Tag - Mathematical biology

I will discuss some questions of interest in neuroscience, seen through the lens of mathematics. No prior knowledge of neuroscience is needed for this talk. Two of the most basic visual capabilities of primates are orientation selectivity, i.e., the ability of neurons to discern orientations of edges, and direction selectivity (DS), their ability to detect the direction of motion. Both properties are enjoyed by neurons in the primary visual cortex (V1), where visual information enters the cerebral cortex, but not by cells that provide input to V1. I wanted to understand the biological origins of these visual functions, especially that of DS, which had remained a mystery for half a century. Building on well known results of Hubel and Wiesel, I will propose ideas that are grounded in biology, tested in large-scale dynamical models of the visual cortex, and shown to be consistent with experimental data. Most of all I would like to share the little bits of mathematical insight that contributed to these results in visual neuroscience.

We extend Parareal, a parallel-in-time method, to simulate the fluid flow around bio-inspired, dynamic structures over a period of time. Examples of these structures include microscopic, slender, and elastic rods modeled to imitate bacterial flagella, cilia or tails of sperm. After they have been discretized, their dynamics can be described by a system of nonlinear Ordinary Differential Equations (ODEs); and when applying an ODE solver to it, we need to invoke a solver for fluid-structure interactions, such as the Method of Regularized Stokeslets (MRS), at every time step to calculate the flow field.

Parareal aims to accelerate the solution of ODEs by assigning different 'slices' of the time domain to different processors. It is an iterative method that alternates between a 'serial sweep' and a 'parallel sweep': in the serial sweep, the ODEs over the entire time domain are solved in serial using a coarse solver; and in the parallel sweep that follows, a more accurate but slower fine solver is applied to solve the ODEs restricted to the time slices concurrently.

Our main contributions include demonstrating the applicability of Parareal to the simulation of
biofluids and developing novel coarse solvers for the serial sweep. We propose to construct novel
non-intrusive coarse solvers by extrapolating a parametrized family of existing solvers. Compared to the existing solvers, they either allow the use of larger time steps, have a higher order of accuracy in time, or both. They are also straightforward to implement and parallelize. Our numerical experiments show that when the number of biological structures is small or when the number of computer cores employed is sufficiently large, the Parareal equipped with the proposed coarse solver can achieve a significantly higher parallel speedup than the more commonly used spatial parallelization.

Plant morphology emerges from cellular growth. The turgor-driven diffuse growth of a cell can be highly anisotropic: significant in the longitudinal direction and negligible in the radial direction. This anisotropy arises from cellulose microfibrils (CMF) reinforcing the cell wall. To maintain the cell's integrity during growth, new material including CMF must be continually deposited into the cell wall. In this talk, I will present a mathematical model which describes the cell as a pressurised cylindrical vessel and the cell wall as a fibre-reinforced viscous sheet, explicitly including the mechano-sensitive angle of CMF deposition. The model incorporates interactions between turgor, external forces, CMF reorientation during wall extension, and matrix stiffening. I will explain the general formulation of the model and summarise the technical steps towards obtaining evolution equations for the cell's length and twist. A generalised Lockhart equation will be derived. I will discuss how the handedness of twisting cell growth depends on external torque and intrinsic wall properties, and interpret numerical results in light of recent experimental findings. Overall, the model provides a meaningful step towards a unified mechanical framework for understanding left- and right-handed growth as seen in many plants.

Filaments (slender, microscopic elastic bodies) are prevalent in biological and industrial settings. In the biological case, the filaments are often active, in that they are driven internally by motor proteins, with the prime examples being cilia and flagella. For cilia in particular, which can appear in dense arrays, their resulting motions are coupled through the surrounding fluid, as well as through surfaces to which they are attached. In this talk, I present numerical simulations exploring the coordinated motion of active filaments and how it depends on the driving force, density of filaments, as well as the attached surface. In particular, we find that when the surface is spherical, its topology introduces local defects in coordinated motion which can then feedback and alter the global state. This is particularly true when the surface is not held fixed and is free to move in the surrounding fluid. These simulations take advantage of a computational framework we developed for fully 3D filament motion that combines unit quaternions, implicit geometric time integration, quasi-Newton methods, and fast, matrix-free methods for hydrodynamic interactions and it will also be presented.

Many microorganisms and cells function in complex (non-Newtonian) fluids, which are mixtures of different materials and exhibit both viscous and elastic stresses. For example, mammalian sperm swim through cervical mucus on their journey through the female reproductive tract, and they must penetrate the viscoelastic gel outside the ovum to fertilize. In micro-scale swimming the dynamics emerge from the coupled interactions between the complex rheology of the surrounding media and the passive and active body dynamics of the swimmer. We use computational models of swimmers in viscoelastic fluids to investigate and provide mechanistic explanations for emergent swimming behaviors. I will discuss how flexible filaments (such as flagella) can store energy from a viscoelastic fluid to gain stroke boosts due to fluid elasticity. I will also describe 3D simulations of model organisms such as C. Reinhardtii and mammalian sperm, where we use experimentally measured stroke data to separate naturally coupled stroke and fluid effects. We explore why strokes that are adapted to Newtonian fluid environments might not do well in viscoelastic environments.

Microorganisms can swim in a variety of environments, interacting with chemicals and other proteins in the fluid. In this talk, we will highlight recent computational methods and results for swimming efficiency and hydrodynamic interactions of swimmers in different fluid environments. Sperm are modeled via a centerline representation where forces are solved for using elastic rod theory. The method of regularized Stokeslets is used to solve the fluid-structure interaction where emergent swimming speeds can be compared to asymptotic analysis. In the case of fluids with extra proteins or cells that may act as friction, swimming speeds may be enhanced, and attraction may not occur. We will also highlight how parameter estimation techniques can be utilized to infer fluid and/or swimmer properties.