2011 Workshop for Young Researchers in Mathematical Biology (WYRMB): Abstracts and Lecture Materials
Plenary Speakers
Phylogenetics is the branch of biology concerned with inferring evolutionary relationships between currently extant species. For instance, are humans more closely related to chimpanzees or to gorillas on an evolutionary tree? A typical phylogenetic analysis from molecular data might consist of sampling gene sequences from a number of species, aligning them, and performing a statistical analysis to choose a tree that best displays the evolutionary relationships of taxa.
While phylogenetic analyses are usually undertaken with standard statistical approaches such as Maximum Likelihood or MCMC in a Bayesian framework, these require formulating a probabilistic model of the DNA substitution process on a tree. Because many of these models are naturally given by polynomial parameterizations, by considering the algebraic varieties these maps define, the viewpoint of algebraic geometry can be used to gain theoretical understanding of the limits and advantages of such models.
The talk begins with an introduction to phylogenetics, and then addresses how algebraic techniques are being used to advance the theoretical end of this field. Surprising connections will be made between seemingly disparate areas of mathematics.
Infectious diseases have been spreading across vast distances for milenia as a result of the movement of both human and animal hosts. In the past, both types of hosts had limited movement ranges, and one observed travelling waves of infection slowly expanding across space. Nowadays, the movement of humans has considerably accelerated and expanded, so that one observes another kind of spread, which appears less coherent.
In this talk, I will discuss the mechanisms that give rise to the spatialization of an infectious disease. I will then present metapopulation models, one of the methods that can be used to describe the spatio-temporal spread of infections between distant locations. I will review some mathematical properties of these models, and will illustrate with a stochastic application in the context of the spread of infections via the global air transportation network.
Discrete time matrix models for the dynamics of structured populations provide one way to study the dynamic consequences of different life history strategies. One fundamental strategy is semelparity. Mathematically, semelparity can be associated with a high co-dimensional bifurcation at R0 = 1 which results in a dynamic dichotomy between persistence equilibrium states (lying in the interior of the positive cone) and synchronous cycles and cycle chains (lying on the boundary of the cone). Biologically, the dynamic alternative is between equilibration with overlapping generations and periodic oscillations with non-overlapping generations. I will describe what has been proved about the bifurcation at R0 = 1 for lower dimensional models. It remains a difficult mathematical challenge to describe the nature of the bifurcation at R0 = 1 for higher dimensional models. Time permitting I will discuss the bifurcation at R0 = 1 for matrix models extended to an evolutionary setting (by evolutionary game theory).
Even understanding the way we perceive very simple images presents a major challenge for both neurophysiologists and computer scientists. In this talk I will discuss two visual effects. In one random dots are superimposed on themselves following a linear transformation. In the second, a rotating disk with radial spokes is viewed under stroboscopic illumination, where the frequency and duration of the stroboscopic flash are varied. Though these phenomena are very different, in both correlation plays a major role in defining the structure of the image. In this talk, I will give demonstrations of these phenomena and discuss related experimental and theoretical work by ourselves and others. In particular, I focus on recent theory that uses the theory of forced nonlinear oscillations to predict the percept of rotating disks during stroboscopic illumination over a wide range of disk rotation speeds and strobe frequencies.
Finally, I suggest that the anatomical structure of the human visual system plays a major role in enabling the amazingly rapid and accurate computation of spatial and time dependent correlation functions carried out by the visual system.
Mathematics within the pharmaceutical industry is, indeed, applied. Applied mathematics and, more generally, quantitative sciences are seen as important capabilities having the potential to address the current scientific and economic challenges being encountered by this industry. In this presentation we give an insider's view of the pharmaceutical drug development process, the pressure points stemming from economic and scientific pressures and where critical applications of mathematics must be achieved.
As a case study, we describe the mathematics brought to bear modeling Hepatitis type C virus (HCV). A diverse set of techniques are applied including ordinary and stochastic differential equations, asymptotic analysis, nonlinear mixed effects models and partial differential equations. These mathematical tools help draw a picture of the treatment and serious side-effects from attempting to cure HCV with the standard of care. To wrap up, we briefly describe some of the mathematics used in financial modeling within the research and development environment. We extend the HCV case study into this financial realm.
Short Talks
The degradation of blood clots is a tightly regulated process. If the mesh of fibrin fibers securing the clot degrades too slowly, thrombi can form, leading to heart attack or stroke. If the fibrin degrades too quickly, excessive bleeding may occur. We study fibrinolysis (the degradation of fibrin by the main fibrinolytic enzyme, plasmin) using a multi-scale mathematical model intended to answer the following question: Why do coarse clots composed of thick fibers lyse more quickly than fine clots composed of thin fibers, despite the fact that individual thin fibers lyse more quickly than individual thick fibers? We use stochastic methods to model lytic processes on scales ranging from individual fiber cross section to whole clot. We find that while fiber number does have an effect on lysis rate, it is not simply "fewer fibers equals faster lysis", as many biologists suggest. In fact, the number of tissuetype plasminogen activator molecules (tPA, an enzyme that converts plasminogen to plasmin) relative to the clot surface area exposed to the tPA strongly influences lysis speeds. We also predict how many plasmin molecules can be produced by a single tPA molecule, how long it takes a given number of plasmin molecules to degrade a single fibrin fiber, and how patterns and speeds of lysis (both on an individual fiber and clot scale) vary under a range of conditions. This last point is of particular interest for development of treatments for occlusive blood clots. Often, a bolus of tPA is injected near the thrombus, in an attempt to initiate therapeutic lysis. Our model predicts other potential targets for future research on effective therapeutic strategies for degrading blood clots.
Necrotizing enterocolitis is a severe inflammatory disease in premature infants that is characterized by wounds in the intestinal wall. The ongoing dynamics of the disease depend upon a complex interplay between the immune system, intestinal bacteria, and intestinal epithelium. We have developed a three-dimensional computational model that examines this complex interplay and its dependence on the spatial structure of the intestine. The model reproduces expected physiological results and shows that the spatial structure of intestinal wounds may affect the outcome of necrotizing enterocolitis.
A novel bacterial defense system against invading viruses, known as Clustered Regularly Interspaced Short Palindromic Repeats (CRISPR), has recently been described. Unlike other bacterial defense systems, CRISPRs, are virus-specific and heritable, producing a form of adaptive immune memory. Specific bacterial DNA regions, CRISPR loci, incorporate on average 25 copies of unique short (30 base pair) regions of viral DNA which allow the bacteria to detect, degrade and have immunity against viruses with matching sub-sequences. Ideally, the number of unique viral-copied regions a CRISPR loci contains would grow indefinitely to allow immunity to accumulate to a large number of viruses. However, the number of these viral-copied regions in the CRISPR loci of any bacteria is limited in length and number. We use a birth-death master equation model to explore the growth and decay of the length of the CRISPR loci and thus the number of viral-copied regions. Additionally, we use a simple probabilistic model to determine bounds on the length of viral-copied region within the CRISPR locus.
Over the past two decades, modeling the ecological impacts of harvesting wild plants, as source of food and medicine, has used stationary population growth rate as the metric to measure effects of harvest. In this talk, I show that using asymptotic rather than the transient growth rates may underestimate the effect of harvest and of other disturbances. The transient growth rate and its variation between population-level harvest intensities (high versus low) were smaller than their asymptotic equivalent. Patterns of elasticity of transient growth rates to perturbation of vital rates were different from those of the asymptotic elasticity. Asymptotic growth rates were more elastic to perturbation of late life stages; however, transient growth rates were more elastic to early life perturbations. These results suggest that the more than fifty published studies on the effects of harvest on wild plant population dynamics using only asymptotic growth rates may have been underestimating such effects in the short-term.
Government investment in public health management can elicit strong responses from individuals within communities. These responses can reduce and even reverse the expected benefits of the policies. Therefore, projections of individual responses to policy can be important ingredients into policy design. Yet our foresight of individual responses to public health investment remains limited. This paper formulates a population game to explore how individual investment through behavior and government investment through taxation impact the health commons. We model the problem of infectious disease management through reductions in transmission risk for a disease that does not elicit immunity in a population without demographic structure. We identify three common modes of government and individual investments and describe how each mode relates to policy responses and health outcomes. We also provide general bounds on the magnitude of practical investment by individuals. The methods we present can be extended to address specific policy problems where public responses are expected to impose key feedbacks.
Work done in collaboration with Darla Lindburg, Rachel A. Smith, and Timothy C. Reluga.
Transforming Growth Factor β (TGFβ) signaling pathway is a prominent regulatory signaling pathway controlling various important cellular processes. It can be induced by ionizing radiation and regulated by Smad in a negative feedback loop through promoting the nuclear import of the regulatory Smad and subsequent expression of inhibitory Smad that forms ubiquitin ligase with Smurf to target active TGFβ receptors for degradation. In this work, we propose a mathematical model to study the Smad-regulated TGFβ signaling pathway. By modularization, we are able to analyze each component subsystem and recover the nonlinear dynamics of the entire network system. Meanwhile the excitability, a common feature observed in the biological systems, along the TGFβ signaling pathway is discussed and supported as well by numerical simulation.
Determining whether a habitat with fragmented or concentrated resources is a benefit or hindrance to a species' well-being is a natural question to ask in Ecology. Such fragmentation may occur naturally or as a consequence of human activities related to development or conservation. In a certain mathematical formulation of this problem, one is led to study an indefinite weight eigenvalue problem, the principal eigenvalue of which is a function of the habitat's makeup and indicates the threshold for which the species either persists or becomes extinct. For a particular but general class of fragmentation profiles, this threshold can be calculated implicitly and optimized to reveal an definitive strategy for minimizing the persistence threshold and thereby allowing the species to persist for the largest range of physical parameters.
This relates to work contained in the publication: A.E. Lindsay, M.J.Ward, (2010) An Asymptotic Analysis of the Persistence Threshold for the Diffusive Logistic Model in Spatial Environments with Localized Patches Discrete and Continuous Dynamical Systems Series B, Volume: 14, Number: 3, pp.1139-1179
Mathematical models can provide a fundamental tool to understand the dynamics of HIV/AIDS and Tuberculosis (TB) transmission. Indeed, once HIV infection has occurred, a long and variable incubation period can elapse before the development of clinically apparent AIDS. It is therefore not apparent when an individual is infected.
There is insufficient information on trends in both HIV prevalence and incidence. Yet only by knowing these trends can we attempt to predict the long-range impact of HIV infection. Transmission dynamics of HIV/AIDS are difficult to describe because the numerous and complex variables and parameters are very hard to control and estimate. In our work we apply Lie Symmetry methods to a Mathematical Model which describes HIV/AIDS and TB coinfection in the presence of treatment. Lie group analysis is a powerful tool to find the general solution of ordinary differential equations. An HIV/AIDS and TB coinfection model which considers antiretroviral therapy for AIDS cases and treatment of all forms of TB, that is latent and active forms of TB, is presented.
Working memory is a process for the short-term storage and manipulation of information necessary for complex cognitive tasks. During the performance of working memory tasks, the prefrontal cortex (PFC) exhibits sustained persistent activity and is believed to play a key role in the process. Experiments have demonstrated that working memory performance is modulated by dopamine, which is known to be altered in certain pathological conditions, including schizophrenia.
A number of models have been proposed for the maintenance of persistent activity in the PFC, often based on either intrinsic cellular bistability or recurrent excitatory connections formed via synaptic adaptation. Consistent with the observation that inhibitory connections dominate the PFC, we present a new approach: a network driven by excitatory-inhibitory interactions where the response to inhibition is modulated by intracellular calcium. Individual neurons fire irregularly, but our model network exhibits emergent properties, such as a clear gamma rhythm. The network is robust to noise and distracters. Only general assumptions about connection probabilities are assumed; the model can represent novel, unlearned stimuli.
Dopamine modulates ion channel activity and synaptic conductances. We study the effects of this modulation on cellular and network behavior, and find the experimentally-observed inverted-U shaped relation between dopamine expression and working memory performance.
This talk will introduce a new modeling framework for gene regulatory networks that incorporates state dependent delays and that is able to capture the cell-to-cell variability. This framework will be presented in the context of finite dynamical systems, where each gene can take on a finite number of states, and where time is also a discrete variable. The state dependent delays represent the time delays of activation and degradation. One of the new features of this framework is that it allows a finer analysis of discrete models and the possibility to simulate cell populations. Applications presented will use one of the best known stochastic regulatory networks, that is involved in controlling the outcome of lambda phage infection of bacteria.
The specification of cell identities during development is orchestrated by signaling molecules named morphogens that establish spatial patterns of gene expression within a field of cells. In the classical view, the interpretation of morphogen gradients depends on the equilibrium morphogen concentrations, but the dynamics of gradient formation are generally ignored. The problem of whether or not morphogen gradient dynamics contribute to developmental patterning has not been explored in detail, in part, because genetic experiments that selectively affect signaling dynamics while maintaining unchanged the steady-state morphogen profile are difficult to design and interpret. Here, I present a mathematical approach to identify genetic mutations in developmental patterning that may affect the transient, but leave invariant the steady-state signalling gradient. As a case study, I illustrate how these tools can be used to explore the dynamic properties of Hedgehog signalling in the developing wing of the fruit fly, Drosophila melanogaster. This analysis provides insights into how different properties of the Hedgehog gradient dynamics, such as the duration of exposure to the signal or the width of the gradient prior to reaching the equilibrium, can be genetically perturbed without affecting the local steady-state distribution of the gradient. I propose that this method can be generally applicable as a tool to design experiments to probe the role of transient morphogen gradients in developmental patterning and discuss potential applications of these ideas in other problems.
A a new SIS model for malaria that incorporates mosquito demography is developed and studied. This model differs from standard SIS models in that the mosquito population involved in disease transmission (adult female mosquitoes questing for human blood) are identified and accounted for. The main focus of this model is disease control. In the presence of the disease, we identified a trivial steady state solution, a nontrivial disease-free steady state solution and an endemic steady state solution and showed that the endemic steady state solution can be driven to instability via a Hopf bifurcation as a parameter is varied in parameter space. The model therefore captures natural oscillations known to exist in malaria prevalence without recourse to external seasonal forcing and/or delays. Besides the basic reproduction number, we also identified a second threshold parameter that is associated with mosquito demography. These two threshold parameters can be used for purposes of disease control. Analysis of our model also indicates that the basic reproduction number for malaria can be smaller than previously thought and that the model exhibits a backward bifurcation. Hence, simply reducing the basic reproduction number below unity may not be enough for disease eradication. The discovery of oscillatory dynamics and the re-interpretation of the basic reproduction number for malaria presents a novel and plausible framework for developing and implementing control strategies. Model results therefore indicate that accounting for mosquito demography is important in explaining observed patterns in malaria prevalence as well as in designing and evaluating control strategies, especially those interventions that are related to mosquito control.
We analyze a closed-loop locomotor model in which a central pattern generator drives a single-joint limb and receives afferent feedback. Transitions associated with changes in ground reaction force or motoneuron outputs abruptly alter the vector field in the limb dynamics phase plane. The positions of the locomotor oscillation trajectory relative to these transient vector fields and their critical points explain the model's ability to replicate an experimentally observed locomotor asymmetry. A contraction argument relying on these transitions provides conditions for existence of a periodic orbit in a reduced model.
Work done in collaboration Sergey Markin, Boris Prilutsky, Ilya Rybak, and Jonathan Rubin.
Blebbing occurs when the cytoskeleton detaches from the cell membrane, resulting in the pressure-driven flow of cytosol towards the area of detachment and the local expansion of the cell membrane. Recent interest has focused on cells that use blebbing for migrating through three dimensional fibrous matrices. In particular, metastatic cancer cells have been shown to use blebs for motility. A dynamic computational model of the cell is presented that includes mechanics of and the interactions between the intracellular fluid, the actin cortex, and the cell membrane. The computational model is used to explore the relative roles in bleb formation time of cytoplasmic viscosity and drag between the cortex and the cytosol. A regime of values for the drag coefficient and cytoplasmic viscosity values that match bleb formation time scales is presented. The model results are then used to predict the Darcy permeability and the volume fraction of the cortex.
Posters
The honeybee antennal lobe (AL) provides an ideal system to study the olfactory system because it is anatomically and genetically simpler than the olfactory bulb of mammals. While anatomical structures within the AL are relatively well known, their functional roles in sensory processing remain poorly understood.
Several studies showed that the dynamic processing in the AL may serve to decorrelate sensory representations through early transients rather than by reaching a stable attractor (Mazor et al., Neuron, 2005). Recently, Broome et al. (Neuron, 2006) showed that when two odors are presented with some time gaps, there is a smooth divergence from detecting the first odor and then a smooth convergence to detecting the second odor. Fernandez et al. (J. Neurosci., 2009) showed that there is a smooth transition in the time-dependent neural representation of the cells in response to a smooth transition in the ratios of components in the binary mixtures.
We propose a mathematical model for this olfactory system to reproduce several experimental results in the AL such as a smooth transition and a smooth divergence of firing patterns in response to mixtures. We find that synaptic/non-synaptic plasticity helps to discriminate odorants and to reproduce realistic results mentioned above.
The influenza virus continuously mutates via antigenic drift, resulting in continuous creation of new variants able to re-infect hosts that have become immune to earlier types. In large part because of this continuous drift property, influenza presents a significant morbidity and mortality burden. On first world countries of the temperate zones, control is largely achieved through mass-production and dissemination of vaccines, how ever this requires prediction of the dominant circulating strain well in advance of the up-coming season. The dynamics of influenza patterns for the tropics are less understood and few mathematical models have try to address this topic. To aid the control of the disease, models are needed that can accurately simulate the spread of influenza over the course of several years, with strain-specific dynamics, including the process of drift and pre-existing immunity to newly circulating strains. Ideally the model should be flexible enough to reproduce the dynamics of both temperate zones and the tropics. We describe a novel SIR deterministic model for the spread of influenza within a population. The model incorporates appearance of new strains due to antigenic drift, and partial immunity within the population to circulating strains due to prior infection. Because seasonality is important to consider in temperate regions, the model includes optional seasonal forcing of the transmission rate. Our model successfully reproduces several key features in influenza data observed in tropical and temperate regions.
Many recent studies have shown that the initiation of human cancer is due to the malfunction of some genes at the restriction(R) checkpoint in G1-S transition period of cell cycle progression. Identifying and studying these genes has a paramount advantage in controlling and possibly in the treatment of human cancer. It is also shown experimentally that there is mutual activation between Cdc25A and CyclinE/Cdk2 while mutual inhibition between CyclinE/Cdk2 and P27kip1. In this study, a new mathematical model for this cancer subnetwork concentration dynamics is developed. Positive steady states are determined and rigorously analyzed. The work is not yet completed. The following are in our list of to do to get biologically meaningful results:
- Including Cdk inhibitor family in the subnetwork.
- Determining parameter values for those which are not experimentally determined.
- Numerical simulation of the model.
We study blood ow in medium-to-large arteries, modeled by the Navier-Stokes equations for an incompressible viscous uid coupled with equations describing the mechanics of the arterial walls. Considering recent in vivo studies that identified significant viscoelastic wall properties and longitudinal wall displacement, we model arterial walls using the linearly viscoelastic Koiter shell model involving both radial and longitudinal displacement. The numerical scheme we use is an extension of the kinematically-coupled scheme described in [1]. Our algorithm is tested using physiological parameters. Results show comparable longitudinal and radial displacements, implying that longitudinal displacement cannot be neglected.
1. G. Guidoboni, R. Glowinski, N. Cavallini, and S. Canic, 2009 Stable loosely-coupled- type algorithm for uid-structure interaction in blood ow, J. Comput. Phys., 228(18): 6916-6937.
Coordination of swarming events resulting in fingering phenomena at the edge of a swarm of Pseudomonas aeruginosa are studied using combination of simulation and experiments. A three-dimensional multiscale model is introduced which combines continuum submodels and a cell-based stochastic discrete submodel. With this multiscale model, a thin liquid film submodel is employed to describe the hydrodynamics of the liquid layer where bacteria live in. Convection-diffusion equations are used to describe dynamics of chemical compounds of quorum sensing. The cell-based stochastic discrete submodel describes cellular dynamics of Pseudomonas aeruginosa. Different motilities are incorporated into this submodel. A simplified Quorum Sensing model is employed to regulate rhamnolipid production of each cell depending on local cell density and nutrient level. Experiment observation showed that a slight decrease of the agar concentration used in swarm plates yields increases in swarm motility of Pseudomonas aeruginosa. Cell colony spreads faster and shows finger patterns at the swarm edge on 0.4% agar, while on 0.6% agar, no finger formation is observed. Swarming on low agar liquid surface is accompanied with a significant amount of liquid production as well as fingering instabilities. The simulation results also demonstrated that swarming on low agar concentration surfaces has a faster expansion compared with swarming on high agar concentration surface. Without liquid production, the expansion rate of cell colony is extremely slow, cells are compactly assembled together; while with water production, the cell colony grows fast and fingering instability at the edge amplifies. Also accumulation of rhamnolipid concentration and cell density is observed in the form of a "finger-nail" domain near the edge of the swarm in numerical simulations.
We present a new algorithm for finding the maximal Lyapunov exponent (MLE), one that is able to distinguish between chaotic and stable periodic attractors. Our algorithm is based on the two-particle method of computing the MLE, but it also uses orthogonal projections onto normal planes to prevent one particle from "lagging" behind. Additionally, implementing a lower threshold trigger to compute the MLE allows for the algorithm to return a negative exponent in the event of a stable periodic attractor, rather than a 0 as most other methods would. We provide a detailed description of the algorithm, proofs of the convergence of the algorithm for three separate types of attractors, and a discussion of how this procedure is being used to study discrete and continuous systems at Ohio University.
Oxygen exchange between blood vessels and tissue occurs primarily in the microcirculation. Irregular microvascular geometries lead to a high degree of heterogeneity in tissue oxygen levels. By combining a model that describes oxygen transport in a heterogeneous blood vessel network with one that describes the regulation of blood flow by active control of arteriolar diameters, we show that flow regulation is able to counteract this heterogeneity, resulting in more uniform oxygen delivery.
Type 2 diabetes is defined by elevated blood glucose levels. Insulin, secreted by pancreatic β cells, is the major hormone that regulates glucose concentration. The pathway to overt disease requires both the resistance of skeletal muscle to insulin and the eventual failure of β cells to compensate for such resistance. However, the actual development of these causative factors is incompletely understood; nevertheless, oxidative stress due to accumulation of reactive oxygen species is thought to play an important role. Here, we use nonlinear ordinary differential equations to investigate the mechanisms underlying the progressive dysfunction of the glucose-insulin regulatory system and the impact of oxidative stress on this process. In particular, we provide a theoretical framework to describe the long-term dynamics of glucose and insulin coupled with those of skeletal muscle and β -cell metabolic function.
Many biological systems are modeled qualitatively with discrete models, such as prob- abilistic Boolean networks, logical models, Petri nets, and agent-based models. Simulation is a common practice for analyzing discrete models, but many systems are far too large to capture all the relevant dynamical features through simulation alone. We convert discrete models into alge- braic models and apply tools from computational algebra to analyze their dynamics. Based on extensive experimentation with both discrete models arising in systems biology and randomly gen- erated networks, we found that the algebraic algorithms presented in this manuscript are fast for systems with the structure maintained by most biological systems, namely sparseness, i.e., while the number of nodes in a biological network may be quite large, each node is a ected only by a small number of other nodes, and robustness, i.e., small number of attractors. All algorithms and methods are available in our package Analysis of Dynamic Algebraic Models (ADAM), a user friendly web-interface that allows for fast analysis of large models, without requiring understanding of the underlying mathematics or any software installation. ADAM is available as a web tool, so it runs platform independent on all systems.
Chemotaxis is the process by which cells undergo directed motion toward an external signal. In Eukaryotic cells, a precursor to such motion is a symmetry breaking event where proteins responsible for cytoskeletal remodelling and motility self organize to form a front and back. A model developed in collaboration with an experimental group of these regulatory proteins and their associated kinetics is presented. It is shown that this model accounts for observed characteristics not found in other models and provides new insights into the physiologically responsible processes. Novel psuedo-analytic methods for analysing such models will be briefly discussed and connections with experimental observations will be highlighted.
Magnetoencephalography (MEG) is a completely non-invasive brain-mapping modality which uses measurements of the magnetic field outside the head induced by electrical brain activity to localize and characterize the activity inside the brain.
Combined with the more traditional EEG, MEG has shown a great potential in localizing the foci of the onset of epileptic seizures, which is vital information for planning surgery for patients suffering from refractive epilepsy.
A great challenge in the MEG inverse problem is that the data are severely corrupted by noise due to both external and biological sources originating from within the brain itself. Therefore, we wish to know if it is possible to separate signals originating from different types of sources, and also if it is possible to identify deep focal sources.
We address the MEG source separation problem within the Bayesian framework.
More specifically, we investigate the possibility of using mixed prior distributions to separate interfering sources with different spatial statistics. Within the same framework, we also propose a novel, depth scan algorithm capable to identify and localize deep focal sources, overcoming the tendency of MEG inverse solution methods to explain all data with cortical sources. Realistic geometry is used to simulate data corresponding to a variety of sources used to illustrate the performance of our methodology.
Hybrid species are known for sharing some common phenotypes from their parents. The rate of variation of the trait between hybrid and its parents is rarely studied by comparative analysis. An improved phylogenetic comparative method (PCMs) is proposed to allow for data sets that involve hybrids. Instead of using phylogenetic trees, the new method analyzes comparative data by incorporating phylogenetic networks where the hybrid can be explicitly identified by joining the two nodes of its parents. A real data set: The body length in cichilds is analyzed by this new method to investigate whether the rate of variation of the hybrid trait falls between its parents. Results show that the rate of variation of this hybrid trait might not differ significantly from usual species in this case. Simulation studies for accessing the robustness of the new method indicate that the new method is sensitive to the timing of the ancient hybridization. This method has increased power and decreased bias when the hybridization event occurs more recently.
There is evidence that cancer develops when cells acquire a sequence of mutations. This sequence determines a hierarchy among the cells, based on how many more mutations they need to accumulate in order to become cancerous. When cells divide, they exhibit telomere loss and differentiate, which defines another cell hierarchy, on top of which is the stem cell. We propose a mutation-generation model, which combines the mutation-accumulation hierarchy with the differentiation hierarchy of the cells, allowing us to take a step further in examining cancer acquisition and growth. The results of the model support the hypothesis of the cancer stem cell: a very small fraction of the cancer cell population is responsible for the cancer growth and development.
We propose a model of ecological niche construction where consumers can hold different strategies in different proportion, investing primarily in consumption or in sustaining the resource. The model has a form of a parametrically distributed system of ODEs, which is then reduced to a low-dimensional non-autonomous system. The evolution of distribution of the parameter that describes the shift in strategies is tracked. We demonstrate that if one wants to predict where a system will evolve, just knowing the rules that govern its dynamics is not enough to make an accurate decision. One will also need to know the composition of the population that is playing by these rules.
Work in collaboration with Faina Berezovskaya and Georgy Karev.
Due to the distinguishing characteristics of circadian clocks, which are entrainable, self-sustained and temperature compensated, circadian rhythms have been a richly explored area within mathematical biology. We have developed a comprehensive mammalian circadian clock model that accurately represents the molecular dynamics of a single cell in SCN with parameters based on published time profile and half-life of core clock protein and mRNA, and phase response curves (PRC) of kinase inhibition and light. With this model, we explain the unusual shape of the PRC to kinase inhibition. In addition, we reveal the mechanism under the period change to perturbations of clock protein levels.
We studied and compared fibrin networks, with and without cells, formed under wild type and hemophilic conditions. The three dimensional structure of each fibrin network was reconstructed from two-dimensional z-stacks of confocal microscopy sections using novel image analysis algorithms. These images were used to establish microstructure-based models for studying the relationship between the structural features and the mechanical properties of the fibrin networks. The mechanical properties were assessed by analyzing the networks' responses to uniaxial tensile and shear stresses, simulating the impact of blood flow on the fibrin network. The elasticity of the fiber network predicted by the model agrees well with prior experimental data.
Work done in collaboration with Eunjung Kim, Kellie R. Machlus, Xiaomin Liu, Timur Kupaev, Joshua Lioi, Alisa S. Wolberg, Danny Z. Chen, Elliot D. Rosen, Zhiliang Xu and Mark Alber.
In response to infection, immune cells are recruited to ght o invading microbes. Chemotaxis, the process by which cells move in response to chemical gradients, plays a prominent role in this defense mechanism. A number of chemical signals, called chemoattractants, are produced at or proximal to sites of infection and diuse into the surrounding tissue. Immune cells sense these chemoattractants and move in the direction where their concentration is greatest, thereby locating the source of attractants and the associated targets. Leading the assault against new infections is a specialized class of white blood cells called neutrophils, which normally circulate in the blood stream. Upon activation, neutrophils adhere to and squeeze through the vascular endothelium, crawling to sites of infection and inammation. There they phagocytose bacteria and release a number of proteases and reactive oxygen intermediates with antimicrobial activity.
Neutrophils respond to many di erent chemoattractants. One open question is how they locate their targets in complex environments consisting of many dierent chemoattractant species and sources. Successful nav- igation under such conditions necessitates a robust mechanism for interpreting and prioritizing the signals received. Recent studies have implicated an intrinsic signaling hierarchy among the known chemoattrac- tants, resulting in their classication as either end target or endogenous species. In particular, the end target variants, produced exclusively at or near the site of infection, have been found to consistently take prece- dence over the endogenous species. Endogenous chemoattractants, on the other hand, induce an apparently counter-intuitive chemotactic response within the cells, where distant sources are favored over proximal sources regardless of their type.
To date, several attempts to explain this preferential bias toward distant sources have involved temporal mechanisms such as sensory adaptation or an inherent delay in the cell response. However, the possibility of spatial eects has remained unaddressed. Here, we present a mathematical model of neutrophil chemotaxis to show that the observed bias can also be recovered by a purely spatial mechanism - the movement of the cell through changing environments naturally leads to changes in sensitivity to dierent concentrations of the chemoattractants. Moreover, we show that this mechanisms enhances the ability of neutrophils to locate target infections within complex environments. This corroborates a recent hypothesis that neutrophils may migrate in a stepwise fashion between chemokine sources as a means to approach distant end targets. Finally, we present experimental results to validate the predictions of our model.
Inspired by networks from neuroscience, we investigate the response properties of small networks of coupled circle flows to time-dependent inputs. Derived as reductions of excitable neuron models, these systems are characterized by a sink and a saddle in each model neuron. We ask when such networks present reliable responses to stimuli: repeatable behavior from trial to trial with distinct initial conditions. Such properties are closely linked to Lyapunov exponents, and are relevant to neural signal processing.
The loggerhead sea turtle (Caretta caretta) is an endangered species with genetically-distinct nesting populations in the Gulf of Mexico and the western North Atlantic Ocean. In this work, we analyze how these populations are affected by localized oil spill catastrophes caused by offshore drilling. The model consists of a system of spatial, stage-classified matrices that describe the annual change in the population native to each nesting region. Oil spills are simulated deterministically in each nesting region, with oil-induced mortality ranging from 25% to 100% and affecting turtle stage classes either proportionally or equally with respect to their age. The results of this study are intended to provide insights into the population dynamics of the Atlantic loggerhead turtles and suggest conservation techniques appropriate in each oil spill case.
The availability of high-throughput parallel methods for sequencing microbial communities is increasing our knowledge of the microbial world at an unprecedented rate. Though most attention has focused on determining lower-bounds on the α-diversity i.e. the total number of different species present in the environment, tight bounds on this quantity may be highly uncertain because a small fraction of the environment could be composed of a vast number of different species. To better assess what remains unknown, we propose instead to predict the fraction of the environment that belongs to unsampled classes. Modeling samples as draws with replacement of colored balls from an urn with an unknown composition, and under the sole assumption that there are still undiscovered species, we show that conditionally unbiased predictors and exact prediction intervals (of constant length in logarithmic scale) are possible for the fraction of the environment that belongs to unsampled classes. Our predictions are based on a Poissonization argument, which we have implemented in what we call the Embedding algorithm. In fixed i.e. non-randomized sample sizes, the algorithm leads to very accurate predictions on a sub-sample of the original sample. We quantify the effect of fixed sample sizes on our prediction intervals and test our methods and others found in the literature against simulated environments, which we devise taking into account datasets from a human-gut and -hand microbiota. Our methodology applies to any dataset that can be conceptualized as a sample with replacement from an urn; in particular, it may find applications in other problems e.g. related to meta-genomics, genome and transcriptome research.
The rat kidneys morphological and transepithelial transport properties may change in response to different physiologic conditions. To better understand those processes, we used a non-linear optimization technique to estimate parameter sets that maximize key measures that assess the effectiveness and efficiency of a mathematical model of the rat urine concentrating mechanism (UCM). We considered two related measures of UCM effectiveness: the urine-to-plasma osmolality (U/P) ratio and free-water absorption rate (FWA). The optimization algorithm sought parameter sets that separately maximize FWA, maximize U/P with the constraint that the predicted urine flow rate is consistent with reported experimental value (denoted by U=Pq), and maximize the ratio U/P to the total NaCl active transport (TAT) (denoted by (U/P)/TAT). When the principal need of the animal is to maximize the impact of its UCM on blood plasma osmolality, the kidney likely undergoes changes that increase FWA. By selecting parameter values that increase model urine ow rate (while maintaining a sufficiently high urine osmolality), the optimization algorithm identified a set of parameter values that increased FWA by 95.6base-case eciency. If, on the other hand, water must be preserved, then the animal may seek to optimize U/P instead. To study that scenario, the optimization algorithm separately sought parameter sets that attained maximum U=Pq and (U/P)/TAT. Those parameter sets increased urine osmolality by 55.4% and 44.5%, respectively, above base-case value; the outer-medullary concentrating capability was increased by 64.6% and 35.5%, respectively, above base case; and the inner-medullary concentrating capability was increased by 73.1% and 70.8%, respectively, above base case. The corresponding urine ow rate and the concentrations of NaCl and urea are all within or near reported experimental ranges.
Work don in collaboration with Anita Layton.
Cardiac arrhythmias, such as ventricular tachycardia (VT) and ventricular fibrillation (VF), are among the leading causes of death in the industrialized world. These are associated with the formation of spiral and scroll waves of electrical activation in cardiac tissue; single spiral and scroll waves are believed to be associated with VT whereas their turbulent analogs are associated with VF. Thus, the study of these waves is an important biophysical problem. We present a systematic study of the combined effects of musclefiber rotation and inhomogeneities on scrollwave dynamics in the TNNP (ten Tusscher Noble Noble Panfilov) model for human cardiac tissue. In particular, we use the threedimensional TNNP model with fiber rotation and consider both conduction and ionic inhomogeneities. We find that, in addition to displaying a sensitive dependence on the positions, sizes, and types of inhomogeneities, scrollwave dynamics also depends delicately upon the degree of fiber rotation. We find that the tendency of scroll waves to anchor to cylindrical conduction inhomogeneities increases with the radius of the inhomogeneity. Furthermore, the filament of the scroll wave can exhibit drift or meandering, transmural bending, twisting, and breakup. If the scrollwave filament exhibits weak meandering, then there is a fine balance between the anchoring of this wave at the inhomogeneity and a disruption of wavepinning by fiber rotation. If this filament displays strong meandering, then again the anchoring is suppressed by fiber rotation; also, the scroll wave can be eliminated from most of the layers only to be regenerated by a seed wave. Ionic inhomogeneities can also lead to an anchoring of the scroll wave; scroll waves can now enter the region inside an ionic inhomogeneity and can display a coexistence of spatiotemporal chaos and quasiperiodic behavior in different parts of the simulation domain. 1
X-ray mammography is currently the most widespread noninvasive diagnostic imaging modality for the early detection of breast tumors. Until a reliable noninvasive methodology is able to decide if the tumor is benign or malignant, the decision is based on the results of biopsy, a procedure which is costly and invasive. The conductivity of breast tumors differs significantly from that of healthy breast tissue, and the impedance of the tissue can provide important information for identifying malignancies. The impedance distribution inside the breast given current-voltage measurements on the surface can be determined with Electrical Impedance Tomography (EIT). The inherently low resolution of this modality is the main obstacle to use it as a reliable diagnostic tool. We propose a Bayesian framework that combines the spatial information provided by X-ray mammography and the information about the conductivity distribution obtained with EIT into a single, composite image. The assimilation of the data is based on the Bayesian prior models paradigm. A computationally efficient and robust algorithm for the X-ray guided solution of the EIT inverse problem for breast cancer diagnostics is presented, and a few computed examples illustrating its performance are also shown.
Electrophysiological recordings from neurons demonstrate that there are two main types of sodium (Na+) currents: those that inactivate rapidly (transient) and those with slow inactivation (persistent). The distinct kinetics of transient and persistent Na+ currents allow them to play very different roles in neurons. Transient currents contribute primarily to the timing of single action potentials, while persistent currents may shape longer-lasting events such as plateau potentials. Transient and persistent Na+ currents can be encoded by different genes or splice variants of the same gene. Neurons may have a fairly homogeneous population of Na+ channels, expressing a single gene or a number of splice variants with similar kinetics. Or, neurons may express a mixture of Na+ channel genes and splice variants with very different kinetics. In both cases, Na+ current kinetics can be post-translationally altered by neuromodulation. Here we use a biophysical model to ask several questions regarding the role of transient and persistent Na+ currents in neurons. Within a homogeneous population of Na+ channels, we explore the effects of ongoing changes in the kinetics of Na+ channels, as would be the case with neuromodulation. Alternatively, we examine the effects of changing the ratio of transient to persistent channels, as would be the case with differential gene or splice variant expression. Our results show systematic shifts in the excitability profiles of neurons that are consistent with experimental results, especially from motor neurons. In addition, this work raises concerns about assuming the involvement of persistent inward currents when activation/deactivation and firing hysteresis are observed. These results highlight the importance of taking into consideration the different dynamical systems which can produce equivalent behaviors when attributing aspects of neural activity to channel expression or function. Overall, this work contributes to understanding the role that specific ionic currents, their differential expression, and their neuromodulation play in shaping neuronal activity.
The recent chromosome conformation capture (3C) technology and its derivatives give us new tools to study the organization of chromosomes and the long-range gene regulation in 3D. Current methods for analyzing 3C data are either descriptive or based on strict model assumptions. We recently develop a Bayesian mixture model to detect the long-range gene regulation and two other models to detect the change of the intensity of long-range gene regulation in two samples. All models perform well in simulated data and real data.
The United States has observed a steady decline in the number of reported Tuberculosis (TB) cases in the past fifty years, but many states, such as Arizona, have had rates consistently above the US average. TB has been regarded as a disease of the disadvantaged, where poverty, overcrowding and undernourishment are responsible for much of the continued spread. Accordingly, the majority of TB cases in Arizona occur in the foreign-born population, whose households usually fall below the poverty line and have less access to adequate health care. Within this population, undocumented immigrants are the most socially and economically disadvantaged. Therefore, immigration laws, including some of the provisions in SB 1070, are likely to cause further marginalization as the increased fear of deportation will discourage undocumented individuals from seeking work and healthcare. Such laws could potentially exacerbate the spread of TB among undocumented immigrants and the low-income communities in which they reside, eventually extending to all socioeconomic classes. To observe the spread of TB in Arizona this paper employs a TB epidemic model that considers low and high income groups and accounts for different degrees of interaction within and between these socioeconomic classes. We also adjust the model parameters to simulate changes in behavior of undocumented immigrants before and after the implementation of an immigration law such as SB 1070. The simulations revealed that over a relatively brief amount of time, the disease's prevalence in both income groups will significantly increase, as expected, and dramatically more in the low-income group. Ultimately, we investigate just one of the many repercussions of an immigration law, such as SB 1070. Though TB is of interest because of the direct increase of foreign-born TB cases, TB is not the necessarily the greatest health threat. Furthermore, this model may be adapted to other communicable diseases ranging from measles to in uenza.
The implantation of medical devices into a host often triggers an inflammatory response. Few methods are currently available to systematically predict and model the cellular dynamics involved in these reactions. This study introduces a kinetics-based predictive tool for analyzing outcomes of reactions of the involved cells, proteins and enzymes. This tool serves useful in studying the transient behavior during the implant healing period. Using a system of biochemical reaction-diffusion equations in two special dimensions we investigate the time dynamics and the special variation of fibrotic reaction kinetics. Upon testing the model to experimental data we confirmed the reliability of this model to systematical predict the many facets of the reaction.
S.epidermidis infections on medically implanted devices are a common problem in modern medicine due to the abundance of the bacteria. Once inside the body, S. epidermidis gather in communities called biofilms and can become extremely hard to eradicate, causing the patient serious complications. We simulate the complex S. epidermidis-Neutrophils interactions in order to determine the optimum conditions for the immune system to be able to contain the infection and avoid implant rejection. Our cellular automata model can also be used as a tool for determining the optimal amount of antibiotics for combating biofilm formation on medical implants.
Roughly 70% of all malaria cases in Southeast Asia occur in India. While mortality due to Plasmodium infection is low in India relative to the total morbidity, malaria still poses an enormous burden to the country. Several factors, including the biology and epidemiology of the disease, emerging drug-resistance of parasites, insecticide resistance of mosquitoes, and socioeconomic barriers, have proven to be difficult obstacles to overcome in the ongoing pursuit of malaria control. Moreover, two malaria parasites are endemic to India: Plasmodium falciparum and Plasmodium vivax, making malaria control in India an even greater challenge. The primary focus of both malaria control and malaria research has been on P. falciparum, the most severe of the four Plasmodium species causing disease in humans. As a consequence, the burden of vivax malaria has been downplayed and the disease has been for the most part neglected. India has implemented several malaria control programs since the 1940's, with little lasting success. We developed a mathematical model describing the dynamics of P. vivax and P. falciparum in the human and mosquito population of India, parameterized using clinical case data, to understand how enhanced control measures affect the competition between the two Plasmodium species. The Enhanced Malaria Control Project was the first of several programs leading to a dramatic increase in the amount of funding for malaria control in India around 1997. Our model predicts that if India had not improved its control strategy, the two species of Plasmodium would continue to coexist. During the enhanced control period, our model predicts that the controlled reproduction numbers for P. vivax and P. falciparum are both below one. However, the reproduction numbers are extremely close to the invasion boundaries. Consequently, we cannot be confident that the diseases are indeed headed towards extinction. Moreover, if P. vivax and P. falciparum are headed for extinction, it will take a very long time for malaria to die out in India and it is unlikely that India will be able to reach its malaria control benchmarks without changing or improving the current strategy.
Over half of the human genome is composed of repetitive elements, nearly 20% from the L1 sub-family alone. When active, these elements can have dramatic effects on the functioning of the genome including, in extreme cases, inserting into and disrupting genes. While most L1 elements are quiescent, increasingly evidence suggests that a number of sites may produce RNA. I have isolated and characterized cDNA sequences corresponding to L1-initiated transcripts in a panel of human tissues, both normal and malignant (chronic lymphocytic leukemia B cells). Splicing from the L1 into flanking gene regions is common, sometimes resulting in an alternate open reading frame of that gene. In some cases, multiple transcript isoforms are detected originating from the same L1 locus. While some L1-initiated transcripts are expressed in two or more tissues, different tissues also show unique expression patterns. This study indicates an unexpected level of complexity in the complement of transcripts originating from L1 elements. I hypothesize that regulation of these transcripts involves interplay between both locus- and tissue-specific factors.
A major focus of synthetic biology is the design and construction of de novo cellular components which perform functions analogous to traditionally engineered devices, for example, toggle switches, counters, and oscillators. In most cases, these synthetic biological constructs contain only genetic elements. An exception is the "metabolator," a synthetic oscillating circuit comprising both genetic and metabolic elements which was developed by the Liao lab at UCLA in 2005. This project explores dynamical behavior in the space of possible hybrid metabolic-genetic circuit designs, taking the metabolator as the starting point for modeling and analysis. Among our results are: (1) Analytical conditions for the onset and frequency of oscillations in a simpli ed ODE model of the metabolator. (2) An alternate design for a metabolic-genetic oscillator, which we call the "reverse metabolator" because the directions of some regulatory connections are switched, which exhibits oscillatory behavior. (3) Bifurcation analysis of the reverse metabolator showing that limit cycle oscillations may occur via Hopf bifurcations and by folds of limit cycles. (4) Analytical and numerical demonstration that a reduced version of the reverse metabolator is capable of oscillations despite having half as many regulatory components as the original metabolator.
Work done in collaboration with William Erik Sherwood and Daniel Segre.
We explore several fundamental mechanisms that determine how correlations and synchrony propagate in networks of neurons. We show that single-cell dynamics and synaptic variability significantly reduce correlations from input to output, but that synaptic convergence dramatically amplifies correlations downstream. Perhaps surprisingly, we find that synaptic convergence is the primary mechanism responsible for the synchronization of feedforward chains, and that synaptic divergence plays a comparatively minor role.
Dengue fever has been a burden to public health officials for decades. Despite strong efforts to 'control' the spread of the disease it keeps spreading at an alarming rate mostly in tropical countries where healthcare is a privilege. Different mathematical and computational models will be discussed to highlight public health challenges and their role in the spread of the disease. Possible 'intervention/control' strategies will be discussed.
We describe evidence that the central pattern generator controlling feeding in Aplysia californica can be better described by a heteroclinic channel than a homogeneous limit cycle. We explore three one-parameter families of limit cycles, where the as the parameter approaches zero the flow approaches a an attracting heteroclinic cycle. The first model is a neuromechanical model, which we use to obtain empirical results about the expected skewness of burst time distributions and the effects of proprioception and noise. The second and third models are simplified two-dimensional models, with which we are able to obtain an analytical expression for the infinitesimal phase resetting curves as the cycle approaches the heteroclinic bifurcation. We then examine in vivo and in vivo recordings from Aplysia californica and show that the recordings are more consistent with the models near the heteroclinic limit.
A spring model is applied to simulation the skeleton structure of the red blood cell membrane and to study the red blood cell rheology in Poiseuille flows with an immersed boundary method. The lateral migration properties of the cells in Poiseuille flows have been investigated. The simulation results show that the rate of migration toward the center of channel depends on the area reduction and the deformability of the cells. We also have combined the above methodology with a fictitious domain method to study the motion of RBCs in a microchannel with a constriction, which can enhance the cellfree layer adjacent to the boundary.
Stable periodic oscillations have been shown to exist in mathematical models for the CTL response to HTLV-I infection in vivo. These periodic oscillations can be the result of mitosis of infected target CD4+ cells, of a general form of response function, or of time delays in the CTL response. In this study, we show that time delays in the CTL response process to HTLV-I infection can lead to the coexistence of multiple stable periodic solutions, which differ in amplitude and period, with their own basins of attraction. Our results imply that the dynamic interactions between the CTL immune response and HTLV-I infection are very complex. Different routes or initial dosages of the viral infection may lead to quantitatively and qualitatively different outcomes.
Over ninety percent of pancreatic cancers contain a mutation in the protein K-Ras. K-Ras is part of a network of proteins that relays transient signals that instruct the cell to proliferate. The mutant K-Ras proteins found in cancer continuously send signals that instruct the cancer cell to proliferate. This constant signal results from disruption of the processes that normally ensure low levels of Ras signal except in response to transient stimulation. My research has utilized mathematical models of these biochemical processes to study mutant Ras proteins. Each process had been well-studied in isolation, but the complexity of the system made it difficult to fully understand how they function collectively. Mathematical models allowed for these and other data to be integrated to study what may be happening within the cancer cell. Model based investigations uncovered non-obvious behaviors in the Ras signaling network that could explain the types of Ras mutants found in cancer and that uncovered a source of proliferation signals that were not appreciated by the experimental Ras community. Experiments suggested by the model have confirmed these and other model predictions. The model is now being expanded to include different proteins that, when mutated, have similar effects promoting cancer. The model is also being expanded to include processes that are targeted by drugs designed to inhibit Ras signaling.
The transmission of West Nile virus, a mosquito-borne disease, naturally depends on weather conditions, since the biology of the vector mosquito changes with ambient temperature and precipitation. Developing a weather-driven West Nile virus transmission model is important: it can reveal the mechanism of the weather impact on virus transmission and also can be used to project the future virus transmission under global warming background. In this paper, the impacts of temperature and precipitation on mosquito population, mosquito biting rate and mortality rate were involved in a SIR compartmental model for mosquito-bird-human system. The observed West Nile virus transmission was reproduced by this model; the sensitivity of the weather impact on the virus transmission is analysed; and the virus transmission driven by a climate change scenario is projected.
In a 2006 paper by Kholodenko, a simple cycle motif in which a signalling protein is modified by two opposing enzymes, such as a phosphatase and kinase or a guanine nucleotide exchange factor and GTPase-activating protein, is described. A slew of different dynamical behaviors can emerge from a simple cycle with the addition of feedback. A bistable switch can come about in a simple fashion in four different ways depending on which process the enzymes inhibit or promote. This research will determine whether nature favors certain motifs over others by using data mining techniques and statistics.
Climate change is causing many species to shift their ranges poleward in latitude or upward in elevation. We analyze an integrodifference equation that combines growth, dispersal, and a constant-speed shift in habitat to assess the impact of climate change on population persistence. For our model, if the population's range shifts too rapidly, the population goes extinct. The critical speed for extinction depends on both growth parameters and on the shape of the dispersal kernel. We demonstrate how to calculate the critical speed numerically, as well as how to approximate the critical speed analytically.
