Progress in systems biology relies on the use of mathematical and statistical models for system level studies of biological processes. Several different modeling frameworks have been used successfully, including traditional differential equations based models, a variety of stochastic models, agent-based models, and Boolean networks, to name some common ones. This talk will focus on several types of discrete models, and will describe a common mathematical approach to their comparison and analysis, which relies on computer algebra. Hence, we refer to such models as "algebraic models." The talk will present specific examples of biological systems that can be modeled and analyzed in this way.
This talk consists of two parts: Pattern formation in families of microtubules under the action of kinesin and the detailed motion of kinesin along a microtubule.
Microtubules are long cylindrical structures (lengths being tens of microns and diameter approximately 25 nm) comprised of tubulin dimers, which self-assemble, 13 protofilaments being required side-to-side to form the circular cross section. In the first set of results, microtubules are represented as stiff, polar rods which are subject to diffusion in position and orientation and also subject to pair-wise interaction, mediated by kinesin molecular motors. The concentration of kinesin is represented by a parameter that feeds into the probability of an interaction occurring when two microtubules collide. The probability of an interaction also depends on the location of the collision point along the lengths of the microtubules, because kinesin accumulates at the positive end of each microtubule. With collision rules in place, Monte-Carlo simulations for large numbers of freely moving microtubules are performed, adjusting parameters for concentration of kinesin and polarity of the microtubules. From these studies, a phase diagram is produced, indicating thresholds for phase change to occur. Simulation results are compared to those from in vitro experiments.
The second part of the talk involves modeling the fine scale dynamics of a kinesin motor as it walks along a microtubule. The two heads of the kinesin molecule alternately bind and unbind to the microtubule with certain mechanisms providing a directional bias to the Brownian motion expected. One bias is the shape of the head and the shape of the binding site, along with the companion electrostatic charges. The second bias is that, utilizing ATP capture and transferal of phosphors for energy, part of the polymeric leg (neck-linker) of the bound head becomes attached towards the front of that head (the lqlq zipped
q state). The trailing head detaches from the microtubule. It then becomes subject to the biased entropic force due to the zipped state of the leading head and also preferentially (because of shape orientation) attaches in front of the currently attached head at which time ADP is released and a conformational change occurs, strengthening the binding. This motion is modeled using stochastic a differential equation. Simulations are performed with different lengths of neck-linkers and the mean speeds of progression obtained. These are compared with experimental results
Soliton like structures called "stable droplets" are found to exist within a paradigm reaction diffusion model which can be used to describe the patterning in a number of fish species. It is straightforward to analyse this phenomenon in the case when two non-zero stable steady states are symmetric, however the asymmetric case is more challenging. We use a recently developed perturbation technique to investigate the weakly asymmetric case.
Body size has been shown to be a significant factor in shaping the structure of food webs, which are network models of the flow of energy in an ecosystem. Recent studies have shown that body size constraints can influence food web dynamics through prey preference and foraging behavior, and can thereby influence the stability of these ecosystem models. Because of its significance, we use body size as the species strategy in an evolutionary game theory approach to studying the influence of predation at individual trophic levels on evolutionarily stable strategies (ESS) in food webs.
We systematically construct small (3-5 species) food webs, and combine ecological and evolutionary dynamics using differential equation models to show how the addition of each trophic level impacts the equilibrium strategies of other species. The strategy in our model influences the intrinsic growth rate and carrying capacity of the basal (plant) species, and the interaction rates across species. We show that when a consumer is introduced, the equilibrium strategy of the basal species evolves toward a value that increases the intrinsic growth rate; however, the strength of this effect is mediated by predator species at the third trophic level. We also show how size-based prey preference can influence strategy dynamics and population sizes over long time scales. These results suggest that understanding evolution of body size is important for understanding the trophic interactions that form the basis for large-scale food web structure and function.
The neuronal networks of the olfactory system transduce and transform complex mixtures of odorant molecules into patterns of the neural activity representing smells. We explore two important aspects of how this process works, at the cellular and the neural circuit level, in modeling studies that produce experimental testable predictions.
1) It has long been known that (in rodents) initial synaptic olfactory processing occurs in the olfactory bulb (OB) glomeruli, but the roles of various juxtaglomerular neurons is still not well understood. Recent experimental studies indicate that endogenously bursting external tufted (ET) cells -- which connect olfactory receptor neurons (ORN; OB input) to mitral cells (MC; OB output) -- play a central role in coordinating intraglomerular activity. We develop a biophysically realistic, Hodgkin-Huxley-style ET cell model that includes membrane currents known to be essential for bursting. We use specialized smooth optimization methods to study the (local) sensitivity of its functional characteristics (e.g. burst duration, interburst interval) to parameters, and statistical analyses to characterize the (global) influence of different currents.
2) Odorant-evoked input to and output from the OB is temporally dynamic, and these dynamics are important in shaping odor perception. Inhalation-evoked input bursts of ORN activity occur with durations, latencies, and amplitudes that vary across glomeruli (for the same odorant) and also in individual glomeruli for different odorants, and similarly diverse activity patterns occur at the MC level. We investigate these dynamics using biophysical models of the ORN-MC and ORN-ET-MC circuits. The modelsâ€™ inputs are taken from recordings of ORN calcium signals of head-fixed rats exposed to odorants and closely reproduce signals received by the real neurons. With this data-driven dynamical modeling approach, we are able to explore how the circuitsâ€™ response dynamics vary for different odorants, synaptic strengths, and intrinsic cellular parameters.
Metabolic rate, heart rate, and lifespan depend on body size according to scaling relationships that extend over ~21 orders of magnitude and that represent diverse taxa and environments. These relationships for body mass have long been approximated by power laws, but there has been intense debate about the values of exponents (e.g., 1/4 versus 1/3). I will show for mammals that these scaling relationships exhibit systematic curvature in logarithmic space. This curvature explains why different studies find different power-law exponents. I will also show how existing optimal network theory can be modified using finite-size corrections and hydrodynamical considerations to predict curvature. I will distinguish among potential physiological mechanisms by comparing model predictions for the direction and magnitude of the curvature with results from empirical data. For the final half of the talk, I will develop modified network models to describe tumor angiogenesis and vascular structure. These new models will help to compare tumor with normal vasculature, to understand different phases (pre- and post-angiogenesis) of tumor growth, and to describe the formation of a necrotic core.