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Applications of Analysis to Mathematical Biology


David Anderson, University of Wisconsin
Using Random Forcing to Study the Out-of-Equilibrium Dynamics of Biochemical Reaction Systems
There are two different natural contexts in which stochastic dynamics arises in the study of biochemical reaction networks. áIn 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. áThe 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. áHere the randomness is a tool used to study the out-of-equilibrium 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. áIn 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 re-map 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 two-dimensional environments which differ in the number of obstacles (or holes) constraining the region accessible to a freely-foraging 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 position-tracking 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 N-obstacle 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 high-dimensional, non-physical 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 micron-scale 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 (~24-hour) 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 load-bearing 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 (solid-fluid) and triphasic (solid-fluid-ion) 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 up-regulation of the anti-apoptotic protein Bcl-2, which in turn leads to increased production of interleukin-8 (CXCL8). The VEGF-BCL2-CXCL8 pathway suggests new targets for the development of anti-angiogenic strategies including short interfering RNA (siRNA) that silence the CXCL8 gene and small molecule inhibitors of Bcl-2.? In this talk I will discuss our efforts to develop and validate mathematical models of tumor-induced angiogenesis and vascular tumor growth that are able to predict the effect of the therapeutic blockage of VEGF, CXCL8, and Bcl-2 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, time-independent steady-state 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 time-variation 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 Schr÷dinger, 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 advection-reaction 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, light-headedness, 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 sit-to stand. Key elements of the heart rate model include a time-delay 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 integrate-and-fire 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 real-valued 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 frequency-dependent 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 non-trivial mathematical questions.

Scott Stevens, Penn State University
Modeling Idiopathic Intracranial Pressure with a Semi-Collapsible 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 semi-collapsible sinus is described by a refined downstream Starling-like 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 Glucose-insulin System
Several mathematical models have been proposed during the last decade to model the glucose-insulin 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 glucose-insulin 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 far-Turing regime is the existence of isolated, localized states of activator concentration. Specifically designed experiments show to profoundly support this prediction.