David Anderson,
University of Wisconsin
Using Random Forcing to Study the OutofEquilibrium Dynamics of
Biochemical Reaction Systems
There are two different natural contexts in which
stochastic
dynamics arises in the study of biochemical reaction networks. In
the first, the stochastic chemical dynamics arises from the
randomness inherent in the formation and breaking of chemical bonds.
This is the randomness present in the Gillespie algorithm. If one
scales up the volume and number of molecules while keeping the
initial concentrations constant, then this intrinsic stochasticity
becomes negligible on the scale of concentrations and the dynamical
system reduces to a collection of deterministic, coupled
differential equations. The second type of stochasticity, which is
the focus of this talk, arises naturally in this scaling limit.
In this second context, one wants to investigate the response of a
biochemical system (whose dynamics are well described by
differential equations) to external excitation. Here the randomness
is a tool used to study the outofequilibrium dynamics of the
biochemical system and the object of study is a set of differential
equations that is forced by a stochastic process in one or a small
number of components. In this talk, I will describe how we take two
distinct approaches in our study of stochastically forced
biochemical systems. In the first, we study how fluctuations
propagate through relatively simple systems and study the effect of
network topology. In the second, we apply fluctuations to in silico
representations of specific biological networks.
Carina Curto,
Rutgers University
From Spikes to Space: Reconstructing Features
A common stimulus reconstruction paradigm involves first computing the
receptive fields of recorded neurons (using both spike trains and the
presented stimuli), and then using the receptive field information together
with neuronal activity in order to "predict" the pattern of stimuli
based on local stimulus features. Some brain regions (such as hippocampus)
undergo quick remapping of receptive fields, depending on context. How
do downstream neurons integrate the mosaic of individual neuronal responses,
with potentially varying receptive fields, to extract global
characteristics of presented stimuli? In rodent hippocampus spatial
information is encoded by place cells, i.e. pyramidal cells that fire in a
restricted convex area of the spatial environment, and are mostly silent
outside. The place fields for the same neurons remap from one spatial
environment to another. To say that place cells "encode" the animal's
location in space suggests that a downstream structure could effectively
decode position information using the activity of place cells alone.
Without information about the place fields themselves, the task of the
downstream neurons is no longer so straightforward. In fact, it is not
obvious that anything at all may be learned about the environment by
looking at the spiking activity of place cells alone.
In this work we show how certain global features of a spatial environment
can be computed from hippocampal spiking activity alone. In particular,
we consider a variety of twodimensional environments which differ in the
number of obstacles (or holes) constraining the region accessible to a
freelyforaging rat. Assuming only basic properties of hippocampal place
fields, we construct an algorithm that distinguishes between these
different environments by computing standard topological invariants
(homology groups) from population spiking activity. These invariants
precisely determine the numbers of obstacles/holes in the environment 
and can be computed without ever using any positiontracking information,
or any other feature of the rat's trajectory through space. In particular,
we never compute place fields or any other correlations between cell
firing and external stimuli.
We tested the algorithm using simulated data, staying as close as possible
to physiological parameters seen in real data. For each of five
distinct environments, open field and Nobstacle environments for N=1,..,4,
we simulated place cell spike trains corresponding to a random walk.
(Place fields in different environments were completely unrelated.)
The algorithm correctly identified each environment from the population
spiking data. Furthermore, on shuffled data sets, the computed homology
groups reflected highdimensional, nonphysical environments. We have
thus shown that global features of the spatial environment can be
reconstructed from hippocampal place cell spiking activity alone.
Greg Forest,
University of North Carolina
Active and Passive Microrheology
A new emphasis in rheology (the study of how materials
flow and deform under loads) has been ignited by biology.
Namely, the need to understand the rheological properties of
cells, tissues, membranes, and biological liquids, to name a few,
has led to new techniques in rheology. One such method combines
advanced microscopy with micronscale embedded particles to
track their position data, driven either to thermal fluctuations
or to some applied force or stress. The lecture will highlight
the basic models and analysis used together with experimental
protocols to inverse characterize biological materials, and then
to directly simulate behavior under different conditions.
Daniel Forger,
University of Michigan
What Makes a Genetic Network a Clock?
Biological clocks time many events in nature including sleep patterns,
developmental events, and hormone release. Recent experimental work has
shown that these clocks consist of feedback networks of interacting genes
and proteins within cells. By analyzing mathematical models (ordinary
differential equations), we derive strict requirements for when
oscillations emerge in genetic feedback networks. We show how a bistable
clock can be built, where external signals can start or stop rhythmicity,
and discuss recent experimental work to build synthetic clocks. Amazingly,
the period of these clocks depends only on how quickly molecules are
removed, and whether rhythms are close to sinusoidal. From this general
theory, more accurate mammalian models of circadian (~24hour) rhythms
will be considered, and we will explain why a family in Utah always wakes
up early.
Monsoor Haider,
North Carolina State University
Continuum Mixture Models of the Cellular Microenvironment in Articular Cartilage
Articular cartilage is the primary loadbearing soft tissue in diarthrodial joints
such as the knee, shoulder and hip.
Cartilage can be modeled as a multiphasic continuum mixture of collagen,
proteoglycan and interstitial fluid with dissolved ions.
Degradation of cartilage leads to osteoarthritis, a painful condition that is,
predominantly, associated with aging. Since cartilage is avascular and aneural,
alterations in physical and chemical variables in its cellular microenvironment
can strongly influence cell metabolic activity. In this talk, biphasic (solidfluid)
and triphasic (solidfluidion) models of mechanical and chemical loading in
cartilage will be presented. Solutions of associated microscale boundary value
problems will be presented and implications for modeling osteoarthritis will be
discussed.
Trachette Jackson,
University of Mitchgan
Modeling the Molecular, Cellular, and Tissue Interactions Associated with
Tumor Induced Angiogenesis
Vascular endothelial growth factor (VEGF) is one of the most potent,
specific and intensively studied tumor angiogenic factors. Recent
experiments show that VEGF is the crucial mediator of downstream
events that ultimately lead to enhanced endothelial cell survival and
increased vascular density within many tumors. The newly discovered
pathway involves upregulation of the antiapoptotic protein Bcl2,
which in turn leads to increased production of interleukin8 (CXCL8).
The VEGFBCL2CXCL8 pathway suggests new targets for the development
of antiangiogenic strategies including short interfering RNA (siRNA)
that silence the CXCL8 gene and small molecule inhibitors of Bcl2.?
In this talk I will discuss our efforts to develop and validate
mathematical models of tumorinduced angiogenesis and vascular tumor
growth that are able to predict the effect of the therapeutic
blockage of VEGF, CXCL8, and Bcl2 at early middle and late stages of
tumor progression.
James Keener,
University of Utah
How Cells Make Measurements
A fundamental problem of cell biology is to understand how
cells make measurements and then make behavioral decisions in
response to these measurements. The full answer to this question is
not known but there are some underlying principles that are coming to
light. The short answer is that the rate of molecular diffusion
contains quantifiable information that can be transduced by
biochemical feedback to give control over physical structures.
In this talk, this principle will be illustrated by two specific examples
of
how
rates of molecular diffusion contain information that is used to make
a measurement and a behavioral decision.
Example 1: Bacterial populations of P. aeruginosa are known to make a
decision to secrete polymer gel on the basis of
the size of the colony in whch they live. This process is called
quorum sensing and only recently has the mechanism for this been
sorted out. It is now known that P. aeruginosa produces a chemical
whose rate of diffusion out of the cell provides information about the
size of the colony which when coupled with positive feedback gives rise
to a hysteretic biochemical switch.
Example 2: Salmonella employ a mechanism that combines molecular
diffusion with a negative feedback chemical network to "know" how long
its flagella are. As a result, if a flagellum is cut off, it will
be regrown at the same rate at which it grew initially.
Harold Layton,
Duke University
Irregular Flow Oscillations in the Nephrons of Spontaneously Hypertensive Rats
Single nephron proximal tubule pressure in spontaneously hypertensive rats (SHR)
can exhibit highly irregular oscillations similar to deterministic chaos.
We used a mathematical model of tubuloglomerular feedback (TGF) to investigate
potential sources of the irregular oscillations and the associated complex power
spectra in SHR. A bifurcation analysis of the TGF model equations, for nonzero
thick ascending limb (TAL) NaCl permeability, was performed by finding roots of
the
characteristic equation, and numerical simulations of model solutions were condu
cted
to assist in the interpretation of the analysis.
These techniques revealed
a complex parameter region, consistent with TGF gain and delays in SHR, where fo
ur
qualitatively different model solutions are possible:
(1) a regime having one stable, timeindependent steadystate solution;
(2) a regime having one stable oscillatory solution only, with fundamental frequency f;
(3) a regime having one stable oscillatory solution only, with frequency
~2f;
and (4) a regime having two possible stable oscillatory solutions,
of frequencies ~f and ~2f.
In addition, we
conducted simulations in which TAL flow was assumed to vary as a function of time and
simulations in which two or three nephrons were assumed to have coupled TGF systems.
Four potential sources of spectral complexity in SHR were identified:
(1) bifurcations that produce qualitative changes in solution type,
leading to multiple spectrum peaks
and their respective harmonic peaks; (2) continuous lability in delay parameters,
leading to broadening of peaks and their harmonics; (3) episodic lability in delay
parameters, leading to multiple peaks and their harmonics; and (4) coupling of small
numbers of nephrons, leading to broadening of peaks, multiple peaks, and their
harmonics.
We conclude that the complex power spectra in SHR may be explained by the
inherent complexity of TGF dynamics, which may include
solution bifurcations, modest timevariation in TGF parameters,
and coupling between small numbers of neighboring nephrons
Michael Mackey,
McGill University
Mathematics, Biology & Physics: Interactions and Interdependence
This talk will examine the contributions of mathematics and physics to
biology and medicine by individuals ranging from Galvani and Volta through
Helmholtz, Huxley, Neher, Wiener, Hardy and many more. The spectacular
successes that mathematics and physics have had in neurophysiology in the
nineteenth and twentieth centuries is likely to be repeated in molecular
biology in the twenty first century. Some of the famous pioneers that
started this latter movement are Schrödinger, Delbruck, and Brenner as well
as a host of others.
Laura Miller,
University of North Carolina
Fluid Dynamics Within the Developing Embryonic Heart
Colleen Mitchell,
University of Iowa
Neural Timing in Highly Convergent System
In order to study how the convergence of many variable neurons on a single
target can sharpen
timing information, we investigate the limit as the number of input neurons
and the number of
incoming spikes required to fire the target both get large with the ratio
fixed. We prove that the
standard deviation of the firing time of the target cell goes to zero in
this limit and we derive the
asymptotic forms of the density and the standard deviation near the limit.
We use the theorems to
understand the behavior of octopus cells in the mammalian cochlear nucleus.
Fred Nijhout,
Duke University
The Systems Biology of Body Size Regulation: A Multidimensional Mechanism
Michael Oberguggenberger,
University of Innsbruck
Delta Waves for Semilinear Hyperbolic Systems
Age dependent population dynamics, population models with
migration and interaction (as well as general nonlinear
advectionreaction models) are often formulated as systems
of semilinear hyperbolic partial differential equations. The
initial value problem consists in finding the time dependent
population density, given an initial distribution with respect
to age or space. Concentration of the initial population at
discrete points can be modelled by point masses (delta
functions). In this situation, classical or even weak
solutions will generally fail to exist. However, regularizing
the initial data by means of convolution with a mollifier and
solving the equations with regularized data may still lead to
a family of classical solutions which converge to a weak
limit. If this is the case, this limit is called a delta wave.
In this talk, I will trace the development of delta waves from
their first appearance in the eighties, touch upon similar
phenomena in nonlinear stochastic partial differential
equations discovered in the nineties, and arrive at recent
regularity results in terms of asymptotic properties of the
regularized solutions.
Mette Olufsen,
NC State University
Heart Rate Regulation During Postural Change from Sitting to Standing
During posture change from sitting to standing, blood pools
in the lower extremities of the body, leading to a decrease
in blood pressure in the upper body and the brain and an
increase in blood pressure in the legs. In subjects with
orthostatic intolerance, postural change may cause dizziness,
lightheadedness, or even fainting. The exact function of the
cardiovascular and respiratory regulatory mechanisms is not well
understood, and our work aims to develop mathematical models that
can help describe these regulatory mechanisms in more detail. One
of these regulatory factors is heart rate, which is increased
during postural change from sitting to standing. Changes in
heart rate are mediated partly due to parasympathetic
withdrawal, partly from a delayed sympathetic activation,
and partly from musclesympathetic stimulation. In this work
we present a model that predict heart rate regulation during
sitto stand. Key elements of the heart rate model include a
timedelay between sympathetic and parasympathetic nervous
responses and an impulse function that predicts the vestibular
response accounting for subject\u2019s physical preparation for
standing. Based on these responses we predict chemical
changes of noradrenaline and acetylcholine and use an
integrateandfire model to predict heart rate. This model
has been validated against data from groups of healthy young,
healthy elderly, and hypertensive elderly subjects.
Charles S. Peskin,
Courant Institute of Mathematical Sciences
Variations on a Theme of Immersed Boundaries
The immersed boundary (IB) method was originally introduced to study
flow patterns around heart valves, which are thin, flexible, elastic
but essentially massless membranes immersed in blood, a viscous
incompressible fluid. Subsequent generalizations, the subject of this
talk, include the representation of immersed elastic bodies of nonzero
thickness, for which the IB method turns out to be more accurate than
in the thin boundary case; the inclusion of boundary mass in excess of
that of the fluid displaced, which is especially important for
aerodynamic applications; the simulation of immersed elastic filaments
like bacterial flagella or DNA that resist both bend and twist; a
stochastic version of the IB method that includes Brownian motion and
is suitable for intracellular biofluid mechanics, including nanoscale
osmotic effects; and finally two electrophysiological versions of the
IB method, one for the simulation of electrodiffusion/advection of
ions coupled to fluid flow within biological cells, and the other for
the bidomain equations of cardiac electrophysiology.
Jeff Rauch, University of Michigan
The Time Integrated Far Field for Maxwell's and D'Alembert's
Equations
For x large consider the electric field, E(t, x) and its
temporal Fourier Transform,
F(E)(˙˙, x). The D.C. component F(E)(0, x) is equal to the time
integral of the electric field. Experi
mentally, one observes that the D.C. component is negligible
compared to the field. In this paper
we show that this is true in the far field for all solutions of
Maxwell˙˙s equations. It is not true for
typical solutions of the scalar wave equation. The difference is
explained by the fact that though
each component of the field satisfies the scalar wave equation, the
spatial integral of ˙˙t E(t, x)
vanishes identically. For the scalar wave equation the spatial
integral of ˙˙t u(t, x) need not vanish.
This conserved quantity gives the leading contribution to the time
integrated far field. We also
give explicit formulas for the far field behavior of the time
integrals of the intensity.
Barry Simon, California Institute of Technology
The Lost Proof of Loewner's Theorem
A realvalued function, F, on an interval (a,b) is called matrix monotone if
F(A) < F(B) whenever A and B are finite matrices of the same order with
eigenvalues in (a,b) and A < B. In 1934, Loewner proved the remarkable
theorem that F is matrix monotone if and only if F is real analytic with
continuations to the upper and lower half planes so that Im F > 0 in the
upper half plane.
This deep theorem has evoked enormous interest over the years and a number
of alternate proofs. There is a lovely 1954 proof that seems to have been
``lost'' in that the proof is not mentioned in various books and review
article presentations of the subject, and I have found no references to the
proof since 1960. The proof uses continued fractions.
I'll provide background on the subject and then discuss the lost proof and a
variant of that proof which I've found, which even avoids the need for
estimates, and proves a stronger theorem.
James Sneyd,
University of Auckland
Calcium oscillations: using math to do physiology
In almost every cell type, the concentration of intracellular
free calcium controls a variety of important processes, such as secretion,
synaptic communication, muscular contraction, and cellular differentiation.
Often, this control is exerted in a frequencydependent manner via
oscillations in the calcium concentration. Thus, over the last 15 years or
so, the study of intracellular calcium dynamics has become an active
research area for theoreticians and experimentalists alike.
I will discuss how very simple mathematical approaches can lead to
significant physiological insight into the mechanisms underlying calcium
oscillations, and how these simple models can, in their turn, pose
nontrivial mathematical questions.
Scott Stevens, Penn State University
Modeling Idiopathic Intracranial Pressure with a SemiCollapsible Sinus.
Idiopathic intracranial hypertension (IIH) is a syndrome of unknown cause
characterized by elevated intracranial pressure (ICP). Imaging of IIH
patients often reveals a stenosis of one or both of the transverse sinuses.
However, the role of this feature in the etiology of IIH has been in dispute.
Many patients with chronic daily headache and migraine have been found to
actually be suffering from a milder form of IIH without papilledema (IIWOP).
These patients often demonstrate pathological, hypertensive spikes and
plateaus upon continuous ICP monitoring. Our previous modeling studies
explained many of the features of IIH and suggested that the sinus stenosis
and hypertension of IIH
are physiological manifestations of a stable state of elevated pressures
that can exist when the transverse sinus is sufficiently collapsible.
However, the prevalence of pathological ICP waveforms observed in IIHWOP
remained unresolved.
The current model resolves this issue. Here, a semicollapsible sinus is
described by a refined downstream Starlinglike resistor based more
precisely on experimental data. The qualitative behavior of this model
is presented in terms of
the collapsibility of the transverse sinus. For a sufficiently rigid sinus
there is a unique stable state of normal pressures. As the degree of
collapsibility increases, a Hopf bifurcation occurs, the normal state
becomes unstable,
low frequency, high amplitude ICP waves prevail, and small perturbations
can lead to hypertensive ICP spikes. This result provides insight into
the prevalence of the low frequency, high amplitude waves observed in IIHWOP.
As collapsibility increases further, so does the duration of the waves
until they are replaced by two stable states, one of normal pressures and
one of elevated pressures. In this parameter domain the model behaves
much like our previous
models where temporary perturbations can cause permanent transitions
between stable states. Clinical implications of these results will be
discussed.
Haiyan Wang, Arizona State University
Mathematical Modeling and Qualitative Analysis of the Glucoseinsulin System
Several mathematical models have been proposed during the last
decade to model the glucoseinsulin system. These currently
existing models assume that insulin degradation is proportional
to insulin production. In this talk, we will introduce a new
model for the glucoseinsulin regulatory system by revisiting
insulin degradation. We will provide mathematical analysis of
the new model. Some applications in insulin therapies will be
discussed. Numerical simulations show that the proposed model
is more realistic.
Arik Yochelis, UCLA
Morphogenesis beyond the Turing Onset: From Periodic to Localized Patterns
Turing proposed that morphogenesis be viewed as the instability of the
homogeneous solution to a set of partial differential equations involving
only activator and inhibitor morphogens. While the Turing paradigm has
been convincingly demonstrated in chemistry, its extension to biology is
more problematic, especially because in biology, the observed patterns are
far from equilibrium. Here, we use nonlinear theory to extend the original
Turing mechanism to patterns that emerge beyond the initial homogeneity. An
important prediction in this farTuring regime is the existence of
isolated, localized states of activator concentration. Specifically designed
experiments show to profoundly support this prediction.
