Application of Mathematics to Medicine and Biology
Org:
Sivabal Sivaloganathan (University of Waterloo)
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 SUE ANN CAMPBELL, University of Waterloo
 CORINA DRAPACA, Pennsylvania State University
 JANE HEFFERNAN, York University
 THOMAS HILLEN, University of Alberta
 BRIAN INGALLS, University of Waterloo
 KAMRAN KAVEH, Harvard University
Evolution in heterogeneous and random environments [PDF]

Many theoretical models of evolution assume that all competing individuals experience the same environment. Here, we consider the more realistic scenario of evolution in heterogeneous environments. We introduce a novel formalism to approach any form of spatial fitness heterogeneity in an evolutionary graph. We calculate the condition for natural selection to favor the mutant type relative to the resident on a complete graph structure. Environmental heterogeneity elucidates an interesting asymmetry between the mutant and resident types. Mutant heterogeneity suppresses fixation probability, and if strong enough, it can completely offset the effects of natural selection. In contrast, resident heterogeneity can amplify a mutantâ€™s fixation probability if population is small and has no effect on mutant fixation probability otherwise. Our results hold for any environmental heterogeneity and selection intensity. We address generalization of the above observations to other graph structures, as well as heterogeneous evolutionary games.
 ALI MAHDIPOUR, University Health Network
Cell Cycle Significance in the Evolutionary Dynamics of Cancer [PDF]

Proliferation has known as one of the main building blocks of almost any evolutionary mechanism in living species. More specifically, during cell replication, each cell undergoes various phases within the cell cycle. The average time of successful divisions and the frequency of cells within each of the cell cycle phases determine their fitness within a population. In continuation to recent researches on the role of various cellcyclecompartments in the evolution of biological systems, we suggest a general framework which highlights the significance of cell cycle in a heterogeneous system. More precisely, we investigate how the cell cycle mechanism may affect the dynamics of malignancy in such a system. We find the speed of initiation/progression of malignancy and its survival rate in diverse cellcycle phases. Our findings may provide a better understanding of malignancy development in infectious diseases and different types of cancer where various phenotypes behave differently during mitosis and replication. This research may also affect current treatment schedules in order to provide more intense therapies.
 LAWRENCE OPREA, McGill University
 KATRIN ROHLF, Ryerson University
Reactive Multiparticle Collision (RMPC) dynamics for stochastic simulations of biochemical systems [PDF]

Stochastic simulation methods are popular means to simulate reaction mechanisms, and can be used to explore biochemical systems for which traditional wellmixed chemical kinetics rate laws no longer apply. The Gillespie algorithm, or other related stochastic simulation algorithms, have had a lot of success in capturing both the wellmixed system dynamics in agreement with wellmixed chemical rate laws and reactiondiffusion mechanisms, as well as effects beyond the applicability of ODE/PDE models. Coupling of the reactive dynamics to fluid flow, however, is a challenge in this framework, and other simulation methods, such as Reactive Multiparticle Collision dynamics (RMPC) can allow for a means to model chemically reactive systems coupled to flow conditions.
This talk will introduce the Reactive Multiparticle Collision (RMPC) dynamics, as well as its generalized Master equation. Simulation results for the Selkov reaction mechanism, as well as those of an intracellular signaling pathway for bacterial chemotaxis in E. coli will be presented. As part of the talk, the theoretical calculation for the selfdiffusion coefficient for the different chemical species will be derived from the RMPC dynamics. The talk will conclude with current work, and future studies for which RMPC can be an important simulation tool.
 FRANCES SKINNER, Krembil Research Institute, UHN and University of Toronto
Mathematically modeling theta oscillations in the hippocampus [PDF]

It is clear that oscillatory brain activities are a ubiquitous feature of brain recordings. In particular, theta rhythms (312Hz) in the hippocampus play fundamental roles in memory processing. Is it possible to have a cellularbased understanding of these activities? In this talk, I will describe some of our recent modeling work from the perspective of a particular cell type in the hippocampus and its contribution to theta rhythms by virtue of its biophysical characteristics.
 ROBERT SMITH?, The University of Ottawa
Comparing malaria surveillance with periodic spraying in the presence of insecticideresistant mosquitoes [PDF]

There is an urgent need for more understanding of the effects of surveillance on malaria control. Indoor residual spraying has had beneficial effects on global malaria reduction, but resistance to the insecticide poses a threat to eradication. We develop a model of impulsive differential equations to account for a resistant strain of mosquitoes that is entirely immune to the insecticide. The impulse is triggered either due to periodic spraying or when a critical number of malaria cases are detected. For small mutation rates, the mosquitoonly submodel exhibits either a single mutantonly equilibrium, a mutantonly equilibrium and a single coexistence equilibrium, or a mutantonly equilibrium and a pair of coexistence equilibria. Bistability is a likely outcome, while the effect of impulses is to introduce a saddlenode bifurcation, resulting in persistence of malaria in the form of impulsive periodic orbits. If certain parameters are small, triggering the insecticide based on number of malaria cases is asymptotically equivalent to spraying periodically.
 MADJID SOLTANI, KN toose University of Technology, Tehran
 ADAM STINCHCOMBE, University of Toronto
 KATHLEEN WILKIE, Ryerson Polytechnic University
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