Multiscale Structural Mechanics of Viruses:Stretching the Limits of Continuum Modeling

William S. Klug

Mechanical & Aerospace Engineering Department

Acknowledgements:

Melissa Gibbons, Lin Ma,Chuck Knobler, Robijn Bruinsma

UCLA

Gijs Wuite, Irena Ivanovska, Wouter RoosVrije Universiteit, Amsterdam

Christoph SchmidtGeorg-August Univerität, Göttingen

Virus Mechanics in Biology and Materials DesignEvilevitch, et al. (2003)

Packaged genomecreates forces/pressure

Protein interactions:assembly and stability

Virus-based synthetic materials(batteries, liquid crystals, solar

cells, fuel cells)

A. Belcher, MIT

Baker, et al. (2000)

Icosahedral Capsid Structure

CCMV

Speir, et al.

• Capsids self-assemble frommultiple copies of similaror identical proteinsubunits

http://en.wikipedia.org/wiki/Icosahedron

3

2 5

5

2 3

52 3

5

2 3

• Spherical capsids assemble with2-, 3-, & 5-fold symmetries of anicosahedron

The Caspar-Klug Construction

Nguyen, et al.

• Form a closed shell byaligning an icosahedrontemplate onto a hexahedralsheet of “capsomers”

• Triangulation number:

(3,1) T=7

Cowpea Chlorotic Mottle Virus (CCMV)

Speir, et al.(1995)

• T=3 Capsid assembles from 180 copies of the sameprotein subunit with 2-, 3-, & 5-fold symmetries

5

2 3

52 3

5

2 3 Johnson & Speir (1997)

• Complex “structural” phase diagram

pH = 5native CCMV

d = 28 nm

pH = 7.5swollen CCMVexpands by 10%

Pores 2 nm in size

Structural phase transition:pH-induced conformational change

Bancroft, Hills, Markham, Virology (1967)Liu, et al., J. Struc. Biol. (2003)

AFM: a probe for capsid mechanics

minute virus of mice (MVM)Carrasco, et al. (2006)

murine leukemia virus Kol, et al. (2006)

ϕ29Ivanovska, et al. (2004)

cowpea chlorotic mottle virus (CCMV)Michel, et al. (2006) (pH 5)Klug, et al. (2006) (pH 6)

Fundamental revelations:

Can sustain nanoNewton-sized forces

Deform elastically (reversibly)up to 5-70% of initial height

Large linear regime in elasticresponse

Excessive force usually leadsto failure/breakage

Properties affected by

• presence of genome,orientation, pH, proteinmutations, maturation

AFM nanoindentation of CCMV at varied pH

Observations:

• Capsids are linearly elastic even at ratherlarge deformation

• Stiff and brittle at pH 5,

• 3 times softer and perfectly elastic at pH 6

• No apparent difference in structure

Can we account for these features withmodeling and simulation?

pH 6

pH 5

~3.5 nm

~20 nm

Michel, et al., PNAS, (2006)Klug, et al., PRL, (2006)

J.-P. Michel, C. Knobler (UCLA)I. Ivanovska, G. Wuite, C. Schmidt, (Vrije Universiteit Amsterdam)

Loading

Unloading

Z Indentation

Z

Questions for theoryand simulation:

• Why is capsid force response linear for such largeindentations?

• Why do some capsids fail and others not? (Geometric?Constitutive?)

• How do local protein structure and conformationalchanges affect the global mechanical response of theshell?

Strategy: coarse-grained Modeling

• Systematically throw away as many DOF aspossible while keeping the essential physics

Push the limits of continuum modeling

Multi-scale simulation

Simple Continuum Model: Spherical ShellAfter all, aren’t capsids more spherical than cows?

The Finite Element Method (FEM)

Quarter capsid w/symmetry boundary

conditions

Rigid sphericalindentor

Rigid plate

• Discretize shape into mesh of simple polyhedralelements (tetrahedra, hexahedra,etc.)

• Approximate displacements locally on elementdomains by interpolation simple polynomialbasis functions

• Minimize energy with respect to nodal fieldvalues (Ritz Method)

Spherical CCMVSimulate indentation using different constitutive laws

• Parameterize Young’s modulus to fit experiment: E = 280 MPa• Indentation response insensitive to constitutive law.• Proper treatment of finite deformations and rotations is crucial!

Gibbons & Klug, PRE (2007)

• Hookean:

• Neo-Hookean and Mooney-Rivlin:(Nonlinear rubber elasticity)

Spherical CCMVVarying shell thickness

• Thicker shells show Hertzianstiffening nonlinearity

• Thinner shells show softeningnonlinearity

• Almost perfectly linear for nominalCCMV thickness t = 3 nm

Key Lessons:• Shell response insensitive to constitutive law.• Linearity of force-indentation curve explained with shell mechanics.• Geometry more important than constitutive behavior.

Gibbons & Klug, PRE (2007)

Questions for theoryand simulation:

• Why is capsid force response linear for such largeindentations?

• Why do some capsids fail and others not? (Geometric?Constitutive?)

• How do local protein structure and conformationalchanges affect the global mechanical response of theshell?

Buckling of elastic shells with disclinations• 2-D Föppl-von Kármán shell model

• Add/remove a slice from hexagonalsheet → stretching needed to keep itflat. Buckling may alleviate stretching.

• Stabilitiy of planar sheet controlled byFöppl-von Kármán number:

• Buckled disclinations implicated indetermining the “facetedness” ofcapsids

Seung & Nelson, Phys. Rev. A (1988)

2-D Young’smodulus

bendingmodulus

5-fold disclination

Lidmar, Mirny, & Nelson, Phys. Rev. E (2003)

Baker, et al. (2000)

Can AFM indentation trigger buckling?

pH-sensitivity of CCMV andchanging FvK number?

pH 6

pH 5

Klug, et al., Phys. Rev. Lett., 97, 228101 (2006)

• Explain change in stiffness as change inmaterial properties.

• Stretching modulus Y affects slope andstability

F! "Y#

R

pH = 5native CCMV

d = 28 nm

pH = 7.5swollen CCMVexpands by 10%

How does swelling transition affectmechanical response?

pH 6

pH 5

?

Note: capsids appear structurally identicalat pH 5 and 6, and differ only in mechanicalresponse.

Amplitude and direction of motion (a) structural data (b) normal mode 24

Tama and Brooks, J. Mol. Bio. (2002)

Normal-mode Analysis of pH-InducedSwelling of CCMV

Ginzburg-Landau theory of swelling transitionas a structural phase transition

Klug, et al., Phys. Rev. Lett. (2006)Guérin & Bruinsma, Phys. Rev. E (2007)

Order parameter: = amplitude of soft swelling mode

Free energy:

Linearizing with respect to order parameter:

• Increasing pH softens welling mode and reduces effectiveYoung’s Modulus Y* and (to lowest order) does not affect κ.

• pH 5 failure may be initiated by buckling (geometric failure).

Maturation of Bacteriophage HK97• Translations and rotations of the subunits (conformational

change) make the final stable mature capsid possible

Chainmail in HII

Different phases during HK97 maturation (Wikoff et. al. 2006) HK97

Cross section of PII, R~24 nm

Parameters for HK97

Simulated result

Experimental structures

Two equilibrium configurations.Both are stable.

Phase Transition Triggered byIndentation

Model Predictions:• Contraction transition not reversed upon unloading.• Considerable hysteresis.• Experiments in progress…

Jump from swollen tocontracted

Questions for theoryand simulation:

• Why is capsid force response linear for such largeindentations?

• Why do some capsids fail and others not? (Geometric?Constitutive?)

• How do local protein structure and conformationalchanges affect the global mechanical response of theshell?

CCMV (native and swollen)

HK97 (procapsid and mature) Hepatitis B

ϕ29

• Model capsid as homogeneous elastic shell (geometricheterogeneity only, for now)

• Obtain geometry from structural biology data: X-ray crystallography

→ all atom coordinates (RCSB Protein Data Bank) Cryo-electron microscopy

→ electron density maps (Electron Microscopy Data Bank)

• Triangulate molecular surfaces

• Build 3-D meshes of interior(tetrahedra fill in space between inner and outer surfaces)

Nonuniform Finite Element Models

Gibbons and Klug, Biophys J, 95(8), October 15, 2008.

Tetrahedral Finite-Element Meshes of CCMV

Native (pH 5) Swollen (pH 7)

• Meshes given same mass and constitutive properties(hyperelastic with E=215MPa)

• 10x fewer nodes than atoms

Indentation Simulations

• Same mass and constitutive properties (E=215MPa)• Local changes in geometry affect response:

Swollen roughly twice as soft as native Swollen more nonlinear than native

SwollenNative

ContactFroces (pN)

Softening from bending“arms” between capsomers

Contact formed withmultiple capsomers

Buckling event

• Model Predictions: Orientation-dependent nonlinearities 3-, 5-fold: softening from local

deformation mode (arm bending) 2-fold: Buckling at high indentation

Conclusions andUnanswered Questions

• Coarse-grained finite element modeling:→ local heterogenieties affect global mechanical response

Shell geometry key to global response 5-fold disclinations focus stress and may facilitate failure Single-protein conformational change Nonuniform capsid topography

• Test predictions via indentation experiments on pH 7 CCMVand HK97 (in progress with Wuite and Schmidt).

• 3-D model of pH 5 capsid doesn’t buckle - is failureactually consitutitve (e.g., fracture/bond-breaking)?

• How important is constitutive heterogeneity? (Need info from atomic interactions)

Constitutive Heterogeneity:FEM + rigidity percolation

(Ongoing work with M. Thorpe, ASU)

• Compute flexibility map withconstraint theory using FIRSTsoftware

• Assign local elastic moduli to FEmodel based on flexibility map

Constitutive Heterogeneity:Steering MD with FEM

(Ongoing work with P. Freddolino, A. Arkhipov, & K. Schulten, UIUC)

1. Steer atoms according to FE interpolation

2. MD relaxation

3. Project atomic forces onto FE nodes

4. Time or descent step with FE nodal DOF

Thanks for your attention