Exact Calculations of Membrane Areas with Simple Models

TL;DR

This paper provides exact calculations of membrane areas using simple models, highlighting the impact of constraints on total and projected areas, with implications for membrane simulations.

Contribution

It introduces exact methods for calculating membrane areas with constraints using self-avoiding walk models and generating functions.

Findings

Total area and projected area differ significantly under constraints.

Exact formulas relate membrane areas to activities q and r.

High temperature divergences are suppressed in the model.

Abstract

The distinction between the true total area and the projected area is elucidated with soluble models which represent the membrane as a self-avoiding string on a plane. Constraining the total area to a predetermined value changes the averages very significantly. The latter are calculated exactly from the generating functions of self-avoiding walks and are shown as functions of activities $q$ and $r$ related to temperature $T=\pm 1/\log (q)$ and lateral force $f=-\log (r)$. The constraint makes the partition functions and averages valid for all $q,r >0$ and reduces the ratio of $A_{tot}$ to the projected area $L$. High temperature divergences are supressed. Possible applications to simulated bilayers/membranes are discussed.