Derivation and analysis of a system modeling the chemotactic movement of hematopoietic stem cells

It has been shown that hematopoietic stem cells migrate in vitro and in vivo following the gradient of a chemotactic factor produced by stroma cells. In this paper, a quantitative model for this process is presented. The model consists of chemotaxis equations coupled with an ordinary differential equation on the boundary of the domain and subjected to nonlinear boundary conditions. The existence and uniqueness of a local solution is proved and the model is simulated numerically. It turns out that for adequate parameter ranges, the qualitative behavior of the stem cells observed in the experiment is in good agreement with the numerical results. Our investigations represent a first step in the process of elucidating the mechanism underlying the homing of hematopoietic stem cells quantitatively.      

A kinetic model for the activated sludge process which considers diffusion and reaction in the microbial floc

A mathematical model has been developed which describes substrate removal, oxygen utilization, and biomass production in an aggregated microbial suspension containing the substrate as a soluble biodegradable material and a uniform floc size. It is applicable to both steady-state and transient conditions. The model, consisting of three partial differential equations and two ordinary differential equations, takes into account the flow pattern in the reactor, intraparticle mass transport of oxygen and substrate, and biochemical reaction by individual cells embedded in the floc. Efficient numerical solution of the coupled nonlinear equations is obtained using an implicit finite difference approach for both the reactor and floc equations. A convergent solution is realized through block interation utilizing the tridiagonal algorithm. Results indicate that a unifying theory of activated sludge dynamics will have to consider coupling between floc chemical kinetics and changes in the bulk liquid characteristics. Floc size emerges as an important influence on system performance. It appears necessary to distinguish between a system response caused by diffuslonal resistances and nutrient limitations within the floc and a response caused by physiological adaption when analyzing the transient behavior of an activated sludge process. Future research should be devoted to rigorous laboratory determinations of model parameters along with extensions to include limitations of nutrients other than orgabnic carbon and oxygen.     

Localized bacterial infection in a distributed model for tissue inflammation

Phagocyte motility and chemotaxis are included in a distributed mathematical model for the inflammatory response to bacterial invasion of tissue. Both uniform and non-uniform steady state solutions may occur for the model equations governing bacteria and phagocyte densities in a macroscopic tissue region. The non-uniform states appear to be more dangerous because they allow large bacteria densities concentrated in local foci, and in some cases greater total bacteria and phagocyte populations. Using a linear stability analysis, it is shown that a phagocyte chemotactic response smaller than a critical value can lead to a non-uniform state, while a chemotactic response greater than this critical value stabilizes the uniform state. This result is the opposite of that found for the role of chemotaxis in aggregation of slimemold amoebae because, in the inflammatory response, the chemotactic population serves as an inhibitor rather than an activator. We speculate that these non-uniform steady states could be related to the localized cell aggregation seen in chronic granulomatous inflammation. The formation of non-uniform states is not necessarily a consequence of defective phagocyte chemotaxis, however. Rather, certain values of the kinetic parameters can yield values for the critical chemotactic response which are greater than the normal response.Numerical computations of the transient inflammatory response to bacterial challenge are presented, using parameter values estimated from the experimental literature wherever possible.     

Three steady state situation in an open chemical reaction system. I

In an isothermal continuously stirred tank reactor (open chemical reaction system) fed by sulphuric acid solutions of bromate, bromide and cerium)(III) bistability (three steady state situation) is experimentally observed. This remarkable behavior, based on the instability of one steady state, has important consequences for the understanding of excitability and biochemical control mechanisms. The mass-balance equations for the reactor and the chemical mechanism of the reaction are combined into a simple mathematical model. The behavior of the resulting nonlinear differential equations is examined analytically and by a graphical integration procedure (method of isoclines). Using realistic kinetic data, the model shows the same behavior as observed in the experiment.     

Morphologically accurate reduced order modeling of spiking neurons

Accurately simulating neurons with realistic morphological structure and synaptic inputs requires the solution of large systems of nonlinear ordinary differential equations. We apply model reduction techniques to recover the complete nonlinear voltage dynamics of a neuron using a system of much lower dimension. Using a proper orthogonal decomposition, we build a reduced-order system from salient snapshots of the full system output, thus reducing the number of state variables. A discrete empirical interpolation method is then used to reduce the complexity of the nonlinear term to be proportional to the number of reduced variables. Together these two techniques allow for up to two orders of magnitude dimension reduction without sacrificing the spatially-distributed input structure, with an associated order of magnitude speed-up in simulation time. We demonstrate that both nonlinear spiking behavior and subthreshold response of realistic cells are accurately captured by these low-dimensional models.     

Subthreshold oscillatory responses of the Hodgkin-Huxley cable model for the squid giant axon

The classical cable equation, in which membrane conductance is considered constant, is modified by including the linearized effect of membrane potential on sodium and potassium ionic currents, as formulated in the Hodgkin-Huxley equations for the squid giant axon. The resulting partial differential equation is solved by numerical inversion of the Laplace transform of the voltage response to current and voltage inputs. The voltage response is computed for voltage step, current step, and current pulse inputs, and the effect of temperature on the response to a current step input is also calculated.The validity of the linearized approximation is examined by comparing the linearized response to a current step input with the solution of the nonlinear partial differential cable equation for various subthreshold current step inputs.All the computed responses for the squid giant axon show oscillatory behavior and depart significantly from what is predicted on the basis of the classical cable equation. The linearization procedure, coupled with numerical inversion of the Laplace transform, proves to be a convenient approach which predicts at least qualitatively the subthreshold behavior of the nonlinear system.     

Steps toward numerical mode analysis of organizing systems

A well established method to analyze dynamical systems described by coupled nonlinear differential equations is to determine their normal modes and reduce the dynamics, by adiabatic elimination of stable modes, to a much smaller system for the amplitudes of unstable modes and their nonlinear interactions. So far, this analysis is possible only for idealized symmetric model systems. We aim to build a framework in which realistic systems with less symmetry can be analyzed automatically. In this paper we present a first example of mode analysis with the assistance of numerical computation. Our method is illustrated using a model system for the ontogenesis of retinotopy, and the results reproduce those from theoretical analysis precisely. Aspects of organization generalized from this model system are discussed. This research was supported by EU projects Daisy and SECO, and the Hertie Foundation.     

Traveling waves in a chemotactic model

A model for chemotaxis in a bacteria-substrate mixture introduced by Keller and Segel, which is described by nonlinear partial differential equations, is studied analytically. The existence of traveling waves is shown for the system in which the substrate diffusion is taken into account and the chemotactic coefficient is greater than the motility one, and the instability of traveling waves is discussed.     

Nonlinear equations of reaction-diffusion type for neural populations

Several simplified differential equations are derived from the Wilson and Cowan model describing the dynamics of excitatory and inhibitory neurons. It is shown, by expansions of the convolution integrals and the input-output functions, that the basic integrodifferential equations can be reduced to two coupled nonlinear partial differential equations of reaction-diffusion type. Further simplification leads to the coupled partial differential equations mathematically equivalent to the FitzHugh-Nagumo equations for the nerve impulse. Through a brief stability analysis in relation to the existing investigations on the bifurcation phenomena, an attempt is made to clarify the consequence due to the approximations introduced in this paper.     

Three-Dimensional Flow of an Oldroyd-B Nanofluid towards Stretching Surface with Heat Generation/Absorption

This article addresses the steady three-dimensional flow of an Oldroyd-B nanofluid over a bidirectional stretching surface with heat generation/absorption effects. Suitable similarity transformations are employed to reduce the governing partial differential equations into coupled nonlinear ordinary differential equations. These nonlinear ordinary differential equations are then solved analytically by using the homotpy analysis method (HAM). Graphically results are presented and discussed for various parameters, namely, Deborah numbers and , heat generation/absorption parameter Prandtl parameter , Brownian motion parameters, thermophoresis parameter and Lewis number . We have seen that the increasing values of the Brownian motion parameter and thermophoresis parameter leads to an increase in the temperature field and thermal boundary layer thickness while the opposite behavior is observed for concentration field and concentration boundary layer thickness. To see the validity of the present work, the numerical results are compared with the analytical solutions obtained by Homotopy analysis method and noted an excellent agreement for the limiting cases.     

一类两种群均有收获率捕食系统的细焦点与极限环

研究一类具有HollingⅡ类功能反应且两种群均为非常数收获率的捕食系统,其中食饵种群具有非线性密度制约.利用微分方程定性与稳定性理论及分支理论,得到系统平衡点的性态及极限环存在与否的充分条件,利用Hopf分支理论得到存到多个极限环的充分条件.     

Stochastic Collective Movement of Cells and Fingering Morphology: No Maverick Cells

The classical approach to model collective biological cell movement is through coupled nonlinear reaction-diffusion equations for biological cells and diffusive chemicals that interact with the biological cells. This approach takes into account the diffusion of cells, proliferation, death of cells, and chemotaxis. Whereas the classical approach has many advantages, it fails to consider many factors that affect multicell movement. In this work, a multiscale approach, the Glazier-Graner-Hogeweg model, is used. This model is implemented for biological cells coupled with the finite element method for a diffusive chemical. The Glazier-Graner-Hogeweg model takes the biological cell state as discrete and allows it to include cohesive forces between biological cells, deformation of cells, following the path of a single cell, and stochastic behavior of the cells. Where the continuity of the tissue at the epidermis is violated, biological cells regenerate skin to heal the wound. We assume that the cells secrete a diffusive chemical when they feel a wounded region and that the cells are attracted by the chemical they release (chemotaxis). Under certain parameters, the front encounters a fingering morphology, and two fronts progressing against each other are attracted and correlated. Cell flow exhibits interesting patterns, and a drift effect on the chemical may influence the cells' motion. The effects of a polarized substrate are also discussed.     

Effects of initial-fixed charge density on pH-sensitive hydrogels subjected to coupled pH and electric field stimuli: a meshless analysis

In this paper, we study the effects of initial fixed-charge density on the response behavior of pH-sensitive hydrogels subjected to coupled stimuli, namely, solution pH and externally applied electric field. This is the first instance in which a coupled stimuli numerical analysis has been carried out for these polymer gels, which are used as active sensing/actuating elements in advanced biomicroelectromechanical systems devices. In this work, a chemo-electro-mechanical formulation, termed the multi-effect-coupling pH-stimulus (MECpH) model, is first presented. This mathematical model takes into account the ionic species diffusion, electric potential coupling, and large mechanical deformation. In addition, a correlation between the diffusive hydrogen ions and fixed-charge groups on the hydrogel polymeric chains is established based on the Langmuir absorption isotherm, and incorporated accordingly into the MECpH model. To solve the resulting highly nonlinear and highly coupled partial differential equations of this mathematical model, the Hermite-Cloud method, a novel true meshless technique, is employed. To demonstrate the accuracy and robustness the MECpH model, computed numerical results are compared with experimental data available from literature. Following this validation, several numerical studies are carried out to investigate the effects of initial fixed-charge density on the volumetric variations of these pH-stimulus-responsive hydrogels when immersed in buffered solutions.     

A new interpretation of the Keller-Segel model based on multiphase modelling

In this paper an alternative derivation and interpretation are presented of the classical Keller-Segel model of cell migration due to random motion and chemotaxis. A multiphase modelling approach is used to describe how a population of cells moves through a fluid containing a diffusible chemical to which the cells are attracted. The cells and fluid are viewed as distinct components of a two-phase mixture. The principles of mass and momentum balance are applied to each phase, and appropriate constitutive laws imposed to close the resulting equations. A key assumption here is that the stress in the cell phase is influenced by the concentration of the diffusible chemical. By restricting attention to one-dimensional cartesian geometry we show how the model reduces to a pair of nonlinear coupled partial differential equations for the cell density and the chemical concentration. These equations may be written in the form of the Patlak-Keller-Segel model, naturally including density-dependent nonlinearities in the cell motility coefficients. There is a direct relationship between the random motility and chemotaxis coefficients, both depending in an inter-related manner on the chemical concentration. We suggest that this may explain why many chemicals appear to stimulate both chemotactic and chemokinetic responses in cell populations. After specialising our model to describe slime mold we then show how the functional form of the chemical potential that drives cell locomotion influences the ability of the system to generate spatial patterns. The paper concludes with a summary of the key results and a discussion of avenues for future research.     

植被系统中植物与环境因子相互作用的动态模拟

本文在综合考虑现有植被发展动态模型的基础上提出包含植被,环境因子,人类或其它外力干扰三者间相互作用情况下的植被系统一般动态模型,并从控制论的角度阐述了植被与环境因子间的耦合,反馈作用。最后以线性逼近和微分拟合的方法求取模型的局部解,进而作出外延预测。实例分析表明,模型预测对实际观测有较好的跟踪性能。     

An Efficient, Nonlinear Stability Analysis for Detecting Pattern Formation in Reaction Diffusion Systems

Reaction diffusion systems are often used to study pattern formation in biological systems. However, most methods for understanding their behavior are challenging and can rarely be applied to complex systems common in biological applications. I present a relatively simple and efficient, nonlinear stability technique that greatly aids such analysis when rates of diffusion are substantially different. This technique reduces a system of reaction diffusion equations to a system of ordinary differential equations tracking the evolution of a large amplitude, spatially localized perturbation of a homogeneous steady state. Stability properties of this system, determined using standard bifurcation techniques and software, describe both linear and nonlinear patterning regimes of the reaction diffusion system. I describe the class of systems this method can be applied to and demonstrate its application. Analysis of Schnakenberg and substrate inhibition models is performed to demonstrate the methods capabilities in simplified settings and show that even these simple models have nonlinear patterning regimes not previously detected. The real power of this technique, however, is its simplicity and applicability to larger complex systems where other nonlinear methods become intractable. This is demonstrated through analysis of a chemotaxis regulatory network comprised of interacting proteins and phospholipids. In each case, predictions of this method are verified against results of numerical simulation, linear stability, asymptotic, and/or full PDE bifurcation analyses.     

Global analysis of a model of plasmid-bearing,plasmid-free competition in a chemostat with inhibitions

A model of competition between plasmid-bearing and plasmid-free organisms in a chemostat was proposed in a paper of Stephanopoulis and Lapidus. The model was in the form of a system of nonlinear ordinary differential equations. Such models were relevant to commercial production by genetically altered organisms in continuous culture. The analysis there was local. The rigorous global analysis was done in a paper of Hsu, Waltman and Wolkowicz in the case of the uninhibited specific growth rates. This paper provides a mathematically rigorous analysis of the global asymptotic behavior of the governing equations in the cases of combinations of inhibited and uninhibited specific growth rates.Research Supported by the National Council of Science, Republic of China     

Neurobiological models of two-choice decision making can be reduced to a one-dimensional nonlinear diffusion equation

The response behaviors in many two-alternative choice tasks are well described by so-called sequential sampling models. In these models, the evidence for each one of the two alternatives accumulates over time until it reaches a threshold, at which point a response is made. At the neurophysiological level, single neuron data recorded while monkeys are engaged in two-alternative choice tasks are well described by winner-take-all network models in which the two choices are represented in the firing rates of separate populations of neurons. Here, we show that such nonlinear network models can generally be reduced to a one-dimensional nonlinear diffusion equation, which bears functional resemblance to standard sequential sampling models of behavior. This reduction gives the functional dependence of performance and reaction-times on external inputs in the original system, irrespective of the system details. What is more, the nonlinear diffusion equation can provide excellent fits to behavioral data from two-choice decision making tasks by varying these external inputs. This suggests that changes in behavior under various experimental conditions, e.g. changes in stimulus coherence or response deadline, are driven by internal modulation of afferent inputs to putative decision making circuits in the brain. For certain model systems one can analytically derive the nonlinear diffusion equation, thereby mapping the original system parameters onto the diffusion equation coefficients. Here, we illustrate this with three model systems including coupled rate equations and a network of spiking neurons.     

A model for cell migration in non-isotropic fibrin networks with an application to pancreatic tumor islets

Cell migration, known as an orchestrated movement of cells, is crucially important for wound healing, tumor growth, immune response as well as other biomedical processes. This paper presents a cell-based model to describe cell migration in non-isotropic fibrin networks around pancreatic tumor islets. This migration is determined by the mechanical strain energy density as well as cytokines-driven chemotaxis. Cell displacement is modeled by solving a large system of ordinary stochastic differential equations where the stochastic parts result from random walk. The stochastic differential equations are solved by the use of the classical Euler–Maruyama method. In this paper, the influence of anisotropic stromal extracellular matrix in pancreatic tumor islets on T-lymphocytes migration in different immune systems is investigated. As a result, tumor peripheral stromal extracellular matrix impedes the immune response of T-lymphocytes through changing direction of their migration.