Instability and pattern formation in reaction-diffusion systems: a higher order analysis

We analyze the condition for instability and pattern formation in reaction-diffusion systems beyond the usual linear regime. The approach is based on taking into account perturbations of higher orders. Our analysis reveals that nonlinearity present in the system can be instrumental in determining the stability of a system, even to the extent of destabilizing one in a linearly stable parameter regime. The analysis is also successful to account for the observed effect of additive noise in modifying the instability threshold of a system. The analytical study is corroborated by numerical simulation in a standard reaction-diffusion system.     

Exact on-lattice stochastic reaction-diffusion simulations using partial-propensity methods

Stochastic reaction-diffusion systems frequently exhibit behavior that is not predicted by deterministic simulation models. Stochastic simulation methods, however, are computationally expensive. We present a more efficient stochastic reaction-diffusion simulation algorithm that samples realizations from the exact solution of the reaction-diffusion master equation. The present algorithm, called partial-propensity stochastic reaction-diffusion (PSRD) method, uses an on-lattice discretization of the reaction-diffusion system and relies on partial-propensity methods for computational efficiency. We describe the algorithm in detail, provide a theoretical analysis of its computational cost, and demonstrate its computational performance in benchmarks. We then illustrate the application of PSRD to two- and three-dimensional pattern-forming Gray-Scott systems, highlighting the role of intrinsic noise in these systems.     

An accelerated algorithm for discrete stochastic simulation of reaction-diffusion systems using gradient-based diffusion and tau-leaping

Stochastic simulation of reaction-diffusion systems enables the investigation of stochastic events arising from the small numbers and heterogeneous distribution of molecular species in biological cells. Stochastic variations in intracellular microdomains and in diffusional gradients play a significant part in the spatiotemporal activity and behavior of cells. Although an exact stochastic simulation that simulates every individual reaction and diffusion event gives a most accurate trajectory of the system's state over time, it can be too slow for many practical applications. We present an accelerated algorithm for discrete stochastic simulation of reaction-diffusion systems designed to improve the speed of simulation by reducing the number of time-steps required to complete a simulation run. This method is unique in that it employs two strategies that have not been incorporated in existing spatial stochastic simulation algorithms. First, diffusive transfers between neighboring subvolumes are based on concentration gradients. This treatment necessitates sampling of only the net or observed diffusion events from higher to lower concentration gradients rather than sampling all diffusion events regardless of local concentration gradients. Second, we extend the non-negative Poisson tau-leaping method that was originally developed for speeding up nonspatial or homogeneous stochastic simulation algorithms. This method calculates each leap time in a unified step for both reaction and diffusion processes while satisfying the leap condition that the propensities do not change appreciably during the leap and ensuring that leaping does not cause molecular populations to become negative. Numerical results are presented that illustrate the improvement in simulation speed achieved by incorporating these two new strategies.     

Multiresolution stochastic simulations of reaction-diffusion processes

Stochastic simulations of reaction-diffusion processes are used extensively for the modeling of complex systems in areas ranging from biology and social sciences to ecosystems and materials processing. These processes often exhibit disparate scales that render their simulation prohibitive even for massive computational resources. The problem is resolved by introducing a novel stochastic multiresolution method that enables the efficient simulation of reaction-diffusion processes as modeled by many-particle systems. The proposed method quantifies and efficiently handles the associated stiffness in simulating the system dynamics and its computational efficiency and accuracy are demonstrated in simulations of a model problem described by the Fisher-Kolmogorov equation. The method is general and can be applied to other many-particle models of physical processes.     

Particle dynamics simulations of Turing patterns

The direct simulation Monte Carlo method is used to reproduce Turing patterns at the microscopic level in reaction-diffusion systems. In order to satisfy the basic condition for the development of such a spatial structure, we propose a model involving a solvent, which allows for disparate diffusivities of individual reactive species. One-dimensional structures are simulated in systems of various lengths. Simulation results agree with the macroscopic predictions obtained by integration of the reaction-diffusion equations. Additional effects due to internal fluctuations are observed, such as temporal transitions between structures of different wavelengths in a confined system. For a structure developing behind a propagating wave front, the fluctuations suppress the induction period and accelerate the formation of the Turing pattern. These results support the ability of reaction-diffusion models to robustly reproduce axial segmentation including the formation of early vertebrae or somites in noisy biological environments.     

K-leap method for accelerating stochastic simulation of coupled chemical reactions

Leap methods are very promising for accelerating stochastic simulation of a well stirred chemically reacting system, while providing acceptable simulation accuracy. In Gillespie's tau-leap method [D. Gillespie, J. Phys. Chem. 115, 1716 (2001)], the number of firings of each reaction channel during a leap is a Poisson random variable, whose sample values are unbounded. This may cause large changes in the populations of certain molecular species during a leap, thereby violating the leap condition. In this paper, we develop an alternative leap method called the K-leap method, in which we constrain the total number of reactions occurring during a leap to be a number K calculated from the leap condition. As the number of firings of each reaction channel during a leap is upper bounded by a properly chosen number, our K-leap method can better satisfy the leap condition, thereby improving simulation accuracy. Since the exact stochastic simulation algorithm (SSA) is a special case of our K-leap method when K=1, our K-leap method can naturally change from the exact SSA to an approximate leap method during simulation, whenever the leap condition allows to do so.     

Overcoming stiffness in stochastic simulation stemming from partial equilibrium: a multiscale Monte Carlo algorithm

In this paper the problem of stiffness in stochastic simulation of singularly perturbed systems is discussed. Such stiffness arises often from partial equilibrium or quasi-steady-state type of conditions. A multiscale Monte Carlo method is discussed that first assesses whether partial equilibrium is established using a simple criterion. The exact stochastic simulation algorithm (SSA) is next employed to sample among fast reactions over short time intervals (microscopic time steps) in order to compute numerically the proper probability distribution function for sampling the slow reactions. Subsequently, the SSA is used to sample among slow reactions and advance the time by large (macroscopic) time steps. Numerical examples indicate that not only long times can be simulated but also fluctuations are properly captured and substantial computational savings result.     

铂电极BZ反应系双电层稀疏区中的Turing结构

将扩散流作为场函数, 考虑φ电势的空间分布, 建立了铂电极BZ反应系在双电层稀疏区的动力学演化机制, 确立了纳入稀疏区φ电势效应的反应-扩散型演化方程. 采用Boltzmann分布近似, 解决了演化方程中含φ电势的流项的线性化问题; 导出了可在算法上实现的三变量体系线性化算子本征值的解析形式. 分别以静态铂电极BZ反应系双电层稀疏区和对应的纯粹BZ反应系作为参考模型系, 分析了经空间对称性破缺产生Turing结构的参数范围. 数值模拟发现, φ电场的存在使铂电极BZ反应系的输运过程在静态双电层稀疏区趋于电化学平衡时, 在对应的纯粹BZ反应体系中可呈现的Turing结构已趋于消失; 而在电流强度不太大的恒流不可逆铂电极BZ反应体系双电层稀疏区中, 鲜明稳定的Turing结构又重新出现在原参数区间内. 同时, 在静态双电层稀疏区不出现Turing结构的参数范围内也可找到类似的恒流稳定空间结构.     

Mobility-induced instability and pattern formation in a reaction-diffusion system

Ions undergoing a reaction-diffusion process are susceptible to electric field. We show that a constant external field may induce a kind of instability on the state stabilized by diffusion in a reaction-diffusion system giving rise to formation of pattern even when the diffusion coefficients of the reactants are equal. The origin of the pattern is due to the difference in mobilities of the two species and is thus markedly different from that of deformed Turing pattern in presence of the field. While this differential flow instability had been shown earlier to result in traveling waves, we realize in the context of stationary pattern formation in a typical reaction-diffusion-advective system. Our analysis is based on a numerical simulation of a generic model on a two-dimensional domain.     

A reduction method for multiple time scale stochastic reaction networks

In this paper we develop a reduction method for multiple time scale stochastic reaction networks. When the transition-rate matrix between different states of the species is available, we obtain systems of reduced equations, whose solutions can successively approximate, to any degree of accuracy, the exact probability that the reaction system be in any particular state. For the case when the transition-rate matrix is not available, one needs to rely on the chemical master equation. For this case, we obtain a corresponding reduced master equation with first-order accuracy. We illustrate the accuracy and efficiency of both approaches by simulating several motivating examples and comparing the results of our simulations with the results obtained by the exact method. Our examples include both linear and nonlinear reaction networks as well as a three time scale stochastic reaction-diffusion model arising from gene expression.     

金属胶束催化体系中经Turing分支形成的空间有序结构与催化活性

摘要基于金属胶束催化反应的三元复合物动力学模型, 分析了金属胶束作为一类特殊的超分子体系, 其参与催化反应的特殊动力学行为与体系中可能出现的自组织结构的关系. 研究结果表明, 在一定的边界条件下, 即使基元反应是线性化学反应步骤, 由于扩散步骤与化学反应的耦合, 以及非理想性的组分热力学行为, 也将不可避免地出现经Turing分支形成的浓度场空间自组织. 计算结果表明, 这种宏观有序结构的形成将使金属胶束催化体系中呈现出以存在反应活性极高的位点为特征的反应活性的空间有序分布; 从自组织理论的角度对这类胶束体系催化活性的时空有序特征给予一定程度的阐释.     

Persistent ATP-Concentration Gradients in a Hydrogel Sustained by Chemical Fuel Consumption

We show the formation of macroscopic ATP-concentrations in an agarose gel and demonstrate that these gradients can be sustained in time at the expense of the consumption of a chemical fuel. The approach relies on the spatially controlled activation of ATP-producing and ATP-consuming reactions through the local injection of enzymes in the matrix. The reaction-diffusion system is maintained in a stationary non-equilibrium state as long as chemical fuel, phosphocreatine, is present. The reaction-diffusion system is coupled to a supramolecular system composed of monolayer protected gold nanoparticles and a fluorescent probe. As a result of this coupling, fluorescence signals emerge spontaneously in response to the ATP-concentration gradients. We show that the approach permits the rational formation of complex fluorescence patterns that change over time as a function of the evolution of the ATP-concentrations present in the system.     

Madaline网络分离重叠光谱峰在灰色体系分析中的应用

将Ｍａｄａｌｉｎｅ网络分离重叠光谱峰法应用于组成范围已知的灰色体系进行分析。模拟数据研究表明：当测量误差在「－０．００５,＋０．００５」范围内,且所含组分峰形相同但峰位不同时,最低检测限是峰位差不小于１．５ｎｍ。１５相实际样品分析表明：本法能给出比较满意的分析结果。     

Reaction-diffusion travelling waves in the acidic nitrate-ferroin reaction

A two-variable model proposed for the acidic nitrate-ferroin reaction is considered in the reaction-diffusion context. An initial-value problem in which an amount of nitrate is introduced locally into ferroin at uniform concentration is treated both analytically and numerically. It is shown that the large time structure is a reaction-diffusion travelling wave of permanent form propagating with constant speed. This asymptotic wave speed is shown to be the minimum possible wave speed and the asymptotic approach to this value is estimated. Properties of the permanent-form travelling waves are derived and solutions valid for small and large values of a parameter , involved in the kinetic mechanism, are obtained.     

Two grid refinement methods in the lattice Boltzmann framework for reaction-diffusion processes in complex systems

This paper studies the optimisation of a numerical model and a computer code to solve numerically reaction-diffusion processes in environmental or biological systems with complicated geometries and mixtures of reactions including time and spatial scales extending over several order of magnitude. In particular, we consider different grid refinement techniques in the framework of a lattice Boltzmann solver for reaction-diffusion systems. Two new grid refinement methods are proposed, which are both quantitatively good. The first method is based on the matching of the concentration profiles and fluxes across two adjacent sub-domains, while the second method is based on nested subgrids. The focus of our study is the trade off between accuracy and CPU time. We show how the different parameters of the method, such as the refinement factors, the location of the boundary between different grids or coupling methods at the interface affect the quality of the numerical solution and the efficiency of the method.     

重新审视滴定分析

Volumetric titration analysis is an important content of chemical analysis; but it is traditionally considered to use large amount sample, take long analysis time, have low sensitivity, and involve harsh analytical conditions. This paper provides perspectives for the above issues. Examples are introduced to illustrate that the sample amount can be easily reduced with improved sample throughput by the use of automatic titration analysis. The non-traditional end-point indication method improves the sensitivity with simplified analytical condition. We hope to introduce the different titration analysis and the breadth of applications to students, and thus, to stimulate their interest.     

An efficient numerical technique for solving nonlinear singularly perturbed reaction diffusion problem

Journal of Mathematical Chemistry - In this paper, a numerical method based on Bernstein polynomial for nonlinear singularly perturbed reaction-diffusion problems is proposed. The solution of this...     

On-line cell lysis and DNA extraction on a microfluidic biochip fabricated by microelectromechanical system technology

Integrating cell lysis and DNA purification process into a micrototal analytical system (microTAS) is one critical step for the analysis of nucleic acids. On-chip cell lysis based on a chemical method is realized by sufficient blend of blood sample and the lyzing reagent. In this paper two mixing models, T-type mixing model and sandwich-type mixing model, are proposed and simulation of those models is conducted. Result of simulation shows that the sandwich-type mixing model with coiled channel performs best and this model is further used to construct the microfluidic biochip for on-line cell lysis and DNA extraction. The result of simulation is further verified by experiments. It asserts that more than 80% mixing of blood sample and lyzing reagent which guarantees that completed cell lysis can be achieved near the inlet location when the cell/buffer velocity ratio is less than 1:5. After cell lysis, DNA extraction by means of a solid-phase method is implemented by using porous silicon matrix which is integrated in the biochip. During continuous flow process in the microchip, rapid cell lysis and PCR-amplifiable genomic DNA purification can be achieved within 20 min. The potential of this microfluidic biochip is illustrated by pretreating a whole blood sample, which shows the possibility of integration of sample preparation, PCR, and separation on a single device to work as portable point-of-care medical diagnostic system.     

Pre-Validation and Performance Prediction Using Pressure Monitoring to Evaluate HPLC Method Development Changes

HPLC methods for pharmaceutical analysis evolve from method development to commercial product testing. The majority of development and validation takes place in an R&D setting, followed by transfer to commercial sites that execute test methods on a routine basis. There is a growing need to increase confidence and probability that developed methods will be successful both during validation and long-term use with products, particularly when introducing new or unfamiliar technologies. An HPLC method pre-validation and performance prediction plan is presented using model pharmaceutical ingredients in prototype liquid formulations. The pre-validation includes limited linearity, repeatability in sample matrix, precision, carry-over, and selectivity. The performance prediction involves injecting large numbers of worst-case samples and monitoring five critical parameters: retention time, tailing factor, resolution, efficiency, and system pressure. Parameter results are assessed with respect to precision and/or system performance. In a proof of principle model, the performance prediction data demonstrate that system suitability parameters remain constant during injections of prepared samples; however, the system pressure increases over time. These underlying data indicate a potential trend prior to method validation. As a result, sample preparation and HPLC condition modifications are evaluated using pressure plots as a key performance metric.

     

A graph-theoretic method for detecting potential Turing bifurcations

The conditions for diffusion-driven (Turing) instabilities in systems with two reactive species are well known. General methods for detecting potential Turing bifurcations in larger reaction schemes are, on the other hand, not well developed. We prove a theorem for a graph-theoretic condition originally given by Volpert and Ivanova [Mathematical Modeling (Nauka, Moscow, 1987) (in Russian), p. 57] for Turing instabilities in a mass-action reaction-diffusion system involving n substances. The method is based on the representation of a reaction mechanism as a bipartite graph with two types of nodes representing chemical species and reactions, respectively. The condition for diffusion-driven instability is related to the existence of a structure in the graph known as a critical fragment. The technique is illustrated using a substrate-inhibited bifunctional enzyme mechanism which involves seven chemical species.