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

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1 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 Roos Vrije Universiteit, Amsterdam Christoph Schmidt Georg-August Univerität, Göttingen

2 Virus Mechanics in Biology and Materials Design Evilevitch, et al. (2003) Packaged genome creates forces/pressure A. Belcher, MIT Protein interactions: assembly and stability Virus-based synthetic materials (batteries, liquid crystals, solar cells, fuel cells)

3 Baker, et al. (2000)

4 Icosahedral Capsid Structure Capsids self-assemble from multiple copies of similar or identical protein subunits Spherical capsids assemble with 2-, 3-, & 5-fold symmetries of an icosahedron CCMV Speir, et al.

5 The Caspar-Klug Construction Form a closed shell by aligning an icosahedron template onto a hexahedral sheet of capsomers Triangulation number: Nguyen, et al. (3,1) T=7

6 Cowpea Chlorotic Mottle Virus (CCMV) T=3 Capsid assembles from 180 copies of the same protein subunit with 2-, 3-, & 5-fold symmetries Complex structural phase diagram Speir, et al.(1995) 2 3 Johnson & Speir (1997) 5

7 Structural phase transition: ph-induced conformational change ph = 5 native CCMV d = 28 nm ph = 7.5 swollen CCMV expands by 10% Pores 2 nm in size Bancroft, Hills, Markham, Virology (1967) Liu, et al., J. Struc. Biol. (2003)

8 AFM: a probe for capsid mechanics ϕ29 Ivanovska, et al. (2004) Fundamental revelations: Can sustain nanonewtonsized forces Deform elastically (reversibly) up to 5-70% of initial height cowpea chlorotic mottle virus (CCMV) Michel, et al. (2006) (ph 5) Klug, et al. (2006) (ph 6) Large linear regime in elastic response Excessive force usually leads to failure/breakage Properties affected by presence of genome, orientation, ph, protein mutations, maturation murine leukemia virus Kol, et al. (2006) minute virus of mice (MVM) Carrasco, et al. (2006)

9 AFM nanoindentation of CCMV at varied ph J.-P. Michel, C. Knobler (UCLA) I. Ivanovska, G. Wuite, C. Schmidt, (Vrije Universiteit Amsterdam) Z Indentation ~20 nm Observations: ph 6 Capsids are linearly elastic even at rather large deformation Stiff and brittle at ph 5, 3 times softer and perfectly elastic at ph 6 No apparent difference in structure ph 5 Z Loading Can we account for these features with modeling and simulation? ~3.5 nm Unloading Michel, et al., PNAS, (2006) Klug, et al., PRL, (2006)

10 Questions for theory and simulation: Why is capsid force response linear for such large indentations? Why do some capsids fail and others not? (Geometric? Constitutive?) How do local protein structure and conformational changes affect the global mechanical response of the shell?

11 Strategy: coarse-grained Modeling Systematically throw away as many DOF as possible while keeping the essential physics Push the limits of continuum modeling Multi-scale simulation

12 Simple Continuum Model: Spherical Shell After all, aren t capsids more spherical than cows?

13 The Finite Element Method (FEM) Discretize shape into mesh of simple polyhedral elements (tetrahedra, hexahedra,etc.) Approximate displacements locally on element domains by interpolation simple polynomial basis functions Rigid spherical indentor Minimize energy with respect to nodal field values (Ritz Method) Quarter capsid w/ symmetry boundary conditions Rigid plate

14 Spherical CCMV Simulate indentation using different constitutive laws Hookean: Neo-Hookean and Mooney-Rivlin: (Nonlinear rubber elasticity) 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)

15 Spherical CCMV Varying shell thickness Thicker shells show Hertzian stiffening nonlinearity Thinner shells show softening nonlinearity Almost perfectly linear for nominal CCMV 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)

16 Questions for theory and simulation: Why is capsid force response linear for such large indentations? Why do some capsids fail and others not? (Geometric? Constitutive?) How do local protein structure and conformational changes affect the global mechanical response of the shell?

17 Buckling of elastic shells with disclinations 2-D Föppl-von Kármán shell model 5-fold disclination Add/remove a slice from hexagonal sheet stretching needed to keep it flat. Buckling may alleviate stretching. Stabilitiy of planar sheet controlled by Föppl-von Kármán number: Seung & Nelson, Phys. Rev. A (1988) 2-D Young s modulus bending modulus Buckled disclinations implicated in determining the facetedness of capsids Lidmar, Mirny, & Nelson, Phys. Rev. E (2003)

18 Baker, et al. (2000)

19 Can AFM indentation trigger buckling?

20 ph-sensitivity of CCMV and changing FvK number? F! "Y # R ph 6 Explain change in stiffness as change in material properties. Stretching modulus Y affects slope and stability ph 5 Klug, et al., Phys. Rev. Lett., 97, (2006)

21 How does swelling transition affect mechanical response? ph = 5 native CCMV d = 28 nm ph = 7.5 swollen CCMV expands by 10% ph 6? ph 5 Note: capsids appear structurally identical at ph 5 and 6, and differ only in mechanical response.

22 Normal-mode Analysis of ph-induced Swelling of CCMV Amplitude and direction of motion (a) structural data (b) normal mode 24 Tama and Brooks, J. Mol. Bio. (2002)

23 Ginzburg-Landau theory of swelling transition as a structural phase transition Order parameter: = amplitude of soft swelling mode Free energy: Linearizing with respect to order parameter: Increasing ph softens welling mode and reduces effective Young s Modulus Y* and (to lowest order) does not affect κ. ph 5 failure may be initiated by buckling (geometric failure). Klug, et al., Phys. Rev. Lett. (2006) Guérin & Bruinsma, Phys. Rev. E (2007)

24 Maturation of Bacteriophage HK97 Translations and rotations of the subunits (conformational change) make the final stable mature capsid possible Different phases during HK97 maturation (Wikoff et. al. 2006) HK97 Cross section of PII, R~24 nm Chainmail in HII

25 Parameters for HK97 Experimental structures Two equilibrium configurations. Both are stable. Simulated result

26 Phase Transition Triggered by Indentation Jump from swollen to contracted Model Predictions: Contraction transition not reversed upon unloading. Considerable hysteresis. Experiments in progress

27 Questions for theory and simulation: Why is capsid force response linear for such large indentations? Why do some capsids fail and others not? (Geometric? Constitutive?) How do local protein structure and conformational changes affect the global mechanical response of the shell?

28 Nonuniform Finite Element Models Model capsid as homogeneous elastic shell (geometric heterogeneity only, for now) Obtain geometry from structural biology data: X-ray crystallography all atom coordinates (RCSB Protein Data Bank) Cryo-electron microscopy CCMV (native and swollen) electron density maps (Electron Microscopy Data Bank) ϕ29 Triangulate molecular surfaces Build 3-D meshes of interior (tetrahedra fill in space between inner and outer surfaces) HK97 (procapsid and mature) Gibbons and Klug, Biophys J, 95(8), October 15, Hepatitis B

29 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

30 Native Indentation Simulations Swollen 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

31 Contact formed with multiple capsomers Buckling event Model Predictions: Orientation-dependent nonlinearities 3-, 5-fold: softening from local deformation mode (arm bending) 2-fold: Buckling at high indentation Softening from bending arms between capsomers Contact Froces (pn)

32 Conclusions and Unanswered 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 CCMV and HK97 (in progress with Wuite and Schmidt). 3-D model of ph 5 capsid doesn t buckle - is failure actually consitutitve (e.g., fracture/bond-breaking)? How important is constitutive heterogeneity? (Need info from atomic interactions)

33 Constitutive Heterogeneity: FEM + rigidity percolation (Ongoing work with M. Thorpe, ASU) Compute flexibility map with constraint theory using FIRST software Assign local elastic moduli to FE model based on flexibility map

34 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

35 Thanks for your attention