Mathematical Biology Seminar - Spring 2022

March 2

Daniel Cooney, University of Pennsylvania

Location: WebEx

A PDE Model to Study Natural Selection Across Multiple Levels of Organization

Natural selection in complex biological and social systems can simultaneously operate across multiple levels of organization, ranging from genes and cells to animal groups and complex human societies. Such scenarios typically present an evolutionary conflict between the incentive of individuals to cheat and the collective incentive to establish cooperation within a group. In this talk, we will explore this conflict by considering a game-theoretic model of multilevel selection in a group-structured population featuring two types of individuals, cooperators and defectors. Assuming that individuals compete based on their payoff and groups also compete based on the average payoff of group members, we consider how the distribution of cooperators within groups changes over time depending on the relative strength of within-group and between-group competition. In the limit of infinitely many groups and of infinite group size, we can describe the state of the population through the probability density of the fraction of cooperators within groups, and derive a hyperbolic partial differential equation for the changing composition of the population. We show that there exists a threshold relative selection strength such that defectors will take over the population for sufficiently weak between-group competition, while cooperation persists in the long-time population when the strength of between-group competition exceeds the threshold. Surprisingly, when groups are best off with an intermediate level of within-group cooperation, individual-level selection casts a long shadow on the dynamics of multilevel selection: no level of between-group competition can erase the effects of the individual incentive to defect. This is joint work with Yoichiro Mori.

March 9

Dobri Dotov, McMaster University

Location: WebEx

From HKB to cross-frequency coupling of neural dynamics: Trying to explain complex patterns of sensorimotor coordination

Repetitive sensorimotor rhythms afford planning and coordination of processes within and between individuals. The classical HKB model has been successful in explaining in-phase and anti-phase coordination in a wide variety of scenarios. Yet, there have been limited attempts to extend the framework to patterns such as tapping at 90 degrees phase, despite the ubiquity of such patterns in human motor and musical behaviors. We made progress by converting the problem of phase-locked coordination between same-frequency oscillators to one of cross-frequency coupling. We adapted the Farey tree levels as a metric of complexity by expressing tapping phase as the ratio of a unit cycle. To test this idea, we conducted a bimanual tapping study. Each trial, participants had to tap at the same overall frequency with both hands but maintain a given target phases between the two hands. We sampled target phases that could be expressed as ratios of a unit cycle and thus could be mapped to levels of the Farey tree. Our proposed metric of complexity was found to be associated with an increase of tapping variability. Furthermore, we observed a range of hidden coordination modes in the sense that they were unstable yet statistically measurable. We also found that a rank-frequency scaling law fits the distribution of these modes. The phase-attractive circle map has wider basins of attractions for such ratios and suggests an interpretation of the present results in terms of hierarchical cross-frequency coupling. We also consider the tendency for small-integer attractors in the single-hand repeated tapping of three-interval rhythms reported in the literature. The model-based interpretation of our behavioral data motivates the question whether cross-frequency coupling of neural dynamics serves as a low-level prior for timing and coordination and thus enables phenomena as diverse as attractor states in bimanual coordination and the simple interval ratios often found in musical rhythms.

March 30

Gokberk Kabacaoglu, Department of Mechanical Engineering, Bilkent University, Ankara, Turkey

Location: WebEx

Neural Network Based Reduced Model for Stokesian Particulate Flows

Stokesian particulate flows describe the hydrodynamics of rigid or deformable particles in the zero Reynolds number regime. Due to highly nonlinear fluid-structure interaction dynamics, moving interfaces, and multiple scales, numerical simulations of such flows are challenging and expensive. I will present our machine-learning-augmented reduced model[1] for fast simulations of such flows. Besides, I will show how the reduced model enables us study optimal microfluidic device design for dense suspensions of deformable particles.

Our goal is to design a deterministic lateral displacement (DLD) device to sort same-size biological cells by their deformability, in particular to sort red blood cells (RBCs) by their viscosity contrast between the fluid in the interior and the exterior of the cells. A DLD device optimized for efficient cell sorting enables rapid medical diagnoses of several diseases such as malaria since infected cells are stiffer than their healthy counterparts. In this context, I will first describe an integral equation formulation[2] that delivers optimal complexity solvers for this type of problems. Despite its excellent theoretical properties, our integral equation solver remains prohibitively expensive for optimization and uncertainty quantification. I will then summarize our efforts to reduce the computational costs, starting from low-resolution discretization, domain truncation, and model reduction.

Model reduction is used to accelerate the action of specific and very expensive nonlinear operators. The final scheme blends ultra low-resolution solvers (who on their own cannot resolve the flow), several regression neural networks, and an operator time-stepping scheme, which we introduced to specifically enable the use of surrogate models. We have used our methodology successfully for flows that are completely different from the flows in the training dataset.

This is a joint work with George Biros at the University of Texas at Austin.

Gokberk joined Mechanical Engineering Department at Bilkent University, Turkey in January 2022. From 2019 to 2021, he was a postdoctoral researcher at the Flatiron Institute, an internal research division of the Simons Foundation in New York. There, he was a member of the Centers for Computational Biology and Computational Mathematics led by Mike Shelley and Leslie Greengard, respectively. He received his bachelor’s degree in Mechanical Engineering from Bilkent University (2014) and his doctoral degree in Mechanical Engineering from the University of Texas at Austin (2019) under the supervision of George Biros. His research efforts focus on computational biological fluid dynamics and fast numerical methods for fluid-structure interaction problems.

April 13

Hao Lin, Rutgers University New Brunswick

Location: WebEx

Universal statistics and mechanical properties in planar tessellated networks

From solar supergranulation to salt flats in Bolivia, from veins on leaves to cells on Drosophila wing disks, polygon-based networks exhibit great complexities, yet similarities and consistent patterns emerge. In this talk I will present two vignettes that explores the universal statistics and features in these systems. In the first, I will demonstrate the ubiquity of a Boltzmann-like, exponential distribution in the squared norm of the deformation tensor. This distribution directly leads to gamma distributions in the polygon aspect ratio, as recently reported by Atia et al. [Nat. Phys. 14, 613 (2018)]. Further I will explain the origin of such distribution in relation to the central limit theorem. The Boltzmann-like feature allows the consistent definition of a pseudo-temperature that promises utility in a thermodynamic ensemble framework. In the second I show that for a mechanically-equilibrated network in the rigid regime, the microscopic, cell-wise stress and strain follows a simple relation, which arises as the natural consequence of energy minimization. The results indicate that the microscopic stress and strain are aligned in the principal directions, and further, all microscopic deformations are affine. The analysis directly leads to a simple analytical expression for the macroscopic shear modulus which unifies all prior results.

Professor Hao Lin is currently a professor in the Mechanical and Aerospace Engineering Department at Rutgers University, where he is also the Undergraduate Director for the Packaging Engineering Program.  He received his BS degree in Mechanics from Peking University in 1996, and PhD in Mechanical Engineering from University of California, Berkeley in 2001.  He was a postdoctoral fellow at Stanford University from 2001 to 2005. He joined the faculty of Mechanical and Aerospace Engineering at Rutgers University in 2005.  He is a recipient of the NSF CAREER Award (2008) and the Presidential Early Career Award for Scientists and Engineers (PECASE, 2010). His research interests span electrohydrodynamics and transport on the micro- and nano-scales, engineered functional surfaces, electroporation-mediated molecular delivery, the mechanics of soft material such as vesicles, cells, and cell aggregates, and most recently, delivery platforms for nucleic acid-based medicine such as COVID vaccines and therapeutics. He is also interested in research related to AI-driven automation of packaging engineering processes in his capacity as the director of the very unique Packaging Engineering Program at Rutgers.

April 27

Joel Tabak, Exeter University

Location: WebEx

Flexible coordination between neuronal rhythms by the duration of synaptic influence

The organisation of neuronal circuits into oscillatory components enables a wide range of neuronal activity patterns, by tuning the coordination between different oscillators. Switching from trot to gallop, for example, can be done by adjusting the coordination between neuronal oscillators that control different limb muscles. How this tuning is done in general remains to be understood. Here, we demonstrate that changing the duration of the directional synaptic signals that link two neuronal oscillators is an effective way to control their coordination. We show that when the synaptic signals are short, the driven oscillator is phase-advanced relative to the driver oscillator. As the synaptic signals lengthen, the phase-advance progressively decreases and can turn into a phase-delay. This is true for all types of neuronal oscillators, provided there is a clear separation between the time scales of the positive and negative feedback mechanisms underlying the oscillations. To illustrate this finding, we show that changing the duration of the synaptic influence can reverse the direction of propagation in a chain of identical oscillators coupled directionally.