Closed, Two Dimensional Surface Dynamics

David V Svintradze

doi:10.3389/fphy.2018.00136

Abstract

We present dynamic equations for two dimensional closed surfaces and analytically solve it for some simplified cases. We derive final equations for surface normal motions by two different ways. The solution of the equations of motions in normal direction indicates that any closed, two dimensional, homogeneous surface with time invariable surface energy density adopts constant mean curvature shape when it comes in equilibrium with environment. In addition, we show that the shape equation is an approximate solution to our equation of motion in the normal direction and is valid for stationary or near to stationary shapes. As an example, we apply the formalism to analyze equilibrium shapes of micelles and explain why they adopt spherical, lamellar and cylindrical shapes. Theoretical calculation for micellar optimal radius is in good agreement with all atom simulations and experiments.

Highlights

Biological systems exhibit a variety of morphologies and experience large shape deformations during a motion
We derive generic equations of motions for closed two-dimensional surfaces and without any a priori symmetric assumptions, we show that constant mean curvature shapes are equilibrium solutions
Our equations of motions (20–25) are generic and exact. It advances our understanding of fluid dynamics because generalizes ideal magneto-hydrodynamic and NaiverStokes equations [5] and in contrast to Navier-Stokes, as we demonstrate in this paper, are trivially solvable for equilibrium shapes

Summary

Introduction

Biological systems exhibit a variety of morphologies and experience large shape deformations during a motion. Such “choreography” of shape motility is characteristic for all living organisms and cells [1] and for proteins, nucleic acids, and to all biomacromolecules in general. Shape motility, which is a motion of two-dimensional surfaces, may be a result of active (by consuming energy) or passive (without consuming energy) processes. The time scale for shape dynamics may vary from slow (nanometer per nanoseconds) to very fast (nanometer per femtosecond) [2, 3]. We derived fully generic equations of motions for three manifolds [5], but purposefully omitted lengthy discussion about motion of two-dimensional surfaces, which is a topic for this paper

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