Research Areas
My
overarching research interest is applying stochastic modeling and
analysis to characterize the behavior of complex systems, particularly
linking phenomena at a formally macroscopic or collective level to the
dynamics prescribed at an individual or formally microscopic
level. Statistical Mechanics of Swimming MicroorganismsOne
of the most vigorous fields of recent inquiry regarding the connection
of collective behavior to the rules of individuals is in the
quantitative modeling of animal swarming or flocking. In
fact for terrestial-scale organisms, for which much of this work is
focused, the individual mobility and interaction rules are not yet
precisely understood, and various hypotheses are under active
exploration. Swimming microorganisms, on the other hand, mainly
because of their rather limited cognitive powers, have comparatively
very well understood mechanisms for motion as well as physical and
biological rules for interaction. A growing range of experiments
on suspensions of swimming microorganisms moreover reveals a range of
remarkable collective behavior, such as patterned flows and changes to
effective fluid properties such as viscosity. From a
mathematical point of view, therefore, because of the sound theoretical
foundation of individual dynamics and interactions and the range of
well-controlled experiments on populations, swimming microorganisms
serve as an excellent model system for developing analytical techniques
for connecting collective dynamics to the individual dynamical
rules. Moreover, beyond this theoretical motivation, the
understanding of suspensions of swimming microorganisms has strong
potential for applications to driving microfluidic systems.
Together with Patrick Underhill (Chemical Engineering), recent graduate
student Kajetan Sikorski, and current graduate
student Yuzhou Qian (Chemical Engineering), we have been working on
incorporating both
statistical correlations and stochastic effects into a theory for
predicting the effective behavior of a swimming microorganisms in a
suspension, given the well-defined physical equations for individual
motion and hydrodynamic interaction. The stochastic aspect here
arises predominantly not from thermal fluctuations, but from
coarse-graining of the detailed dynamics of the swimming process.
An important part of this project is clarifying, on quantitative
physical grounds, the source and impact of these stochastic components.
Intracellular transport
The
fundamental actors in transporting organelles within cells are families
of protein molecules, predominantly dynein and kinesin, which act as
molecular-level engines, converting chemical fuel (ATP) into some sort
of useful work, such as directed transport. Mathematical models
for these molecular motors are typically stochastic, as thermal
fluctuations play a strong role in influencing their dynamics, both by
the noisy buffeting by the solvent molecules as well as the random
diffusive transport of ATP to the proper binding location to activate
the next chemomechanical cycle for progress. A great deal of
mathematical and physical study of the mechanisms and performance of
natural and synthetic molecular motors has taken place over the last 20
years. With Juan Latorre (Freie Universitaet Berlin) and
Grigorios Pavliotis (Imperial College), we contributed to this body of
work by applying homogenization theory and various techniques from
stochastic analysis to map out how the transport effectiveness of a
fundamental ``flashing ratchet" model depends on the governing
parameters, and to evaluate the regimes in which simplified Markov
chain models adequately represented the effective transport of the more
detailed model. Over the last five years, a rapidly growing
amount of theoretical and experimental work has been focusing on how
the dynamics of a cargo is governed by multiple molecular motors bound
to it simultaneously, as is expected to be relevant in vivo.
Within a working group at the Statistics and Applied Mathematical
Sciences Institute (SAMSI) we critically examined this literature, and
with John Fricks (Penn State, statistics), Scott McKinley (Univ.
Florida) and Avanti Athreya (Johns Hopkins), we formulated a new
modeling framework for molecular motors cooperatively bound to a cargo
which both accounted for stochastic spatial fluctuations in a more
physically consistent manner than most of the existing models and
allowed for analytical techniques to connect single-motor properties to
those of the motor-cargo complex. We are now, together with
experimentalist William Hancock (Penn State, bioengineering), working
on developing our basic model to incorporate more realistic features,
with the broad aim of linking detailed dynamical understanding of
molecular motors at an individual level to larger scale experimental
observations. During discussions of this effort at
the Zentrum fuer Interdisziplinaere Forschung in Bielefeld, Germany,
some of the other research fellows in residence, namely Leonid Koralov
(Maryland), Yuri Makhnovskii (Topchiev Institute), and Leonid Bogachev
(Leeds) have begun collaborating with me on developing some of the more
technically challenging mathematical aspects needed to capture the
relevant physics. Stochastic Dynamics in Neuronal and Other Networks
A
grand challenge in theoretical neuroscience is to explain how
higher-level brain functions, such as memory and object recognition,
can be explained in terms of the known biophysics of the constituent
neurons and their connections. The overwhelming complexity of the
brain naturally frustrates a direct mathematical attack on this
problem, so mathematical neuroscience has in recent years focused on
more tractable analogue questions, whose resolution can bring insight
into these broader conceptual issues. I became involved in
mathematical neuroscience through co-advising former graduate student
Katherine Newhall (now at UNC Chapel Hill) with colleague
Gregor Kovacic and collaborator David Cai (Shanghai Jiao-Tong), where
we were concerned with a project of characterizing the presence of
strongly synchronous behavior in a neuronal network model in terms of
the individual neuron dynamics and the network model describing their
connections. We employed the basic stochastically driven
integrate-and-fire model for the neurons, and both simple and,
with Maxim Shkarayev (Iowa State) complex scale-free network
model architectures. Randomness enters into these models
both to describe generic unstructured input signals (typically via
canonical Poisson point processes) and to generate complex networks,
for which only certain statistics are prescribed. Katherine
Newhall, Peter Mucha (UNC), Amanda Traud (NC State), and I are
exploring generalizations of these ideas to more general dynamical
network models with cascades. During a recent visit to the
Mathematical Biosciences Institute (MBI), Janet Best (Ohio State) and I
formed a research group, together with Fatih Olmez, Jung-Eun Kim (UNIST
Korea), and Deena Schmidt (UN Reno), to pursue a related question of
how the architecture
of a metastable neuronal network model for a sleep-wake system affects
the distribution of times spent in the sleep or wake states.
Undergraduate research student Anthony Trubiano has been assisting this
program through examination
of how to effectively characterize sleep and wake states, and
transitions between them, in the neuronal network model.
Stochastic network modeling in
epidemiologyKristin
Bennett have been working with graduate student Lei Yao to develop a
mathematical model and statistical technique for
inferring unobserved transmission events from data of treated
tubercolosis patients in New York State. The objective is
to apply individual-based stochastic dynamical models for the spread of
tubercolosis with computational methods from network tomography and
phylogenetics to integrate temporal information from the data into
Bennett's clustering methods.
Statistical Aspects of Multiscale ComputingDuring
a visit to the Statistical and Applied Mathematical Sciences Institute
(SAMSI) for their program on ``Stochastic Dynamics," a
cross-disciplinary project emerged between computational scientist
Sorin Mitran (University of North Carolina), myself, and statisticians
M. Susie Bayarri (Universitat de Valencia), James Berger (Duke),
Murali Haran (Penn State, ), and Hans Rudolf Kuensch (ETH Zurich) from
a contemporaneous program. We have been exploring, with undergraduate
student Ben Walker, how to strengthen, using systematic techniques from
the statistics community, the representation and interaction of the
microscale and mesoscale elements of a novel multiscale simulation
method (time-parallel continuum-kinetic-molecular) recently introduced
by Mitran.Stochastic modeling in
ecology
Brad
Lister (biology) and I have co-advised a number of undergraduate
students on theoretical ecology modeling projects as part of the
``Computational Science Training in the Mathematical Sciences''
program, and out of this experience, we have been developing
mathematical models for ecosystems which incorporate more biophysical
realism. In particular, we are working on an size-structured
model for daphnia with explicit coupling to nutrient concentrations,
applying a novel probabilistic model for computing the transfer of
individuals between size compartments. We are also developing
quantitative frameworks for how foraging decisions are affected by
partial information accumulated over time concerning predators.
Graduate student Karen Cumings has begun working with us on
understanding recently discovered mechanisms for how plants interact
with each other. Multidisciplinary graduate student Christine
Goodrich is in the process of formulating a research project concerning
a quantitative study of how human activites affect the services which
the environment provides to human living. Nonlinear Wave Turbulence
With
graduate student Michael Schwarz, Gregor Kovacic and I have been
investigating the application of turbulence and kinetic theories to the
dispersionless form of the one-dimensional Majda-McLaughlin-Tabak
model. With recent graduate student Warren Towne (now at Lincoln
Laboratories), Yuri L'vov (Rensselaer) and I
have been developing an approach to simulating weakly nonlinear
statistical systems which mitigates artifacts of periodic boundary
conditions commonly employed.
Some of the above projects are supported by NSF grants DMS-1211665 and DMS-1344962 (RTG).
Any opinions, findings, and conclusions or recommendations expressed
in this material are those of the author and do not necessarily
reflect the views of the National Science Foundation.
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