Applied and Computational Mathematics Seminar

Abstract: Sea ice, which covers a significant portion of the earth's surface, is an interestingly complicated material consisting of a mixture of solid ice and liquid brine phases which are coupled by thermodynamic considerations, Among other things, sea ice plays an important role in regulating macroscale heat transport between the ocean and the atmosphere. It also is a platform for microbial life, lots of it in fact, that uses the ice as a sort of shelter though eventually becoming part of the local food chain. A model will be presented that hypothesizes that, in turn, the resident microbial population might impact sea ice structure and, in particular, its transport properties including heat transport.

Abstract: Phylogenetic networks are a generalization of phylogenetic trees allowing for the representation of speciation and reticulate evolutionary events such as hybridization or horizontal gene transfer. The inference of phylogenetic networks from biological sequence data is a challenging problem, with many theoretical and practical questions still unresolved. In this talk, I will give an overview of the state of the art in phylogenetic network inference. I will then discuss a novel divide-and-conquer approach for inferring level-1 networks under the network multispecies coalescent model. I will end by discussing some open problems and avenues for future research.
Parts of this talk are based on joint work with Elizabeth Allman, Hector Baños, and John Rhodes.

Dynamical systems can undergo qualitative, topological changes in their orbit structure called bifurcations when underlying parameters cross a threshold: the "shape" of their behavior alters fundamentally. The development of data-driven tools for modeling these changes holds special promise in the life sciences, from the design of gene regulatory networks to the prediction of catastrophic oscillations in neural circuits. In this talk, I describe an ongoing research program which tackles this challenge by focusing on the realistic case where governing equations are unknown and dynamical behavior must be predicted from prior knowledge given noisy, sparse data. Building on classical work in so-called model manifold theory, our approach learns a shared feature landscape where diverse systems coalesce within a unified embedding space, revealing their underlying qualitative structure. I first describe work which uses such learned universal embeddings of low-dimensional dynamical systems to classify circuits by their function. Next, I demonstrate how a simple autoencoder can learn an implicit notion of topological conjugacy which functions as a robust detector of Hopf bifurcations in single-cell RNA sequencing data from the pancreas. Finally, we generalize to the case of spatiotemporal dynamics, where I outline recent work on building reduced-order parametric models ofpartial differential equations with applications to spatial patterning in the ocellated lizard. We conclude with some future directions, notably extensions to high-dimensional systems and applications to synthetic biology, where engineered organisms and tissues could be designed for stable, predictable functions in dynamic environments.

Most physiological mechanisms exhibit high variability due to the fundamental stochasticity of biochemical reaction pathways. Quantifying the impact of stochastic effects is necessary for a deeper understanding of physiological processes and their regulation, and helps in the choice of an efficient approach for their computational modeling. This is especially true in the case of synaptic neurotransmitter release, which is caused by the fusion of the secretory vesicle membrane with the cell membrane in response to calcium ion binding. Although stochastic calcium channel gating is one of the primary source of this stochasticity, it can be implemented in a computationally inexpensive way in combination with deterministic simulation of the downstream calcium diffusion and binding reactions. Another fundamental reason for the high variability of synaptic response is that only a small number of calcium ions enter the synaptic terminal through a single channel during an action potential. This fact entails large fluctuations due to calcium diffusion and its binding to calcium buffers and vesicle release sensors, leading to a widely-held view that solving continuous deterministic reaction-diffusion equations does not provide high accuracy when modeling calcium-dependent cell processes. 

However, several comparative studies show a surprising close agreement between deterministic and trial-averaged stochastic simulations of calcium dynamics, as long as calcium channel gating is not calcium-dependent. This result deserves careful investigation. This talk will focus on further analysis and comparison of stochastic and mass-action modeling of vesicle release, showing that the discrepancy between deterministic and stochastic approaches remains small even when only as few as 40-50 ions enter per single channel-vesicle complex. The reason for the close agreement between stochastic and mass-action simulations is that the discrepancy between the two approaches is determined by the size of the correlation between the local calcium concentration and the state of the vesicle release sensor, rather than fluctuation amplitude. Whereas diffusion and buffering increases fluctuation size, the same processes appear to de-correlate fluctuations in calcium concentration from fluctuations in calcium sensor binding state. Finally, contrary to naïve intuition, the mass action / mean-field reaction-diffusion description allows an accurate estimate of the probability density of vesicle release latency (first-passage time), rather than providing information about trial-averaged quantities only. These results may help in the choice of appropriate and efficient tools for the modeling of this and other fundamental biochemical cell processes.

We study the problem of noisy data fitting and spectral measure recovery in the classes Stieltjes and Completely monotone functions. The analytical setting is least squares over the convex or conical hull of a space curve in IR^n. I will begin with the case of least squares over convex hulls where the convergence analysis is simple. Here, I will introduce our method of analyzing the support of the optimal spectral measure. I will then move to the case of conical hulls (non-negative least squares). In this context, convergence analysis is not well understood. We introduce a simple assumption on the problem which allows us to overcome the convergence problem. While our assumption does not initially hold in either our motivating examples, I will show they can be rescaled so that is does hold. This rescaling allows certain algorithms for convex minimization to be extended to minimization problems with non-negativity constraints. I will provide several numerical examples which show convergence of algorithms as well as highlight our spectral measure analysis at certain noise levels.