Physical Constraints on Biological Form and Function: Stochastic processes, hydrodynamics, electrokinetics

Biological systems exhibit remarkable diversity of form, yet similar geometries and transport behaviours recur across organisms that differ in scale, lineage, material composition, and function. Such regularities are often interpreted as products of evolutionary adaptation. This thesis affirms the complementary view that biological form and function are constrained, selected, and sometimes generated by physical processes that operate independently of evolutionary intent.

The central claim is that natural morphologies and biological transport processes occupy a restricted subset of all mathematically possible configurations. The constraints considered here arise from stochastic mechanical wear at the macroscale, viscous hydrodynamics at the microscale, and electrokinetic transport at the nanoscale. Each constraint is studied using a combination of simplified experimental analogues, theoretical modelling, and comparison with available observations.

Chapter 1 investigates pointed biological structures, including stingers, spines, teeth, and tusks. Although these structures vary widely in size, their tips commonly exhibit paraboloidal profiles. Rather than attributing this geometry solely to piercing performance or buckling resistance, the chapter proposes stochastic mechanical wear as an alternative route to universality. Pencil tips are used as experimental simulacra, showing that random abrasive interactions drive initially distinct shapes towards the common paraboloidal form. This process is described by mean curvature flow, for which the parabolic tip emerges as a geometric attractor. The experimental model is extended to space-filling spirals, where growth-rate constraints and self-contact restrict the family of realisable spiral geometries.

Chapter 2 examines contact at the microscale. A particle approaching a surface through a viscous fluid must drain the intervening liquid film, which gives rise to a singular hydrodynamic resistance as the gap tends to zero. Under constant forcing, true contact is therefore reached only asymptotically. To achieve finite-time contact, spatially varying forces are studied as a viable alternative strategy. Experiments with ferromagnetic spheres attracted towards a permanent magnet isolate the role of such forces. The results and accompanying theory show that induced dipolar forces accelerate approach but cannot remove the final lubrication singularity, whereas dispersion-like forces that diverge with decreasing gap can drive finite-time contact. These findings suggest that hydrodynamics could restrict the overall rate of cellular processes such as statolith sedimentation, vesicle deposition, and microbial particle capture. Particle-surface contact at cellular scales may therefore require short-range colloidal, electrokinetic, or dispersion interactions.

Chapter 3 studies intercellular signalling in plants through plasmodesmata, nanometric channels that connect neighbouring cells across rigid cell walls. Plasmodesmata contain an axial desmotubule and tether-like proteins that partially occlude the transport pathway. Although these structures hinder diffusion geometrically, their charged surfaces can enhance ionic transport under nanometric confinement. A simplified unit-cell model and reduced Poisson–Nernst–Planck description show that the competition between geometric constriction and electrokinetic enhancement can produce an optimal range of tether spacing consistent with reported biological dimensions. This suggests that plasmodesmal architecture may be constrained by ionic signal regulation in addition to structural requirements.

Together, these studies present biological morphology and function as inevitable outcomes constrained by physical phenomena that are inescapable at the corresponding spatial scales. Stochastic mechanical interactions shape pointed tips alike, growth and space constraints restrict observable spiral geometries, hydrodynamic resistance may delay an organism’s response to stimuli, and electrokinetic confinement can turn apparent obstructions into transport regulators.