Mathematics Dept.
  Events
Oxford University
Thu 18 Jan 2018, 11:00am SPECIAL
Mathematical Biology Seminar / Probability Seminar
Math 126
Modelling mutations: mechanisms and evolutionary consequences
Math 126
Thu 18 Jan 2018, 11:00am-12:00pm

Abstract

 As the source of new genetic variation, mutations constitute a fundamental process in evolution. While most mutations lower fitness, rare beneficial mutations are essential for adaptation to changing environments. Thus, understanding the effects of mutations and estimating their rate is of strong interest in evolutionary biology. The necessity to treat rare mutational events stochastically has also stimulated a rich mathematical literature. Typically, mutations are modelled simply as an instantaneous change of type, occurring at a fixed rate. However, the underlying biology is more complex. I will present two recent projects delving deeper into mutational mechanisms and their consequences. Firstly, mutations can exhibit a multi-generational delay in phenotypic expression. Secondly, individuals within a population can vary in their propensity to mutate. Through analytical and simulation methods, we investigated the impact of these biological complexities on (a) population fitness and capacity to evolve, and (b) our ability to accurately infer mutation rates from data. I will conclude by discussing some future directions to incorporate these insights into more realistic models and to quantify the distribution of mutation rate empirically.

Note for Attendees

 Math 126 is behind a locked glass door. Latecomers without access should knock loudly!
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Kornelia Hera
Eotvos Lorand University, Budapest
Fri 19 Jan 2018, 2:00pm
Harmonic Analysis Seminar
MATH 126
Furstenberg-type estimates for unions of affine subspaces
MATH 126
Fri 19 Jan 2018, 2:00pm-3:00pm

Abstract

A plane set is called a t-Furstenberg set for some t in (0,1), if it has an at least t-dimensional intersection with some line in each direction (here and in the sequel dimension refers to Hausdorff dimension).  Classical results are that every t-Furstenberg set has dimension at least 2t, and at least t + 1/2.

We generalize these estimates for families of affine subspaces. As a result, we prove that the union of any s-dimensional family of k-dimensional affine subspaces is at least k + s/(k+1) -dimensional, and is exactly k + s -dimensional if s is at most 1.

Based on joint work with Tamas Keleti and Andras Mathe.
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Oxford University
Fri 19 Jan 2018, 3:00pm SPECIAL
Department Colloquium
ESB 2012
Stochastic population dynamic models with applications to pathogen evolution
ESB 2012
Fri 19 Jan 2018, 3:00pm-4:00pm

Abstract

Biological populations facing severe environmental change must adapt in order to avoid extinction. This so-called “evolutionary rescue” scenario is relevant to many applied problems, including pathogen evolution of drug resistance during the treatment of infectious diseases. Understanding what drives the rescue process gives rise to interesting mathematical modelling challenges arising from two key features: demographic and evolutionary processes occur on the same timescale, and stochasticity is inherent in the emergence of rare well-adapted mutants. In this talk, I will present recent work on population dynamics in changing environments, merging biological realism with tractable stochastic models. Firstly, I will describe a model of drug resistance evolution in chronic viral infections, which serves as a case study for a novel mathematical approach yielding analytical approximations for the probability of rescue. Secondly, I will present a combined theoretical and experimental investigation of the classical problem of establishment (non-extinction) of new lineages, using antibiotic-resistant bacteria as a model system. Finally, I will discuss some future directions in modelling antibiotic treatment to predict optimal dosing strategies, and in developing a general theoretical framework for evolutionary rescue that unites approaches to distinct applied problems.

Note for Attendees

Refreshments will be served at 2:45 p.m. in ESB 4133, the PIMS Lounge.
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Jay Newby
University of North Carolina, Chapel Hill
Mon 22 Jan 2018, 12:00pm SPECIAL
ESB 4133 (PIMS Lounge)
Seminar Talk in Math Biology, Applied Stochastics: How molecular crowding is changing our understanding of spatial patterning in living cells
ESB 4133 (PIMS Lounge)
Mon 22 Jan 2018, 12:00pm-1:00pm

Details


Molecular crowding has recognized consequences for biological function. However, there are also circumstances in which un-crowding is important that is, when molecules must evacuate from a region before a given process can occur. One example is offered by the T cell, where large surface molecules must evacuate from a region to allow for the T cell to interact with its target, thereby facilitating immune function. Evacuation is fundamentally stochastic and spatial, since diffusion is a major driver. Studies of molecular evacuation present mathematical and computational challenges. For example, in some scenarios, it is a "rare event", making straightforward simulation unfeasible. To obtain a complete picture of diffusional evacuation, we use a combination of perturbation theory and numerical simulation. I will also show evidence of persistent un-crowding in living fungal cells. Based on our understanding of diffusional evacuation, we know that diffusion alone cannot explain these observations. I will discuss our current efforts to quantify and resolve how fungal cells control un-crowding.

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Cornell Statistical Science and Biological Statistics & Computational Biology
Mon 22 Jan 2018, 3:00pm SPECIAL
Institute of Applied Mathematics
ESB 2012
An ODE to Statistics: Inference about Nonlinear Dynamics
ESB 2012
Mon 22 Jan 2018, 3:00pm-4:00pm

Abstract

Ordinary differential equation models are used extensively within mathematics as descriptions of processes in the real world. However, they are rarely employed by statisticians and there is a paucity of methods for combining differential equation models with data. This talk provides a survey of recently developed statistical methods for estimating parameters from data, conducting model criticism and improvement for differential equation models in the light of data, and designing experiments that yield optimal estimates of parameters. It ends with some perspectives on the current state of the field and open problems.

Note for Attendees

Reception before the talk in ESB 4133 (the PIMS lounge). This is in the IAM/PIMS distinguished speaker series.
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Saskatchewan
Mon 22 Jan 2018, 4:00pm
Algebraic Geometry Seminar
MATH 126
Asymptotic geometry of hyperpolygons
MATH 126
Mon 22 Jan 2018, 4:00pm-5:00pm

Abstract

Nakajima quiver varieties lie at the interface of geometry and representation theory.  I will discuss a particular example, hyperpolygon space, which arises from star-shaped quivers.  The simplest of these varieties is a noncompact complex surface admitting the structure of an "instanton", and therefore fits nicely into the Kronheimer-Nakajima classification of ALE hyperkaehler 4-manifolds, which is a geometric realization of the McKay correspondence for finite subgroups of SU(2).  For more general hyperpolygon spaces, we speculate on how this classification might be extended by studying the asymptotic geometry of the variety.  In moduli-theoretic terms, this involves driving the stability parameter for the quotient to an irregular value.  This is joint work in progress with Harmut Weiss, building on previous work with Jonathan Fisher.
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Jay Newby
University of North Carolina, Chapel Hill
Tue 23 Jan 2018, 3:45pm SPECIAL
Department Colloquium
ESB 2012 (PIMS)
Weaker is better: how weak transient molecular interactions give rise to robust, dynamic immune protection
ESB 2012 (PIMS)
Tue 23 Jan 2018, 3:45pm-4:45pm

Abstract

The longstanding view in chemistry and biology is that high-affinity, tight-binding interactions are optimal for many essential functions, such as receptor-ligand interactions. Yet, an increasing number of biological systems are emerging that challenge this view, finding instead that low-affinity, rapidly unbinding dynamics can be essential for optimal function. These mechanisms have been poorly understood in the past due to the inability to directly observe such fleeting interactions and the lack of a theoretical framework to mechanistically understand how they work. In fact, it is only by tracking the motion of effector nanoprobes that afford detection of multiple such interactions simultaneously, coupled with inferences by stochastic modeling, Bayesian statistics, and bioimaging tools, that we recently begin to obtain definitive evidence substantiating the consequences of these interactions. A common theme has begun to emerge: rapidly diffusing third-party molecular anchors with weak, short-lived affinities play a major role for self organization of micron-scale living systems. My talk will demonstrate how these ideas can answer a longstanding question: how mucosal barriers selectively impede transport of pathogens and toxic particles, while allowing diffusion of nutrients.
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UBC
Wed 24 Jan 2018, 3:10pm
Probability Seminar
LSK 460
Heat flow on snowballs
LSK 460
Wed 24 Jan 2018, 3:10pm-4:10pm

Abstract

Quasisymmetric maps are fruitful generalizations of conformal maps. Quasisymmetric uniformization problem seeks for extensions of uniformization theorem beyond the classical context of Riemann surfaces.

The goal of this talk is to show that quasisymmetric uniformization problem is closely related to random walks and diffusions. I will explain how the existence of quasisymmetric maps is equivalent to heat kernel estimates for the simple random walk on a family of planar graphs. The same methods also apply to diffusions on a class of fractals homeomorphic to the 2-sphere.

These ideas will be illustrated using snowballs and their graph approximations. Snowballs are high dimensional analogues of Koch snowflake.

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Courant Institute, NYU
Wed 24 Jan 2018, 3:15pm
Mathematical Biology Seminar
PIMS Lounge, ESB 4133
Mechanical Positioning of Multiple Myonuclei in Muscle Cells
PIMS Lounge, ESB 4133
Wed 24 Jan 2018, 3:15pm-4:15pm

Abstract

 Many types of large cells have multiple nuclei. In long muscle cells, nuclei are distributed almost uniformly along their length, which is crucial for cell function. However, the underlying positioning mechanisms remain unclear. We examine computationally the hypothesis that a force balance generated by microtubules positions the nuclei. Rather than assuming what the forces are, we allow for various types of forces between pairs of nuclei and between the nuclei and the cell boundary. Mathematically, this means that we start with a great number of potential models. We then use a reverse engineering approach by screening the models and requiring their predictions to fit imaging data on nuclei positions from hundreds of muscle cells of Drosophila larva. Computational screens result in a small number of feasible models, the most adequate of which suggests that the nuclei repel each other and the cell boundary with forces that decrease with distance.

This suggests that microtubules growing from nuclear envelopes push on neighboring nuclei and the cell boundary. We support this hypothesis with stochastic microscopic simulations. Using statistical and analytical tools such as correlation and bifurcation analysis, we make several nontrivial predictions: An increased nuclear density near the cell poles, zigzag patterns in wider cells, and correlations between the cell width and elongated nuclear shapes, all of which we confirm by image analysis of the experimental data.

This is joint work with Mary Baylies, Alex Mogilner and Stefanie Windner.

 

 

Note for Attendees

Refreshments: PIMS tea will be served at 2:45 in ESB 4133.
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