Mathematical Modeling of Biological Systems and of Linguistics

Biological Systems

In the last three decades of the twentieth century it became clear that the advances in the physical and biological sciences could only be handled by suitable quantitative as well as qualitative mathematical approaches. The complexity of phenomena ranging from wave propagation in random media to genome expression data processing and modeling calls for sophisticated mathematical techniques which include nonlinear dynamics, harmonic analysis, stochastic differential equations, computational mathematics and inverse problems. Some of the areas that have a direct relation with the practical and theoretical research at IMPA are the following: immunology, competing species, inverse problems in medical imaging, computer vision, genome-wide expression and data processing.

Current efforts at IMPA include the groups of J. Palis and M. Viana in dynamical systems, P.C. Pinto Carvalho in the area of computer graphics and vision, and J. P. Zubelli in the area of medical imaging, computational mathematics, and inverse problems. The interlocutors that are currently connected with the ongoing efforts in these areas include: F. A. Grünbaum (Berkeley), A. Hastings (U. C. Davies), J. Milnor (SUNY), D. Mumford (Harvard and Brown University), G. Papanicolaou (Stanford), K. Sigmund (Vienna), J.-C. Yoccoz (Collège de France).

Team and Interlocutors

L. Bevilacqua, R. Feijoo, A. Galeão, J. Koiller, W. Zin, J. P. Zubelli.

A. Hastings (University of California at Davies), M. Magnasco, (Rockefeller University), K. Sigmund, (University of Vienna).

Invading Species

We first illustrate the kind of questions one might encounter in such fields. Let us consider an ecological community with say N species. The addition of one further specie may result in the expulsion of one or more species, or in the equilibrium of (N+1) such co-habitating species. Question: How to model, predict, and understand such phenomena ? This is relevant also in the settling of mutations, in physical invasion of a certain space by a new population, or even in the dispersion of epidemics.

There are several possible approaches, such as for example: (a) discrete dynamical systems (See: "Bifurcation Analysis of Population Invasion: on-off Intermittency and Basin Riddling," by Oscar de Feo e Regis Ferriere, Intern. J. of Bifurcation and Chaos, vol 10, n 2 (2000), 443-452). (b) Lotka-Volterra type systems (See: "The Invasion Question," Alan Hastings, J. Theoretical Biol. 121 (1986), n.2, 211-220) (c) Game Theory ("Invasion Dynamics of the Finitely Repeated Prisioner's Dilemma," Martin Nowak and Karl Sigmund, Games and Economic Behavior 11, pp 364 - 390 (1995)).

Immunology

The introduction of Lotka-Volterra type models was one of the main mathematical contributions of population dynamics in the XX century. It is a key connection between biology and mathematics. Dynamical systems techniques have been systematically applied to such and other models.

A better comprehension of the HIV dynamics, accomplished in the last decade and which has directly contributed to more effective treatments could be traced in part to the work: "Antigenic Diversity Thresholds and the Development of Aids" by M. Nowak, R. M. Anderson, T. F. W. Wolfs & R. May, Science 254, 963-969 (15 nov 1991). In this work, the authors develop a Lotka-Volterra type model for cells in the immunological system and the viral charge. Such models have proven effective not only for the "in vivo" understanding of HIV dynamics, but also for the improvement of the drug cocktail treatments.

Relating Dynamical Systems to Biological Models

Dynamical systems approaches are essential for understanding ecological food webs. As has been increasingly recognized, the traditional approach of examining conditions leading to stable equilibria is not sufficient for understanding what allows species to coexist in natural systems. There are many open questions concerning the nonlinear dynamics of interacting species. Another related problem that will require similar approaches is understanding the consequences of invading species for the ecosystems they enter. As the effects of these species cascade through the ecosystem, the nonlinear effects have the potential of leading to suprising consequences, and predictions of the effects will again require the tools of dynamical systems.

Unification of genetics and natural selection was achieved, most successfully, in the thirties, through a dynamical system model. Ever since, the of use Dynamical System ideas in analyzing ecological and biological models has led to a remarkable improvement of our understanding of their evolution. Besides the now classical studies of limit cycles in Lotka-Volterra type equations, and the use of homoclinic or heteroclinic cycles (a notion originated from Celestial Mechanics) in the description of invasion processes, we could mention more recent applications of results from chaotic dynamics (strange attractors) in modeling the evolution of animal populations. These ideas are showing to be equally useful in understanding natural evolution, at the level of both species and molecules. An account of recent progress in this direction can be found in "Mathematical challenges from molecular evolution" by P. Shuster, in Mathematics Unlimited - 2001 and beyond, Springer Verlag.

Mathematical Modeling of Biological Phenomena

In the recent years several researchers at IMPA, and in other key institutions in Brazil, have been devoting a substantial amount of their attention to different aspects of mathematics which belong to the interface with areas of relevance in our everyday life. One key area is the study of medical imaging and inverse problems. The basic problem here consists in trying to determine non-invasively the internal properties of objects (or bodies) using only information from external measurements of radiation. A classical example is that of computerized tomography, whose solid mathematical and algorithmical foundations led to important developments. This was recognized by the awardal of the Nobel prize to Cormack and Hounsfield. Another example is the area of optical diffuse tomography, which endeavor to image the interior of objects by means of radiation that diffuses and scatters. An example being infrared radiation inside tissues. In 1989, J. P. Zubelli under the guidance of A. Grunbaum at Berkeley started studying the problem of reconstructing the interior of bodies that diffuse and scatter radiation by means of practical and effective algorithms. The findings and algorithms of Grunbaum, Singer, and Zubelli were published in "Image Reconstruction of the Interior of Bodies that Diffuse Radiation." SCIENCE, # 248, (1990), 990-993.

Hemodynamics of the Cardiovascular System

The main aim of this project is to produce a code designed to give the dynamical answer in space and time of the blood flow in the arterial system. The arterial system is personalized, that is, the geometry of the arteries belonging to a given individual is recovered using the tomographic images. The project consists of two steps: a) To solve the unidimensional pulse propagation problem, including non linear effects both, material and geometric ( elastic and viscoelastic behavior, pressure dependent artery geometry, etc...), with boundary conditions ( peripheral circulation) simulated by nested branching of the arterioles characterized by a fractionary geometry. b)- To solve the flow problem inside an artery segment considered as a tri-dimensional vessel using the boundary and initial conditions as given by the uni-dimensional model. All the flow details along the segment and at the cross-section are obtained including possible pathologic disturbances (sthenosis, aneurysms, etc) or surgical modifications (bypass). The solution to the first problem is obtained by integration along the characteristic lines and to the second through the variational formulation and integration with the finite element method. Special attention has been given to mesh generation procedures, optimization algorithms to solve large equation systems and computer graphics as well to deal with image reconstruction. The results have been validated by comparison with experimental data.

Physiology and Dynamics of the Respiratory System

The problem is to analyze the dynamical response of the lungs to the airflow in the cycle inspiration x expiration. The solutions up to now have succeeded to incorporate well-known rheologic properties in discretized simple models. Both normal lungs and lungs with lesions have been studied. The more realistic modeling considering the successive bifurcation of the flow till the small compartments (alveolus) are reached and the pulmonary membrane representation as a continuum needs interdisicplina cooperation (biophysicists, engineers, mathematicians). Recent results have shown that the spatial distribution of the capillary system is characterized by a fractionary geometry. Along the same line, our group has developed a theory showing that a fractal dimension can characterize densely packed thin membranes. It turns out therefore to solve the dynamics of a continuum characterized by a fractionary geometry. This is a new and challenging problem. Besides, it is possible to put another question: Assuming that nature always optimize vital processes, can we say that there is fractal dimension associated to the optimization of a given biochemical process ? This is a most interesting question outreaching problems far beyond the study of the lung.

Micro-Organisms Motion

Mathematical and computational modeling of microorganisms motion (bacteria and ciliated) and evaluation of the efficient energy use to overcome hydrodynamic resistance. Mathematical techniques are developed to solve analytically and numerically the Stokes equation subjected to non-conventional boundary conditions (ellipsoids, surfaces of revolution). For very complex boundary geometry (surfaces connected to curves modeling flagella) the boundary integral methods need improvement. The research progress to include new biological information s (force generation through internal mechanisms - molecular motors - and the consequent coordination and relationship with the surrounding hydrodynamic medium. C. Peskin has proposed a promising technique to this study.

Linguistics

Non-Equillibrium Statistical Mechanics and Stochastic Modeling in Linguistics

The main focus of the project is the study of critical phenomena in random fields and stochastic processes. This is one of the main domains in the modern theory of probability. In the last decades it played a major role in several mathematical developments with applications in an increasing set of domains of knowledge. In particular this group works in stochastic modeling in linguistics and image-processing. This is a natural extension of the group's main line of research in non equilibrium statistical mechanics.

Interacting Particle Systems

The research team is, A. Galves, P. A. Ferrari, L. R. Fontes (USP), C. Landim (IMPA), A. Toom (UFPE).

External collaborators include top level researchers as T. Liggett, E. Presutti, E. Olivieri, J. Lebowitz, M. Menshikov e P. Picco.

Interacting particle systems are flexible models of evolution of many components in time and space obeying local rules. The initial motivation comes from the study of non equilibrium systems in statistical mechanics. A recent source of problems come from Monte Carlo methods in statistical inference and in Bayesian signal processing.

The complexity of these models is due to the large number of components. The problems currently under study include the existence and characterization of invariant measures, hydrodynamic behavior, convergence to equilibrium, metastability, convergence of algorithms and perfect simulation.

This group has been collaborating for 20 years with international centers, their members participated as speakers in the most important meetings in the area and received support form Brazilian and international agencies. Their members publish in the principal journals of the area.

The group participated actively training young researchers, attracting Ph-D's students and post-doctoral students from Europe and Latin America.

Probabilistic Tools for Pattern Identification Applied to Linguistics

This is a multidisciplinary project with the participation of the probabilists P. Ferrari, L. R. Fontes, A. Galves (USP), C. Landim (IMPA), N. L. Garcia (UNICAMP) the linguist C. Galves (UNICAMP) and the computer scientist A. Mandel (USP). The statistician C. Dorea (UNB) works in the close area of inference in stochastic processes.

The main team collaborates with an important group of foreign researchers including the linguists A. Kroch (U Penn), M. Nespor (Ferrara) and J.-R. Vergnaud (USC), the statistical physicists M. Cassandro (Roma), P. Collet (CNRS), X. Bressaud (Marseille) and R. Fernández (Rouen) and the statistician R. Maronna (La Plata).

This interdisciplinary project has two main objectives. From the point of view of Probability Theory we want to develop the necessary tools to identify patterns in trajectories of stochastic processes. From the point of view of Linguistics, we want to use these tools to identify characteristic rhythmic patterns that are distinct for Brazilian Portuguese and Modern European Portuguese.

From the point of view of Probability Theory, this research is related to the study of chains of infinite order, the so called chains of complete connections. The modeling of rhythmic patterns in natural languages is a leading edge in Linguistics research. A most important technical question appears at this point: the automatic identification of the vowels and consonants. Classical methods in Time series are not suitable to approach this type of problem.

This project is being at work since 1994. It has received support by FAPESP, has been an active participation in the training of young researchers, attracting PhD and Post-doctoral students. The group is currently constructing an electronic corpus of historical Portuguese with free access in the address http:/www.ime.usp.br/~tycho.