Cells generate various biological rhythms that control important aspects of cell physiology including circadian (daily) events, cell division, embryogenesis, DNA damage repair and metabolism. Since these cellular rhythms can determine the fitness or fate of organisms, how cells generate and control rhythms has become a central problem in biology. While recent experimental work has identified many genes and proteins that are involved in biological clocks, identification of entire biochemical network seems far from complete since current experimental techniques require tremendous amount of work. On the other hand, output of the networks, timecourses of genes and proteins can be easily acquired with advances in technology. I will describe how to use these timecourse data to reveal biochemical network structure by using a fixed-point criteria. Moreover, the structures of biochemical networks are tightly related with their functions. I will discuss how the structures of biochemical network play role in maintaining rhythms and regulating period over a wide range of conditions with two examples: circadian rhythms and p53 rhythms.
Rapid climate warming has caused species across the globe to shift their geographic ranges poleward in latitude or upward in elevation. We naturally ask: will species be able to keep up with climate warming? To answer this question, I considered a mathematical model for a single-species population with distinct growth and dispersal stages.
The model is based on an integrodifference-equations framework, and is thus able to accommodate a diverse assortment of dispersal mechanisms. I incorporated climate warming by letting the niche curve, a curve describing environmental suitability for population growth on a spatial gradient, shift in one direction. The equation thus becomes non-autonomous. This equation can prescribe climate-warming scenarios and environmental heterogeneity in a versatile way. I compared different warming scenarios, and in this talk I will show that acceleration of climate warming imposes extra burden on the species compared with constant-speed warming, even if the amount of warming is the same over the same period of time. There is also a bifurcation phenomenon in this problem: under constant-speed warming, the population may fail to persist, and go extinct, if climate warming is too rapid. The threshold speed for persistence, or the critical speed, can be viewed as the species ability to keep up with climate warming. I will show that this critical speed depends both on the species’ growth and its dispersal.
This talk will cover some recent progress on numerical homotopy method to solve systems of nonlinear partial differential equations (PDEs) arising from biology and physics. This new approach, which is used to compute multiple solutions and bifurcation of nonlinear PDEs, makes use of polynomial systems (with thousands of variables) arising by discretization. Examples from hyperbolic systems, tumor growth models, and a blood clotting model will be used to demonstrate the ideas.
Inverse problems of partial differential equations often require the use of ``regularization" tricks, particularly when they are ill-posed or ill-conditioned. Recent work has elucidated the connection between the commonly used techniques of Tikhonov regularization and Bayesian Gaussian random fields. This interpretation of regularization within the Bayesian framework suggests that one should use regularizing terms that are consistent with apriori knowledge of the desired field.
The resulting method, using the path-integral-formulation of random fields, is amenable to deterministic perturbative approximation. Using the path-integral formulation as a computational tool, one is able to quantify uncertainty in the solution to inverse problems. Such an approach is desirable compared to standard computational methods of inversion which typically involve the use of Markov-Chain Monte-Carlo methods. In this talk, I will present the path-integral method for inverse problems. Electrostatic inverse problems will be provided for illustrative purposes.
Poisson abundance models have been widely used in ecology to model species abundance patterns. Also, certain information-based objects (for example Shannon entropy or mutual information) have been adapted to address problems like comparison of diversity or overlap between populations. Application of similar methods to questions related to immunology or genomics is a challenge due to severe under-sampling and sampling errors.
In the present talk we propose estimators of general measures of entropy based on survey sampling techniques as well as conditional expectations. We analyze consistency and asymptotic normality of such estimators and demonstrate their performance. We also propose a general discrete Poisson mixture modeling framework for T-cell receptor repertoire data and discuss its advantages as well as challenges.
There are many intracellular signalling pathways where the spatial distribution of the molecular species cannot be neglected. These pathways often contain negative feedback loops and can exhibit oscillatory dynamics in space and time. One such class of pathways is those involving transcription factors (e.g. Hes1, p53-Mdm2, NF-kB, heat-shock proteins).
In this talk, results from recent mathematical models which have been used to study the spatio-temporal dynamics of intracellular systems will be presented. Using the Hes1 gene regulatory network as an exemplar, both phenomenological partial differential equation models and mass action based spatial stochastic models will be considered. The benefits and drawbacks of the two different approaches will be discussed. For the partial differential equation approach, I will show how the diffusion coefficient can act as a bifurcation parameter, which, if in a certain range, can drive oscillatory dynamics.
Dispersal, which refers to the movement of an organism between two successive areas impacting survival and reproduction, is one of the most studied concepts in ecology and evolutionary biology. How do organisms adopt their dispersal patterns? Is there an "optimal", or evolutionarily stable, dispersal strategy that emerges from the underlying ecology? In this talk, we consider a reaction-diffusion model of two competing species for the evolution of conditional dispersal in a spatially varying but temporally constant environment. Two species are different only in their dispersal strategies, which are a combination of random dispersal and biased movement upward along the resource gradient. In the absence of biased movement or advection, A. Hastings (1983) showed that dispersal is selected against in spatially varying environments. When there is a small amount of biased movement or advection, we show that there is a positive random dispersal rate that is both locally evolutionarily stable and convergent stable. Our analysis of the model suggests that a balanced combination of random and biased movement might be a better habitat selection strategy for populations. This is joint work with Y. Lou of Ohio State University.
Mammals process sound signals via mechanotransduction of traveling waves within the cochlea. The passive mechanics of the cochlea, including the dynamics of its fluid and subsequent wave motion of its basilar membrane, can be represented by a linear system of PDEs. These interactions are well understood; however, nonlinear processes also exist within the inner ear, resulting in many unexplained phenomena. Experimentalists now point to the cochlea's outer hair cells and their unique demonstration of electromotility as the source of the nonlinearities; nevertheless, how these cells influence the system remains unclear.
Because of the inner ear's miniscule size and its sensitivity to surgical insult, mathematical models prove critical in determining the cochlear micromechanics. Here we develop a comprehensive, three-dimensional model for the active cochlea and use our formulation to explain experimental observations such as amplification and sharpening of the basilar membrane displacement peaks. We introduce a novel model for the outer hair cell force production and, by including this forcing, arrive at nonlinear equations of motion. Asymptotic methods and a hybrid analytic-numeric algorithm are used to obtain an approximate solution, and we ultimately find that our results replicate many of the expected nonlinearities.
Feedback control is important for biological systems which are both relatively insensitive to stochastic fluctuation of their parameters and able to adapt to changes in their environment. However, how feedback control enhances the robustness in spatial dynamics is still unclear. In this talk, I will discuss several spatial models arising from the studies of cell polarization, tissue patterning and stem cell lineage. Our mathematical and computational results show that feedback control can achieve robust cell polarization and tissue patterning against different stochastic effects; feedback control on the stem cell cycle is necessary for forming temporary "stem cell niche" that forms an important part of tissue stratification. All our findings are consistent with the experimental results our collaborators have observed and provide a stepping-stone for making new hypotheses in biology.