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.
Pathogen and parasite interactions may affect the developmental course of an infection within individuals and as well as having epidemiological consequences. I will discuss theoretical results that explore the dynamics of the human immunodeficiency virus, HIV, in the presence of a co-infection. At the individual level, a simple mathematical model can be used to predict the viral battle outcome of using an engineered virus to fight HIV. This model can also be modified to predict the dynamics of a long-term infection. I will also illustrate a more complex scenario by linking within-host and between-host dynamics using a natural co-infection and assess its impact in accelerating the spread of HIV
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.
Although extensively applied to model biochemical reactions inside cells, the law of mass action assumes that reactions occur in homogeneous, well-stirred volumes. This is inconsistent with the heterogeneous environment that characterizes intracellular reactions.
In fact, reaction kinetics can be either classical (following the law of mass action) or non-classical (following Kopelman's fractal-like kinetics). Both cases are likely to occur, but it is not clear under what conditions one obtains these reaction types. In this talk, I will discuss the use of Monte-Carlo simulations to investigate kinetic laws in in vivo conditions. Within the context of a two dimensional lattice-based simulation of the Michaelis-Menten mechanism in diffusion-limited conditions, I will also discuss the effect of obstacle density and size on reactant diffusion and rate coefficients. We show that obstacle density and size affect diffusion, first- and second- order rates. We also show that particle rotations and weak force interactions among particles lead to a significant decay in the fractal-like kinetics. All together, these results suggest that the effects of fractal-like kinetics disappear under less restricted conditions than previously believed.
Infectious diseases are a major public and environmental health concern, and are ubiquitous in nature. The evolutionary dynamics of pathogen virulence is driven by fitness differences among strains, and the selective forces that shape those fitness differences. A major contributing factor in these virulence dynamics is the well known transmission-virulence tradeoff, in which individuals infected with highly virulent strains (i.e., strains that induce high mortality) also on average yield fewer new infections due to a shorter window of infectiousness. This work is motivated by empirical observation of a second (and seemingly novel) tradeoff, between host movement and virulence, observed during the empirical study of the pathogenic bacterium Mycoplasma gallisepticum in North American House Finch populations.
In this talk, I will first discuss the motivating biological system, and introduce a partial differential equations (PDE) model of pathogen spread throughout a susceptible host population. I will then discuss an extension of that model which includes a novel movement-virulence tradeoff between pathogen strains, whereby sickness behaviour reduces the mobility of this otherwise highly mobile host. This second tradeoff can significantly alter the spatiotemporal virulence dynamics that arise in the classical case that only includes transmission-virulence tradeoff. I'll also present a Price Equation reformulation of this model to clarify how different ecological and evolutionary forces shape the overall dynamics. These results have important implications for the observation and interpretation of spatio-temporal epidemic data from a number of emerging wildlife and human diseases, and may help explain transient virulence dynamics in spatially spreading emerging infectious diseases.
Neurons can have extensive spatial geometries, but they are often modeled as single-compartment objects that ignore the spatial anatomy of the cell. This simplification is made for mathematical tractability and computational efficiency. However, many neurons are not electrotonically compact, and single-compartment models cannot be expected to fully capture their behavior. Dendritic properties can have substantial effects on the dynamics of single neurons, as well as the activity in neuronal networks. We study the influences of thin and general diameter passive dendrites on the dynamics of single neurons. For sufficiently thin dendrites and general somatic dynamics, we elucidate the mechanisms by which dendrites modulate the firing frequency of neurons. We find that the average value of the somatic oscillator's phase response curve indicates whether or not the dendrite will cause an increase or decrease in firing frequency. For general diameter dendrites and idealized somatic dynamics, we find that the neuron displays bistable behavior between periodic firing and quiescence. Furthermore, we identify the mechanism that causes this bistability to occur as somato-dendritic ping-pong. This mechanism was previously only described in models that contain active dendritic conductances.
Cancer immunoediting is to study the balance between tumor cells and immune system. There are many important factors and mechanisms that play significant roles in cancer immunoediting, such as CD200-CD200R and Interleukin-12 (IL-12) cytokine family.
CD200 is a cell membrane protein that interacts with CD200 receptor (CD200R) of myeloid lineage cells. During tumor initiation and progression, CD200-positive tumor cells can interact with M1 and M2 macrophages through CD200-CD200R-compex to silence macrophages. However, this mechanism has been shown to have apparently two contradictory experimental results in tumor growth: inhibition and promotion. In this talk, I will introduce a system of partial differential equations that we constructed to explain why these two opposite experimental results can both take place depending on the "affinity" of M1 and M2 macrophages to form the complex CD200-CD200R with tumor.
The Interleukin-12 (IL-12) cytokine family, which is composed of heterodimeric cytokines, includes IL-12, IL-23, IL-27, and IL-35. IL-12 and IL-23 are mainly pro-inflammatory cytokines with key roles in the development of the TH1 and TH17 subsets of helper T cells, respectively. IL-27 has been regarded as an immunoregulatory cytokine, due to both of pro-inflammatory and anti-inflammatory functions in anti-tumor activity. Notably, recent experiments in transgenic mice indicate that IL-27 significantly enhances the survival of activated tumor antigen specific CD8+ T cells and hence promotes tumor rejection. IL-35, the most recently identified member of IL-12 cytokine family, is a potent inhibitory cytokine pro-duced by regulatory T cells. Recent transgenic mouse experiments demonstrate that IL-35 enhances the tumor growth and angiogenesis. Hence, I will also introduce another two mathematical models for IL-27 and IL-35 that we constructed to study the functions of IL-27 and IL-35. These models qualitatively fit with the above experimental results and provide some hypotheses to develop therapeutic protocols in cancer treatment.
The effects of noise on the dynamics of nonlinear systems is known to lead to many counter-intuitive behaviors. Using simple planar limit cycle oscillators, we show that the addition of moderate noise leads to qualitatively different dynamics. In particular, the system can appear bistable, rotate in the opposite direction of the deterministic limit cycle, or cease oscillating altogether. Utilizing standard techniques from stochastic calculus and recently developed stochastic phase reduction methods, we elucidate the mechanisms underlying the different dynamics and verify our analysis with the use of numerical simulations. Lastly, we show that similar bistable behavior is found when moderate noise is applied to the more biologically realistic FitzHugh-Nagumo model.
Oncolytic virotherapy is a tumor treatment which uses viruses to selectively target and destroy cancer cells. Clinical trials have demonstrated varying degrees of success for the therapy with limitations predominantly due to barriers to viral spread throughout the tumor and the immune response to the virus.
Fusogenic viruses, capable of causing cell-to-cell fusion upon infection of a tumor cell, have shown promise as oncolytic agents in experimental studies. The fusion causes the formation of multinucleated syncytia which enhances viral spread through the tumor and eventually leads to cell death. We formulate a partial differential equations model with a moving boundary to describe the treatment of a spherical tumor with a fusogenic oncolytic virus. In this talk, I will discuss the existence and uniqueness of local solutions to the nonlinear hyperbolic-parabolic system. In a special case, a reduction to an ordinary differential equations system allows for a global stability analysis which provides a prediction of success or failure of the treatment. Numerical simulations demonstrate exponential growth or decay of the tumor depending on viral burst size and rate of syncytia formation.
I will also briefly discuss work in progress on modeling the upregulation of the matricellular protein CCN1 in oncolytic virotherapy of glioma. Overexpression of CCN1 has been shown experimentally to induce an antiviral immune response including the proinflammatory activation of macrophages. Understanding the interactions between the tumor, virus and immune response is critical to improving the efficacy of virotherapy.
Extirpation of vertebrates by human activity results in "empty forests", with disrupted ecological processes, including seed dispersal of plants. Although seed dispersal is typically modeled as monotonically decreasing with distance from the tree, vertebrates disperse seeds in clumps to preferred areas. These seeds must survive the attack of insect seed predators in order to germinate, and clumped seed deposition can greatly alter the number and spatial distribution of germinating plants. I will show how the interaction between seed dispersal by vertebrates and patterns of plant mortality due to insect seed predators shapes the spatial pattern of seed survivorship, and use individual-based models to examine how dispersal disruption modifies these patterns. This basic understanding will help us predict the future of plant communities faced by anthropogenic pressures that include the hunting of seed dispersers.
During forward swimming of crayfish, four pairs of limbs called swimmerets swing rhythmically through power and return strokes. Neighboring limbs move in a back to front metachronal wave with a delay of approximately 25% of the period. Interestingly, this posterior to anterior progression is maintained over the entire range of behaviorally relevant stroke frequencies. Previous work has modeled the neural circuitry coordinating this motion as a chain of nearest neighbor coupled oscillators, and it was shown that the architecture of this circuitry could provide a robust mechanism for this behavior. However, this study ignored the presence of weaker longer range coupling between oscillators, and how the coupling affects this mechanism is unknown.
In this talk, I will discuss the role of the long range coupling in the swimmeret system. An analytical argument using a phase model suggests that the presence of long range coupling speeds up the metachronal wave when the connection strength is sufficiently weak. Numerical simulations show that this effect extends to larger connection strengths. Further, we confirm these predictions in a more detailed conductance-based neuronal model. Finally, we verify the validity of the model by comparing results from the phase model to experiments that probe the effects of long range coupling.
Combined with results from a computational fluid dynamics model, our findings indicate that the long range coupling might exist to ensure that the crayfish’s limb movement during forward swimming is in an optimally efficient regime.
New imaging modalities are continuously developed in basic science research labs across the nation. In spite of the advances in imaging technologies, the signals that are recorded are still plagued with noise. High quality image denoising and segmentation algorithms are a necessity in every basic science lab that uses imaging as it primary tool of investigation. In this talk, I will review total variation minimization based methods for image denoising and segmentation. I will then discuss how one may use the ROF model to adaptively resolved the features in general images across scales, and conclude with an analysis of the convergence rate of the resulting multiscale decomposition scheme.
In my talk I will discuss several simple food web modules that combine population dynamics with evolutionary trait dynamics. Classical food web modules (e.g., trophic chain, exploitative competition, apparent competition, intraguild predation etc.) have proved to be instrumental in our understanding of mechanisms regulating biodiversity. These simple food web modules assume that the interaction strength between species is fixed. In other words, they do not consider various individual adaptations to changing environment. Because decisions an individual has to make often depend on what the others are doing, the optimal strategies require game theoretical framework. The two times scales involved in these models (demographic and evolutionary) lead to two possible scalings that simplify resulting models. The approach that assumes trait dynamics operate on a slower time scale when compared with population dynamics led to the so-called "adaptive dynamics". When trait dynamics operate on much faster time scale, the resulting models are described by control system (where controls describe traits) with state dependent feedbacks. As these feedbacks are often multivalued, the population dynamics are described by a differential inclusion. In my talk I will show how models that combine population with evolutionary dynamics change our understanding of mechanisms of biodiversity in simple food web modules.
In subdivided populations, adaptation to a local environment may be hampered by maladaptive gene flow from other subpopulations. We study a continent-island model in which an ancestral population sends migrants to a colony exposed to a different environment. At an isolated locus, i.e., unlinked to other loci under selection, a locally beneficial mutation can be established and maintained only if its selective advantage exceeds the immigration rate of alternative allelic types. We show that, if a beneficial mutation arises in linkage to a locus at which a locally adapted allele is already segregating in migration-selection balance, the new mutant can invade and be maintained under much higher immigration rates than predicted by one-locus theory. We deduce the maximum amount of gene flow that admits the preservation of the locally adapted haplotype on the strength of recombination and selection. We calculate the selective advantage of recombination-reducing mechanisms, such as chromosome inversions, which often seem to play a role in speciation. Our analysis provides conditions for the evolution of clusters of locally adaptive genes, or islands of divergence, as found by some empirical studies. For an extended model that allows for epistasis, we discuss how much gene flow is needed to inhibit speciation by the accumulation of Dobzhansky-Muller incompatibilities.
Mechanistic mathematical models are important tools for understanding the processes that shape ecological systems. Models have been used to describe life cycles of individuals, population dynamics, behavior, and more. However, in order for these models to reach their full potential as both tools for understanding and for prediction we must be able to link modeled quantities to data and infer model parameters.
However, general methods of parameter inference for many of these models are not available, and we must think carefully about how to link sophisticated models with robust inferential techniques. Here I discuss three examples of ecological models of these types. First is a model of the temperature dependence of malaria transmission, which shows the power of even simple models combined with data. The second example uses an example of an individual-based model (IBM) developed to describe the spread of Chytridiomycosis in a population of frogs.
This case study shows how one can perform inference for IBMs that exhibit certain characteristics with a traditional likelihood-based approach. Third, I present a bioenergetic model of individual growth and reproduction in a dynamic environment. This example highlights how input mis-specification can affect inference, and the consequences for prediction in novel environments.
Sam Karlin was one of the great mathematicians to contribute to evolutionary theory, both in the forward direction (dynamics) and the backward direction (bioinformatics). Karlin introduced two theorems in 1976 to analyze the effect of population subdivision on the protection of genetic diversity. Both could be summarized as the phenomenon that mixing reduces growth, with the consequence that greater dispersal in heterogenous environments reduces the survival of rare alleles.
They provide the basis to prove very generally that populations can always be invaded by genetic variants for information transmission (mutation and recombination) that better preserve information during reproduction --- the Reduction Principle initially discovered by Marc Feldman. The Reduction Principle has made an appearance in recent work on reaction diffusion models of dispersal in continuous space. Could a continuous-space version of Karlin's theorem be at work here? I will describe my recent extension of Karlin's theorem to infinite dimensional Banach spaces. This result unifies the reaction diffusion models showing that the slower disperser wins. It also applies to the generators of strongly continuous semigroups, elliptic operators, Schrivdinger operators, and local and nonlocal diffusions. The phenomenon that mixing reduces growth and hastens decay could be described as a universal phenomenon.
Cholera is a severe water-borne disease that remains a global threat to public health. In this talk, we review some recent studies on cholera dynamics from the viewpoint of mathematical epidemiology. We then present a generalized cholera model which explicitly incorporates both human-to-human and environment-to-human transmission pathways, and which unifies several existing deterministic models. We conduct a stability analysis on the local and global dynamics, and demonstrate the application of this framework with realistic case study. Some results from optimal control simulation are also discussed. In addition, We extend the model to periodic environments to investigate cholera transmission with seasonal variation.
Regulation or management to a constant set–point is fundamental across science and engineering. In conservation management or pest control, population managers aim to regulate the population to a desired density. In order to be useful in applications, set–point regulation should be robust to parametric uncertainty and measurement errors. We address how set–point regulation can be achieved in a robust way. We describe the control theory concept of Integral Control. Integral control is a simple yet powerful technique developed by control engineers, which is ubiquitous in engineering but has not yet (to our knowledge) received attention in population dynamics. One striking feature of integral controllers is that they can be implemented on the basis of both minimal knowledge of the system to be managed or regulated, and in the presence of considerable system uncertainty. This that makes them appealing for population management/conservation, where uncertainty and incomplete measurements are expected. In this talk we discuss the theory of integral controllers and give hypothetical examples.
I will introduce mathematical models describing the influence of external factors in the temporal dynamics of populations. One model incorporates climate factors into the dynamics of seasonal influenza through three ecology-based response functions: response of influenza virus survival and human susceptibility to air temperature as well as influenza virus transmission response to specific humidity. I will discuss numerical simulation results obtained when the model are driven by temperature and specific humidity data. Interestingly, the models reproduce not only the reported double peaks of influenza A cases in subtropicalregion, but also the observed temporal pattern of flu in temperate regions (one winter peak).
Two other models incorporating the effect of insect outbreaks either as a single disturbance in the forest population dynamics or coupled with wildfire disturbances. The results show that 1) the beetle-tree system parameterized model exhibits the well known temporal dynamics of beetle-tree interaction described by the dual equilibria theory. 2) The beetle-tree-fire model reveals the existence of positive feedback between wildfire and insect outbreak disturbances in certain region of fire strength. This result agrees with one of the current theories in the field.
Not only carbon (C) but also nutrient elements such as nitrogen (N) and phosphorous (P) are pivotal for organismal growth, reproduction, and maintenance. Newly emerging mathematical models linking population dynamics with these key elements greatly improve historic trophic interaction models and resolve many existing paradoxes. Most of these models assume strict homeostasis in heterotroph and non-homeostasis in autotroph due to the fact that the stoichiometric variability of heterotroph is much less than that of autotroph. Via bifurcations we study when the "strict homeostasis" assumption is sound and when not. Incorporating light dependence on the growth of autotrophs, the resulting dynamics reveal a series of homoclinic and heteroclinic bifurcations in low light conditions giving the explanation for why microcosm experiments have had unreliable results in low light conditions.
The folding pattern of the human brain has many ridges (gyri) and valleys (sulci), making it difficult to visualize and analyze. The development of these folding patterns is not fully understood and there is debate in the biological and neuroscientific communities as to why folds develop in a particular location. Additionally, there are many diseases involving the folding the patterns of the brain that occur in early development and causes of these diseases are not understood. I will discuss some mathematical and computational models we have developed using a prolate spheroid domain to gain insight into cortical folding pattern formation. I will also discuss how conformal mapping and topology can assist with the study and analysis of diseases in the human brain.
The study of traveling waves of activity in neural tissue can provide deep insights into the functions of the brain during sensory processing or during abnormal states such as epilepsy, migraines or hallucinations. Computational models of these systems usually describe the tissue as a vast interconnected network of excitatory neurons comprised of large number of units with similar properties, for example integrate and fire neurons. It is also widely assumed that while the strength of the connections between neurons changes as a function of distance that separates them, this interaction does not depend on other local parameters. These assumptions allow for formulation of a set of integro-differential equations describing the propagation of the traveling wave fronts in a one-dimensional integrate-and-fire network of synaptically coupled neurons, allowing for investigation of the network dynamics during wave initiation and during the transition toward constant-speed propagation. We further explore how the presence of periodic inhomogeneities affects the propagation dynamics, aiming to derive more precise estimates for the conditions when propagation failure occurs.