In this talk we discuss recently developed topological methods for producing maps of biomedical data, as well as evaluating the structures they produce. The methods are a combination of algebraic techniques as well as combinatorial ones, and have produced some interesting examples, which we will present.
Mathematical models of biochemical reaction networks give rise to dynamical systems that are usually high dimensional, nonlinear, and have many unknown parameters. Nevertheless, it is often possible to use reaction network properties to conclude global injectivity of the associated vector field. Moreover, it turns out that similar criteria for global injectivity hold for general nonlinear maps. We explain how these criteria relate to various problems such as the Jacobian Conjecture in algebraic geometry and the Bezier self-intersection problem in computer graphics.
We investigate the relationship between the structure (connectivity) and a function (stimulus space representation) for a simple class of neural network models. The problem of relating network connectivity to stimulus space properties leads to a number of geometric questions about the set of stable fixed points associated to a particular network. I will discuss some recent results relating to these questions. This is joint work with Anda Degeratu and Vladimir Itskov.
During growth processes many biological and physiological systems develop residual stresses. These stresses are present in the body even in the absence of external or body loadings and are known to play an important role in regulation processes. Residual stress can be observed when the body is cut and part of the stresses are relieved. A fundamental difficulty in elasticity is to describe the mechanics of body with residual stresses. The problem comes from the absence of an obvious choice for an unstressed reference configurations where all kinematic and physical variables can be evaluated. By proper consideration of the manner in which stresses are relieved, one can define a virtual configuration. By borrowing arguments from elasto-plasticity and the theory of dislocations, the geometry of this configuration can be fully characterized. The virtual configuration is, in general, not an Euclidean manifold. It is associated with a metric (the growth metric) and an affine connection. These geometric objects shed some new lights on some of the fundamental assumptions of the theory of growing elastic bodies. It also provides a theoretical framework to compute physical quantities of importance and help us understand the role of stresses in the mechanics of biological structures.
A closed meander of order n is a non-self-intersecting closed curve in the plane which crosses a horizontal line at 2n points. Meanders occur in a variety of settings from combinatorial models of polymer folding to the Temperley-Lieb algebra, yet the exact meander enumeration problem remains open. Building on results for plane trees and noncrossing partitions motivated by the biology of RNA folding, we prove that meanders are connected under appropriately defined local move transformations. The resulting meander graphs have some interesting characteristics and suggest new approaches to the enumeration question. As we will explain, meanders also relate to the challenging biomathematical problem of comparing different possible folds for an RNA sequence.
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 demonstrate interesting mathematical problems that arise in studying cancer initiation and progression. The topics will include: Stem cells and tissue architecture; Cancer and aging, and Drug resistance in cancer.
Microbes live in environments that are often limiting for growth. They have evolved sophisticated mechanisms to sense changes in environmental parameters such as light and nutrients, after which they swim or crawl into optimal conditions. This phenomenon is known as "chemotaxis" or "phototaxis." Using time-lapse video microscopy we have monitored the movement of phototactic bacteria, i.e., bacteria that move towards light. These movies suggest that single cells are able to move directionally but at the same time, the group dynamics is equally important. Following these observations, in this talk we will present a hierarchy of mathematical models for phototaxis: a stochastic model, an interacting particle system, and a system of PDEs. We will discuss the models, their simulations, and our theorems that show how the system of PDEs can be considered as the limit dynamics of the particle system. Time-permitting, we will overview our recent results on particle, kinetic, and fluid models for phototaxis. This is a joint work with Devaki Bhaya, Tiago Requeijo, and Seung-Yeal Ha.
I will over view some results on how stochastic fluctuations pass through particular biochemical networks. I will mainly be interested in how the fluctuate are modified by the structure of the network and how in certain instances larger networks can be approximated by other networks.
I will discuss new computational tools based on topological methods that extracts coarse, but rigorous, combinatorial descriptions of global dynamics of multiparameter nonlinear systems. This techniques are motivated by several observations which we claim can, at least in part, be addressed.
To make the above mentioned comments concrete I will describe the techniques in the context of a simple model arising in population biology.
The sequence alignment problem has been intimately connected to the field of genomics during the past 30 years, and it study continues to be an integral part of the field of computational biology. We will discuss how this problem emerged from mathematics, and in turn we'll review some of the combinatorics that has developed from the study of the problem. In particular, we will see connections to Legendre polynomials, the assymetric exclusion process and various aspects of polyhedral geometry.
The history of rich influences of biology on the field of partial differential equations will be reviewed briefly, including reaction-diffusion systems, reaction-hyperbolic systems, and bio-fluid dynamics. Some current work by young researchers will be discussed. Difficult technical and conceptual challenges in partial differential equations that arise in diverse biological fields will be proposed.
In phylogenetics, stochastic models of biological sequence evolution along trees are used to infer evolutionary relationships between organisms. The form of these models has led to interesting connections with several areas of mathematics.
After a brief introduction, we describe how models used in phylogenetic analyses have natural connections to algebraic geometry, and questions of tensor rank. The statistical issue of identifiability of model parameters is most naturally studied in this framework, and computational algebra tools have been usefully applied to better understand when consistent inference is possible.
Suspensions of motile bacteria can display coherent dynamical structures living on scales much larger than those of an individual bacterium, with the associated mixing dynamics having likely advantages in efficiently moving nutrients throughout the population. I will discuss how such structures can emerge as the result of hydrodynamic coupling between motile particles, how they are related to micro-mechanical aspects of swimming actuation, and how such dynamics is modified by chemo-sensing and taxis.
R.Thomas noticed that for a gene network to admit several stationary states, the associated interaction graph must contain a positive circuit. Once a mathematical model has been chosen for the gene network, this rule becomes a precise mathematical assertion. We shall explain how this theorem was proved for differentiable models, boolean models and discrete models. We shall also present open questions, in particular those having to do with the role of negative circuits.
Recent technological advances have made it feasible to conduct genome-wide scans in large populations to find genetic markers for common diseases and other traits. However, analyzing the data for assocations between traits and combinations of multiple markers is computationally challenging. Data has the form of an n by m table (n individuals, m markers, and a finite number of values allowed in each cell, such as one of 2 haplotypes or one of 3 genotypes). We study a transformation of this table and a decomposition of the Pearson X2 statistic for a contingency table by Irwin (1949) that allows us to efficiently cluster together highly correlated markers. This work is joint with Vineet Bafna and Dumitru Brinza at the University of California, San Diego.
Hyperbolic conservation laws with source terms have attracted much attention in the biosciences because they play an important role in modeling tactically-driven cell migration. In particular, sharp interfaces, or shocks, in the wave form of cell migration are often observed which motivates the study of advection-reaction-diffusion (ARD) models where the diffusion is considered a viscous small perturbation. In this talk, we will give a twist to the existing hyperbolic PDE theory and apply recently developed geometric singular perturbation methods to this class of problems. In particular, we identify the underlying geometric structures that lead to the existence of travelling waves and shocks in ARD models.
If time permits, we will also show that the same geometric approach can be used to analyze complex oscillatory patterns observed in certain cell membrane potentials, such as stellate and pituitary cells.