Mathematical Biology Seminar - Spring 2018

The timing of human sleep is strongly modulated by the 24 h circadian rhythm, and desynchronization of sleep-wake cycles from the circadian rhythm can negatively impact health. We have developed a physiologically-based mathematical model for the neurotransmitter-mediated interactions of sleep-promoting, wake-promoting and circadian rhythm-generating neuronal populations that govern sleep-wake behavior in humans. To investigate the dynamics of circadian modulation of sleep patterns and of entrainment of the sleep-wake cycle with the circadian rhythm, we have reduced the dynamics of the sleep-wake regulatory network model to a one-dimensional map. The map dictates the phase of the circadian cycle at which sleep onset occurs on day n + 1 as a function of the circadian phase of sleep onset on day n. The map is piecewise continuous with discontinuities caused by circadian modulation of the duration of sleep and wake episodes and the occurrence of rapid eye movement (REM) sleep episodes. Analysis of map structure reveals changes in sleep patterning, including REM sleep behavior, as sleep occurs over different circadian phases. In this way, the map provides a portrait of the circadian modulation of sleep-wake behavior and its effects on REM sleep patterning. Using the map, we are analyzing effects of sleep deprivation and bifurcations of the sleep-wake regulatory network model to understand how variations in the homeostatic sleep drive affect human sleep patterning over development.

It has been shown recently that changing the fluidic properties of a drug can improve its efficacy in ablating solid tumors. We model diffusion of the drug in a spherical tumor with leaky boundaries. The death of a cell after a given exposure time depends on both the concentration of the drug and the amount of oxygen available to the cell. Higher oxygen availability leads to cell death at lower drug concentrations. It can be assumed that a minimum concentration is required for a cell to die, effectively connecting diffusion with efficacy. The concentration threshold decreases as exposure time increases, which allows us to compute dose-response curves. Furthermore, these curves can be plotted at much finer time intervals compared to that of experiments. The curves may then be used to produce a dose-response-time surface giving an observer a complete picture of the drug's efficacy for an individual. Finally, since the diffusion and leak coefficients and the availability of oxygen is different for different individuals, we develop statistical models for these parameters to connect the model with a population.

Fast running insects employ a tripod gait with three legs in swing and three in stance, while slower walkers use a tetrapod gait with two legs in swing and four in stance. Fruit flies exhibit a tetrapod to tripod transition at intermediate speeds. In this talk, I discuss the mechanism of such gait transitions in a bursting neuron model where each cell represents a hemi-segmental thoracic circuit of the central pattern generator. Under phase reduction, the 24 ODE bursting neuron model becomes 6 coupled phase oscillators, each corresponding to a network driving one leg. Assuming that left and right legs maintain constant phase relations, it further reduces to a system describing ipsilateral phase differences defined on a torus. I show that bifurcations occur from multiple stable tetrapod gaits to a unique stable tripod gait as speed increases, and illustrate our results using data fitted to freely walking fruit flies. Finally, I comment on how heterogeneity in the intrinsic dynamics of each leg can affect the existence of tetrapod gaits and their transition to tripod gaits.
This is joint work with Vaibhav Srivastava and Philip Holmes

In this talk, we present an immersed boundary method for mass transfer across permeable deformable moving interfaces interacting with the surrounding fluids. One of the key features of our method is the introduction of the mass flux as an independent variable, governed by a vector transport equation. The flux equation, coupled with the mass transport and the fluid flow equations, allows for a natural implementation of an immersed boundary algorithm when the flux across the interfaces is proportional to the jump in concentration. As an example, the oxygen transfer from red blood cells in a capillary vessel is used to illustrate the applicability of the proposed method. We show that our method is capable of handling multi-physics problems involving fluid-structure interaction with multiple deformable moving interfaces and mass transfer simultaneously. If time permits, extension of the current method will be discussed.
This is joint work with S. Takagi, K. Sugiyama, and X. Gong

In this talk, we study the global dynamics of a mean field model of electroencephalographic activity in the brain, which is composed of a system of coupled ODEs and PDEs. We show the existence and uniqueness of weak and strong solutions of this model and discuss the regularity of these solutions. We establish biophysically plausible semidynamical system frameworks for this model, and show that the semigroups of weak and strong solution operators possess bounded absorbing sets. We show that there exist sets of parameter values for which the semidynamical systems do not possess a global attractor due to the lack of the compactness property. In this case, the internal dynamics of the ODE components of the solutions can create asymptotic spatial discontinuities in the solutions, regardless of the smoothness of the initial values and forcing terms. Finally, we demonstrate an application of this model in the computational study of the alpha- and gamma-band rhythmic activity in the neocortex, and address some challenges involved in the numerical computation of the solutions of this model.

The three-dimensional spatiotemporal organization of genetic material inside the cell nucleus remains an open question in cellular biology. During the time between two cell divisions, the functional form of DNA in cells, known as chromatin, fills the cell nucleus in its uncondensed polymeric form, which allows the transcriptional machinery to access DNA. Recent in vivo imaging experiments have cast light on the existence of coherent chromatin motions inside the nucleus, in the form of large-scale correlated displacements on the scale of microns and lasting for seconds. To elucidate the mechanisms for such motions, we have developed a coarse-grained active polymer model where chromatin is represented as a confined flexible chain acted upon by active molecular motors, which perform work and thus exert dipolar forces on the system. Numerical simulations of this model that account for steric and hydrodynamic interactions as well as internal chain mechanics demonstrate the emergence of coherent motions in systems involving extensile dipoles, which are accompanied by large-scale chain reconfigurations and local nematic ordering. Comparisons with experiments show good qualitative agreement and support the hypothesis that long-ranged hydrodynamic couplings between chromatin-associated active motors are responsible for the observed coherent dynamics.

Genetic circuits that exhibit robust oscillations are important in synthetic biology --- they are used as genetic clocks, biosensors, etc. In this work we consider models for genetic circuits with robust oscillations driven by delayed dynamics, where the delay arises due to lengthy transcription times. We first consider a (one-dimensional) delay differential equation (DDE) as an approximation for the dynamics and we review prior results on existence, uniqueness and stability of periodic oscillatory solutions for DDEs. We then provide three refinements of the DDE approximation, each considered individually: (a) we impose a natural non-negativity constraint on the solution of the DDE and provide conditions for existence, uniqueness and stability of periodic oscillatory solutions; (b) we add small noise to the system and provide estimates of the exit time of solutions from small neighborhoods of periodic orbits of the deterministic system, from the perspective of large deviations; and (c) we consider a system of coupled DDEs and present conditions for existence, uniqueness and stability of synchronized periodic oscillatory solutions.
This talk includes joint work with R. J. Williams and R. J. Lipshutz.