Mathematical model for transient isoelectric focusing of simple ampholytes

A mathematical model describing transient processes in isoelectric focusing (IEF) of L biprotic ampholytes is presented. The model is a generalization of our previous research on steady slate in IEF and consists of L nonlinear partial differential equations coupled with 2L+2 algebraic equations. Constraints imposed by the mode of operation, viz., constant current. voltage or power, are described. Due to the nonlinearity of the equations, analysis of the model requires computer simulation. Model equations suitable for computer implementation are derived.     

Stochastic simulations on a model of circadian rhythm generation

Biological phenomena are often modeled by differential equations, where states of a model system are described by continuous real values. When we consider concentrations of molecules as dynamical variables for a set of biochemical reactions, we implicitly assume that numbers of the molecules are large enough so that their changes can be regarded as continuous and they are described deterministically. However, for a system with small numbers of molecules, changes in their numbers are apparently discrete and molecular noises become significant. In such cases, models with deterministic differential equations may be inappropriate, and the reactions must be described by stochastic equations. In this study, we focus a clock gene expression for a circadian rhythm generation, which is known as a system involving small numbers of molecules. Thus it is appropriate for the system to be modeled by stochastic equations and analyzed by methodologies of stochastic simulations. The interlocked feedback model proposed by Ueda et al. as a set of deterministic ordinary differential equations provides a basis of our analyses. We apply two stochastic simulation methods, namely Gillespie's direct method and the stochastic differential equation method also by Gillespie, to the interlocked feedback model. To this end, we first reformulated the original differential equations back to elementary chemical reactions. With those reactions, we simulate and analyze the dynamics of the model using two methods in order to compare them with the dynamics obtained from the original deterministic model and to characterize dynamics how they depend on the simulation methodologies.     

On the accuracy of a diffusion approximation to a discrete state–space Markovian model of a population

The traditional Kolmogorov equations treat the size of a population as a discrete random variable. A model is introduced that extends these equations to incorporate environmental variability. Difficulties with this discrete model motivate approximating the population size as a continuous random variable through the use of diffusion processes. The set of cumulants for both the population size and the environmental factors affecting the population size characterize the population–environmental system. The evolution of this set, as predicted by the diffusion approximation, closely matches the corresponding predictions for the discrete model. It is also noted that the simulation estimates of the cumulants against which the predictions of the diffusion model are checked can vary considerably between simulations — despite averaging over a large number of simulation runs. The precision of the simulation estimates–both over time and with differing cumulant order–is discussed.     

On the accuracy of a diffusion approximation to a discrete state–space Markovian model of a population

The traditional Kolmogorov equations treat the size of a population as a discrete random variable. A model is introduced that extends these equations to incorporate environmental variability. Difficulties with this discrete model motivate approximating the population size as a continuous random variable through the use of diffusion processes. The set of cumulants for both the population size and the environmental factors affecting the population size characterize the population–environmental system. The evolution of this set, as predicted by the diffusion approximation, closely matches the corresponding predictions for the discrete model. It is also noted that the simulation estimates of the cumulants against which the predictions of the diffusion model are checked can vary considerably between simulations — despite averaging over a large number of simulation runs. The precision of the simulation estimates–both over time and with differing cumulant order–is discussed.     

Resolving discrepancies between deterministic population models and individual-based simulations

ABSTRACT This work ties together two distinct modeling frameworks for population dynamics: an individual-based simulation and a set of coupled integrodifferential equations involving population densities. The simulation model represents an idealized predator-prey system formulated at the scale of discrete individuals, explicitly incorporating their mutual interactions, whereas the population-level framework is a generalized version of reaction-diffusion models that incorporate population densities coupled to one another by interaction rates. Here I use various combinations of long-range dispersal for both the offspring and adult stages of both prey and predator species, providing a broad range of spatial and temporal dynamics, to compare and contrast the two model frameworks. Taking the individual-based modeling results as given, two examinations of the reaction-dispersal model are made: linear stability analysis of the deterministic equations and direct numerical solution of the model equations. I also modify the numerical solution in two ways to account for the stochastic nature of individual-based processes, which include independent, local perturbations in population density and a minimum population density within integration cells, below which the population is set to zero. These modifications introduce new parameters into the population-level model, which I adjust to reproduce the individual-based model results. The individual-based model is then modified to minimize the effects of stochasticity, producing a match of the predictions from the numerical integration of the population-level model without stochasticity.     

稻萍鱼螺群落食物链的模拟方法

在田间试验和网室内系统测定的基础上,对稻萍鱼螺生态系统的群落结构和食物链行为进行了数学模拟．用插值法解析了稻萍群落模型;并在此基础上用差方方程Q_(t 1)=Q_te~-J_t表达了群落中的萍鱼食物链关系．所得的数学模型通过计算机运行后,得出了优化的群落食物链结构并成功地应用于实践．     

A simulation support system for solving large physiological models on microcomputers

Although physiological modeling and computer simulation have become useful research tools to test new scientific theories and to design and analyze laboratory experiments, developing a new model can be a tedious process because the investigator must often write very complex and specific routines for data input and output. To facilitate the design of new models (as well as the use of existing models), we have developed MODSIM, a FORTRAN-based simulation support system for the IBM PC computer than can accommodate very large dynamic models having up to several thousand equations. It provides the investigator with utilities for continuous on-line graphical and/or tubular output, as well as facilities for dynamic interaction with the model. The user must only supply a model as a list of mathematical equations written in FORTRAN, along with the initial values of the model variables and parameters. The model is precompiled, compiled, and then linked to the MODSIM utilities. Without further programming, the user can then solve the model, select variables for graphical output, and stop the model at any time to analyze the data or to change a parameter before resuming the simulation. This simulation system makes it very easy to develop new models that actively interact with the experimental research of the investigator.     

Modeling adaptive biological systems

During the evolution of many systems found in nature, both the system composition and the interactions between components will vary. Equating the dimension with the number of different components, a system which adds or deletes components belongs to a class of dynamical systems with a finite dimensional phase space of variable dimension. We present two models of biochemical systems with a variable phase space, a model of autocatalytic reaction networks in the prebiotic soup and a model of the idiotypic network of the immune system. Each model contains characteristic meta-dynamical rules for constructing equations of motion from component properties. The simulation of each model occurs on two levels. On one level, the equations of motion are integrated to determine the state of each component. On a second level, algorithms which approximate physical processes in the real system are employed to change the equations of motion. Models with meta-dynamical rules possess several advantages for the study of evolving systems. First, there are no explicit fitness functions to determine how the components of the model rank in terms of survivability. The success of any component is a function of its relationship to the rest of the system. A second advantage is that since the phase space representation of the system is always finite but continually changing, we can explore a potentially infinite phase space which would otherwise be inaccessible with finite computer resources. Third, the enlarged capacity of systems with meta-dynamics for variation allows us to conduct true evolution experiments. The modeling methods presented here can be applied to many real biological systems. In the two studies we present, we are investigating two apparent properties of adaptive networks. With the simulation of the prebiotic soup, we are most interested in how a chemical reaction network might emerge from an initial state of relative disorder. With the study of the immune system, we study the self-regulation of the network including its ability to distinguish between species which are part of the network and those which are not.     

Test of the Inverse Monte Carlo Method for the Calculation of Interatomic Potential Energies in Atomic Liquids

Abstract

The principle purpose of this paper is to demonstrate the use of the Inverse Monte Carlo technique for calculating pair interaction energies in monoatomic liquids from a given equilibrium property. This method is based on the mathematical relation between transition probability and pair potential given by the fundamental equation of the “importance sampling” Monte Carlo method. In order to have well defined conditions for the test of the Inverse Monte Carlo method a Metropolis Monte Carlo simulation of a Lennard Jones liquid is carried out to give the equilibrium pair correlation function determined by the assumed potential. Because an equilibrium configuration is prerequisite for an Inverse Monte Carlo simulation a model system is generated reproducing the pair correlation function, which has been calculated by the Metropolis Monte Carlo simulation and therefore representing the system in thermal equilibrium. This configuration is used to simulate virtual atom displacements. The resulting changes in atom distribution for each single simulation step are inserted in a set of non-linear equations defining the transition probability for the virtual change of configuration. The solution of the set of equations for pair interaction energies yields the Lennard Jones potential by which the equilibrium configuration has been determined.     

A Multidimensional Multispecies Continuum Model for Heterogeneous Biofilm Development

We propose a multidimensional continuum model for heterogeneous growth of biofilm systems with multiple species and multiple substrates. The new model provides a deterministic framework for the study of the interactions between several spe1cies and their effects on biofilm heterogeneity. It consists of a system of partial differential equations derived on the basis of conservation laws and reaction kinetics. The derivation and key assumptions are presented. The assumptions used are a combination of those used in the established one dimensional model, due to Wanner and Gujer, and for the viscous fluid model, of Dockery and Klapper. The work of Wanner and Gujer in particular has been extensively used through the years, and thus this new model is an extension to several spatial dimensions of an already proven working model. The model equations are solved using numerical techniques, for purposes of simulation and verification. The new model is applied to two different biofilm systems in several spatial dimensions, one of which is equivalent to a system originally studied by Wanner and Gujer. Dimensionless formulations for these two systems are given, and numerical simulation results with varying initial conditions are presented. An erratum to this article can be found at     

Analytical Expressions for the Steady-State Concentrations of Glucose, Oxygen and Gluconic Acid in a Composite Membrane for Closed-Loop Insulin Delivery

The mathematical model of Abdekhodaie and Wu (J Membr Sci 335:21–31, 2009) of glucose-responsive composite membranes for closed-loop insulin delivery is discussed. The glucose composite membrane contains nanoparticles of an anionic polymer, glucose oxidase and catalase embedded in a hydrophobic polymer. The model involves the system of nonlinear steady-state reaction–diffusion equations. Analytical expressions for the concentration of glucose, oxygen and gluconic acid are derived from these equations using the Adomian decomposition method. A comparison of the analytical approximation and numerical simulation is also presented. An agreement between analytical expressions and numerical results is observed.     

Mathematical modeling of the dynamics of postradiation injury and recovery of intestinal epithelium

A mathematical model for the dynamics of the crypt-villus system in irradiated mammals has been developed. The model involves a chalones mechanism of regulation crypt cell reproduction rate and represents a system of four nonlinear differential equations. The simulation results are in a good agreement with the experimental data obtained within a wide range of doses.     

N-linked oligosaccharide analysis of glycoprotein bands from isoelectric focusing gels

Glycoproteins often display a complex isoelectric focusing profile because of the presence of negatively charged carbohydrates, such as sialic acid, phosphorylated mannose, and sulfated GalNAc. Until now, understanding the role of these charged carbohydrates in determining the isoelectric focusing profile has been limited to observing pattern shifts following complete removal of the sugars in question. We have developed a simple and sensitive method for analyzing N-linked oligosaccharides from the individual isoelectric focusing bands of a glycoprotein using recombinant human thyroid-stimulating hormone as a model system. N-linked oligosaccharides were released and profiled from individual bands following electroblotting of isoelectric focusing gels. As might be predicted, high-pH anion-exchange chromatography-pulsed amperometric detection and matrix-assisted laser desorption/ionization-time of flight analyses indicated that the bands that migrated closer to the positive electrode contained more sialylated N-linked oligosaccharides. The sialic acid content of these bands correlated with that predicted from the corresponding oligosaccharide analyses.     

An operator splitting method for solving the bidomain equations coupled to a volume conductor model for the torso

In this paper we present a numerical method for the bidomain model, which describes the electrical activity in the heart. The model consists of two partial differential equations (PDEs), which are coupled to systems of ordinary differential equations (ODEs) describing electrochemical reactions in the cardiac cells. Many applications require coupling these equations to a third PDE, describing the electrical fields in the torso surrounding the heart. The resulting system is challenging to solve numerically, because of its complexity and very strict resolution requirements in time and space. We propose a method based on operator splitting and a fully coupled discretization of the three PDEs. Numerical experiments show that for simple simulation cases and fine discretizations, the algorithm is second-order accurate in space and time.     

Mathematical modeling of the formation of apoptosome in intrinsic pathway of apoptosis

Caspase-9 is the protease that mediates the intrinsic pathway of apoptosis, a type of cell death. Activation of caspase-9 is a multi-step process that requires dATP or ATP and involves at least two proteins, cytochrome c and Apaf-1. In this study, we mathematically model caspase-9 activation by using a system of ordinary differential equations (an ODE model) generated by a systems biology tool Simpathica—a simulation and reasoning system, developed to study biological pathways. A rudimentary version of “model checking” based on comparing simulation data with that obtained from a recombinant system of caspase-9 activation, provided several new insights into regulation of this protease. The model predicts that the activation begins with binding of dATP to Apaf-1, which initiates the interaction between Apaf-1 and cytochrome c, thus forming a complex that oligomerizes into an active caspase-9 holoenzyme via a linear binding model with cooperative interaction rather than through network formation.     

Exact epidemic models on graphs using graph-automorphism driven lumping

The dynamics of disease transmission strongly depends on the properties of the population contact network. Pair-approximation models and individual-based network simulation have been used extensively to model contact networks with non-trivial properties. In this paper, using a continuous time Markov chain, we start from the exact formulation of a simple epidemic model on an arbitrary contact network and rigorously derive and prove some known results that were previously mainly justified based on some biological hypotheses. The main result of the paper is the illustration of the link between graph automorphisms and the process of lumping whereby the number of equations in a system of linear differential equations can be significantly reduced. The main advantage of lumping is that the simplified lumped system is not an approximation of the original system but rather an exact version of this. For a special class of graphs, we show how the lumped system can be obtained by using graph automorphisms. Finally, we discuss the advantages and possible applications of exact epidemic models and lumping.     

Novel isoelectric precipitation of proteins in a pressurized carbon dioxide-water-ethanol system

A novel isoelectric precipitation of proteins in a pressurized carbon dioxide-water-ethanol system was developed where carbon dioxide was used as a volatile acid. The pH-pressure curves of the system with the absence and presence of proteins were investigated. By introducing the pressurized carbon dioxide to a solution containing protein, the pH value in the solution was decreased to the isoelectric region of the model protein BSA. Addition of ethanol could lower the buffer capacity of the protein, which made the precipitation concentration of protein go beyond the limits in a system without ethanol and well exploited the application field of the technique. In addition, ethanol in solution played the role of aiding precipitation in the process. Another model protein, hen egg white lysozyme, was also studied but could not be precipitated in the above system. All of these phenomena prove that isoelectric precipitation is the key point in the pressurized carbon dioxide-water-ethanol system.