of our previous study, and we illustrate how to solve a system of coupled reaction–diffusion equations on a growing domain. This system of equations can be used to study the spatial and temporal distributions of different generations of cells within a population that diffuses and proliferates within a growing tissue. The exact solution is obtained by applying an uncoupling transformation, and the uncoupled equations are solved separately before applying the inverse uncoupling transformation to give the coupled solution. We present several example calculations to illustrate different types of behaviour. The first example calculation corresponds to a situation where the initially–confined population diffuses sufficiently slowly that it is unable to reach the moving boundary at x = L(t). In contrast, the second example calculation corresponds to a situation where the initially–confined population is able to overcome the domain growth and reach the moving boundary at x = L(t). In its basic format, the uncoupling transformation at first appears to be restricted to deal only with the case where each generation of cells has a distinct proliferation rate. However, we also demonstrate how the uncoupling transformation can be used when each generation has the same proliferation rate by evaluating the exact solutions as an appropriate limit.     

Modelling a Wolbachia Invasion Using a Slow–Fast Dispersal Reaction–Diffusion Approach

This paper uses a reaction–diffusion approach to examine the dynamics in the spread of a Wolbachia infection within a population of mosquitoes in a homogeneous environment. The formulated model builds upon an earlier model by Skalski and Gilliam (Am. Nat. 161(3):441–458, 2003), which incorporates a slow and fast dispersal mode. This generates a faster wavespeed than previous reaction–diffusion approaches, which have been found to produce wavespeeds that are unrealistically slow when compared with direct observations. In addition, the model incorporates cytoplasmic incompatibility between male and female mosquitoes, which creates a strong Allee effect in the dynamics. In previous studies, linearised wavespeeds have been found to be inaccurate when a strong Allee effect is underpinning the dynamics. We provide a means to approximate the wavespeed generated by the model and show that it is in close agreement with numerical simulations. Wavespeeds are approximated for both Aedes aegypti and Drosophila simulans mosquitoes at different temperatures. These wavespeeds indicate that as the temperature decreases within the optimal temperature range for mosquito survival, the speed of a Wolbachia invasion increases for Aedes aegypti populations and decreases for Drosophila simulans populations.     

Reaction–diffusion model of atherosclerosis development

Atherosclerosis begins as an inflammation in blood vessel walls (intima). The inflammatory response of the organism leads to the recruitment of monocytes. Trapped in the intima, they differentiate into macrophages and foam cells leading to the production of inflammatory cytokines and further recruitment of white blood cells. This self-accelerating process, strongly influenced by low-density lipoproteins (cholesterol), results in a dramatic increase of the width of blood vessel walls, formation of an atherosclerotic plaque and, possibly, of its rupture. We suggest a 2D mathematical model of the initiation and development of atherosclerosis which takes into account the concentration of blood cells inside the intima and of pro- and anti-inflammatory cytokines. The model represents a reaction-diffusion system in a strip with nonlinear boundary conditions which describe the recruitment of monocytes as a function of the concentration of inflammatory cytokines. We prove the existence of travelling waves described by this system and confirm our previous results which suggest that atherosclerosis develops as a reaction-diffusion wave. The theoretical results are confirmed by the results of numerical simulations.     

Evolutionarily Stable and Convergent Stable Strategies in Reaction–Diffusion Models for Conditional Dispersal

We consider a mathematical model of two competing species for the evolution of conditional dispersal in a spatially varying, but temporally constant environment. Two species are different only in their dispersal strategies, which are a combination of random dispersal and biased movement upward along the resource gradient. In the absence of biased movement or advection, Hastings showed that the mutant can invade when rare if and only if it has smaller random dispersal rate than the resident. When there is a small amount of biased movement or advection, we show that there is a positive random dispersal rate that is both locally evolutionarily stable and convergent stable. Our analysis of the model suggests that a balanced combination of random and biased movement might be a better habitat selection strategy for populations.     

Theoretical Analysis of Time-to-Peak Responses in Biological Reaction Networks

Processing of information by signaling networks is characterized by properties of the induced kinetics of the activated pathway components. The maximal extent of pathway activation (maximum amplitude) and the time-to-peak-response (position) are key determinants of biological responses that have been linked to specific outcomes. We investigate how the maximum amplitude of pathway activation and its position depend on the input and wiring of a signaling network. For this purpose, we consider a simple reaction A→B that is regulated by a transient input and extended this to include back-reaction and additional partners. In particular, we show that a unique maximum of B(t) exists. Moreover, we prove that the position of the maximum is independent of the applied input but regulated by degradation reactions of B. Indeed, the time-to-peak-response decreases with increasing degradation rate, which we prove for small models and show in simulations for more complex ones. The identified dependencies provide insights into design principles that facilitate the realization dynamical characteristics like constant position of maximal pathway activation and thereby guide the characterization of unknown kinetics within larger protein networks.     

Mode Transitions in a Model Reaction–Diffusion System Driven by Domain Growth and Noise

Pattern formation in many biological systems takes place during growth of the underlying domain. We study a specific example of a reaction-diffusion (Turing) model in which peak splitting, driven by domain growth, generates a sequence of patterns. We have previously shown that the pattern sequences which are presented when the domain growth rate is sufficiently rapid exhibit a mode-doubling phenomenon. Such pattern sequences afford reliable selection of certain final patterns, thus addressing the robustness problem inherent of the Turing mechanism. At slower domain growth rates this regular mode doubling breaks down in the presence of small perturbations to the dynamics. In this paper we examine the breaking down of the mode doubling sequence and consider the implications of this behaviour in increasing the range of reliably selectable final patterns.     

Parameter Sensitivity Analysis of Stochastic Models Provides Insights into?Cardiac Calcium Sparks

We present a parameter sensitivity analysis method that is appropriate for stochastic models, and we demonstrate how this analysis generates experimentally testable predictions about the factors that influence local Ca²⁺ release in heart cells. The method involves randomly varying all parameters, running a single simulation with each set of parameters, running simulations with hundreds of model variants, then statistically relating the parameters to the simulation results using regression methods. We tested this method on a stochastic model, containing 18 parameters, of the cardiac Ca²⁺ spark. Results show that multivariable linear regression can successfully relate parameters to continuous model outputs such as Ca²⁺ spark amplitude and duration, and multivariable logistic regression can provide insight into how parameters affect Ca²⁺ spark triggering (a probabilistic process that is all-or-none in a single simulation). Benchmark studies demonstrate that this method is less computationally intensive than standard methods by a factor of 16. Importantly, predictions were tested experimentally by measuring Ca²⁺ sparks in mice with knockout of the sarcoplasmic reticulum protein triadin. These mice exhibit multiple changes in Ca²⁺ release unit structures, and the regression model both accurately predicts changes in Ca²⁺ spark amplitude (30% decrease in model, 29% decrease in experiments) and provides an intuitive and quantitative understanding of how much each alteration contributes to the result. This approach is therefore an effective, efficient, and predictive method for analyzing stochastic mathematical models to gain biological insight.     

An Efficient Wavelet Based Approximation Method to Steady State Reaction–Diffusion Model Arising in Mathematical Chemistry

The mathematical model of Rahamathunissa and Rajendran (J Math Chem 44:849–861, 2008) in an amperometric biosensor response is discussed. In this paper, we have applied the shifted second kind Chebyshev wavelets (CW) to obtain the numerical solutions of reaction–diffusion equations containing a nonlinear term related to Michaelis–Menton kinetics of the enzymatic reaction. The application of the shifted second kind CW operational matrices for solving initial and boundary value problems is presented. The obtained numerical results demonstrate efficient and applicability of the proposed method. The power of the manageable method is confirmed. Moreover the use of shifted second kind CW method is found to be simple, efficient, accurate, small computation cost, and computationally attractive.     

Application of Amphipols for Structure–Functional Analysis of TRP Channels

Amphipathic polymers (amphipols), such as A8-35 and SApol, are a new tool for stabilizing integral membrane proteins in detergent-free conditions for structural and functional studies. Transient receptor potential (TRP) ion channels function as tetrameric protein complexes in a diverse range of cellular processes including sensory transduction. Mammalian TRP channels share ~20 % sequence similarity and are categorized into six subfamilies: TRPC (canonical), TRPV (vanilloid), TRPA (ankyrin), TRPM (melastatin), TRPP (polycystin), and TRPML (mucolipin). Due to the inherent difficulties in purifying eukaryotic membrane proteins, structural studies of TRP channels have been limited. Recently, A8-35 was essential in resolving the molecular architecture of the nociceptor TRPA1 and led to the determination of a high-resolution structure of the thermosensitive TRPV1 channel by cryo-EM. Newly developed maltose-neopentyl glycol (MNG) detergents have also proven to be useful in stabilizing TRP channels for structural analysis. In this review, we will discuss the impacts of amphipols and MNG detergents on structural studies of TRP channels by cryo-EM. We will compare how A8-35 and MNG detergents interact with the hydrophobic transmembrane domains of TRP channels. In addition, we will discuss what these cryo-EM studies reveal on the importance of screening different types of surfactants toward determining high-resolution structures of TRP channels.     

Physicochemical Aspects of Reaction of Ozone with Galactolipid and Galactolipid–Tocopherol Layers

The impact of reaction of galactolipids with ozone on the physicochemical properties of their monolayers was examined. In Megli and Russo (Biochim Biophys Acta, 1778:143–152, 2008), Cwiklik and Jungwirth (Chem Phys Lett, 486:99–103, 2010), Jurkiewicz et al. (Biochim Biophys Acta, 1818:2388–2402, 2012), Khabiri et al. (Chem Phys Lett, 519:93–99, 2012), and Conte et al. (Biochim Biophys Acta, 1828:510–517, 2013), the properties of layers formed from model mixtures composed of chosen lipids and selected oxidation products were studied, whereas in this work, question was raised as to how the oxidation reactions taking place in situ affect the physical properties of the galactolipid layers. So, set experiment should take into account the effect of all reaction products. The mechanical characteristics of monolayers of monogalactosyldiacyl-glycerol (MGDG) and digalactosyldiacylglycerol (DGDG) were determined by Langmuir trough technique, and the electrical properties of liposomes formed from these lipids by measuring their electrophoretic mobility. Considerable loss of galactolipid molecules forming monolayers was found at ozone concentrations (in aqueous medium) higher than 0.1 ppm with a stronger effect measured for MGDG. That goes along with the greater amounts of MDA found in the extracts of oxidized MGDG films compared with DGDG. Based on this, it was concluded that an additional galactose group present in DGDG molecules acts protectively under oxidative conditions. The surface tension of the solutions (of small volume) contacting the oxidized galactolipids films was significantly reduced, indicating the presence of soluble in polar media, surface active reaction products. The presence of α-tocopherol in mixtures with tested galactolipids at a molar ratio of lipid to tocopherol equal to 1.7:1 caused some inhibition of lipid oxidation, reducing the decrease of amount of lipid particles forming the monolayer. Here, also protective effect of α-tocopherol was greater for the MGDG compared to DGDG.     

Formation of Microspherules of Lunar Regolith in Plasma–Dust Processes Initiated by Meteoroid Impacts

Plasma Physics Reports - The possibility of the formation of microspherules in plasma-dust processes initiated by meteoroids impacting the lunar surface is discussed. It is demonstrated that...     

A Nonlinear Stability Analysis of Vegetative Turing Pattern Formation for an Interaction–Diffusion Plant-Surface Water Model System in an Arid Flat Environment

The development of spontaneous stationary vegetative patterns in an arid flat environment is investigated by means of a weakly nonlinear diffusive instability analysis applied to the appropriate model system for this phenomenon. In particular, that process can be modeled by a partial differential interaction–diffusion equation system for the plant biomass density and the surface water content defined on an unbounded flat spatial domain. The main results of this analysis can be represented by closed-form plots in the rate of precipitation versus the specific rate of plant density loss parameter space. From these plots, regions corresponding to bare ground and vegetative patterns consisting of parallel stripes, labyrinth-like mazes, hexagonal arrays of gaps, irregular mosaics, and homogeneous distributions of vegetation, respectively, may be identified in this parameter space. Then those theoretical predictions are compared with both relevant observational evidence involving tiger and pearled bush patterns and existing numerical simulations of similar model systems as well as placed in the context of the results from some recent nonlinear vegetative pattern formation studies.