Wednesday, January 30, 2013 - 11:05 , Location: Skiles Bld Room 005 , Andrew Vlasic , Indiana University , Organizer:
For many evolutionary dynamics, within a population there are finitely many types that compete with each other. If we think of a type as a strategy, we may consider this dynamic from a game theoretic perspective. This evolution is frequency dependent, where the fitness of each type is given by the expected payoff for an individual in that subpopulation. Considering the frequencies of the population, the logarithmic growth is given by the difference of the respective fitness and the average fitness of the population as a whole. This dynamic is Darwinian in nature, where Nash Equilibria are fixed points, and Evolutionary Stable Strategies are asymptotically stable. Fudenberg and Harris modified this deterministic dynamic by assuming the fitness of each type are subject to population level shocks, which they model by Brownian motion. The authors characterize the two strategy case, while various other authors considered the arbitrary finite strategy case, as well as different variations of this model. Considering how ecological and social anomalies affect fitness, I expand upon the Fudenberg and Harris model by adding a compensated Poisson term. This type of stochastic differential equation is no longer continuous, which complicates the analysis of the model. We will discuss the approximation of the 2 strategy case, stability of Evolutionary Stable Strategies and extinction of dominated strategies for the arbitrary finite strategy case. Examples of applications are given. Prior knowledge of game theory is not needed for this talk.
Wednesday, November 28, 2012 - 11:00 , Location: Skiles Bldg Rm.005 , Igor Belykh , Georgia State , firstname.lastname@example.org , Organizer:
This talk focuses on mathematical analysis and modeling of dynamical systems and networks whose coupling or internal parameters stochastically evolve over time. We study networks that are composed of oscillatory dynamical systems with connections that switch on and off randomly, and the switching time is fast, with respect to the characteristic time of the individual node dynamics. If the stochastic switching is fast enough, we expect the switching system to follow the averaged system where the dynamical law is given by the expectation of the stochastic variables. There are four distinct classes of switching dynamical networks. Two properties differentiate them: single or multiple attractors of the averaged system and their invariance or non-invariance under the dynamics of the switching system. In the case of invariance, we prove that the trajectories of the switching system converge to the attractor(s) of the averaged system with high probability. In the non-invariant single attractor case, the trajectories rapidly reach a ghost attractor and remain close most of the time with high probability. In the non-invariant multiple attractor case, the trajectory may escape to another ghost attractor with small probability. Using the Lyapunov function method, we derive explicit bounds for these probabilities. Each of the four cases is illustrated by a specific technological or biological network.
Wednesday, September 19, 2012 - 11:05 , Location: Skiles 005 , Dmitry Korkin , Informatics Institute and Department of Computer Science, University of Missouri-Columbia , Organizer:
We have recently witnessed the tremendous progress in evolutionary and regulatory genomics of eukaryotes fueled by hundreds of sequenced eukaryotic genomes, including human and dozens of animal and plant genomes and culminating in the recent release of The Encyclopedia of DNA Elements (ENCODE) project. Yet, many interesting questions about the functional and structural organization of the genomic elements and their evolution remain unsolved. Computational genomics methods have become essential in addressing these questions working with the massive genomic data. In this presentation, I will talk about two interesting open problems in computational genomics. The first problem is related to identifying and characterizing long identical multispecies elements (LIMEs), the genomic regions that were slowed down through the course of evolution to their extremes. I will discuss our recent findings of the LIMEs shared across six animal as well as six plant genomes and the computational challenges associated with expanding our results towards other species. The second problem is finding genome rearrangements for a group of genomes. I will present out latest approach approach that brings together the idea of symbolic object representation and stochastic simulation of the evolutionary graphs.
A NEW PARADIGM OF CANCER PROGRESSION AND TREATMENT DISCOVERED THROUGH MATHEMATICAL MODELING: WHAT MEDICAL DOCTORS WON’T TELL YOUWednesday, April 25, 2012 - 11:05 , Location: Skiles 006 , Leonid Khanin , Idaho State University , email@example.com , Organizer:
Normal 0 false false false EN-US X-NONE X-NONE Over the last several decades, cancer has become a global pandemic of epic proportions. Unfortunately, treatment strategies resulting from the traditional approach to cancer have met with only limited success. This calls for a paradigm shift in our understanding and treating cancer. In this talk, we present an entirely mechanistic, comprehensive mathematical model of cancer progression in an individual patient accounting for primary tumor growth, shedding of metastases, their dormancy and growth at secondary sites. Parameters of the model were estimated from the age and volume of the primary tumor at diagnosis and volumes of detectable bone metastases collected from one breast cancer and 12 prostate cancer patients. This allowed us to estimate, for each patient, the age at cancer onset and inception of all detected metastasis, the expected metastasis latency time and the rates of growth of the primary tumor and metastases before and after the start of treatment. We found that for all patients: (1) inception of the first metastasis occurred very early when the primary tumor was undetectable; (2) inception of all or most of the surveyed metastases occurred before the start of treatment; (3) the rate of metastasis shedding was essentially constant in time regardless of the size of the primary tumor, and so it was only marginally affected by treatment; and most importantly, (4) surgery, chemotherapy and possibly radiation bring about a dramatic increase in the rate of growth of metastases. Although these findings go against the conventional paradigm of cancer, they confirm several hypotheses that were debated by oncologists for many decades. Some of the phenomena supported by our conclusions, such as the existence of dormant cancer cells and surgery-induced acceleration of metastatic growth, were first observed in clinical investigations and animal experiments more than a century ago and later confirmed in numerous modern studies.
Wednesday, April 18, 2012 - 13:05 , Location: Skiles 006 , Gemma Huguet , NYU , Organizer:
In this talk we will present a numerical method to perform the effective computation of the phase advancement when we stimulate an oscillator which has not reached yet the asymptotic state (a limit cycle). That is we extend the computation of the phase resetting curves (the classical tool to compute the phase advancement) to a neighborhood of the limit cycle, obtaining what we call the phase resetting surfaces (PRS). These are very useful tools for the study of synchronization of coupled oscillators. To achieve this goal we first perform a careful study of the theoretical grounds (the parameterization method for invariant manifolds and the Lie symmetries approach), which allow to describe the isochronous sections of the limit cycle and, from them, to obtain the PRSs. In order to make this theoretical framework applicable, we design a numerical scheme to compute both the isochrons and the PRSs of a given oscillator. Finally, we will show some examples of the computations we have carried out for some well-known biological models. This is joint work with Toni Guillamon and R. de la Llave
Wednesday, April 18, 2012 - 11:00 , Location: Skiles 006 , David Murrugarra , Virginia Tech , Organizer: Christine Heitsch
Modeling stochasticity in gene regulation is an important and complex problem in molecular systems biology. This talk will introduce a stochastic modeling framework for gene regulatory networks. This framework incorporates propensity parameters for activation and degradation and is able to capture the cell-to-cell variability. It 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 a discrete variable. 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. A background to stochastic modeling will be given and applications will use two of the best known stochastic regulatory networks, the outcome of lambda phage infection of bacteria and the p53-mdm2 complex.
Multi-scale Model of CRISPR-induced Coevolutionary Dynamics: Diversification at the Interface of Lamarck and DarwinWednesday, March 7, 2012 - 11:00 , Location: Skiles 006 , Lauren Childs , Biology, Georgia Tech , Organizer: Christine Heitsch
The CRISPR (Clustered Regularly Interspaced Short Palindromic Repeats) system is a recently discovered immune defense in bacteria and archaea (hosts) that functions via directed incorporation of viral DNA into host genomes. Here, we introduce a multi-scale model of dynamic coevolution between hosts and viruses in an ecological context that incorporates CRISPR immunity principles. We analyze the model to test whether and how CRISPR immunity induces host and viral diversification and maintenance of coexisting strains. We show that hosts and viruses coevolve to form highly diverse communities through punctuated replacement of extant strains. The populations have very low similarity over long time scales. However over short time scales, we observe evolutionary dynamics consistent with incomplete selective sweeps of novel strains, recurrence of previously rare strains, and sweeps of coalitions of dominant host strains with identical phenotypes but different genotypes. Our explicit eco-evolutionary model of CRISPR immunity can help guide efforts to understand the drivers of diversity seen in microbial communities where CRISPR systems are active.
Wednesday, February 29, 2012 - 11:00 , Location: Skiles 006 , Yi Jiang , GSU , firstname.lastname@example.org , Organizer:
Angiogenesis, growth of new blood vessels from existing ones, is animportant process in normal development, wound healing, and cancer development. Presented with increasingly complex biological data and observations, the daunting task is to develop a mathematical model that is useful, i.e. can help to answer important and relevant questions, or to test a hypothesis, and/or to cover a novel mechanism. I will present two cell-based multiscale models focusing on biochemical (vescular endothelial growth factors) and biomechanical (extra-cellular matrix) interactions. Our models consider intracellular signaling pathways, cell dynamics, cell-cell andcell-environment interactions. I will show that they reproduced someexperimental observations, tested some hypotheses, and generated more hypotheses.
Wednesday, February 8, 2012 - 14:00 , Location: Skiles 005 , Nathan Baker , Pacific Northwest National Laboratory , Organizer: Christine Heitsch
Implicit solvent models are important components of modern biomolecular simulation methodology due to their efficiency and dramatic reduction of dimensionality. However, such models are often constructed in an ad hoc manner with an arbitrary decomposition and specification of the polar and nonpolar components. In this talk, I will review current implicit solvent models and suggest a new free energy functional which combines both polar and nonpolar solvation terms in a common self-consistent framework. Upon variation, this new free energy functional yields the traditional Poisson-Boltzmann equation as well as a new geometric flow equation. These equations are being used to calculate the solvation energies of small polar molecules to assess the performance of this new methodology. Optimization of this solvation model has revealed strong correlation between pressure and surface tension contributions to the nonpolar solvation contributions and suggests new ways in which to parameterize these models. **Please note nonstandard time and room.**
Wednesday, January 25, 2012 - 11:00 , Location: Skiles 006 , Anne Shiu , University of Chicago , Organizer: Christine Heitsch
Chemical reaction networks taken with mass-action kinetics are dynamical systems governed by polynomial differential equations that arise in systems biology. In general, establishing the existence of (multiple) steady states is challenging, as it requires the solution of a large system of polynomials with unknown coefficients. If, however, the steady state ideal of the system is a binomial ideal, then we show that these questions can be answered easily. This talk focuses on systems with this property, are we say such systems have toric steady states. Our main result gives sufficient conditions for a chemical reaction system to admit toric steady states. Furthermore, we analyze the capacity of such a system to exhibit multiple steady states. An important application concerns the biochemical reaction networks networks that describe the multisite phosphorylation of a protein by a kinase/phosphatase pair in a sequential and distributive mechanism. No prior knowledge of chemical reaction network theory or binomial ideals will be assumed. (This is joint work with Carsten Conradi, Mercedes P\'erez Mill\'an, and Alicia Dickenstein.)