Mathematical Biology

The principal function of the nucleus is to facilitate storage, retrieval, and maintenance of the genetic information encoded into DNA and RNA sequences. A unique feature of nucleoplasm—the fluid of the nucleus—is that it contains chromatin (DNA) and RNA.

In contrast to other important biological polymer hydrogels, such as mucus and extracellular matrix, the nucleic acid polymers have a sequence. Recent experiments have shown that during the growth phase of the cell cycle, chromatin condenses in a sequence specific manner into regions within the nucleoplasm, possibly so that functionally related genes are grouped together spatially even though they might be far apart in terms of sequence distance.

At the same time, we are becoming increasingly aware of the role of liquid-liquid phase separation (LLPS) in cellular processes in the nucleus and the cytoplasm. Complex molecular interactions over a wide range of timescales can cause large biopolymers (RNA, protein, etc) to phase separate from the surrounding nucleoplasm into distinct biocondensates (spherical droplets in the simplest cases).

I will discuss recent work modelling the role of nuclear biocondensates in neurodegenerative disease and several ongoing projects related to
modelling and microscopy image analysis.

Understanding how cells change their identity and behaviour over time in living systems is a key question in many fields of biology. Measurement of cell states is inherently destructive, and so the relationship of the current state of a cell to some future state, or ‘fate’, cannot be observed experimentally. Trajectory inference refers to the general problem of trying to estimate various aspects of the state-fate relationship. We discuss optimal transport as a useful analytical tool for trajectory inference, and we develop a mathematical framework for recovering trajectories in both non-equilibrium as well as equilibrium systems.

For cells to divide, they must undergo mitosis: the process of spatially organizing their copied DNA (chromosomes) to precise locations in the cell. This procedure is carried out by stochastic components that manage to accomplish the task with astonishing speed and accuracy. New advances from our collaborators in the New York Dept of Health provide 3D spatial trajectories of every chromosome in a cell during mitosis. Can these trajectories tell us anything about the mechanisms driving them? The structure and context of this cutting-edge data makes utilizing classical tools from data science or particle tracking challenging. I will discuss my progress with Alex Mogilner on developing analysis for this data and mathematical modeling of emergent phenomena.

To initiate movement, cells need to form a well-defined "front" and "rear" through the process of cellular polarization. Polarization is a crucial process involved in embryonic development and cell motility and it is not yet well understood. Mathematical models that have been developed to study the onset of polarization have explored either biochemical or mechanical pathways, yet few have proposed a combined mechano-chemical mechanism. However, experimental evidence suggests that most motile cells rely on both biochemical and mechanical components to break symmetry. I will describe a mechano-chemical mathematical model for emergent organization driven by both cytoskeletal dynamics and biochemical reactions. We have identified one of the simplest quantitative frameworks for a possible mechanism for spontaneous symmetry breaking for initiation of cell movement. The framework relies on local, linear coupling between minimal biochemical stochastic and mechanical deterministic systems; this coupling between mechanics and biochemistry has been speculated biologically, yet through our model, we demonstrate it is a necessary and sufficient condition for a cell to achieve a polarized state.

Cellular polarization plays a critical during cellular differentiation, development, and cellular migration through the establishment of a long-lived cell-front and cell-rear. Although mechanisms of polarization vary across cells types, some common biochemical players have emerged, namely the RhoGTPases Rac and Rho. The low diffusion coefficient of the active form of these molecules combined with their mutual inhibitory interaction dynamics have led to a prototypical pattern-formation system that can polarizes cell through a non-Turing pattern formation mechanism termed wave-pinning. We investigate the effects of paxillin, a master regulator of adhesion dynamics, on the Rac-Rho system through a positive feedback loop that amplifies Rac activation. We find that paxillin feedback onto the Rac-Rho system produces cells that (i) self-polarize in the absence of any input signal (i.e., paxllin feedback causes a Turing instability) and (ii) become arrested due to the development of multiple protrusive regions. The former effect is a positive finding that can be related to certain cell-types, while the latter outcome is likely an artefact of the model. In order to minimize the effects of this artefact and produce cells that can both self-polarize as well as migrate for extended periods of time, we revisit some of model's parameter values and use lessons from previous models of polarization. This approach allows us to draw conclusions about the biophysical properties and spatiotemporal dynamics of molecular systems required for autonomous decision making during cellular migration.