Vertex Models of Epithelial Morphogenesis

The dynamic behavior of epithelial cell sheets plays a central role during numerous developmental processes. Genetic and imaging studies of epithelial morphogenesis in a wide range of organisms have led to increasingly detailed mechanisms of cell sheet dynamics. Computational models offer a useful means by which to investigate and test these mechanisms, and have played a key role in the study of cell-cell interactions. A variety of modeling approaches can be used to simulate the balance of forces within an epithelial sheet. Vertex models are a class of such models that consider cells as individual objects, approximated by two-dimensional polygons representing cellular interfaces, in which each vertex moves in response to forces due to growth, interfacial tension, and pressure within each cell. Vertex models are used to study cellular processes within epithelia, including cell motility, adhesion, mitosis, and delamination. This review summarizes how vertex models have been used to provide insight into developmental processes and highlights current challenges in this area, including progressing these models from two to three dimensions and developing new tools for model validation.     

Vertex Models of Epithelial Morphogenesis

The dynamic behavior of epithelial cell sheets plays a central role during numerous developmental processes. Genetic and imaging studies of epithelial morphogenesis in a wide range of organisms have led to increasingly detailed mechanisms of cell sheet dynamics. Computational models offer a useful means by which to investigate and test these mechanisms, and have played a key role in the study of cell-cell interactions. A variety of modeling approaches can be used to simulate the balance of forces within an epithelial sheet. Vertex models are a class of such models that consider cells as individual objects, approximated by two-dimensional polygons representing cellular interfaces, in which each vertex moves in response to forces due to growth, interfacial tension, and pressure within each cell. Vertex models are used to study cellular processes within epithelia, including cell motility, adhesion, mitosis, and delamination. This review summarizes how vertex models have been used to provide insight into developmental processes and highlights current challenges in this area, including progressing these models from two to three dimensions and developing new tools for model validation.     

Implementation of Contraction to Electrophysiological Ventricular Myocyte Models,and Their Quantitative Characterization via Post-Extrasystolic Potentiation

Heart failure (HF) affects over 5 million Americans and is characterized by impairment of cellular cardiac contractile function resulting in reduced ejection fraction in patients. Electrical stimulation such as cardiac resynchronization therapy (CRT) and cardiac contractility modulation (CCM) have shown some success in treating patients with HF. Computer simulations have the potential to help improve such therapy (e.g. suggest optimal lead placement) as well as provide insight into the underlying mechanisms which could be beneficial. However, these myocyte models require a quantitatively accurate excitation-contraction coupling such that the electrical and contraction predictions are correct. While currently there are close to a hundred models describing the detailed electrophysiology of cardiac cells, the majority of cell models do not include the equations to reproduce contractile force or they have been added ad hoc. Here we present a systematic methodology to couple first generation contraction models into electrophysiological models via intracellular calcium and then compare the resulting model predictions to experimental data. This is done by using a post-extrasystolic pacing protocol, which captures essential dynamics of contractile forces. We found that modeling the dynamic intracellular calcium buffers is necessary in order to reproduce the experimental data. Furthermore, we demonstrate that in models the mechanism of the post-extrasystolic potentiation is highly dependent on the calcium released from the Sarcoplasmic Reticulum. Overall this study provides new insights into both specific and general determinants of cellular contractile force and provides a framework for incorporating contraction into electrophysiological models, both of which will be necessary to develop reliable simulations to optimize electrical therapies for HF.     

Additive Damage Models for Cellular Pharmacodynamics of Radiation–Chemotherapy Combinations

Many cancer patients receive combination treatments with radiation and chemotherapy. Available mathematical models for cellular pharmacodynamics have limited ability to represent observed in vitro responses to radiochemotherapy. Here, a family of additive damage models is proposed to describe cell kill resulting from radiochemotherapy with fixed schedule and variable doses. The pathways by which the agents produce cellular damage are assumed to converge in a single cell death process, so that survival depends on total damage, which can be represented as a sum of contributions from the various damage pathways. Heterogeneity in response across the cell population is ascribed to variations in the damage threshold for cell kill. The family of proposed models includes effects of one or two pathways of damage for each agent, saturation in drug responses, and cooperative or antagonistic interactions between agents. Models from this family with 4–7 unknown parameters are tested for their ability to fit 218 in vitro literature data sets for a range of drugs and cell lines. Overall, the additive damage models are found to outperform models based on the existing concept of independent cell kill, according to the corrected Akaike Information Criterion. The results are used to assess the importance of the various effects included in the models. These additive damage models have potential applications to the optimization of treatment and to the analysis and interpretation of in vitro screening data for new drug–radiation combinations.     

Modelling of simple and complex calcium oscillations. From single-cell responses to intercellular signalling.

This review provides a comparative overview of recent developments in the modelling of cellular calcium oscillations. A large variety of mathematical models have been developed for this wide-spread phenomenon in intra- and intercellular signalling. From these, a general model is extracted that involves six types of concentration variables: inositol 1,4,5-trisphosphate (IP3), cytoplasmic, endoplasmic reticulum and mitochondrial calcium, the occupied binding sites of calcium buffers, and the fraction of active IP3 receptor calcium release channels. Using this framework, the models of calcium oscillations can be classified into 'minimal' models containing two variables and 'extended' models of three and more variables. Three types of minimal models are identified that are all based on calcium-induced calcium release (CICR), but differ with respect to the mechanisms limiting CICR. Extended models include IP3--calcium cross-coupling, calcium sequestration by mitochondria, the detailed gating kinetics of the IP3 receptor, and the dynamics of G-protein activation. In addition to generating regular oscillations, such models can describe bursting and chaotic calcium dynamics. The earlier hypothesis that information in calcium oscillations is encoded mainly by their frequency is nowadays modified in that some effect is attributed to amplitude encoding or temporal encoding. This point is discussed with reference to the analysis of the local and global bifurcations by which calcium oscillations can arise. Moreover, the question of how calcium binding proteins can sense and transform oscillatory signals is addressed. Recently, potential mechanisms leading to the coordination of oscillations in coupled cells have been investigated by mathematical modelling. For this, the general modelling framework is extended to include cytoplasmic and gap-junctional diffusion of IP3 and calcium, and specific models are compared. Various suggestions concerning the physiological significance of oscillatory behaviour in intra- and intercellular signalling are discussed. The article is concluded with a discussion of obstacles and prospects.     

Stochastic Models for Phototaxis

This work studies two mathematical models for describing the motion of phototactic bacteria, i.e., bacteria that move toward light. Based on experimental observations, we conjecture that the motion of the colony toward light depends on certain group dynamics. These group dynamics are hypothesized to be coordinated through an individual property of each bacterium, which we refer to as excitation. The excitation of each individual bacterium is assumed to change based on the excitation of the neighboring bacteria. Under these assumptions, we propose a (discrete) cellular automaton model and derive an analogous stochastic model for describing the evolution in time of the location of bacteria, the excitation of individual bacteria, and a surface memory effect. We provide simulation results and discuss in detail the role of the various model parameters in controlling the emerging dynamics.     

Models of cardiac excitation-contraction coupling in ventricular myocytes

Mathematical and computational modeling of cardiac excitation-contraction coupling has produced considerable insights into how the heart muscle contracts. With the increase in biophysical and physiological data available, the modeling has become more sophisticated with investigations spanning in scale from molecular components to whole cells. These modeling efforts have provided insight into cardiac excitation-contraction coupling that advanced and complemented experimental studies. One goal is to extend these detailed cellular models to model the whole heart. While this has been done with mechanical and electophysiological models, the complexity and fast time course of calcium dynamics have made inclusion of detailed calcium dynamics in whole heart models impractical. Novel methods such as the probability density approach and moment closure technique which increase computational efficiency might make this tractable.     

A computational modeling and analysis in cell biological dynamics using electric cell-substrate impedance sensing (ECIS)

In this paper, a study of computational modeling and multi-scale analysis in cell dynamics is presented. Our study aims at: (1) deriving and validating a mathematical model for cell growth, and (2) quantitatively detecting and analyzing the biological interdependencies across multiple observational scales with a variety of time and frequency resolutions. This research was conducted using the time series data practically measured from a novel on-line cell monitoring technique, referred to as electric cell-substrate impedance sensing (ECIS), which allows continuously tracking the cellular behavior such as adhesion, proliferation, spreading and micromotion. First, comparing our ECIS-based cellular growth modeling analysis results with those determined by hematocytometer measurement using different time intervals, we found that the results obtained from both experimental methods consistently agreed. However, our study demonstrated that it is much easier and more convenient to operate with the ECIS system for on-line cellular growth monitoring. Secondly, for multi-scale analysis our results showed that the proposed wavelet-based methodology can effectively quantify the fluctuations associated with cell micromotions and quantitatively capture the biological interdependencies across multiple observational scales. Note that although the wavelet method is well known, its application into the ECIS time series analysis is novel and unprecedented in computational cell biology. Our analyses indicated that the proposed study on ECIS time series could provide a hopeful start and great potentials in both modeling and elucidating the complex mechanisms of cell biological systems.     

Models of the inositol trisphosphate receptor

The inositol (1,4,5)-trisphosphate receptor (IPR) plays a crucial role in calcium dynamics in a wide range of cell types, and is often a central feature in quantitative models of calcium oscillations and waves. We review deterministic and stochastic mathematical models of the IPR, from the earliest ones of the 1970s and 1980s, to the most recent. The effects of IPR stochasticity on Ca2+ dynamics are briefly discussed.     

Multiscale bio-chemo-mechanical model of intimal hyperplasia

We consider a computational multiscale framework of a bio-chemo-mechanical model for intimal hyperplasia. With respect to existing models, we investigate the interactions between hemodynamics, cellular dynamics and biochemistry on the development of the pathology. Within the arterial wall, we propose a mathematical model consisting of kinetic differential equations for key vascular cell types, collagen and growth factors. The luminal hemodynamics is modeled with the Navier–Stokes equations. Coupling hypothesis among time and space scales are proposed to build a tractable modeling of such a complex multifactorial and multiscale pathology. A one-dimensional numerical test-case is presented for validation by comparing the results of the framework with experiments at short and long timescales. Our model permits to capture many cellular phenomena which have a central role in the physiopathology of intimal hyperplasia. Results are quantitatively and qualitatively consistent with experimental findings at both short and long timescales.

     

Development of a two-dimension manifold to represent high dimension mathematical models of the intracellular Mammalian circadian clock

A new focus for mathematical models of the circadian pacemaker involves the encapsulation within the models of detailed biological processes responsible for generating those circadian rhythms. Representing greater biological detail requires more mathematical equations, which pose a greater challenge for the analysis of such systems. Development of a method that retains the predominant dynamics while still providing biologically detailed information is advantageous. Two high-dimension mathematical models of intracellular mammalian circadian pacemakers, Leloup-Goldbeter and Forger-Peskin, with 19 and 73 differential equations, respectively, have been published. The authors projected each of these high-dimension models onto their respective manifold using proper orthogonal functions (POFs) obtained from the empirical decomposition of the model's phase space to obtain a 2-dimension model. The resulting 2-dimension model, represented by 2 differential equations, predicts most of the salient characteristics of a biological clock including approximately 24-h oscillations, entrainment to an LD cycle, phase response curves, and the amplitude recovery dynamics that emerge following amplitude suppression. The manifold representation simplifies the mathematical analysis, since only 2 variables need to be observed and analyzed to understand the behavior of the biological clock. This reduced model derived from a model based on biological variables can be used for the development and analysis of mathematical models of the coupled mammalian oscillators to understand the dynamics of the integrated circadian pacemaker.     

Analysis and Simulation of Division- and Label-Structured Population Models

In most biological studies and processes, cell proliferation and population dynamics play an essential role. Due to this ubiquity, a multitude of mathematical models has been developed to describe these processes. While the simplest models only consider the size of the overall populations, others take division numbers and labeling of the cells into account. In this work, we present a modeling and computational framework for proliferating cell populations undergoing symmetric cell division, which incorporates both the discrete division number and continuous label dynamics. Thus, it allows for the consideration of division number-dependent parameters as well as the direct comparison of the model prediction with labeling experiments, e.g., performed with Carboxyfluorescein succinimidyl ester (CFSE), and can be shown to be a generalization of most existing models used to describe these data. We prove that under mild assumptions the resulting system of coupled partial differential equations (PDEs) can be decomposed into a system of ordinary differential equations (ODEs) and a set of decoupled PDEs, which drastically reduces the computational effort for simulating the model. Furthermore, the PDEs are solved analytically and the ODE system is truncated, which allows for the prediction of the label distribution of complex systems using a low-dimensional system of ODEs. In addition to modeling the label dynamics, we link the label-induced fluorescence to the measure fluorescence which includes autofluorescence. Furthermore, we provide an analytical approximation for the resulting numerically challenging convolution integral. This is illustrated by modeling and simulating a proliferating population with division number-dependent proliferation rate.     

Models for Bounded Systems with Continuous Dynamics

Summary .  Models for natural nonlinear processes, such as population dynamics, have been given much attention in applied mathematics. For example, species competition has been extensively modeled by differential equations. Often, the scientist has preferred to model the underlying dynamical processes (i.e., theoretical mechanisms) in continuous time. It is of both scientific and mathematical interest to implement such models in a statistical framework to quantify uncertainty associated with the models in the presence of observations. That is, given discrete observations arising from the underlying continuous process, the unobserved process can be formally described while accounting for multiple sources of uncertainty (e.g., measurement error, model choice, and inherent stochasticity of process parameters). In addition to continuity, natural processes are often bounded; specifically, they tend to have nonnegative support. Various techniques have been implemented to accommodate nonnegative processes, but such techniques are often limited or overly compromising. This article offers an alternative to common differential modeling practices by using a bias-corrected truncated normal distribution to model the observations and latent process, both having bounded support. Parameters of an underlying continuous process are characterized in a Bayesian hierarchical context, utilizing a fourth-order Runge–Kutta approximation.     

Dynamics of Calcium Fluxes in Nonexcitable Cells: Mathematical Modeling

Mathematical models simulating the dynamics of calcium redistribution (elicited by experimental interference with the pathways of calcium fluxes) in cellular compartments have been developed, based on a minimal scheme of the pathways of calcium fluxes in nonexcitable cells suspended in calcium-free medium. The models are consistent with available experimental data. All parameters are quantitatively related to the intrinsic properties of calcium adenosine triphosphatases (ATPases) and cellular membranes; there is no interdependence between the parameters. The models can be used as the basis for quantitative analysis and interpretation of experimental data. The activities of plasma membrane and sarcoendoplasmic reticulum calcium ATPases (PMCA and SERCAs) are governed by different mechanisms. PMCA is likely to undergo transitions from inactive to active to “dormant” (not identical to the initial) and back to inactive states, the mean duration of the cycle lasting for minutes or longer. The sequence of the transitions is initiated, presumably, by an increase in cytosolic calcium concentration. The transition of PMCA from inactive to active (at least at low rates of increase in cytosolic calcium concentration) is likely to be slower than that from active to dormant. SERCA, presumably, transits from inactive to active state in response to increases in calcium leakage from calcium stores. Whereas PMCA extrudes excess calcium (a definite quantity of it) in a short pulse, SERCA retakes calcium back into the stores permanently at a high rate. The models presented here may be the best means for the moment to quantitatively relate the dynamics of calcium fluxes in nonexcitable cells with known or putative properties of the mechanisms underlying activation of calcium ATPases.     

由微分方程所描述的微生物连续培养动力系统(Ⅰ)

陆续介绍微生物连续培养(Chemostat)的基本原理,以单种微生物连续培养模型为基础,较详细地介绍几类由微分方程所描述的微生物连续培养动力系统模型,涉及的问题有解的稳定性,系统的持久性,周期解和Hopf分支等．     

具体长结构的鱼类种群的数学模型及其应用

本文报道了一个新的鱼类种群的数学模型,它是一阶变系数线性偏微分方程。该模型可用于预测某一水域中某种鱼类种群不同体长的鱼的数量随时间变化的情况。       

Stochastic E2F Activation and Reconciliation of Phenomenological Cell-Cycle Models

The transition of the mammalian cell from quiescence to proliferation is a highly variable process. Over the last four decades, two lines of apparently contradictory, phenomenological models have been proposed to account for such temporal variability. These include various forms of the transition probability (TP) model and the growth control (GC) model, which lack mechanistic details. The GC model was further proposed as an alternative explanation for the concept of the restriction point, which we recently demonstrated as being controlled by a bistable Rb-E2F switch. Here, through a combination of modeling and experiments, we show that these different lines of models in essence reflect different aspects of stochastic dynamics in cell cycle entry. In particular, we show that the variable activation of E2F can be described by stochastic activation of the bistable Rb-E2F switch, which in turn may account for the temporal variability in cell cycle entry. Moreover, we show that temporal dynamics of E2F activation can be recast into the frameworks of both the TP model and the GC model via parameter mapping. This mapping suggests that the two lines of phenomenological models can be reconciled through the stochastic dynamics of the Rb-E2F switch. It also suggests a potential utility of the TP or GC models in defining concise, quantitative phenotypes of cell physiology. This may have implications in classifying cell types or states.     

CompuCell, a multi-model framework for simulation of morphogenesis

MOTIVATION: CompuCell is a multi-model software framework for simulation of the development of multicellular organisms known as morphogenesis. It models the interaction of the gene regulatory network with generic cellular mechanisms, such as cell adhesion, division, haptotaxis and chemotaxis. A combination of a state automaton with stochastic local rules and a set of differential equations, including subcellular ordinary differential equations and extracellular reaction-diffusion partial differential equations, model gene regulation. This automaton in turn controls the differentiation of the cells, and cell-cell and cell-extracellular matrix interactions that give rise to cell rearrangements and pattern formation, e.g. mesenchymal condensation. The cellular Potts model, a stochastic model that accurately reproduces cell movement and rearrangement, models cell dynamics. All these models couple in a controllable way, resulting in a powerful and flexible computational environment for morphogenesis, which allows for simultaneous incorporation of growth and spatial patterning. RESULTS: We use CompuCell to simulate the formation of the skeletal architecture in the avian limb bud. AVAILABILITY: Binaries and source code for Microsoft Windows, Linux and Solaris are available for download from http://sourceforge.net/projects/compucell/     

Assessing Mathematical Models of Influenza Infections Using Features of the Immune Response

The role of the host immune response in determining the severity and duration of an influenza infection is still unclear. In order to identify severity factors and more accurately predict the course of an influenza infection within a human host, an understanding of the impact of host factors on the infection process is required. Despite the lack of sufficiently diverse experimental data describing the time course of the various immune response components, published mathematical models were constructed from limited human or animal data using various strategies and simplifying assumptions. To assess the validity of these models, we assemble previously published experimental data of the dynamics and role of cytotoxic T lymphocytes, antibodies, and interferon and determined qualitative key features of their effect that should be captured by mathematical models. We test these existing models by confronting them with experimental data and find that no single model agrees completely with the variety of influenza viral kinetics responses observed experimentally when various immune response components are suppressed. Our analysis highlights the strong and weak points of each mathematical model and highlights areas where additional experimental data could elucidate specific mechanisms, constrain model design, and complete our understanding of the immune response to influenza.     

The Squid Giant Axon: Mathematical Models

The voltage clamp results of Hodgkin and Huxley have been reanalyzed in terms of alternative mathematical models. The model used for the potassium conductance changes is similar to that of the HH model except that an empirical functional relationship replaces the fourth power Law used by HH and the twenty-fifth power law used by Cole and Moore. The model used for the sodium conductance changes involves the explicit use of one variable only rather than the two variables m and h of HH. The rise and fall of the sodium conductance during a depolarizing voltage clamp is obtained by specifying that this one variable satisfies a second order differential equation which results from the coupling of two first order equations. Not only can the adjustable parameters of these models be made to give good fit to the clamp conductance data but the models can also then be used to compute action potential curves. Theoretical interpretations can also be given to these mathematical models.