Applied Mathematics Colloquia Series
Archive

Spring 2007

January

01/17
George Oster
Departments of Molecular & Cellular Biology and ESPM
University of California, Berkeley

Arthur Winfree Memorial Lecture: From biochemistry to morphogenesis in myxobacteria

Many aspects of metazoan morphogenesis find parallels in the communal behavior of microorganisms. The cellular slime mold D. discoideum has long provided a metaphor for multicellular embryogenesis. However, the spatial patterns in D.d. colonies are generated by an intercellular communication system based on diffusible morphogens, whereas the interactions between embryonic cells are more often mediated by direct cell contact. For this reason, the myxobacteria have emerged as a contending system in which to study spatial pattern formation, for their colony strutures rival those of D.d. in complexity, yet communication between cells in a colony is carried out by direct cell contacts.

Myxococcus xanthus are Gram-negative bacteria that glide on solid surfaces, periodically reversing their direction of movement. When starved, M. xanthus cells organize their movements into waves of cell density that sweep over the colony surface. These waves are unique: although they appear to interpenetrate, they actually reflect off one another when they collide, so that each wave crest oscillates back and forth with no net displacement. Since the waves reflect the coordinated back and forth oscillations of the individual bacteria, we call them accordion waves. The spatial oscillations of individuals are a manifestation of an internal biochemical oscillator, probably involving the Frz chemosensory system. These internal ‘clocks’—each of which is quite variable—are synchronized by collisions between individual cells utilizing a contact mediated signal transduction system. The result of collision signaling is that the collective spatial behavior is much less variable than the individual oscillators. In this work we present new experimental observations in which individual cells marked with GFP can be followed in groups of unlabelled cells in monolayer cultures. This data, together with a new agent-based computational model demonstrates that the only properties required to explain the ripple patterns are an asymmetric biochemical limit cycle that controls direction reversals, and asymmetric contact induced signaling between cells: head-to-head signaling is stronger than head-to-tail signaling. Together, the experimental and computational data provide new insights into how populations of interacting oscillators can synchronize and organize spatially to produce morphogenetic patterns that may have parallels in higher organisms.

01/19
Arjen Doelman

Semi-Strong Pulse Interactions

In this talk we consider localized solutions of pulse type of reaction-diffusion equations. We briefly discuss the existence and stability of solitary pulses, but our main interest lies in the dynamics of interacting pulses. We introduce the concept of semi-strong pulse interactions, that goes beyond the limitations of the weak interaction case, in which the pulses are assumed to be 'sufficiently far apart'. Unlike in the weak interaction case, semi-strong interacting pulses change in shape and character. We derive equations for the evolution of semi-strong interacting pulses and develop a renormalization group technique to establish the nonlinear asymptotic stability of semi-strong pulse interactions in a Gierer-Meinhardt system.

01/26
Robert Sims

Lieb-Robinson bounds and related results

Locality in quantum spin systems can be described in terms of a bound onthe commutator of local observables with disjoint supports. The first proof of such estimates was given by Lieb and Robinson in 1972. In mytalk, I will discuss a generalization of these results and a variety ofapplications. For example, these new results imply the existence of thedynamics for a wider class of interactions, a bound on the propagation ofcorrelations through the system, and a proof of exponential decay of (spatial)correlations in the ground state of gapped Hamiltonians. Moreover,combining the above mentioned applications, we were recently able to provea multi-dimensionalLieb-Schultz-Mattis theorem concerning the existence of low-lying excitations for thespin-1/2 antiferromagnetic Heisenberg model.

February

02/16
Arnd Scheel

The impossible period-doubling of a spiral wave

What happens when dynamical systems change from apparently simple to complicated dynamics? In this lecture, we examine the period-doubling of a spiral wave in a reaction-diffusion system. Period-doublings are one of the universal routes to chaos in low-dimensional dynamics. However, we will argue that spiral waves should *not* undergo period-doublings, since spiral waves are merely equilibria, not periodic orbits, when viewed in a corotating frame. In order to reconcile this conflict between theoretical prediction and experimental observation, we will then take a closer look at coherent structures in general, and spiral waves in particular. Our analysis will reveal a list of instability mechanisms that is quite different from what finite-dimensional dynamics would have predicted. We will show dizzying direct numerical simulations that may illustrate the phenomena -- or just hypnotize you.

02/23
David Cai

One model, one regime, and many phenomena

We will present our large-scale computational modeling of the primary visual cortex (V1). In particular, we will discuss network mechanisms underlying spatiotemporal dynamics associated with spontaneous on-going activity of the V1 and the line-motion illusion --- which is the illusory motion sensation from a static cue of a flashed stationary square quickly followed by a stationary bar. Related issues, such as kinetic theory of neuronal network dynamics, will also be addressed.

March

03/09
Stephane Lafortune

Stability of persisting periodic solutions to a complex Ginzburg-Landau perturbation of NLS

It was shown in [Cruz-Pacheco, Levermore, and Luce (2004)] that some periodic solutions to the nonlinear Shrodinger equation (NLS) persist when the NLS is subject to a perturbation leading to the Complex Ginzburg Landau equation (CGL). In this presentation, I will show how one can use methods coming from the theory of integrability together with the Evans function to study the spectral stability of these persisting solutions. In particular I show that the solutions of NLS are spectrally stable with respect to periodic perturbations. However, the solutions become unstable when NLS is perturbed to CGL.

03/30
Natalia Komarova

Cancer as a micro-evolutionary process

Even though much progress has been made in main stream experimental cancer research at the molecular level, traditional methodologies alone are insufficient to resolve many important conceptual issues in cancer biology. For example, for the most part, it is still unknown how cancer originates, what drives its progression, and how treatment failure can be prevented. In this talk, I will describe novel mathematical tools which help obtain new insights into these processes. I will also show how the mathematical insights are combined with experimental studies through collaborations with cancer biologists. The main idea is to study cancer as an evolutionary dynamical system on a selection-mutation network. I will discuss the following topics: Stem cells and tissue architecture; Cancer and aging, and Drug resistance in cancer.

April

04/06
Wayne Hayes

Outer Solar System Surfing the Edge of Chaos

The existence of chaos among the system of Jovian planets (Jupiter, Saturn, Uranus, and Neptune) is not yet firmly established. Many investigators (Sussman, Wisdom, Holman, Murray, among many others) have consistently measured a Lyapunov time of between 5 and 12 million years. Murray and Holman demonstrated that the chaos arises from the overlap of three-body resonances, and Guzzo has corroborated their theory across a wide range of system parameters. Conversely, other investigators (Laskar, Newman, Grazier, and Varadi, among several others) have compelling evidence against chaos. Namely, Newman et al. have convincingly demonstrated that a sympletic integration using the Wisdom + Holman symplectic mapping with a 400-day timestep reproduces the chaos seen by others, but that the chaos disappears and the orbit converges to being regular as the timestep decreases.

Using high-precision integrations and convergence testing, I demonstrate that the resolution of the apparent paradox is simple. The orbital positions of the Jovian planets is known only to a few parts in $10^7$. It turns out that, within the observational error ellipsoid, there exist both chaotic and regular solutions. Thus, some investigators legitimately find chaos, while others legitimately find no chaos. The question of whether the Outer Solar System is really chaotic cannot be answered using current observations.

04/13
Jane Wang

Efficient Locomotion in Fluids: Paper Falling and Dragonfly Flight

Falling leaves and insect flight are two examples of interactions between moving boundaries and unsteady flows. Such interactions enable most of the living species, including bacteria, insects, fish, and birds, to travel in fluids. In this talk, I will first describe the computational tools that we use to solve this class of problems, including the immersed interface method for simulating multiple (rigid or flexible) wing interactions. I will then discuss the general mechanisms of force generation in flapping flight and show the efficient strategies used by dragonflies.

04/20
Peter Miller

The Semiclassical Modified Nonlinear Schroedinger Equation: Facts and Artifacts

I will discuss some recent work (joint with J. DiFranco) on semiclassical Cauchy problems for an integrable perturbation of the focusing nonlinear Schrodinger equation. The perturbation is singular in the sense of inverse-scattering and also in the more practical sense that the modified equation admits solutions with surprisingly different properties than the unmodified equation. "Facts and Artifacts" is a reference to a similarly-titled paper by E. V. Doktorov, whose lecture on the subject in Edinburgh in 2004 originally piqued our interest.

04/27
Boris Shraiman

Physical aspects of growth control in development

Development of multicellular organisms involves an intricate interplay of tissue growth and patterning. What are the mechanisms that determine the final size of limbs and organs and ensure proper proportions? The talk will review our current understanding on the example of fly wing development and describe a possible mechanism of growth coordination based on mechanical interactions within the tissue.