Interest in stochastic modeling of biochemical processes has increased over the past two decades due to advancements in computing power and an increased understanding of the underlying biology. The Gillespie algorithm is an exact simulation technique for reproducing sample paths from a continuous-time Markov chain. However, when spatial and temporal time scales vary within a given system, a purely stochastic approach becomes intractable. In this work, we discuss two types of hybrid approximations, namely piecewise-deterministic and quasi-equilibrium approximations. These approaches yield strong approximations for either the entire biochemical system or a subset of the system, provided the purely stochastic system is appropriately rescaled.
We first develop a class of stochastic models to study the roll of stem cells' division patterns in tumor's initiation. Then, we investigate the process of tumor's initiation from the site of chronic inflammation and its progression. We propose that metastases could be the result of wound healing process by adapted immune cells in the sites of inflammation. From this hypothesis, we predict that any treatments generating necrotic cells anywhere in body including in the tumor microenvironment, which is orchestrated by adapted immune, stromal, and/or epithelial cells may not be a good solution.
Invasive aspergillosis is a condition caused by inhalation of the ubiquitous fungus Aspergillus fumigatus. In immuno-compromised patients, mortality rates are as high as 30%, with the fungus remaining resistant to therapeutic treatments. A three-dimensional agent-based simulation of the immune response to A. fumigatus is presented, focusing on the first 96 hours after inoculation. The simulation incorporates macrophages, neutrophils, epithelial cells, and cytokine diffusion. Free iron is highly regulated by the immune system, but the fungus requires iron in order to grow. Thus, the model emphasizes the battle for iron between host and pathogen. The model is built from the bottom up based on literature and basic mechanistic rules. Results from parameter sensitivity are discussed, not only in terms of robustness but also to shed some light on key mechanisms in the development of invasive aspergillosis. The agent-based simulation serves as an in silico laboratory, allowing for investigation of key parameters and cell behaviors that may provide insight into the future of therapeutic intervention.
Understanding genetic foundations of disease and drug response is crucial for development of patient-specific therapies. Improvement of genomic technologies has enabled identification of numerous candidate variants associated with disease risk or altering drug responses; however, the mechanisms how genetic factors influence individual differences remain poorly understood. The vast majority of variants identified in genome-wide association studies (GWAS) reside outside the protein coding region influencing gene transcription. SNPs localized in transcribed regions can also affect RNA processing, alternative splicing, and translation, in addition to directly altering the encoded amino acid sequence. Despite the importance of translation and the role of alternative splicing in cellular responses to environmental stimuli, the role of genetic factors in modulating RNA-centered processes is less well understood. The goal my study is to identify clinically relevant genetic variants affecting gene expression. In my work I use RNAseq-based high throughput approach that enables transcriptome-wide analysis of expression levels of various classes of RNA. A small pilot study has already revealed numerous clinically relevant regulatory variants that affect translation. I will expand this study to identify genetic variants influencing gene expression in various biological systems. Understanding the molecular and genetic mechanisms of gene expression is essential for understanding candidate risk alleles identified in large-scale association studies, leading to predictive models of disease risk and treatment response, key factors in personalized medicine.
Retroviral capsids are fullerene-like, consisting of capsid protein hexamers and pentamers. Mathematical models for the lattice structure help understand the underlying biological mechanisms in the formation of viral capsids. It is known that viral capsids could be categorized into three major types: icosahedron, tube, and cone. While the model for icosahedral capsids is established and well-received, models for tubular and conical capsids need further investigation. This talk will give an overview of current methods for defining icosahedral capsids and propose new models for the tubular and conical capsids based on an extension of the Caspar-Klug quasi-equivalence theory.
Parameters play important roles in biological systems. In this talk, I will present some recent numerical methods on parameter study of nonlinear partial differential equations (PDEs) arising in biology. These numerical methods include homotopy continuation, bootstrapping and reduced basis methods. They make use of polynomial systems (with thousands of variables) arising by discretization, and can be used to compute bifurcation points as well as multiple solutions. This talk will also cover the applications of these numerical methods to several biological models including atherosclerosis where the critical parameters are the LDL and HDL serum concentration, and blood clotting model where high dimensional parameters are explored.
Diabetes affects millions of Americans, and correctly identifying individuals afflicted with this disease, especially those in early stages or in progression towards diabetes, remains an active area of study. A crucial aspect of diabetes research lies in appropriately describing the dynamic relationship between glucose and insulin. Multiple mathematical constructs have been suggested for investigating glucose-insulin interactions, but perhaps the most well-known and widely used was suggested by Richard Bergman, Claudio Cobelli, and colleagues by the early 1980s. Their so-called /minimal model /utilizes a system of ordinary differential equations to describe glucose clearance during an intravenous glucose tolerance test (IVGTT). The usual methods for parameterizing the rate coefficients of this system rely heavily upon numerical ODE solvers; however, such solutions do not yield a straightforward result for the glucose as a function of time.
We have developed an alternative approach for resolving the Bergman-Cobelli minimal model, which hinges upon explicitly solving the glucose subsystem via analytic techniques. We use of an auxiliary intermediary function to represent the glucose concentration exactly, and we achieve patient-specific parameterization by using IVGTT data and calculating a constrained optimization problem. Because we explicitly solve for the glucose dynamics, our proposed method is not only easier to fit with an individual's data but is also significantly more transparent regarding the time evolution of the glucose concentration. Our approach also permits the development of other results, such as predicted time until baseline return, that are not possible with current ODE minimal model solvers.
Heterogeneity in contact network structure is known to play an important role in the spread of epidemics. Models taking full network structure into account quickly become intractable as the size of the network increases. Pair approximation techniques have been widely used and an alternative edge-based compartmental model has been proven to be the large graph limit of the SIR stochastic system on a class of random graphs with a specified degree distribution. We determine a sufficient condition on the degree distribution under which the edge-based and pair approximation models agree. Based on this result, we formulate a tractable hybrid stochastic-deterministic model which allows for efficient parameter estimation. We generalize to the case of a dynamic, multilayer graph and prove a law of large numbers showing convergence of the stochastic system to a deterministic pairwise model. The motivations for this framework in Ebola modeling will also be discussed.
The mitotic cell cycle is commonly depicted as movement around a circle starting from cell division, where the cell traverses various growth phases before dividing and starting over at the beginning. Mathematicians would call this an oscillator, and when many cells in a large population are able to affect one another’s progression through the cell cycle, we have a system of coupled oscillators. Systems of coupled oscillators have a rich history that goes back to the seventeenth century, and they provide a natural setting to investigate the phenomenon of synchronization. What then when such a system is out of synch – as observed in autonomous oscillations in yeast? A type of coupled oscillator system called Response / Signaling, where the interaction terms are phase dependent, provides an answer in the form of clustering. Existence, uniqueness, and stability of clustered solutions can all be established in certain cases, and numerical simulations confirm a more surprising result: a fractal structure on the parameter space determines the number of clusters that will form. Adding a variable to represent a chemical which mediates the intercellular signaling results in a cascade of bifurcations that biologists would call Quorum Sensing, as increasing population density changes the stability of certain clustered solutions. Passing to the limit as the number of cells approaches infinity results in a nonlocal partial differential equation requiring new mathematical ideas to solve.
We live in an environment that is constantly changing. On a large time scale, climate change has a global effect on the dynamics of plant and animal populations. On a smaller scale, there are seasonal changes of local habitats, for example, flooding and drying of wetland habitats. In this talk, I will present a spatial perspective of the effects of environmental changes. What happens when the suitable habitat of a population changes its location, or its size over time? Are there limits of the population’s ability to cope with these spatial changes? What if the changes have different trends on different time scales? I will present a set of mathematical models aiming at answering these questions.
Abstract not submitted
Biology occurs in concentrated mixtures of ions that are far from ideal. Solutions are remarkably concentrated where they are most important. Multi-molar solutions (often >10M) are found in and near DNA & RNA, proteins, active sites of enzymes, transporters, and ion channels, as well as electrodes of batteries. Poisson Boltzmann treatments fail dramatically to deal with the upper bound of concentration produced by the finite size of ions and so cannot describe ionic solutions of life, or ions where they are most important.
Jinn-Liang Liu derived the Fermi distribution for mixtures of spheres of any diameter and density. We combined this distribution with electrodynamics and diffusion equations to produce PNPF (Poisson-Nernst-Planck-Fermi). PNPF yields a fourth order pde equation (due to Santangelo) for concentrated ionic solutions with polarizable spherical water, and interstitial defects. This model accounts for the measured properties of calcium channels (so important in biology in general and the heart in particular). It accounts for the activity of bulk solutions of sodium and calcium using one unchanging parameter. The model computes efficiently in three dimensions (because of numerical methods developed for semiconductors applied to polarizable spherical water with an essential stabilization from the interstitial defects).
It is essential of course to use a fully consistent approach in which electrodynamics equations are always satisfied. Sadly, Markov and mass action models are local in their essence and so cannot deal with the global nature of current flow: Interrupting current far from an electrochemical reaction changes the local flow on an atomic scale as guaranteed by electrodynamics that demands perfect conservation of current everywhere always at any one time. We hope PNPF may be a useful replacement for Markov and mass action models.
We do not know the full range of utility (let alone validity) of the PNPF approach, but so far it deals easily with systems that we could not solve any other way.
Collagen type 1 is the most abundant extracellular matrix protein in adult tissues. The collagen fiber assembly is a complex multi-step process involving several intermediate stages. My research focus is to understand the collagen fiber structure and regulation at the molecular level and how it affects cell-matrix interactions and mechanical properties of the underlying tissue. In particular we are studying how discoidin domain receptors (DDR1 and DDR2) interact with collagen type 1. DDRs are receptor tyrosine kinases expressed in a variety of mammalian cells. We have elucidated that by binding to collagen DDRs inhibit the fibrillogenesis and native structure of collagen fibers. These are critical findings as the quantity and quality of collagen fibers can be altered in a number of pathologies. Our ongoing work aims to elucidate the functional consequences of an altered collagen fiber ultrastructure. In particular we aim to undersatnd how the collagen fiber ultrastructure impacts fiber mechanics, collagen mineralization and cell-matrix interactions.
The past two decades have witnessed increasing interest in geometric partial differential equations (PDEs). However, much attention is paid to the use of second-order geometric PDEs as low-pass filters in signal, image and data analysis. My talk focuses on some non-conventional aspects of geometric PDEs. First, I discuss the construction of arbitrarily high-order geometric PDEs and their utility for image and surface analysis. Additionally, the design of nonlinear high-pass filters from a coupled PDE system is illustrated. Appropriate combination of geometric PDEs gives rise to the PDE transform. Like the wavelet transform, the PDE transform is able to decompose signal, image and data into functional modes with controllable time-frequency localizations. The inverse PDE transform results in a perfect reconstruction. Finally, I will discuss the applications of PDE transform to signal, image, surface and data analysis, as well as multiscale modeling of biomolecular systems, such as protein solvation and ion channel transport.
The shape and function of an axon is dependent on its cytoskeleton, including microtubules, neurofilaments and actin. Neurofilaments accumulate abnormally in axons in many neurological disorders. An early event of such accumulation is a striking radial segregation of microtubules and neurofilaments. This segregation phenomenon has been observed for over 30 years now, but the underlying mechanism is still poorly understood. I will present a stochastic multiscale model that explained these phenomena and generated testable predictions. I will also present our progress in deriving a continuum PDE model from the stochastic model. The PDE model is analytically more tractable and computationally more efficient. This is joint work with Professor Anthony Brown's lab from the Department of Neuroscience at the Ohio State University.
Cell-to-cell communication is fundamental to biological processes which require cells to coordinate their functions. A simple strategy adopted by many biological systems to achieve this communication is through cell signaling, in which extracellular signaling molecules released by one cell are detected by other cells via specific mechanisms. These signal molecules activate intracellular pathways to induce cellular responses such as cell motility or cell morphological changes. Proper communication thus relies on precise control and coordination of all these actions.
The budding yeast /Saccharomyces//cerevisiae/, a unicellular fungi, has been a model system for studying cell-to-cell communication during mating because of its genetic tractability. In this work, we performed for the first time computer simulations of the yeast mating process. Our computational framework encompassed a moving boundary method for modeling cell shape changes, the extracellular diffusion of mating pheromones, a generic reaction-diffusion model of yeast cell polarization, and both external and internal noise. Computer simulations revealed important robustness strategies for mating in the presence of noise. These strategies included the polarized secretion of pheromone, the presence of the alpha-factor protease Bar1, and the regulation of sensing sensitivity; all were consistent with data in the literature. In summary, we constructed a framework for simulating yeast mating and cell-cell interactions more generally, and we used this framework to reproduce yeast mating behaviors qualitatively and to identify strategies for robust mating.
In this talk, I will introduce hybrid methods that we developed recently using solution and domain decompositions, finite element, and finite difference techniques to fast calculate the electrostatics of protein in ionic solvent. In particular, they have been applied to the numerical solutions of three dielectric continuum models – the Poisson-Boltzmann equation, a size modified Poisson-Boltzmann equation, and a nonlocal modified Poisson-Boltzmann equation. Their performance will be discussed and compared with the corresponding finite element solvers. These hybrid techniques can also be extended to solve a class of nonlinear interface boundary value problems. This project is a joined work with my student, Jinyong Ying, under the support by NSF award DMS-1226259.
In the first part of the talk, I will present an efficient numerical algorithms for solving 3D steady-state Poisson-Nernst-Planck (PNP) equations with excess chemical potentials described by the classical density functional theory. Numerical results for ion and electron transport in solid electrolyte are shown to be in good agreement with the experimental data. Biomolecules exhibit conformational fluctuations near equilibrium states, including uncertainty in various biological properties in a dynamic way. In the second part of the talk, I will present our recent work in employing compressive sensing for quantifying high-dimensional conformational uncertainty in biomolecular solvation. To enhance the sparsity in high-dimensional Hermite polynomial expansions, we consider data-driven rotation-based linear mapping strategies to identify new bases for random variables, and greatly improve the efficiency and accuracy of the compressive sensing-based uncertainty quantification method. We demonstrate the effectiveness of the new method with applications in quantifying the conformational uncertainty in biomolecular solvation.
A norm in subcellular biochemistry is the changes in macromolecular conformations over the time scales longer than reaction cycle. The so-called dynamic disorder bears physiological consequences in a living cell by way of enzymatic catalysis and regulation of gene expression as well. In particular, we study the impact of dynamic disorder in the molecules of central dogma on the cell-to-cell variability of an otherwise isogenic population in a uniform environment. We discuss its evolutionary implications and related engineering challenges in the recent efforts of synthetic biology.
One of the most active areas of network science, with an explosion of publications during the last few years, is the study of "multilayer networks," in which heterogeneous types of entities can be connected via multiple types of ties that change in time. Multilayer networks can include multiple subsystems and "layers" of connectivity, and it is important to take multilayer features into account to try to improve our understanding of complex systems. In this talk, I'll give an introduction to multilayer networks and will discuss applications in areas such as transportation, sociology, neuroscience, and ecology.
Consider a set of communities (patches), connected to one another by a network. When can disease invade this network? This talk will consist of two parts relating to this question. In the first, we will show how the ability of disease to invade depends both upon the properties of the communities, as well as on the network structure. In particular, we make this dependence explicit for a broad class of disease models, through the rooted spanning trees of the network and a generalized inverse of the graph Laplacian. This first part is joint work with Zhisheng Shuai, Marisa Eisenberg, and Pauline van den Driessche. In the second part, I will discuss how network structure for graphs with "decay" can be described in terms of this generalized inverse. This generalized inverse is connected to transient random walks on the graph, and can be used to derive a natural distance metric, centrality measures, and community detection algorithms. I will describe some of these measures, together with implications for disease dynamics. This second part is joint work with Karly Jacobsen.
Quadratic harnesses (QHs) are stochastic processes with a very natural conditional structure: conditional exectations and conditional variances with respect to the past-future filtration of the process are linear and quadratic, respectively. This class of processes inculdes e.g. the Wiener process, the Poisson process and some other Levy processes, as well as free Brownian motion or quantum Bessel process. In the talk I will describe basic properties of QHs. The main novelty I am going to present is a connection between QHs and asymmetric simple exclusion processes (ASEPs). The ASEPs are related to statistical mechanics and describe behavior of balls tending to move in one direction but not allowed to move if a neighbouring site is occupied. ASEPs are widely studied recently due to relation to the so called KPZ (Kardar, Parisi, Zhang) equation. The connection between QHs and ASEPs allows to study properties of ASEP through some features of QHs. As an example I will consider the Large Deviation Principle for number of occupied sites, a rather difficult issue which, with the help of QHs representation, resolves just to control of (changing in time) state space of the process. This joint work with Wlodek Bryc (Univ. Cincinnati) is available at arXiv:1511.01163