The main theme of this proposed work is on the mathematical and computational investigation of spatial-temporal dynamics of cell signaling, design and implementation of powerful numerical solvers to solve such models in complex and moving domains. The specificity of cellular responses to receptor stimulation is encoded by the spatial and temporal dynamics of downstream signaling networks. In recent years, it has become apparent that distinct spatial-temporal activation profiles of the same repertoire of signaling proteins result in different gene activation patterns and diverse physiological responses. In many cases, spatially localized scaffold proteins that bind and organize multiple proteins into complexes within a pathway have merged as essential factors in shaping the quantitative response behavior of a pathway. In order for a better exploration of spatially localized scaffold proteins in mitogen-activated protein kinase (MAPK) cascades, the first part of the project is to explore the roles of substrate sequestration in combination with multisite phosphorylation for the modulation of ultrasensitivity and multistabilities. For the second part of the project, guided by known experimental observations, the PI proposes to develop mathematical models to further computationally investigate how a spatially localized and moving scaffold interacts with other components in a cascade and promotes specific cellular responses with spatial-temporal dynamics, such as long-range signals, graded/binary responses, noise reduction and traveling waves. Furthermore, in order to meet the computational challenges and demands which arise from the mathematical models of above complicated biological systems, the PI will design and implement more efficient and more accurate numerical methods than are currently done for convection and reaction-diffusion coupled equations with complex and moving geometries in high-spatial dimensions. Through mathematical modeling and computational analysis, the PI hopes that this proposed work may be able to shed lights on such a fundamental question: What do localized scaffold proteins really do?
Errors in cellular information processing are responsible for a variety of life-threatening or chronic diseases, such as cancer, autoimmunity and diabetes. This project seeks to provide better understanding on proper signal propagation across the cell in space and time, and such quantitative studies closely combined with experiments will deepen and advance our understanding of signal transduction inside the cell and thus may lead to design new drugs for a better treatment of above-mentioned diseases. In addition, the computational tools developed in this work will make computational exploration of complex biological systems more efficient, by reducing simulation time and at the same time producing more accurate solution through the integrated use of fast numerical solvers, front tracking method and adaptive mesh refinement. The developed mathematical and computational methods are also expected to have a broad impact on the studies of a large class of many other biological systems when interactions and transport of many bio-chemical species are involved with complex and moving geometries, and other special target applications are but not limited to protein trafficking and embryonic development. In addition, a critical ingredient for the success of this and related projects is the education and training of the next generation of mathematicians with expertise in mathematical biology and computation. Therefore this research project will provide and enhance multi-disciplinary training at the interface among mathematics, scientific computing and biology for both graduate and undergraduate students.
The theme of this project is on the mathematical and computational investigation of cell signaling involving scaffolds, design and implementation of powerful numerical solvers to solve the proposed models. Overall the significant progress has been made. One graduate student supported by this project, Dr. Kanadpriya Basu, completed his Ph.D. thesis under the supervision of the PI in the summer of 2012. Currently the PI is supervising another Ph.D. graduate student, Mr. Sameed Ahmed. In addition, ten (10) journal papers have been published since 2010 supported by this award, and the new findings were presented at more than 30 professional conferences, workshops or department seminars. Moreover, The PI together with his colleagues have developed a new initiative course, "Modeling and computation for complex biological systems" (Math-728A, B), for both graduate and advanced undergraduate students with diverse backgrounds who are mainly interested in mathematical modeling and interdisciplinary research. This course has exposed the students with start-of-the-art mathematical modeling and computational tools to study complex systems that arise from a great variety of biological, physical and engineering applications. A list of the main findings that are directly related to this proposal is summarized as follows. 1: Scaffold proteins can enhance protein phosphorylation by facilitating an interaction between a protein kinase enzyme and its target substrate. In this work we consider a simple mathematical model of a scaffold protein and show that under specific conditions, the presence of the scaffold can substantially raise the likelihood that the resulting system will exhibit bistable behavior. Using deficiency theory and other methods, we also provide a number of necessary conditions for bistability, such as the presence of multiple phosphorylation sites and the dependence of the scaffold binding/unbinding rates on the number of phosphorylated sites (PLoS Computational Biology). 2: For reaction-diffusion-advection equations, the stiffness from the reaction and diffusion terms often requires very restricted time step size, while the nonlinear advection term may lead to a sharp gradient in localized spatial regions. It is challenging to design numerical methods that can efficiently handle both difficulties. In this paper, we couple IIF/cIIF with WENO methods using the operator splitting approach to solve reaction-diffusion-advection equations. In particular, we apply the IIF/cIIF method to the stiff reaction and diffusion terms and the WENO method to the advection term in two different splitting sequences. Calculation of local truncation error and direct numerical simulations for both splitting approaches show the second order accuracy of the splitting method, and linear stability analysis and direct comparison with other approaches reveals excellent efficiency and stability properties (Journal of Computational Physics). 3: The phosphorylation of a protein on multiple sites has been proposed to promote the switch-like regulation of protein activity. Recent theoretical work, however, indicates that multisite phosphorylation, by itself, is less effective at creating switchlike responses than had been previously thought. Here, using simple ordinary differential equations to represent phosphorylation, dephosphorylation, and binding/sequestration, we demonstrate that the combination of multisite phosphorylation and regulated substrate sequestration can produce a response that is both a good threshold and a good switch. Several strategies are explored, including both stronger and weaker sequestration with successive phosphorylations, as well as combinations that are more elaborate (Biophysical Journal). 4: Implicit integration factor (IIF) method, a class of efficient semi-implicit temporal scheme, was introduced recently for stiff reaction-diffusion equations. To reduce cost of IIF, compact implicit integration factor (cIIF) method was later developed for efficient storage and calculation of exponential matrices associated with the diffusion operators in two and three spatial dimensions for Cartesian coordinates with regular meshes. Unlike IIF, cIIF cannot be directly extended to other curvilinear coordinates, such as polar and spherical coordinates, due to the compact representation for the diffusion terms in cIIF. In this paper, we present a method to generalize cIIF for other curvilinear coordinates through examples of polar and spherical coordinates. In addition, we present a method for integrating cIIF with adaptive mesh refinement (AMR) to take advantage of the excellent stability condition for cIIF. Excellent performance of the new methods is observed (Journal of Computational Physics). 5: When developing efficient numerical methods for solving parabolic types of equations, severe temporal stability constraints on the time step are often required due to the high-order spatial derivatives and/or stiff reactions. In this paper, by treating the discretization matrices in diagonalized forms, we develop an efficient cIIF method for solving a family of semilinear fourth-order parabolic equations, in which the bi-Laplace operator is explicitly handled and the computational cost and storage remain the same as to the classic cIIF for second-order problems. In particular, the proposed method can deal with not only stiff nonlinear reaction terms but also various types of homogeneous or inhomogeneous boundary conditions. Numerical experiments are finally presented to demonstrate effectiveness and accuracy of the proposed method (Discrete and Continuous Dynamical Systems-B).
