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Mathematical modeling and computer simulations have become an integral part of modern biological research. The strength of theoretical approaches is in the simplification of complex biological systems. We here consider the general problem of receptor-ligand binding in the context of antibody-antigen binding. On the one hand, we establish a quantitative mapping between macroscopic binding rates of an ordinary differential equation model and their microscopic equivalents as obtained from simulating the spatio-temporal binding kinetics by agent-based models. On the other hand, we investigate the impact of various receptor properties - such as their dimensionality of motion, morphology and binding valency - on the receptor-ligand binding kinetics. To this end, we implemented an algorithm that simulates antigen binding by B cell derived receptors with a Y-shaped morphology that can move in different dimensionalities, i.e. either as membrane-anchored receptors or as soluble receptors. The mapping of the

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Abstract , References , Similar Articles , Additional Information Abstract: This paper considers the problem of stabilizing a control system governed by a combination of partial and ordinary differential equations. The partial differential equations govern the evolution of the system in the interior of some spatial domain, and the ordinary differential equations describe the evolution of the boundary data; the control enters through the boundary ordinary differential equations in a bilinear fashion. We provide sufficient conditions for feedback stabilization of such ``hybrid systems. Two examples to wave equations with dynamic boundary conditions are provided ...

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TY - JOUR. T1 - Tumor growth dynamics with nutrient limitation and cell proliferation time delay. AU - Alsheri, Ahuod. AU - Alzahrani, Ebraheem O.. AU - Asiri, Asim. AU - El-Dessoky, Mohamed M.. AU - Kuang, Yang. PY - 2017/12/1. Y1 - 2017/12/1. N2 - It is known that avascular spherical solid tumors grow monotonically, often tends to a limiting final size. This is repeatedly confirmed by various mathematical models consisting of mostly ordinary differential equations. However, cell growth is limited by nutrient and its proliferation incurs a time delay. In this paper, we formulate a nutrient limited compartmental model of avascular spherical solid tumor growth with cell proliferation time delay and study its limiting dynamics. The nutrient is assumed to enter the tumor proportional to its surface area. This model is a modification of a recent model which is built on a two-compartment model of cancer cell growth with transitions between proliferating and quiescent cells. Due to the limitation of ...

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The reconstruction of gene regulatory networks from time series gene expression data is one of the most difficult problems in systems biology. This is due to several reasons, among them the combinatorial explosion of possible network topologies, limited information content of the experimental data with high levels of noise, and the complexity of gene regulation at the transcriptional, translational and post-translational levels. At the same time, quantitative, dynamic models, ideally with probability distributions over model topologies and parameters, are highly desirable. We present a novel approach to infer such models from data, based on nonlinear differential equations, which we embed into a stochastic Bayesian framework. We thus address both the stochasticity of experimental data and the need for quantitative dynamic models. Furthermore, the Bayesian framework allows it to easily integrate prior knowledge into the inference process. Using stochastic sampling from the Bayes posterior distribution,

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ABSTRACT:We consider a minimal cascade model previously proposed ~1 for the mitotic oscillator driving the embryonic cell division cycle. The model is based on a bicyclic phosphorylation-dephosphorylation cascade involving cyclin and cdc2 kinase. By constructing stability diagrams showing domains of periodic behavior as a function of the maximum rates of the kinases and phosphatases involved in the two cycles of the cascade, we investigate the role of these converter enzymes in the oscillatory mechanism. Oscillations occur when the balance of kinase and phosphatase rates in each cycle is in a range bounded by two critical values. The results suggest ways to arrest the mitotic oscillator by altering the maximum rates of the converter enzymes. These results bear on the control of cell proliferation. The original paper reference is cited below: Arresting the mitotic oscillator and the control of cell proliferation: insights from a cascade model for cdc2 kinase activation, Goldbeter, A. and ...

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ABSTRACT:We consider a minimal cascade model previously proposed ~1 for the mitotic oscillator driving the embryonic cell division cycle. The model is based on a bicyclic phosphorylation-dephosphorylation cascade involving cyclin and cdc2 kinase. By constructing stability diagrams showing domains of periodic behavior as a function of the maximum rates of the kinases and phosphatases involved in the two cycles of the cascade, we investigate the role of these converter enzymes in the oscillatory mechanism. Oscillations occur when the balance of kinase and phosphatase rates in each cycle is in a range bounded by two critical values. The results suggest ways to arrest the mitotic oscillator by altering the maximum rates of the converter enzymes. These results bear on the control of cell proliferation. The original paper reference is cited below: Arresting the mitotic oscillator and the control of cell proliferation: insights from a cascade model for cdc2 kinase activation, Goldbeter, A. and ...

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Title: Stochastic and deterministic spatial models for complex systems Abstract: Interacting particle models are often employed to gain understanding of the emergence of macroscopic phenomena from microscopic laws of nature. These individual-based models capture fine details, including randomness and discreteness of individuals, that are not considered in continuum models such as partial differential equations (PDE) and integral-differential equations. The challenge, which is fundamental in any multi-scale modeling approach for complex systems, is how to simultaneously retain key information in microscopic models as well as efficiency and robustness of macroscopic models. In this talk, I will discuss how this challenge can be overcome by elucidating the probabilistic connections between particle models and PDE, in particular, why naively adding diffusion terms to ordinary differential equations might fail to account for spatial dynamics in population models. These connections also explain how ...

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Physiologically-based pharmacokinetic (PBPK) modelling is traditionally employed to predict drug concentration-time profiles in plasma and tissues using information from physiology / biology, in vitro experiments and in silico predictions. Model-based analysis of population pharmacokinetic (PK) data is rarely performed in such a mechanistic framework, as empirical compartmental models are mainly utilised for this purpose. However, the combination of traditional PBPK methodologies with parameter estimation techniques and non-linear mixed effects modelling is an approach with progressively increasing impact due to the significant advantages it offers. Therefore, the general aim of this thesis is to illustrate, explore and thus further facilitate the application of physiologically-based pharmacokinetic models in the context of population data analysis.In order to pursue this aim, this work firstly particularly focuses on the population pharmacokinetics of simvastatin (SV) and its active metabolite, ...

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TY - BOOK. T1 - Recent progress on reaction-diffusion systems and viscosity solutions. AU - Du, Yihong. AU - Ishii, Hitoshi. AU - Lin, Wei Yueh. PY - 2009/1/1. Y1 - 2009/1/1. N2 - This book consists of survey and research articles expanding on the theme of the International Conference on Reaction-Diffusion Systems and Viscosity Solutions, held at Providence University, Taiwan, during January 3-6, 2007. It is a carefully selected collection of articles representing the recent progress of some important areas of nonlinear partial differential equations. The book is aimed for researchers and postgraduate students who want to learn about or follow some of the current research topics in nonlinear partial differential equations. The contributors consist of international experts and some participants of the conference, including Nils Ackermann (Mexico), Chao-Nien Chen (Taiwan), Yihong Du (Australia), Alberto Farina (France), Hitoshi Ishii (Japan), N Ishimura (Japan), Shigeaki Koike (Japan), Chu-Pin ...

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In this paper we investigate additional regularity properties for global and trajectory attractors of all globally defined weak solutions of semi-linear parabolic differential reaction-diffusion equations with discontinuous nonlinearities, when initial data uτ ∈ L2(Ω). The main contributions in this paper are: (i) sufficient conditions for the existence of a Lyapunov function for all weak solutions of autonomous differential reaction-diffusion equations with discontinuous and multivalued interaction functions; (ii) convergence results for all weak solutions in the strongest topologies; (iii) new structure and regularity properties for global and trajectory attractors. The obtained results allow investigating the long-time behavior of state functions for the following problems: (a) a model of combustion in porous media; (b) a model of conduction of electrical impulses in nerve axons; (c) a climate energy balance model; (d) a parabolic feedback control problem. ...

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###### A multiscale model reduction method for nonlinear monotone elliptic equations in heterogeneous media

|p style=text-indent:20px;|In this paper, we present a multiscale model reduction framework within Generalized Multiscale Finite Element Method (GMsFEM) for nonlinear elliptic problems. We consider an exemplary problem, which consists of nonlinear p-Laplacian with heterogeneous coefficients. The main challenging feature of this problem is that local subgrid models are nonlinear involving the gradient of the solution (e.g., in the case of scale separation, when using homogenization). Our main objective is to develop snapshots and local spectral problems, which are the main ingredients of GMsFEM, for these problems. Our contributions can be summarized as follows. (1) We re-cast the multiscale model reduction problem onto the boundaries of coarse cells. This is important and allows capturing separable scales as discussed. (2) We introduce nonlinear eigenvalue problems in the snapshot space for these nonlinear harmonic functions. (3) We present convergence analysis and numerical results, which show that

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The (asymptotic) behaviour of the second moment of solutions to stochastic differential equations is treated in mean-square stability analysis. This property is discussed for approximations of infinite-dimensional stochastic differential equations and necessary and sufficient conditions ensuring mean-square stability are given. They are applied to typical discretization schemes such as combinations of spectral Galerkin, finite element, Euler-Maruyama, Milstein, Crank-Nicolson, and forward and backward Euler methods. Furthermore, results on the relation to stability properties of corresponding analytical solutions are provided. Simulations of the stochastic heat equation illustrate the theory.

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Abstract: The main bottleneck when performing computational fluid dynamics (CFD) simulations of combustion systems is the computation and integration of the highly non-linear and stiff chemical source terms. In recent times, machine learning has emerged as a promising tool to accelerate combustion chemistry, involving the use of regression models to predict the chemical source terms as functions of the thermochemical state of the system. However, combustion is a highly nonlinear phenomenon, and this often leads to divergence from the true solution when the neural network representation of chemical kinetics is integrated in time. This is because these approaches minimize the error during training without guaranteeing successful integration with ordinary differential equation (ODE) solvers. In this work, a novel neural ODE approach to combustion modeling, ChemNODE, is developed to address this issue. The source terms predicted by the neural network are integrated during training, and by ...

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###### Analysis and simulations of coupled bulk-surface reaction-diffusion systems on exponentially evolving volumes : Sussex...

In this article we present a system of coupled bulk-surface reaction-diffusion equations on exponentially evolving volumes. Detailed linear stability analysis of the homogeneous steady state is carried out. It turns out that due to the nature of the coupling (linear Robin-type boundary conditions) the characterisation of the dispersion relation in the absence and presence of spatial variation (i.e. diffusion), can be decomposed as a product of the dispersion relation of the bulk and surface models thereby allowing detailed analytical tractability. As a result we state and prove the conditions for diffusion-driven instability for systems of coupled bulk-surface reaction-diffusion equations. Furthermore, we plot explicit evolving parameter spaces for the case of an exponential growth. By selecting parameter values from the parameter spaces, we exhibit pattern formation in the bulk and on the surface in complete agreement with theoretical predictions.. ...

###### A forward particle tracking Eulerian-Lagrangian Localized Adjoint Method for solution of the contaminant transport equation in...

The contaminant transport equation is solved in three dimensions using the Eulerian-Lagrangian Localized Adjoint Method (ELLAM). Trilinear and finite volume test functions defined by the characteristics of the governing equation are employed and compared. Integrations are simplified by forward tracking of integration points along the characteristics. The resulting equations are solved using a preconditioned conjugate gradient method. The algorithm is coupled to a block-centered finite difference approximation of the groundwater flow equation similar to that used in the popular MODFLOW code. The ELLAM is tested by comparison with 1D and 3D analytic solutions. The method is then applied with random, spatially correlated hydraulic conductivities in a simulation of a tracer experiment performed on Cape Cod, Massachusetts. The linear test function ELLAM was found to perform better than the finite volume ELLAM. Both ELLAM formulations were found to be robust, computationally efficient and relatively ...

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###### Reduction of mathematical models of signal transduction networks: simulation-based approach applied to EGF receptor signalling....

en] Biological systems and, in particular, cellular signal transduction pathways are characterised by their high complexity. Mathematical models describing these processes might be of great help to gain qualitative and, most importantly, quantitative knowledge about such complex systems. However, a detailed mathematical description of these systems leads to nearly unmanageably large models, especially when combining models of different signalling pathways to study cross-talk phenomena. Therefore, simplification of models becomes very important. Different methods are available for model reduction of biological models. Importantly, most of the common model reduction methods cannot be applied to cellular signal transduction pathways. Using as an example the epidermal growth factor (EGF) signalling pathway, we discuss how quantitative methods like system analysis and simulation studies can help to suitably reduce models and additionally give new insights into the signal transmission and processing ...

###### DML-CZ - Czech Digital Mathematics Library: On the number of stationary patterns in reaction-diffusion systems

We study systems of two nonlinear reaction-diffusion partial differential equations undergoing diffusion driven instability. Such systems may have spatially inhomogeneous stationary solutions called Turing patterns. These solutions are typically non-unique and it is not clear how many of them exists. Since there are no analytical results available, we look for the number of distinct stationary solutions numerically. As a typical example, we investigate the reaction-diffusion system designed to model coat patterns in leopard and jaguar ...

###### An Algorithm for the Numerical Solution of Two-Sided Space-Fractional Partial Differential Equations : Computational Methods in...

We introduce an algorithm for solving two-sided space-fractional partial differential equations. The space-fractional derivatives we consider here are left-handed and right-handed Riemann-Liouville fractional derivatives which are expressed by using Hadamard finite-part integrals. We approximate the Hadamard finite-part integrals by using piecewise quadratic interpolation polynomials and obtain a numerical approximation of the space-fractional derivative with convergence order ...

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The global sensitivity analysis of a complex numerical model often calls for the estimation of variance-based importance measures, named Sobol indices. Metamodel-based techniques have been developed in order to replace the CPU time-expensive computer code with an inexpensive mathematical function, which predicts the computer code output. The common metamodel-based sensitivity analysis methods are well suited for computer codes with scalar outputs. However, in the environmental domain, as in many areas of application, the numerical model outputs are often spatial maps, which may also vary with time. In this paper, we introduce an innovative method to obtain a spatial map of Sobol indices with a minimal number of numerical model computations. It is based upon the functional decomposition of the spatial output onto a wavelet basis and the metamodeling of the wavelet coefficients by the Gaussian process. An analytical example is presented to clarify the various steps of our methodology. This ...

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###### Development of a Five-Degree-of-Freedom Seated Human Model and Parametric Studies for Its Vibrational Characteristics

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We present a method for orientation-selective edge detection and enhancement based on the irradiance transport equation. The proposed technique distinguishes the sign of the derivative of the intensity pattern along an arbitrarily selected direction. The method is based on the capacity of liquid-crystal displays to generate simultaneously a contrast reverted replica of the image displayed on it. When both images (the original one and its replica) are imagined across a slightly defocused plane, one obtains an image with enhanced first derivatives. Unlike most Fourier methods, the proposed technique works well with a low-coherence light source, and it does not require precise alignment. The proposed method does not involve numerical processing, and thus it could be potentially useful for processing large images in real-time applications. Validation experiments are presented.. © 2010 Optical Society of America. Full Article , PDF Article ...

###### Exercise class: Partial Differential Equations, Lecture 6, Part 1 - Exercise class: Partial Differential Equations - Giovanni...

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###### Mechanistic analysis of multi-omics datasets to generate kinetic parameters for constraint-based metabolic models | BMC...

Constraint-based modeling uses mass balances, flux capacity, and reaction directionality constraints to predict fluxes through metabolism. Although transcriptional regulation and thermodynamic constraints have been integrated into constraint-based modeling, kinetic rate laws have not been extensively used. In this study, an in vivo kinetic parameter estimation problem was formulated and solved using multi-omic data sets for Escherichia coli. To narrow the confidence intervals for kinetic parameters, a series of kinetic model simplifications were made, resulting in fewer kinetic parameters than the full kinetic model. These new parameter values are able to account for flux and concentration data from 20 different experimental conditions used in our training dataset. Concentration estimates from the simplified kinetic model were within one standard deviation for 92.7% of the 790 experimental measurements in the training set. Gibbs free energy changes of reaction were calculated to identify reactions that

###### A bistable reaction-diffusion system in a stretching flow - Nottingham ePrints

We examine the evolution of a bistable reaction in a one-dimensional stretching flow, as a model for chaotic advection. We derive two reduced systems of ordinary differential equations (ODEs) for the dynamics of the governing advection-reaction-diffusion partial differential equations (PDE), for pulse-like and for plateau-like solutions, based on a non-perturbative approach. This reduction allows us to study the dynamics in two cases: first, close to a saddle-node bifurcation at which a pair of nontrivial steady states are born as the dimensionless reaction rate (Damkoehler number) is increased, and, second, for large Damkoehler number, far away from the bifurcation. The main aim is to investigate the initial-value problem and to determine when an initial condition subject to chaotic stirring will decay to zero and when it will give rise to a nonzero final state. Comparisons with full PDE simulations show that the reduced pulse model accurately predicts the threshold amplitude for a pulse ...

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Physical Principles and Techniques of Protein Chemistry, Part A deals with the principles and application of selected physical methods in protein chemistry evaluation. This book is organized into nine chapters that cover microscopic, crystallographic, and electrophoretic techniques for protein conformational perturbations evaluation. This text first presents a general account of electron microscopy, its specimen preparation, optimum conditions for high resolution, measurement of electron micrographs, and illustrative examples of protein study. This book then examines the different types of maps from X-ray methods and the diffraction data from fibrous proteins. The subsequent chapters cover discussions on UV spectroscopy of proteins; luminescence properties of proteins and related compounds; and perturbation and flow methods for evaluation of proteins dynamic properties and rate constants. Other chapters deal with the evaluation of proteins dielectric properties using dielectric relaxation, electric