New Techniques for Modeling in Precision Medicine: From GPUs to Wearables
The talk will focus on two new computational platforms we have built for precision health and the models that can be tested on them. The Wearable Data Analysis Platform has analyzed > 1,000,000 days of minute-by-minute wearable data to predict circadian (daily) phase and sleep from populations undergoing circadian disruption (e.g., shift workers, travelers, and students). We will show how our methods can also be used to pre-detect COVID infection. Our Whole Brain Simulation Platform quickly simulates and visualizes millions of neurons (e.g., all neurons in the mouse cortex) on a desktop GPU to pinpoint the contribution of brain regions to overall brain behaviors. We develop new mathematical techniques to make these possible, including a new Kalman Filter using level set methods, asymmetric particle population density methods to simulate large neural networks, and hybrid approaches using machine learning and physiological modeling.
Emergence of network connectivity from biologically plausible synaptic plasticity rules
Neural networks have non-random patterns of connectivity that are needed to implement different computations and functions in the brain. This non-random connectivity is set up by mechanisms of synaptic plasticity that change the strength of synaptic connections between neurons based on the activity of the neurons. These mechanisms can be formulated into learning rules, which allow us to mathematically characterize their influence on changes in connectivity as a function of neural activity. I will present two types of learning rules, one that changes connections between excitatory neurons and enables the formation of specific connectivity structures called assemblies, and another that changes the connections from inhibitory to excitatory neurons and enables stable and flexible learning.
Shape of data in biology
Biological processes are multi-scale. Spatial structures and patterns vary across levels of organisation, from molecular to multi-cellular to multi-organism. With more sophisticated mechanistic models and data available, quantitative tools are needed to study their evolution in space and time. Topological data analysis (TDA) is a computational field of mathematics that studies the shape of data. The most prominent tool in TDA is persistent homology (PH), which provides a multi-scale summary of data. Here we present extensions to the PH pipeline and highlight its utility with concrete case studies.
Cell-based modeling in cell and developmental biology: from individual cell behavior to collective pattern formation
To form the patterns and behaviors that we see in multicellular development, cells must carefully coordinate their behavior through biophysical and biochemical cues. It will be no surprise that I feel that numerical modeling and theory are essential for analyzing the mechanism of such coordinated, collective cell behavior. To do so, single-cell models must be sufficiently detailed so they correctly capture essential aspects of individual cells and do not oversimplify. At the same time, single-cell models must be sufficiently simple and computationally efficient so they can be upscaled to multicellular systems. Here I will present a series of our recent hybrid cellular Potts models for modeling individual cell behavior, and show how these can be used to study the coordinated cell behavior that is seen in biological development. I will discuss single cell models used to analyze observations such as anomalous cell migration patterns of immune cells, mechanical cell and extracellular matrix interactions, and models of anisotropic force generation. After discussing the evolutionary transition to multicellularity, I will present recent work on collective cell behavior, in particular models of blood vessel development. I will conclude by presenting strategies for experimental falsification and iterative correction of multicellular models. Altogether, I will present the use of cell-based modeling in analyzing how local cell-microenvironment interactions coordinate cell behavior during multicellular patterning.
Structured equations in biology; relative entropy, Monge-Kantorovich distance
Models arising in biology are often written in terms of Ordinary Differential Equations. The celebrated paper of Kermack-McKendrick (1927), founding mathematical epidemiology, showed the necessity to include parameters in order to describe the state of the individuals, as time elapsed after infection. During the 70s, many mathematical studies were developed when equations are structured by age, size, more generally a physiological trait. The renewal, growth-fragmentation are the more standard equations.
The talk will present structured equations, show that a universal generalized relative entropy structure is available in the linear case, which imposes relaxation to a steady state under non-degeneracy conditions. In the nonlinear cases, it might be that periodic solutions occur, which can be interpreted in biological terms, e.g., as network activity in the neuroscience.
When the equations are conservation laws, a variant of the Monge-Kantorovich distance (called Fortet-Mourier distance) also gives a general non-expansion property of solutions.
Mathematical modeling of cellular mechanotransduction
Cell shape and function are tightly coupled at different scales. In this talk, I will focus on two length scales of cell shape and function — (1) changing the shape of the cell locally due to curvature generation, relevant for trafficking and (2) impact of cell shape on mechanotransduction. Using biophysical modeling, I will discuss our recent efforts on generate membrane curvature due to membraneprotein interactions and the energy landscape of these deformations. Next, I will discuss how the assembly of membrane-protein aggregates can impact signaling dynamics of second messengers such as calcium and cAMP in cells. Finally, as a global example of mechanotransduction, I will discuss how YAP/TAZ nuclear translocation can be mediated by cell shape and substrate stiffness.