simulations Extracting coarse-grained elastic behavior of capsid proteins from molecular dynamics Outline Feb. 2009

Transcription

1 Extracting coarse-grained elastic behavior of capsid proteins from molecular dynamics simulations Stephen D. Hicks Christopher L. Henley Cornell University Gordon Conf. on Physical Virology Feb Outline Goal: Whole-capsid elasticity from all-atom MD of single domains 1. Set-up: generalized springs 2. Method to extract statics and dynamics from simulation

2 Motivations for getting elasticity Capsid shape (rounded or faceted?) depends on bend/stretch ratio of elastic constants [Lidmar et al, PRE 2003]. Atomic-force-microscope (AFM) indentation measurements: predict value of spring constant? Guidance in fitting protein units into hypothetical models of the full structure Mutant budding morphologies due to a point mutation: because it changes an elastic or angle parameter relating two capsid units? [Vogt lab] Key ingredient in our old irreversible assembly model [Hicks and Henley PRE 2006]: bend/stretch ratio determined rate of fatal mistakes; thermal fluctuations in shape should govern growth rate.

3 Generalized Springs Treat each protein as one or a few rigid units Study each pairwise interaction by simulating just 2 units small, tractable MD simulations Interaction may be either a flexible covalent linker, or non-covalent docking Determine interactions with no assumptions except small deviations from a free energy minimum) Predict measurements with no adjustable parameters!

4 rag replacements General form of spring interactions stretch shear twist bend Generalize spring to depend on all 6 degrees of freedom Free energy (expand to second order) spring constant becomes 6 6 matrix F = 1 2 ut Kuu u + u T Kuξ ξ ξ T Kξξ ξ translational stiffnesses orientational stiffnesses PSfrag replacements translation-orientation coupling! Actual position u ξ Equilibrium position

5 What we could do with this? Coarse-grained simulations of spring networks tractable 1000-fold reduction in dimension derive lattice spacings, continuum elastic parameters,... [Image: whole-capsid simulation Freddolino et al...]

6 Apply to retrovirus (HIV): interactions? Possible interactions in generic capsid Quasiequivalence allows 3 classes of interactions between capsid proteins (shown as trapezoids) Trimer, Hexamer/Pentamer, and Dimer units and interactions. Mature HIV capsid has 4 interactions CA has N-terminal domain (NTD, blue); C-terminal domain (CTD, yellow) (0) NTD-CTD linker (1) NTD-CTD (Trimer) (2) NTD-NTD (Pent./hex.) (3) CTD-CTD (Dimer)

7 Simulation details Simulation details NAMD software package 1,500,000 2fs = 3ns 2 domains + 4 7A water CTD dimer linker NTD heterodimer

8 Example timeseries: linker Rotational directions sin(θx/2) acements PSfrag replacements PSfrag replacements sin(θy /2) time time time sin(θz/2) Translational directions acements rxpsfrag replacements rypsfrag replacements time time time rz

9 Method: statics We measure the equilibrium fluctuations and use the equipartition theorem to extract entire 6 6 stiffness matrix K: xαxβ = TK 1 αβ

10 Problem: slow relaxations X X X Artifact of isolating one pair Real configuration is equilibrium of forces acting on all sides Cut other interactions drifts towards unrealistic state Unequilibrated modes have large x 2, hence appear as spurious soft directions in K matrix

11 Equilibrated vs. relaxing directions mostly equilibrated after 1ns PSfrag replacements time sin(θx /2) relaxing during entire simulation PSfrag replacements time sin(θz/2)

12 Stochastic dynamics: two key matrices K and Γ d x dt = Γ f (x) + ξ(t), f (x) = force = K x ξ(t) = noise, satisfying ξ i(t)ξj(t ) = Dijδ(t t ) D = 2kBT Γ = noise (diffusion) matrix By fluctuation dissipation relation (Einstein), the same matrix (Γ 1 ) tells the viscous damping force We can measure the noise/damping matrix gij(τ) δxi(τ)δxj(τ) Dijτ, for short τ drift velocity vd = Γ f, tells force to be canceled. Combine into relaxation matrix W = Γ 1/2 KΓ 1/2 ; it has dimensions (time) 1 and its eigenvalues are relaxation rates of the (overdamped) eigenmodes. Check that relaxation times are sufficiently shorter than run time.

13 Results Technicality coordinates have different units (3 position + 3 angle) so can t diagonalize 6 6 matrix... instead, get effective 3 3 matrices by letting the other 3 coordinates vary freely. Averages: Diffusion Stiffness rot (1/s) trans (A 2 /s) rot (kbt ) trans (kbt /A 2 ) NTD NTD CTD CTD Linker (U) Linker (F)

14 Conclusions Generalized springs are a useful and tractable intermediate between full atomistic MD and continuum models Apply stochastic dynamics formalism (overdamped version) to simulation dynamics Dynamics gives (1) criterion for equilibration of simulation (2) time scale for quantitative computations of all-capsid dynamics using the network model (no adjustable parameters) Thanks to DOE funding, Cornell CCMR Computing Facility, and useful discussions with David Roundy

15 Molecular dynamics simulations: more HIV mature capsid (source structures) Full-length CA (CTD NTD linker stiffness) (Ganser-Pornillos et al, 2007) CTD CTD dimer bond (Gamble et al, 1997) NTD NTD hexamer bond (Tang et al, 2002; Mortuza et al, 2004) Simulation details

16 Slow relaxations: more NTD-CTD linker before/after: motion is mainly angular (untwisting)

17 Results: more Stiffness values Ratio of angular/coordinate part is 20 nm comparable with inter-unit distance Noise/dissipation matrix Ratio of coordinate/angular parts is 0.6 nm: comparable to radius of each protein unit. Consistent with hydrodynamic predictions for positional and rotational diffusion constants of a sphere