Mykhailo Potomkin, UC Riverside

Theory of Confined Active Matter

The growing field of active matter studies the dynamics of systems comprised of many identical objects capable of autonomous motion. Examples include living systems such as bacterial colonies, ant swarms, animal flocks, and human crowds, to name a few. Confinements, which are ubiquitous in nature and experimental setups, can significantly influence the transport of living matter. Motile organisms inevitably bump into walls unless they use specific mechanisms to avoid them. This leads to a non-trivial coupling between the parts of an active matter system that accumulate at the walls and the parts moving in the bulk, that is, at some distance from the walls. The theory of confined active matter addresses particular questions such as: How do curvature and corners in confinement affect the distribution and dynamics of an active matter system? What is the most efficient way to computationally predict the transport of an active matter system in confined environments? These questions lead to intriguing mathematical problems involving differential geometry, differential equations, and stochastic processes.

"Boundary accumulations of active rods in microchannels with elliptical cross-section" (with C. Brown and S.D. Ryan), submitted (2024), [arXiv preprint] [supplementary videos]
"A kinetic approach to active rod dynamics in confined domains" (with L. Berlyand, P.-E. Jabin, and E. Ratajczyk), SIAM Multiscale Modeling and Simulation, Vol. 18, Issue 1, pp. 1-18 (2020) [arXiv preprint]
"Self-propulsion and shear flow align active particles in nozzles and channels" (with L. Dominguez Rubio, R. Baker, A. Sen, L. Berlyand, and I. Aranson), Advanced Intelligent Systems, Vol. 3, Issue 2, 2000178 (2020)
"Fight the Flow: The Role of Shear in Artificial Rheotaxis for Individual and Collective Motion" (with R. Baker, J. E. Kauffman, A. Laskar, O. Shklyaev, L. Dominguez-Rubio, H. Shum, Y. Cruz-Rivera, I. Aranson, A. C. Balazs, and A. Sen), Nanoscale, Vol. 11, pp. 10944-10951 (2019)
"Focusing of Active Particles in a Converging Flow" (with A. Kaiser, L. Berlyand, and I. Aranson), New Journal of Physics, Vol. 19, 115005 (2017) [arXiv preprint]

Cell migration

Biological cells are highly complex systems whose structures have developed over nearly 4 billion years. It is fascinating that cell organization, despite consisting of a myriad of intricate biological and biomechanical processes, allows cells to move persistently in a certain direction. The field of cell motility modeling seeks to elucidate how such persistent motility can emerge from basic cellular processes. Additionally, how do cells interact moving in pairs, clusters, and tissues? Although cells don’t make decisions or process information in the same way that humans do, they exhibit a variety of behavioral patterns when interacting with one another. Understanding the “social life” of cells through mathematical modeling enhances our knowledge of wound healing, cancer prevention and treatment, and other biomedical applications.

"Durotaxis and extracellular matrix degradation promote clustering of cancer cells" (with O. Kim, Y. Klymenko, M. Alber, I. Aranson), iScience, Vol. 23, Issue 3, p. 111883 (2025)
"Sharp Interface Limit in a Phase Field Model of Cell Motility" (with V. Rybalko and L. Berlyand), Network and Heterogeneous Media, Vol. 12, Number 4, pp. 551-590 (2017) [arXiv preprint]
"Phase-Field Model of Cell Motility: Traveling Waves and Sharp Interface Limit" (with V. Rybalko and L. Berlyand), Comptes Rendus Mathematique, Vol. 354, Issue 10, pp. 986-992 (2016) [arXiv preprint]

Further ongoing research

Physics of swimming at microscale

We model and analyze how swimming microorganisms exploit flexible appendages like flagella and cilia to navigate and propel themselves in a background flow.

"Dynamics and steady state of squirmer motion in liquid crystal" (with L. Berlyand, H. Chi, and N.K. Yip), European Journal of Applied Mathematics, pp. 1-42 (2023), [arXiv preprint]
"Self-sustained three-dimensional beating of a model eukaryotic flagellum" (with B. Rallabandi and Q. Wang), Soft Matter (2022)
"Surface anchoring controls orientation of a microswimmer in nematic liquid crystal" (with H. Chi, L. Zhang, L. Berlyand, and I. Aranson), Communications Physics, Vol. 3, 162 (2020)
"Flagella bending affects macroscopic properties of bacterial suspensions" (with M. Tournus, L. Berlyand, and I. Aranson), Journal of the Royal Society Interface, Vol. 14, Issue 130, 20161031 (2017) [arXiv preprint]

Nonlinear transport

We focus on discrete-to-continuous limits in models that implement basic rules of transport. This research began by examining how a large number of kinesin proteins move unidirectionally along microtubules in biological cells. This motion is reminiscent of vehicle traffic on a highway, though it includes some biologically specific aspects. Models in continuous limits may effectively predict certain characteristics, such as the tendency to form traffic jams.

"Motor protein transport along inhomogeneous microtubules" (with S. D. Ryan and Z. McCarthy), Bulletin of Mathematical Biology, Vol. 83, Issue 2, pp. 1-29 (2021) [arXiv preprint]

Many particle systems

Many models in physics, astronomy, and biology use an agent-based modeling approach that describes phenomena as systems of many identical point particles interacting with one another. These particles can be molecules, stars, or bacteria. Interactions inevitably lead to a rise in correlations between particles and the emergence of collective patterns. Evaluating these correlations can be a very challenging task. We seek computationally feasible methods to capture these correlations. Additionally, we are interested in characterizing the attractors of many-particle systems to determine whether their long-term behavior is regular or chaotic.

"Continuum approximations to systems of correlated interacting particles" (with L. Berlyand, R. Creese, and P.-E. Jabin), Journal of Statistical Physics, Vol. 174, Issue 4, pp. 808-829 (2018) [arXiv preprint]
"Complexity Reduction in Many Particle Systems with Random Initial Data" (with L. Berlyand and P.-E. Jabin), SIAM/ASA J. Uncertainty Quantification, Vol. 4, Issue 1, pp. 446-474 (2016) [arXiv preprint]
"Asymptotic Behavior of Nonlinear Transmission Plate Problem", Proceedings of IFIP Conference on System Modeling and Optimization, pp. 521-527 (2011)