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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 ,
belykh@math.gsu.edu ,
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.

Wednesday, April 25, 2012 - 11:05 ,
Location: Skiles 006 ,
Leonid Khanin ,
Idaho State University ,
khanin@isu.edu ,
Organizer:

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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.

Wednesday, 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 ,
yjiang12@gsu.edu ,
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.)