Mathematical Biology

There are two rotary motors in biology, ATP synthase and the bacterial flagellar motor. Both are driven by transmembrane ionic currents. We consider an idealized model of such a motor, essentially an electrostatic turbine. The model has a rotor and a stator, which are closely fitting cylinders. Attached to the rotor is a fixed density of negative charge, with helical symmetry. Positive ions move longitudinally by drift and diffusion on the stator. A key assumption is local electroneutrality of the combined charge distribution. With this setup we derive explicit formulae for the transmembrane current and the angular velocity of the rotor in terms of the transmembrane electrochemical potential difference of the positive ions and the mechanical torque on the motor. This relationship between "forces" and "fluxes" turns out to be linear, and given by a symmetric positive definite matrix, as anticipated by non-equilibrium thermodynamics, although we do not make any use of that formalism in deriving the result. The equal off-diagonal terms of this 2x2 matrix describe the electromechanical coupling of the motor. Although macroscopic, the model can be used as a foundation for stochastic simulation via the Einstein relation.

The regulation and maintenance of an organ’s shape is a major outstanding question in developmental biology. The Drosophila wing imaginal disc serves as a powerful system for elucidating design principles of the shape formation in epithelial morphogenesis. Yet, even simple epithelial systems such as the wing disc are extremely complex. A tissue’s shape emerges from the integration of many biochemical and biophysical interactions between proteins, subcellular components, and cell-cell and cell-ECM interactions. How cellular mechanical properties affect tissue size and patterning of cell identities on the apical surface of the wing disc pouch has been intensively investigated. However, less effort has focused on studying the mechanisms governing the shape of the wing disc in the cross-section. Both the significance and difficulty of such studies are due in part to the need to consider the composite nature of the material consisting of multiple cell layers and cell-ECM interactions as well as the elongated shape of columnar cells. Results obtained using iterative approach combining multiscale computational modelling and quantitative experimental approach will be used in this talk to discuss direct and indirect roles of subcellular mechanical forces, nuclear positioning, and extracellular matrix in shaping the major axis of the wing pouch during the larval stage in fruit flies, which serves as a prototypical system for investigating epithelial morphogenesis. The research findings demonstrate that subcellular mechanical forces can effectively generate the curved tissue profile, while extracellular matrix is necessary for preserving the bent shape even in the absence of subcellular mechanical forces once the shape is generated. The developed integrated multiscale modelling environment can be readily extended to generate and test hypothesized novel mechanisms of developmental dynamics of other systems, including organoids that consist of several cellular and extracellular matrix layers.

Human neutrophils and other immune cells sense chemical gradients to navigate to sites of injury, infection, and inflammation in the body. Impressively, these cells can detect gradients that differ by as little as about 1% in concentration across the length of the cell. Abstract models suggest that they may do this by integrating opposing local positive and long-range negative signals generated by receptors. However, the molecular basis for signal processing remains unclear. To investigate models of sensing, we developed experimental tools to control receptors with light while measuring downstream signaling responses with spatial resolution in single cells. We are directly measuring responses to both local and cell-wide receptor activation to determine the wiring of signal processing. While we do not see evidence for long-range negative signals, we do see a subcellular context-dependence of signal transmission. We propose that signal transmission from receptors happens locally, but cell-wide polarity biases sensing to maintain persistent migration and achieve temporal averaging to promote directional accuracy.

Models become notably more complex when stochasticity is introduced. One of the best ways to add frustrating amounts of randomness to your model: incorporate humans. In this talk, I discuss three different ways in which humans have made things messy in my mathematical models, statistical models, and data science work. Despite the fact that humans do, indeed, make things messy, they also make our models so much more realistic, interesting, and intriguing. So while humans make things messy, it is so worth it to bring them into your work.

Modeling "social" interactions within a large population has proven to be a rich subject of study for a variety of scientific communities during the past few decades. Specifically, with the goal of predicting the macroscopic effects resulting from microscopic-scale endogenous as well as exogenous interactions, many emblematic models for the emergence of collective behaviors have been proposed. In this talk we present a dynamical model for generic crowds in which individual agents are aware of their local environment, i.e., neighboring agents and domain boundary features, and may seek static targets. Our model incorporates features common to many other "active matter'' models like collision avoidance, alignment among agents, and homing toward targets. However, it is novel in key respects: the model combines topological and metrical features in a natural manner based upon the local environment of the agent's Voronoi diagram. With only two parameters, it is shown to capture a wide range of collective behaviors that go beyond the more classical velocity consensus and group cohesion. The work presented here is joint with R. Choksi and J.C. Nave at McGill