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Author: Emily Paige Harvey Publisher: ISBN: Category : Calcium Languages : en Pages : 151
Book Description
Oscillations in free intracellular calcium (Ca2+) concentration are known to act as signals in almost all cell types, transmitting messages which control cellular processes including muscle contraction, cellular secretion and neuronal firing. Due to the universal nature of calcium oscillations, understanding the physiological mechanisms that underlie them is of great importance. A key feature of intracellular calcium dynamics that has been found experimentally is that some physiological processes occur much faster than others. This leads to models with variables evolving on very different time scales. In this thesis we survey a range of representative models of intracellular calcium dynamics, using geometric singular perturbation techniques with the aim of determining the usefulness of these techniques and what their limitations are. We find that the number of distinct time scales and the number of variables evolving on each time scale varies between models, but that in all cases there are at least two time scales, with at least two slower variables. Using geometric singular perturbation techniques we identify parameter regimes in which relaxation oscillations are seen and those where canard induced mixed mode oscillations are present. We find that in some cases these techniques are very useful and explain the observed dynamics well, but that the theory is limited in its ability to explain the dynamics when there are three or more distinct time scales in a model. It has been proposed that a simple experiment, whereby a pulse of inositol (1,4,5)- trisphosphate (IP3) is applied to a cell, can be used to distinguish between two competing mechanisms which lead to calcium oscillations [53]. However, detailed mathematical investigation of models has identified an anomalous delay in the pulse responses of some models, making interpretation of the experimental data difficult [14]. In this thesis we find that the response of models to a pulse of IP3 can be understood in part by using geometric singular perturbation techniques. Using recently developed theory for systems with three or more slow variables, we find that the anomalous delay can be due to the presence of folded nodes and their corresponding canard solutions or due to the presence of a curve of folded saddles. This delay due to a curve of folded saddles is a novel delay mechanism that can occur in systems with three or more slow variables. Importantly, we find that in some models the response to a pulse of IP3 is contrary to predictions for all bifurcation parameter values, which invalidates the proposed experimental protocol.
Author: Emily Paige Harvey Publisher: ISBN: Category : Calcium Languages : en Pages : 151
Book Description
Oscillations in free intracellular calcium (Ca2+) concentration are known to act as signals in almost all cell types, transmitting messages which control cellular processes including muscle contraction, cellular secretion and neuronal firing. Due to the universal nature of calcium oscillations, understanding the physiological mechanisms that underlie them is of great importance. A key feature of intracellular calcium dynamics that has been found experimentally is that some physiological processes occur much faster than others. This leads to models with variables evolving on very different time scales. In this thesis we survey a range of representative models of intracellular calcium dynamics, using geometric singular perturbation techniques with the aim of determining the usefulness of these techniques and what their limitations are. We find that the number of distinct time scales and the number of variables evolving on each time scale varies between models, but that in all cases there are at least two time scales, with at least two slower variables. Using geometric singular perturbation techniques we identify parameter regimes in which relaxation oscillations are seen and those where canard induced mixed mode oscillations are present. We find that in some cases these techniques are very useful and explain the observed dynamics well, but that the theory is limited in its ability to explain the dynamics when there are three or more distinct time scales in a model. It has been proposed that a simple experiment, whereby a pulse of inositol (1,4,5)- trisphosphate (IP3) is applied to a cell, can be used to distinguish between two competing mechanisms which lead to calcium oscillations [53]. However, detailed mathematical investigation of models has identified an anomalous delay in the pulse responses of some models, making interpretation of the experimental data difficult [14]. In this thesis we find that the response of models to a pulse of IP3 can be understood in part by using geometric singular perturbation techniques. Using recently developed theory for systems with three or more slow variables, we find that the anomalous delay can be due to the presence of folded nodes and their corresponding canard solutions or due to the presence of a curve of folded saddles. This delay due to a curve of folded saddles is a novel delay mechanism that can occur in systems with three or more slow variables. Importantly, we find that in some models the response to a pulse of IP3 is contrary to predictions for all bifurcation parameter values, which invalidates the proposed experimental protocol.
Author: Nathan Pages Publisher: ISBN: Category : Calcium ions Languages : en Pages : 127
Book Description
This thesis considers models of intracellular calcium ions (Ca2+). We aim to show how mathematical modelling can help us understand Ca2+ dynamics and how the investigation of Ca2+ dynamics models can motivate the development of new mathematical tools. The first part of the thesis presents a model of Ca2+ dynamics in parotid acinar cells. This model is simulated using a finite element method on an anatomically accurate reconstruction of a cluster of cells. Parotid acinar cells are exocrine cells; therefore, the Ca2+ model is coupled with a fluid flow model. From simulations, we gathered three main results. Firstly, the structure of the cell determines which of the possible mechanisms can create the observed Ca2+ concentration oscillations. Secondly, a wave propagation mechanism is needed to transport the Ca2+ oscillation from the apical to the basal region; we propose a mechanism based on calcium-induced calcium-release channels. Finally, there is a strong co-dependence between fluid secretion and Ca2+ dynamics; therefore, it is necessary to model fluid secretion alongside Ca2+ dynamics. Geometric singular perturbation theory (GSPT) in its classical form, which assumes that each variable is associated with a distinct timescale, has previously been used to study Ca2+ dynamics problems with multiple timescales. However, this association is not valid in general and particularly for models of Ca2+ dynamics; instead, a non-standard form of GSPT, which does not rely on the separation of variables by timescale, is more appropriately used for the analysis of Ca2+ models. We applied non-standard GSPT to a simplified canonical model of Ca2+ dynamics to explain the structure of its relaxation oscillations. We linked timescales to distinct physiological processes underlying different terms in the model, making possible a physiological interpretation of the analysis. Our approach overcomes problems that arise when using classical GSPT. Specifically, we were able to study models that exhibit more timescales than variables and in which a variable can be characterised as either fast or slow depending on the position in phase space. Our strategy of identifying timescales in a model based on careful consideration of the underlying physiology is quite general and is expected to be useful for other Ca2+ dynamics models or process-based models with multiple timescales.
Author: Wenjun Zhang Publisher: ISBN: Category : Wave functions Languages : en Pages : 121
Book Description
Oscillations in cytoplasmic calcium concentration are a crucial control mechanism in almost every cell type. Two important classes of oscillation are of particular interest: solitary and periodic waves. Both types of waves are commonly observed in physical experiments and found in mathematical models of calcium dynamics and other excitable systems. In this thesis, we try to understand these two classes of wave solutions. We first investigate wave solutions of the canonical excitable model, the FitzHugh-Nagumo (FHN) equations. We analyze the FHN equations using geometric singular perturbation theory and numerical integration, and find some new codimension-two organizing centres of the overall dynamics. Many analytical results about the FHN model in its classical form have already been established. We devise a transformation to change the form of the FHN equations we study into the classical form to make use of the results. This enables us to show how basic features of the bifurcation structure of the FHN equations arise from the singular limit. We then study waves of a representative calcium model. We analyze the dynamics of the calcium model in the singular limit, and show how homoclinic and Hopf bifurcations of the full system arise as perturbations of singular homoclinic and Hopf bifurcations. We compare the wave solutions in the FHN model and the calcium model, and show that the dynamics of the two models differ in some respects (most importantly, in the way in which diffusion enters the equations). We conclude that the FHN model should not uniformly be used as a prototypical model for calcium dynamics. Motivated by phenomena seen in the FHN and calcium models, we then investigate reduction techniques for excitable systems, including the quasi-steady state approximation and geometric singular perturbation theory, and show that criticality of Hopf bifurcations may be changed when applying these reduction methods to slow-fast biophysical systems. This suggests that great care should be taken when using reduction techniques such as these, to ensure that spurious conclusions about the dynamics of a model are not drawn from the dynamics of a reduced version of the model. Finally, we describe the class of numerical algorithms used to compute features of the detailed bifurcation sets for the FHN and calcium models, and show how these were used to locate a non-structurally stable heteroclinic connection between periodic orbits in a calcium model; this is the first time such a global bifurcation has been computed.
Author: Richard Bertram Publisher: Springer ISBN: 3319181149 Category : Mathematics Languages : en Pages : 120
Book Description
This book contains two review articles on mathematical physiology that deal with closely related topics but were written and can be read independently. The first article reviews the basic theory of calcium oscillations (common to almost all cell types), including spatio-temporal behaviors such as waves. The second article uses, and expands on, much of this basic theory to show how the interaction of cytosolic calcium oscillators with membrane ion channels can result in highly complex patterns of electrical spiking. Through these examples one can see clearly how multiple oscillatory processes interact within a cell, and how mathematical methods can be used to understand such interactions better. The two reviews provide excellent examples of how mathematics and physiology can learn from each other, and work jointly towards a better understanding of complex cellular processes. Review 1: Richard Bertram, Joel Tabak, Wondimu Teka, Theodore Vo, Martin Wechselberger: Geometric Singular Perturbation Analysis of Bursting Oscillations in Pituitary Cells Review 2: Vivien Kirk, James Sneyd: Nonlinear Dynamics of Calcium
Author: Geneviève Dupont Publisher: Springer ISBN: 3319296477 Category : Mathematics Languages : en Pages : 453
Book Description
This book discusses the ways in which mathematical, computational, and modelling methods can be used to help understand the dynamics of intracellular calcium. The concentration of free intracellular calcium is vital for controlling a wide range of cellular processes, and is thus of great physiological importance. However, because of the complex ways in which the calcium concentration varies, it is also of great mathematical interest.This book presents the general modelling theory as well as a large number of specific case examples, to show how mathematical modelling can interact with experimental approaches, in an interdisciplinary and multifaceted approach to the study of an important physiological control mechanism. Geneviève Dupont is FNRS Research Director at the Unit of Theoretical Chronobiology of the Université Libre de Bruxelles; Martin Falcke is head of the Mathematical Cell Physiology group at the Max Delbrück Center for Molecular Medicine, Berlin; Vivien Kirk is an Associate Professor in the Department of Mathematics at the University of Auckland, New Zealand; James Sneyd is a Professor in the Department of Mathematics at The University of Auckland, New Zealand.
Author: Martin Falcke Publisher: Springer ISBN: 9783662143988 Category : Science Languages : en Pages : 300
Book Description
Written as a set of tutorial reviews on both experimental facts and theoretical modelling, this volume is intended as an introduction and modern reference in the field for graduate students and researchers in biophysics, biochemistry and applied mathematics.
Author: Peter E. Kloeden Publisher: Springer ISBN: 3319030809 Category : Mathematics Languages : en Pages : 326
Book Description
Nonautonomous dynamics describes the qualitative behavior of evolutionary differential and difference equations, whose right-hand side is explicitly time dependent. Over recent years, the theory of such systems has developed into a highly active field related to, yet recognizably distinct from that of classical autonomous dynamical systems. This development was motivated by problems of applied mathematics, in particular in the life sciences where genuinely nonautonomous systems abound. The purpose of this monograph is to indicate through selected, representative examples how often nonautonomous systems occur in the life sciences and to outline the new concepts and tools from the theory of nonautonomous dynamical systems that are now available for their investigation.
Author: Emily Paige Harvey
Author: Emily Paige Harvey
Author: Nathan Pages
Author: Wenjun Zhang
Author: Richard Bertram
Author: Geneviève Dupont
Author: Alireza Atri
Author: Ritu Agarwal
Author: Martin Falcke
Author: Marah Townzen Funk
Author: Peter E. Kloeden