Funded Grants

Merging dynamical systems modeling and analysis at different levels of biological organization

Biological systems are much more complicated than physical systems. Thus a description of biological systems requires that we focus on particular aspects at the expense of neglecting details relating to non-focal aspects, particularly when the description relates to the formulation of a model to be used for quantitative analysis and prediction. Examples include epidemiological models used to understand processes driving the spread of disease and to evaluate control strategies such as vaccination, case isolation and quarantine. These models generally divide the population into various disease, demographic, and risk classes. They ascribe identical properties to all individuals in the same class, thereby ignoring important sources of variability among individuals such as host genetics, pathogen genetics, and the contact (or pairing) behavior of hosts. Some of these difficulties have inspired the formulation of so-called individual-based and network models. Even these models do not provide a proper treatment of processes that take place within individual hosts, such as evolution of drug-resistant pathogen strains—a problem that is now seriously affecting our ability to control AIDS, tuberculosis, and malaria. A full analysis of drug resistance in epidemics at a minimum requires the merging of models of within-host pathogen-strain dynamics and among host disease dynamics. Within-host disease models typically consider different time and spatial scales from epidemic models. Most importantly the two types of models represent descriptions of biological systems at different hierarchical levels of organization.

One of the most important challenges today in the general area of biological systems analysis is how to formulate models at consecutive levels of organization in a way that facilitates merging level of analysis. In this proposal, I develop approaches to modeling consecutive levels of biological systems in ways that facilitates the analysis of the affects of processes at one level on processes and states at the other level. My current research program includes several different problems that require merging two levels of analysis in biological systems. Questions that my collaborators and I propose to address include:

  1. The drug resistance problem described above.
  2. How the foraging behavior of individuals over daily times scales produces an average population level consumer-resource feeding rate response at monthly or even annual time scales.
  3. How the dynamics of bovine tuberculosis in herds of African buffalo (time scale is years, no spatial structure) and the movements of individual buffalo between herds (time scale is weeks) produce spatio-temporal patterns (time scale is decades) at the spatial scale of a large wildlife reserve (in this case the Kruger National Park which is the size of New Jersey).
  4. How processes controlling contact among individuals at daily time scales affect the course of an epidemic, especially in sexually transmitted diseases such as gonorrhea and AIDS.
  5. How plant defensive traits (chemicals, mechanical barriers) expressed at a cost to the reproductive rate of different strains co-evolves with insect traits that influence their ability to exploit the different plant strains. This analysis includes modeling individual host-plant choices using neural networks, and will shed light on the question of the evolution of generalists versus specialists in guilds of herbivorous insects that exploit the same general group of plants that vary according to their evolving defensive strategy.
  6. How communities of different bacterial species assemble (number of ecotypes, genetic variability within ecotypes) and form trophic food webs in specific resource environments (e.g. human skin, a particular class of rotting carcasses) based on the metabolic needs of individual cells, transport of resources into individual cells and excretion of products out of these cells. Of particular interest is the community that assembles in acid mine drainage systems, such as the federally identified Iron Mountain toxic waste site in northern California. Such communities are fueled through their oxidation of iron pyrites, in the process releasing high concentrations of sulfuric acid and other pollutants into mountain streams.

Problems 1, 3, 4, and 6 are currently funded in the context of specific system studies. The funding provided under this grant will support the development of general methods that apply to problems requiring the formulation of models at two consecutive levels of description: a lower “constitutive” level and an upper “integrative” level. These models are formulated in terms of processes acting at each level and then linked across levels. If the time constants of the constitutive level are much faster than the integrative level then the integrative level dynamics can be analyzed by assuming the constitutive level is either at an equilibrium or is at some time average value. For systems where the disparity of time constants is not that great, I propose linearizing the constitutive system around its equilibrium or average value and exploring the degree to which this approximation allows results to be obtained at the integrative system level. I also propose extending to the types of two-layer models described above generalized methods of sensitivity analysis that have been developed in biochemical systems theory under the rubric of Metabolic Control Analysis (MCA) and its extension to Hierarchical Control Analysis. In the guise of Trophic Control Analysis, my collaborators and I have recently modified the theory to apply to trophic chains in ecology, but it remains for a control analysis theory to be formulated for linked constitutive-integrative level systems models. This theory should yield theorems stated in the form of matrix relationships that can be used to solve for the relative effects of processes at one level of description on processes and values of variables at another level of description for the biological system at hand. With such theorems in hand, we hope to gain new insights into all six questions enumerated above, and pave the way for others to do the same in systems across the whole spectrum of biology, from biochemistry and genetics, through physiology and ecology, to evolutionary population biology.