Mathematical Biology

Molecular interactions lie at the core of biochemistry and biology, and their understanding is crucial to the advancement of biotechnology, therapeutics, and diagnostics. Most existing tools make “ensemble” measurements and report a single result, typically averaged over millions of molecules or more. These measurements can miss rare events, averaging out the natural variations or sub-populations within biological samples, and consequently obscure insights into multi-step and multi-state reactions. The ability to make and connect robust and quantitative measurements on multiple scales - single molecules, cellular complexes, cells, tissues - is a critical unmet need. In this talk, I will introduce a general method called “CLiC” imaging to image molecular interactions one molecule at time with precision and control, and under cell-like conditions. CLiC works by mechanically confining molecules to the field of view in an optical microscope, isolating them in nanofabricated features, and eliminates the complexity and potential biases inherent to tethering molecules. By imaging the trajectories of many single molecules simultaneously and in a dynamic manner, CLiC allows us to investigate and discover the design rules and mechanisms which govern how therapeutic molecules or molecular probes interact with target sites on nucleic acids; and how molecular cargo is released inside cells from lipid nanoparticles. In this talk, I will discuss applications of our imaging platform to better understand DNA, RNA, protein interactions, as well as emerging classes of genetic medicines, gene editing and drug delivery systems. I will highlight current and potential future applications to connect our observations from the level of single molecule to single cells, and opportunities for collaboration as we set up our labs at UBC.

Recursive distributional equations (RDEs) are ubiquitous in probability. For example, the standard Gaussian distribution can be characterized as the unique fixed point of the following RDE

$$X = (X_1 + X_2) / \sqrt{2}$$

among the class of centered random variables with standard deviation of 1. (The equality in the equation is in distribution; the random variables and must all be identically distributed; and and must be independent.)

Recently, it has been discovered that the dynamics of certain recursive distributional equations can be solved using by using tools from numerical analysis, on the convergence of approximation schemes for PDEs. In particular, the framework for studying stability and convergence for viscosity solutions of nonlinear second order equations, due to Crandall-Lions, Barles-Souganidis, and others, can be used to prove distributional convergence for certain families of RDEs, which can be interpreted as tree- valued stochastic processes. I will survey some of these results, as well as the (current) limitations of the method, and our hope for further interplay between these two research areas.

We develop a general framework for identifying phase reduced equations for finite populations of coupled oscillators that is valid far beyond the weak coupling approximation. This strategy represents a general extension of the theory from [Wilson and Ermentrout, Phys. Rev. Lett 123, 164101 (2019)] and yields coupling functions that are valid to higher-order accuracy in the coupling strength for arbitrary types of coupling (e.g., diffusive, gap-junction, chemical synaptic). These coupling functions can be used to understand the behavior of potentially high-dimensional, nonlinear oscillators in terms of their phase differences. The proposed formulation accurately replicates nonlinear bifurcations that emerge as the coupling strength increases and is valid in regimes well beyond those that can be considered using classic weak coupling assumptions. We demonstrate the performance of our approach through two examples. First, we use diffusively coupled complex Ginzburg-Landau (CGL) model and demonstrate that our theory accurately predicts bifurcations far beyond the range of existing coupling theory. Second, we use a realistic conductance-based model of a thalamic neuron and show that our theory correctly predicts asymptotic phase differences for non-weak synaptic coupling. In both examples, our theory accurately captures model behaviors that weak coupling theories can not.

Speaker Biography

Youngmin Park, Ph.D., is currently a PIMS Postdoc at the University of Manitoba under the supervision of Prof. Stéphanie Portet. He received his PhD in Mathematics from the University of Pittsburgh in 2018, where he applied dynamical systems methods to problems in neuroscience. His first postdoc involved auditory neuroscience research at the University of Pennsylvania in the Department of Otorhinolaryngology, before moving on to his next postdoc researching molecular motor dynamics in the Department of Mathematics at Brandeis University. He is now at Manitoba, continuing to apply dynamical systems methods to biological questions related to molecular motor transport and neural oscillators.

Active matter systems, ranging from liquid crystals to populations of cells and animals, exhibit complex collective behavior characterized by pattern formation and dynamic phase transitions. However, quantitative classification is challenging for heterogeneous populations of varying size, and typically requires manual supervision. In this talk, I will demonstrate that a combination of topological data analysis (TDA) and machine learning can uniquely identify the spatial arrangement of agents by keeping track of clusters, loops, and voids at multiple scales. To validate the approach, I will present 3 case studies: (1) data-driven modeling and analysis of epithelial-mesenchymal transition (EMT) in mammary epithelia, (2) unsupervised classification of cell sorting, and self-assembly patterns in co-cultures, and (3) parameter recovery from animal swarming trajectories.

Growing plant shoots exhibit circumnutations, namely, oscillations that draw three-dimensional trajectories, whose projections on the horizontal plane generate pendular, elliptical, or circular orbits. A large body of literature has followed the seminal work by Charles Darwin in 1880, but the nature of this phenomena is still uncertain and a long-lasting debate produced three main theories: the endogenous oscillator, the exogenous feedback oscillator, and the two-oscillator model. After briefly reviewing the three existing hypotheses, I will discuss a possible interpretation of these spontaneous oscillations as a Hopf-like bifurcation in a growing morphoelastic rod.