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Mathematical analysis of a mesoscale model for multiphase membranes

Jakob Fuchs, Matthias Röger

Abstract

In this paper we introduce a mesoscale continuum model for membranes made of two different types of amphiphilic lipids. The model extends work by Peletier and the second author [Arch. Ration. Mech. Anal. 193, 2009] for the one-phase case. We present a mathematical analysis of the asymptotic reduction to the macroscale when a key length parameter becomes arbitrarily small. We identify two main contributions in the energy: one that can be connected to bending of the overall structure and a second that describes the cost of the internal phase separations. We prove the $Γ$-convergence towards a perimeter functional for the phase separation energy and construct, in two dimensions, recovery sequences for the convergence of the full energy towards a 2D reduction of the Jülicher-Lipowsky bending energy with a line tension contribution for phase separated hypersurfaces.

Mathematical analysis of a mesoscale model for multiphase membranes

Abstract

In this paper we introduce a mesoscale continuum model for membranes made of two different types of amphiphilic lipids. The model extends work by Peletier and the second author [Arch. Ration. Mech. Anal. 193, 2009] for the one-phase case. We present a mathematical analysis of the asymptotic reduction to the macroscale when a key length parameter becomes arbitrarily small. We identify two main contributions in the energy: one that can be connected to bending of the overall structure and a second that describes the cost of the internal phase separations. We prove the -convergence towards a perimeter functional for the phase separation energy and construct, in two dimensions, recovery sequences for the convergence of the full energy towards a 2D reduction of the Jülicher-Lipowsky bending energy with a line tension contribution for phase separated hypersurfaces.
Paper Structure (5 sections, 3 theorems, 34 equations, 1 figure)

This paper contains 5 sections, 3 theorems, 34 equations, 1 figure.

Table of Contents

  1. Introduction
  2. A mesoscale model for two-phase membranes
  3. Gamma-convergence of a mesoscale phase separation energy
  4. Single curve two-phase energy: recovery sequence construction
  5. Discussion

Key Result

Lemma 2.6

Let $Z=(\gamma,\theta,\chi,M)\in\mathcal{Z}_\varepsilon$ as in Definition def:ConfSingleCurve and $M_1(Z), M_2(Z)$ as defined in eq:DefMj be given. We set $a_\varepsilon(0)=\lambda_\varepsilon$ and $a_\varepsilon(1)=1$. Then for $\tilde{F}_\varepsilon$ as in eq:EnSingleCurve the lower estimate holds with

Figures (1)

  • Figure 1: The total width being equal on either side, switching type does not increase the energy.

Theorems & Definitions (13)

  • Remark 2.2
  • Definition 2.3
  • Definition 2.4: Multiphase energy for a single curve
  • Remark 2.5: Additional energy term
  • Lemma 2.6
  • proof
  • Definition 2.7: Mesoscale model
  • Definition 3.1: Phase separation energies
  • Theorem 3.2: $\Gamma$-convergence
  • proof
  • ...and 3 more