Applications of Mathematics in Biology, Physiology, and Medicine

Courant Institute of Mathematical Sciences
New York University, New York, NY 
October 20 - 21, 2006

The Courant Institute is at 251 Mercer Street in Manhattan
between 4th and 3rd Streets.

A Stochastic Immersed Boundary Method for Microscopic Fluid Dynamics : Toward Modeling Cellular Mechanics, Paul J. Atzberger, University of California-Santa Barbara.

Abstract: The immersed boundary method is one modeling approach which has been widely applied to macroscopic systems involving flexible elastic structures which interact with a fluid. At sufficiently small length scales thermal fluctuations become significant and must be taken into account. In this talk, we shall discuss an extension of the immersed boundary method framework which incorporates thermal fluctuations through appropriate stochastic forcing terms in the fluid equations. This gives a system of stiff SPDE's for which standard numerical approaches perform poorly. We discuss a numerical method which exploits stochastic calculus to handle stiff features of the equations. We further show how this numerical method can be applied in practice to model the basic microscopic dynamics of polymers, polymer knots, membrane sheets, and vesicles. We also discuss preliminary work on modeling the mechanics of cellular structures.

Gamma oscillations and attention, Christoph Börgers, Tufts University.

Abstract: Gamma frequency (approximately 40 Hz) oscillations in electrical fields in the brain are known to be correlated with various forms of attention. I will talk about mechanisms underlying gamma oscillations, and about ways in which these oscillations may be useful in vigilance and stimulus selection. This is work done in collaboration with Nancy Kopell and Steven Epstein.

Biofluidmechanics of reproduction, Lisa Fauci, Tulane University.

Abstract: Complex fluid-structure interactions are central to mammalian fertilization. Motile spermatozoa, muscular contractions of the uterus and oviduct, as well as ciliary beating generate forces that drive fluid motion. At the same time, the dynamic shapes of these biostructures are determined by the fluid mechanics. In this talk we will give an overview of the classical work in fluid dynamics that has been applied to reproduction. We will also present recent computational models, based upon an immersed boundary framework, that promise to provide insight into these complex, coupled dynamical systems.

Computational Modeling of Arterial Platelet Thrombosis, Aaron Fogelson, Univerisity of Utah.

Abstract: Arterial blood clots (thrombi) that form as a consequence of artherosclerotic plaque rupture are comprised largely of aggregates of platelets. These thrombi form under conditions in which the flow changes substantially both in space (initially because of the plaque, later also because of the clots) and in time (as the thrombi develop). We present a continuum model that describes platelet thrombosis initiated by a ruptured atherosclerotic plaque in a coronary-artery-sized vessel. It includes full treatment of the fluid dynamics, and the aggregation of platelets in response to the plaque rupture and further chemical signals. In the model, the growing clots influence the fluid motion by a distribution of forces that act on the fluid rather than by an explicit change in fluid domain geometry. Among the behaviors seen with this model are the growth of wall-adherent platelet thrombi to occlude the vessel and stop the flow, and the transient growth and subsequent embolization of thrombi leaving behind a passivated injured surface. The model will be the basis for an exploration of the interactions among flow, biology, and vessel geometry during arterial thrombosis.

Towards a Detailed Understanding of the Mammalian Timekeeping, Daniel Forger, University of Michigan.

Abstract: Charles Peskin and I developed a detailed mathematical model of the circadian (~24-hour) clock within most mammalian cells. In line with much of Peskin's work, this model coupled a careful description of biological data (without unjustified mathematical simplifications) with extensive in silico experiments. This model has general large interest from the biological community, and I will describe experiments carried out in several labs which test predictions from this model. In particular, this model has been used to show that the most well studied (and first identified) mammalian clock mutation behaves in an opposite manner to what had been previously thought, and that genetic mutations in certain clock genes may actually lead to more accurate timekeeping. Inspired by the work of McQueen and Peskin, I will also describe ongoing efforts to develop a multi-scale model of the human circadian system which bridges the molecular biology within a single cell with predictive models of human alertness and performance.

Towards an Electro-Mechano-Fluidic Model of the Heart, Boyce Griffith, Courant Institute, New York University.

Abstract: Although the equations that describe cardiac mechanics (including blood, muscle, and valve mechanics) and electrophysiology are different, in both cases a realistic treatment demands the use of methods that account for anisotropy, inhomogeneity, and complex geometries. We employ a unified theoretical framework, the immersed boundary (IB) method, for both aspects of cardiac physiology. This unified approach not only yields methodological overlap but also allows for substantial software reuse. We also anticipate that this approach will greatly simplify the task of coupling our models of cardiac mechanics and electrophysiology.

This talk will include an overview of recent work on the application of an adaptive version of the IB method to the three-dimensional simulation of blood flow in the heart. I shall also discuss the application of the IB framework to the bidomain equations of cardiac electrophysiology. Computer animations of both fluid mechanical and electrophysiological simulation results will be shown.

This is joint work with Charlie Peskin (CIMS), Dave McQueen (CIMS), and Rich Hornung (LLNL).

The reaction-diffusion master equation and spatially continuous stochastic reaction-diffusion models, Samuel Isaacson, University of Utah.

Abstract: We will present several mathematical models for studying reaction-diffusion processes wherein both noise in the chemical reaction process and diffusion of individual molecules may be important. In particular, we will examine the relation between the reaction-diffusion master equation model of spatially distributed stochastic chemical kinetics and models that track individual particles. Our analysis will demonstrate the importance of modeling point binding, equivalently binding to a small target, in understanding the reaction-diffusion master equation.

Using Simple Stochastic Differential Equations to Solve Complicated Partial Differential Equations, Michael Mascagni, Florida State University.

Abstract: This talk begins with an overview of methods to solve PDEs based on the representation of point solutions of the PDEs as expected values of functionals of stochastic processes defined by the Feynman-Kac formula. The particular stochastic processes that arise in the Feynman-Kac formula are solutions to specific SDEs defined by the characteristics of the differential operator in the PDE. The Feynman-Kac formula is applicable to wide class of linear initial and initial-boundary value problems for elliptic and parabolic PDEs. We then concentrate our attention on elliptic boundary value problems that arise in applications in materials science and biochemistry. These problems are similar in that the PDEs to be solved are rather simple, and hence the associated SDEs that arise in the Feynman-Kac formula are likewise simple. However, the geometry of the problem is often complicated and amenable to several acceleration approaches particular to these simple SDEs. We will specifically describe the walk on spheres, Greens function first passage, last passage, walk on the boundary, and walk on subdomains methods in this context. These methods will be presented in the setting of several applications studied by the author and his research collaborators.

Fluid dynamics and mechanosensing in the developing embryonic heart, Laura Miller, University of North Carolina.

Abstract: The embryonic vertebrate hearts develops from a simple heart tube into a valve and chambered pump through a series of complicated morphological changes. During this transformation, the flow patterns within the heart are constantly changing due to changes in the viscosity of the fetal blood, chamber and valve morphology, and the kinematics of contraction. Cardiac endothelial cells can sense and respond to local variations in shear stress caused by such changes in the larger scale fluid dynamics. A number of recent studies suggest that these fluid dynamic signals are responsible for triggering biochemical cascades within the cell, leading to the transcription of genes necessary for cardiac morphogenesis.

One aspect of heart development that is particularly sensitive to alterations in cardiac flow patterns is the development of the heart valves. The focus of this study is to understand how the cardiac cushions, which later become the valves, are formed through a complex interaction of flow, mechanosensing, biochemical cascades within the cell, and changes in morphology. On the macroscale, flow patterns change as the cardiac cushions begin to bulge out from the endocardial wall. Such changes in cardiac flow patterns cause temporal and spatial variations in shear stress along the endothelial lining of the heart tube. This aspect of the problem was studied using the immersed boundary method to describe how shear depends upon Reynolds number and morphology. The next level of the study is to determine how a range of shear and pressure changes might be detected by cardiac endothelial cells through the extracellular glycocalyx. This component of the problem has been studied using a dynamically scaled flow tank and physical models of the glycocalyx. Flow velocities through the model glycocalyx were measured directly over a range of free stream velocities and possible configurations of the glycocalyx. On both levels, several fluid dynamic transitions occur which may be important for the signaling of the development of the valves.

Three-Dimensional Model of Cellular Electrical Activity, Yoichiro Mori, University of British Columbia.

Abstract: We present a three-dimensional model of cellular electrical activity. This model takes into account the three-dimensional geometry of biological tissue as well as ionic concentration dynamics, both of which are neglected in conventional models of electrophysiology.

We use both asymptotic and analytic methods to study the system of equations. We find in particular that the model possesses multiple spatiotemporal scales.

This modelling methodology is applied to cardiac physiology. Numerical simulations with this model is used to explore an anomalous mode of action potential propagation: cardiac action potential propagation without gap junctions.

Short term regulation during postural change from sitting to standing, Mette Olufsen, North Carolina State University.

Abstract: When standing up, blood is pooled in the legs due to the effect of gravity resulting in a drop in systemic arterial pressure and widening of the blood flow velocity. This can be modeled by increasing the blood pressure in the compartments representing the lower body. To restore blood pressure and blood flow velocity a number of regulatory mechanisms are activated. The most important mechanisms are autonomic reflexes mediated by the sympathetic nervous system and cerebral autoregulation mediated by changes in concentrations of oxygen and carbon dioxide. The response to standing is an increase in nervous activity, which results in increased heart rate and cardiac contractility, vasoconstriction of the systemic arterioles, and changes in unstressed volume and venous compliance. The response by the cerebral autoregulation is to dilate arterioles in the cerebral vascular bed. It is not clear how the autonomic and autoregulation interacts; one theory suggests that vasoconstriction, resulting from increased sympathetic activity, has an effect throughout the body, but that cerebral vasoconstriction gets overridden (possibly with a significant delay) by autoregulation resulting in a net vasodilatation of the cerebral vascular bed. In this work we demonstrate how mathematical modeling can be used to predict the interaction between autonomic and autoregulation, and to identify sensitive model parameters.

Stochastic Processes in Cell Biology, Arjun Raj, MIT.

Abstract: Until rather recently, biologists have thought of the molecular biology of cells as consisting of largely deterministic processes. With the advent of new experimental techniques, however, scientists have discovered that not only are many of these processes stochastic, but that the stochastic behavior can have real consequences for cellular function. In this talk, we describe two different examples of stochastic models of processes in cellular biology.

In the first, we model the separation of sister chromosomes to daughter cells during the anaphase A phase of mitosis. We model the driving force as arising from a Brownian ratchet and investigate the role chromosome flexibility plays in the dynamics of such a molecular motor. In particular, we are able to reproduce the classical experimental results of Nicklas by showing that chromosomes of very different sizes all move at the same speed when chromosomes are very flexible.

In the second, we examine the basic process of gene expression in mammalian cells. We developed an experimental system through which we were able to detect and count individual molecules of a specific mRNA in single cells. Surprisingly, we found that the numbers of mRNA fluctuated widely from cell to cell. These observations are consistent with a model of gene expression in which genes themselves randomly transition between active and inactive states. This large amount of randomness raises interesting questions about how cells are able to function reliably in the presence of such large random fluctuations.

Timing computations in the auditory brain stem, John Rinzel, New York University.

Abstract: Sound localization involves precise temporal processing by neurons in the auditory brain stem. The first neurons in the auditory pathway to receive input from both ears can distinguish interaural time differences (ITDs) in the sub-millisecond range. These cells in the mammalian medial superior olive (MSO) have specialized biophysical features: two dendrites, each receiving input from only one side; very short membrane time constant (as little as 0.5 ms); specialized ionic channel properties, including a low-voltage activated K+ current, I-KLT. This I-KLT contributes to phasic firing (just one spike in response to a step of current, at onset), precise phase-locking, and extremely timing-sensitive coincidence detection. We will describe the temporal feature-selecting properties of MSO cells based on biophysical (HH-like) modeling, in vitro (gerbil) electrophysiology and application of concepts from dynamical systems theory and coding theory.

A Tale of Histone Tails, Tamar Schlick, New York University.

Abstract: Eukaryotic chromatin is the fundamental protein/nucleic acid unit that stores the genetic material. Understanding how chromatin fibers fold and unfold as well as details of their structure and dynamics on a range of spatial and dynamical scales is important for interpreting fundamental biological processes like DNA replication and transcription regulation.

Using a new mesoscopic model of oligonucleosome chains and a tailored configurational-bias Monte Carlo method that efficiently samples the possible conformational states of oligonucleosomes, we elucidate the role of each histone tail in regulating chromatin structure and detail the global folding pattern. Analyses indicate that the H4 histone tails are most important in terms of mediating internucleosomal interactions, especially in highly compact chromatin with linker histones, followed by H3, H2A, and H2B tails in decreasing order of importance. In addition to mediating internucleosomal interactions, the H3 histone tails crucially screen the electrostatic repulsion between the entering/exiting DNA linkers. The H2A and H2B tails distribute themselves along the periphery of chromatin fibers and thus are important for mediating fiber/fiber interactions. A delicate balance between tail-mediated internucleosomal attraction and electrostatic repulsion among DNA linkers allows adjacent DNA linkers to align perpendicular to each other in linker histone-deficient chromatin, leading to the formation of an irregular zigzag-folded fiber with dominant pairwise interactions between nucleosomes i and i+/-4. With linker histone proteins included, the global folding pattern changes markedly, so that the dominant pairwise interactions occur between nucleosomes i and i+/-3.

Ionic and Metabolic Oscillations in Pancreatic Beta-Cells, Arthur Sherman, National Institutes of Health, NIDDK.

Abstract: Insulin is secreted in pulses with a period of about five minutes from the beta-cells of the pancreas. These pulses are in turn driven by oscillations of cytosolic calcium. Two parallel streams of investigation over more than two decades have studied metabolic oscillations and ionic mechanisms as possible sources of the calcium oscillations. We propose that the two are linked by a potassium channel, K(ATP), that senses the ATP and ADP levels in the cell. This directly transduces metabolic oscillations into oscillations of membrane potential and calcium. However, calcium can also affect metabolism both by stimulating ATP-consuming pumps and by depolarizing the mitochondria. A unified model that combines the above elements and can thereby explain a diverse set of experimental observations using only a few simple assumptions will be presented.