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Linear Elastic Model for Generating Wavy Structure in Lipid Membrane

by Paritosh Mahata

Linear Elastic Model for Generating Wavy Structure in Lipid Membrane by Peripheral Proteins

by Paritosh Mahata PRAVARTANA - 2016 Indian Institute of Technology Kanpur Kanpur 208016, India

Linear Elastic Model for Generating Wavy Structure in Lipid Membrane

by Paritosh Mahata

Outline

Introduction

• Lipid Molecule and Cell Membrane
• Mechanisms of Membrane Curvature Generation
• Qualitative Essence of Wavy Structure

Lipid Membrane as Linear Elastic Material

• Assumptions

Computational Model

• Model Parameters

Computations Results

• For Single Monolayer
• For Lipid Bilayer

Conclusions

Linear Elastic Model for Generating Wavy Structure in Lipid Membrane

by Paritosh Mahata

Introduction

Binding of proteins, cholesterol, and other cellular components make the membrane non-homogeneous in structure Peripheral proteins can sense and generate membrane curvature to maintain biological processes like Endocytosis and Exocytosis Malfunctioning of binding interplay between protein and membrane produces several diseases like cancer, Parkinsons, etc

1

http://commons.wikimedia.org/wiki/File: 0312 Animal Cell and Components.jpg

2

www.wcc.hawaii.edu/facstaff/.../005%20Biological%20Molecules.ppt

Lipid Molecule 1 Cell Membrane 2

Linear Elastic Model for Generating Wavy Structure in Lipid Membrane

by Paritosh Mahata

Introduction

Mechanisms of Membrane Curvature Generation

Ref: http://www.endocytosis.org/research/structur/structure.html Ref: T. Baumgart et al. Annu. Rev. Phys. Chem, 62,483 (2011)

Linear Elastic Model for Generating Wavy Structure in Lipid Membrane

by Paritosh Mahata

Introduction

Qualitative Essence of Wavy Structure

Ref: Kabaso et al. Annu. Rev. Phys. Chem 62, 483 (2011) Ref: Arkhipov et al. Biophys. J. 95, 2806 (2008)

Linear Elastic Model for Generating Wavy Structure in Lipid Membrane

by Paritosh Mahata

Lipid Membrane as Linear Elastic Material

Hooke’s Law 3: σij = Cijklǫkl ǫkl = 1 2 ∂uk ∂xl + ∂ul ∂xk

• Equilibrium Equations:

σij,j + fi = 0 Elastic moduli tensor C has total 21 independent constants Assumptions 4:

• Membrane is isotropic in lateral direction and solid in transverse direction
• Plane strain deformation of the membrane in the plane of cross-section
• Two volume stretching-compression moduli are equal

3

• W. S. Slaughter (2002). The linearized theory of elasticity, First Edition

4

Campelo et al. Biophys. J. 95, 2325 (2008)

Linear Elastic Model for Generating Wavy Structure in Lipid Membrane

by Paritosh Mahata

Lipid Membrane as Linear Elastic Material

Independent elastic moduli constants of the membrane material reduce into three - Cxxxx, Cxxyy, and Cxyxy Displacement Equations: Cxxxx ∂2ux ∂x2 + Cxyxy ∂2ux ∂y2 + (Cxxyy + Cxyxy) ∂2uy ∂x∂y = 0 Cxyxy ∂2uy ∂x2 + Cyyyy ∂2uy ∂y2 + (Cxxyy + Cxyxy) ∂2ux ∂x∂y = 0 Displacement equations are solved using Finite Element Method in ABAQUS

Linear Elastic Model for Generating Wavy Structure in Lipid Membrane

by Paritosh Mahata

Computational Model

L

H f = qE

x y p(x) (0,0) Rigid Inclusion

yc

H 3

r

Head Tail

dp

Boundary Conditions: σyy(x, 0) = σxy(x, 0) = 0 (Lower surface) σyy(x, H) = p(x), σxy(x, H) = 0 (Upper surface) ux(L, y) = 0, and σxy(L, y) = 0 (Right surface) ux(L, y) = 0, for y < yc − r =

• r2 − (y − yc)2

for y ≥ yc − r (Left surface) and σxy = 0 To restrict rigid motion: uy(0, yc − r) = 0

Linear Elastic Model for Generating Wavy Structure in Lipid Membrane

by Paritosh Mahata

Model Parameters

H = 2 nm, r = 0.5 nm, and dp = 0.8 nm L is obtained from the binding data of peripheral proteins Elastic Constants 5:

• For Head

Ch

xxxx = 4 × 109 N/m2, Ch xxyy = 3.93 × 109 N/m2,

and Ch

xyxy = 1.5 × 107 N/m2

• For Tail

Ct

xxxx = 1 × 109 N/m2, Ct xxyy = 0.98 × 109 N/m2,

and Ct

xyxy = 1.5 × 107 N/m2 5

Campelo et al. Biophys. J. 95, 2325(2008)

Linear Elastic Model for Generating Wavy Structure in Lipid Membrane

by Paritosh Mahata

Model Parameters

Electrostatic Force: p(x) = f x L (Linear) p(x) = f x L (Convex Parabolic) and p(x) = f x2 L2 (Concave Parabolic) where f = electrostatic force = qE

Ref: Kabaso et al. Annu. Rev. Phys. Chem 62,483(2011).

Linear Elastic Model for Generating Wavy Structure in Lipid Membrane

by Paritosh Mahata

Computations

A B C D O R L

dc

Deformed Upper Surface of Membrane

La

Φ

For different variations of p(x), charge intensity q is computed to get dc for corresponding R Binding data of peripheral proteins are used to get L and R

Linear Elastic Model for Generating Wavy Structure in Lipid Membrane

by Paritosh Mahata

Results

For Monolayer (Binding of Amphiphysin N-BAR):

5 10 15 20 25 30 35 0.03 0.06 0.09 0.12 R (nm) q (C/m2)

(a)

5 10 15 20 25 30 35 0.02 0.04 0.06 0.08 R (nm) q (C/m2)

(b)

5 10 15 20 25 30 35 0.03 0.06 0.09 0.12 R (nm) q (C/m2)

(c)

Hydrophobic and Electrostatic Only Electrostatic Interaction p(x) = f x

L

p(x) = f x

L

p(x) = f x2

L2

Linear Elastic Model for Generating Wavy Structure in Lipid Membrane

by Paritosh Mahata

Results

For Lipid Bilayer (Binding of Amphiphysin N-BAR):

10 15 20 25 30 35 0.01 0.02 0.03 0.04 0.05 0.06 0.07 R (nm) q (C/m2) Monolayer Bilayer p(x) = f x2

L2

Linear Elastic Model for Generating Wavy Structure in Lipid Membrane

by Paritosh Mahata

Results

For Lipid Bilayer (Different depth of penetration of inclusion):

0.2 0.4 0.6 0.8 1 0.02 0.04 0.06 0.08 dp (nm) q (C/m2) 0.2 0.4 0.6 0.8 1 0.01 0.02 0.03 0.04 dp (nm) q (C/m2) 0.2 0.4 0.6 0.8 1 0.01 0.015 0.02 0.025 dp (nm) q (C/m2) 0.2 0.4 0.6 0.8 1 0.005 0.01 0.015 0.02 dp (nm) q (C/m2) Linear Convex Parabolic Concave Parabolic R = 15 nm R = 20 nm R = 25 nm R = 30 nm

Linear Elastic Model for Generating Wavy Structure in Lipid Membrane

by Paritosh Mahata

Results

For Lipid Bilayer (Only Hydrophobic Interaction):

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 −0.7 −0.65 −0.6 −0.55 −0.5 −0.45 −0.4 −0.35 −0.3 −0.25 −0.2 −0.2 dp (nm) dc (nm) L = 7.84 nm L = 8.02 nm L = 8.11 nm L = 8.15 nm Only Hydrophobic Interaction

Linear Elastic Model for Generating Wavy Structure in Lipid Membrane

by Paritosh Mahata

Conclusions

Electrostatic interaction between protein and membrane plays an important role in membrane curvature generation Insertion of amphipathic helices of the BAR protein reduces the positive deformations of the membrane towards binding proteins In comparison to the single monolayer, higher charge intensities are required to bend lipid bilayer membrane Shallow insertion of amphipathic helices produces negative curvature to the membrane

Linear Elastic Model for Generating Wavy Structure in Lipid Membrane

by Paritosh Mahata

Acknowledgment

• Dr. Sovan Lal Das

Associate Professor, Department of Mechanical Engineering, IIT Kanpur

Linear Elastic Model for Generating Wavy Structure in Lipid Membrane

by Paritosh Mahata

THANK YOU