Modeling virus capsid assembly: a multiscale
approach
S. D. Hicks
C. L. Henley
Cornell University
Brown CM Seminar, 11/5/09
Viruses
Model
A portrait of the virus as a young capsid
Irreversible models of capsid assembly
1
2
3
Viruses
Lifecycle
Structure
Quasiequivalence
Model
Overview
Statics
Kinetics
Failure
Results
Pictures
Success rate
Size distribution
Our new work
Results
A tale of two three tensors
Elastic models from
atomistic simulations
4
5
6
Atomistic Simulations
Motivation
HIV capsid
Relative coordinates
Statics
Random walk model
Stiffness/Results
Dynamics
Motivation
Diffusion
Relaxation
Conclusions
Viruses
Model
Virus Lifecycle
Oversimplified theoretical picture
1. Virus enters cell and disassembles
2. Cell mechanisms replicate genome and proteins
3. Genome and proteins assemble into new viruses
4. New viruses exit cell (either budding or after cell death)
Results
Viruses
Model
Results
Virus Structure
Viruses made of
Genome (RNA or DNA)
Capsid (protein coat)
Lipid envelope (optional)
Virus capsid
Protects genome from hostile
extracellular environment
Made from 100s of copies
of same structural protein
Wide variety of shapes
(spheres, tubes, cones, . . . )
Viruses
Model
Quasiequivalence
Results
[Caspar and Klug, 1962]
Icosahedral group allows just 60 symmetry equivalent units
... but capsid can have ≫ 60 (”60T”) copies of same protein
Quasiequivalence:
same bonding
with slight deformations
Three kinds of bond:
(a) dimer (b) trimer
(c) hexamer /pentamer
(need at least 2 kinds)
12 Disclinations
(= 5-fold verices)
Viruses
Model
Results
Quasiequivalence: Example
T = 13 triangular lattice
√
( 13 =
A
p
32 + 12 + 3 · 1)
Rice Dwarf Virus
Reddy, et al, J. Virol. 75, 11943 (2001)
Nakagawa, et al, Structure 11, 1193 (2003)
Viruses
Model
Results
Quasiequivalence: Example
T = 13 triangular lattice
√
( 13 =
A
p
32 + 12 + 3 · 1)
Rice Dwarf Virus
Reddy, et al, J. Virol. 75, 11943 (2001)
Nakagawa, et al, Structure 11, 1193 (2003)
Viruses
Model
Results
Polymorphism
icosahedral
sphere-like
Quasiequivalence permits polymorphism
(shape specified by where you put the
5-fold vertices a.k.a. disclinations)
Experimental polymorphism:
Some viruses make tubes or
different sized icosahedra
depending on pH, salt, mutations...
others are generically irregular...
θ*
cone
1
=
1.2 T
1
0.8
0.6
0.4
0.2
T=3
Tubes
0.2 0.4 0.6 0.8
1
∆Ε/κ
Ionic strength (M)
Out of these practically degenerate
structures, what determines the
actual one assembled?
tube
Dimer
0.8
T=3
0.6
0.4
0.2
Tube
T‡3
4
5
6
pH
7
(Bruinsma et al, 2003)
Viruses
Model
Results
HIV Polymorphism
(Benjamin et al, 2005)
Viruses
Model
A portrait of the virus as a young capsid
Irreversible models of capsid assembly
1
2
3
Viruses
Lifecycle
Structure
Quasiequivalence
Model
Overview
Statics
Kinetics
Failure
Results
Pictures
Success rate
Size distribution
Our new work
Results
A tale of two three tensors
Elastic models from
atomistic simulations
4
5
6
Atomistic Simulations
Motivation
HIV capsid
Relative coordinates
Statics
Random walk model
Stiffness/Results
Dynamics
Motivation
Diffusion
Relaxation
Conclusions
Viruses
Model
Results
Irreversible Growth Model
Ingredients
Discrete configurations: growing capsid = triangular network
(All completed vertices 5- or 6-fold)
Two main components:
(1) elastic energy (statics)
(2) growth rules (kinetics)
(Hicks and Henley, PRE, 2006)
Viruses
Model
Elastic Energy
Elastic Hamiltonian
stretching stiffness X bending stiffness
X
1
2
H=
κ(1 − cos(θij − θ0 ))
2 Y (rij − r0 ) +
hi ,ji
hi ,ji
Dimensionless parameters
1. Preferred dihedral angle (spontaneous curvature) θ0
2. “Foppl-von Karman” length ℓ2F = κ/Y
Results
Viruses
Model
Elastic Energy
H=
X1
hi ,ji
2
Y (rij − r0 )2 +
X
hi ,ji
κ(1 − cos(θij − θ0 ))
More about elastic free energy...
Multiscale idea: Y , κ are cartoons of
stiffnesses extracted from simulating
two proteins (in part II of talk)
local elasticity can transfer
long-range information (disclination
interaction)
phase diagram of equilibrium models
(Bruinsma et al, 2003)
(Lidmar, Mirny, and Nelson,p2003):
buckling transition in ℓF = κ/Y
(sharpness of capsid shape, on right)
Results
Viruses
Model
Results
Growth Rules
Irreversible assembly
Each step, add a triangle to edge(s)
or glue two edges together
Rate for each possible step depends on
(local) geometry of relaxed structure
rate ΓA to add a triangle to one edge
angle-dependent rate ΓI,J (α) to
insert between or to join two edges
α
insert
accrete
No 4-fold or 7-fold vertices allowed
join
Viruses
Model
Growth Rules
Others’ kinetic models: presuppose the final structure
Their aim: what is the pathway?
– “Local rules” models (Schwartz et al 1998)
– MD of toy polyhedra with numerous point-point interactions
engineered to favor T1 or T3 capsids (Rapaport et al 1999)
Kinetic traps? (cf. Zlotnick 1994)
overly fast growth ⇒ monomers used up,
get stuck with all unfinished capsids
our model: infinite monomer reservoir, no trap
(implicit assumption: nucleation-limited kinetics)
Our primary concern about assembly:
How does the choice of parameters determine final structure?
Results
Viruses
Model
Failure
Defn: when partial capsid has no legal way to complete
Why do we care?
To really distinguish mechanisms, study their failures!
... different models make the same symmetric “correct” capsid
Errors are elusive experimentally:
Partially assembled capsids rarely caught in the act (too fast)
... but real viruses have many duds: maybe flawed capsids?
Results
Viruses
Model
Results
Gross Failure Modes
From our model, “crevasses”:
grey triangles should overlap
3
3
5 5
compare: Hagan and
Chandler 2006
Steric avoidance
Failure modes sensitive to particulars of steric avoidance
Levandovsky and Zandi (2009): “steric attraction” reduces failures
Viruses
Small failure mode
Model
Results
Viruses
Model
Small failure mode
Unfillable hole
Required disclination already six-fold
Irreversible growth has failed
“Final structure” includes failure modes
Results
Viruses
Model
A portrait of the virus as a young capsid
Irreversible models of capsid assembly
1
2
3
Viruses
Lifecycle
Structure
Quasiequivalence
Model
Overview
Statics
Kinetics
Failure
Results
Pictures
Success rate
Size distribution
Our new work
Results
A tale of two three tensors
Elastic models from
atomistic simulations
4
5
6
Atomistic Simulations
Motivation
HIV capsid
Relative coordinates
Statics
Random walk model
Stiffness/Results
Dynamics
Motivation
Diffusion
Relaxation
Conclusions
Viruses
Model
Results: examples of assembled capsids
Complete
Failed
Results
Viruses
Model
Success rate
Factors that increased success rate
smaller capsid size (failure is ∼poissonian)
larger ℓF (larger bend/stretch κY )
less accretion, more insert/join steps
keeping both insert and join steps
Results
Viruses
Model
Size distribution
Y /κ = 1
Y /κ = 3.33
Y /κ = 10
Y /κ = 33.3
1/(average radius)
1.0
0.8
0.6
0.4
0.2
0.0
0.0
√
0.2
0.4
0.6
0.8
1.0
spontaneous curvature (2 3 tan(θ0 /2))
More consistent assembly at small stretch/bend Y /κ,
but perhaps stronger capsid at large Y /κ
Might explain maturation transition in e.g. HK97 virus?
(small Y /κ till complete, switch to large Y /κ)
Results
Viruses
Model
Results
Other’s recent growth model
(Levandovsky and Zandi, PRL 2009)
Controversy: does
capsid start at big end
or little end of cone?
L & Z: neither – nucleated
at kink on the side!
Levandovsky and Zandi ran a similar model, but changed
steric term to allow attraction between distant parts. ⇒
network can re-join on the far side of a crevice.
For a special value of the spontaneous curvature,
got shapes v. similar to HIV capsids.
Viruses
Model
New work
Hexagon-based models
“Universality” dogma ⇒ (wrongly) expected same behavior
whether trimers or hexamers
With hexagons, only one kind of partly-filled corner:
less arbitary (more robust?) growth rules
Modeling immature HIV: lattice has only hexamers –
(Holes at 5-fold places, network reconnects on the far side)
Planned: collaborate with experimentalists to analyze
individual configurations measured by cryo-EM-tomography
Results
Atomistic Simulations
Statics
A portrait of the virus as a young capsid
Irreversible models of capsid assembly
1
2
3
Viruses
Lifecycle
Structure
Quasiequivalence
Model
Overview
Statics
Kinetics
Failure
Results
Pictures
Success rate
Size distribution
Our new work
Dynamics
A tale of two three tensors
Elastic models from
atomistic simulations
4
5
6
Atomistic Simulations
Motivation
HIV capsid
Relative coordinates
Statics
Random walk model
Stiffness/Results
Dynamics
Motivation
Diffusion
Relaxation
Conclusions
Atomistic Simulations
Statics
Dynamics
Molecular Dynamics Simulations
Motivation
Connect continuum elasticity to microscopic structures
Computational efficiency / Physical interpretability
Description of overdamped dynamics
Application to real virus: HIV
arXiv:0903.2082 [q-bio.BM], submitted to PRL
Atomistic Simulations
Statics
HIV capsid structure
HIV mature capsid (source structures)
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)
(top)
(side)
modified from Wright et al, EMBO J. 26, 2218 (2007)
Dynamics
Atomistic Simulations
Statics
Dynamics
Simulation details
NAMD software package
1,500,000 steps @ 2fs = 3ns
2 domains + 7Å extra water
linker
NPT→stiffness, NVE→noise
CTD dimer
NTD heterodimer
Atomistic Simulations
Statics
Dynamics
Interactions
1 stretch
4 bends
1 twist
Each domain is rigid body (6 degrees of freedom)
Expect bond to be anisotropic: general interactions
Generalize spring to depend on all 6 degrees of freedom
6×6 stiffness tensor K.
Complication: coordinates have different units
Atomistic Simulations
Statics
A portrait of the virus as a young capsid
Irreversible models of capsid assembly
1
2
3
Viruses
Lifecycle
Structure
Quasiequivalence
Model
Overview
Statics
Kinetics
Failure
Results
Pictures
Success rate
Size distribution
Our new work
Dynamics
A tale of two three tensors
Elastic models from
atomistic simulations
4
5
6
Atomistic Simulations
Motivation
HIV capsid
Relative coordinates
Statics
Random walk model
Stiffness/Results
Dynamics
Motivation
Diffusion
Relaxation
Conclusions
Atomistic Simulations
Statics
Random walk model
Equation of motion
Generalized/coarse grained coordinates xi , i = 1 . . . N (∼ 6)
dx
= Γ f (x, t) + ζ(t)
dt
ζ(t) ⊗ ζ(t ′ ) = 2Dδ(t − t ′ )
Γ = mobility tensor
D = diffusion tensor (detailed balance: D = kB T Γ )
Expand potential to second order
1
U(x) = U0 − f∗ · (x − x∗ ) + (x − x∗ )K(x − x∗ )
2
K = stiffness tensor
Dynamics
Atomistic Simulations
Statics
Dynamics
Stiffness
dx
dt
= −Γ K(x − x∗ ) + ζ(t)
Equipartition theorem
If simulation has reached equilibrium (and f∗ = 0),
x∗ = hx(t)i
kB T K −1 = hx(t) ⊗ x(t)i − x∗ ⊗ x∗
Build up network from measured x∗ and K for each bond type
Atomistic Simulations
Statics
Results: Lattice Geometry
Li et al 2000: 10.7nm lattice constant
Our lattice: 9.1nm
Dynamics
Atomistic Simulations
Statics
Dynamics
Results: Stiffness
AFM indentation (Kol et al 2007): Youngs modulus Y = 115MPa
Simple extension with measured K : Youngs modulus Y = 77MPa
Atomistic Simulations
Statics
A portrait of the virus as a young capsid
Irreversible models of capsid assembly
1
2
3
Viruses
Lifecycle
Structure
Quasiequivalence
Model
Overview
Statics
Kinetics
Failure
Results
Pictures
Success rate
Size distribution
Our new work
Dynamics
A tale of two three tensors
Elastic models from
atomistic simulations
4
5
6
Atomistic Simulations
Motivation
HIV capsid
Relative coordinates
Statics
Random walk model
Stiffness/Results
Dynamics
Motivation
Diffusion
Relaxation
Conclusions
Atomistic Simulations
Statics
Dynamics
Problem
External forces
Calculating K required assumption of equilibration
Proteins taken out of biological context: external forces
Deduce balancing force from drift velocity v̄:
f∗ = −Γ −1 v̄
position (radians)
-0.55
-0.60
-0.65
-0.70
-0.75
-0.80
-0.85
0.0
0.5
1.0
1.5
time (ns)
2.0
2.5
3.0
Atomistic Simulations
Statics
Dynamics
Diffusion
Diffusion tensor
At small |t ′ − t|:
2D|t ′ − t| = [x(t ′ ) − x(t)] ⊗ [x(t ′ ) − x(t)] ≡ G(t ′ − t)
Measure correlation functions G(∆t)
Linear fit at small ∆t (past ballistic scale)
0.08
0.07
G (∆t) (Å2 )
G (∆t) (Å2 )
1.2
1.0
0.8
0.6
0.4
0.2
0.0
0
0.06
0.05
0.04
0.03
0.02
0.01
20
40
60
∆t (ps)
80
100
0.00
0
1
2
3
∆t (ps)
4
5
Atomistic Simulations
Statics
Dynamics
Diffusion
Stokes’ Drag
D (trans) =
kB T
6πηr
D (rot) =
kB T
8πηr 3
One-body Simulations
NPT simulations: D (trans) ≈ 3Å2 /ns: way too small
Langevin thermostat: 10× as much friction as water
Finite size effect: D (trans) decreased by ∼ 1/L
→ simulate different volumes NVE, extrapolate
CTD
NTD
trans (Å2 /ns)
55
35
rot (rad2 /ns)
0.0084
0.0038
Atomistic Simulations
Statics
Dynamics
Relaxation
Relaxation matrix
Transform coordinates s.t. noise is isotropic (Dij → δij )
Transformed stiffness tensor becomes relaxation matrix
R = Γ 1/2 KΓ 1/2
Diagonalize to find relaxation times
2.5
1.5
1.0
0.5
0.0
-0.5
-1.0
-1.5
-2.0
-2.5
0.0
0.5
1.0
1.5
τR ∼ 1.6ns
2.0
2.5
3.0
time (ns)
normalized position
normalized position
2.0
τR ∼ .47ns
2.0
1.5
1.0
0.5
0.0
-0.5
-1.0
0.0
0.5
1.0
1.5
time (ns)
unforced
forced
2.0
2.5
3.0
Atomistic Simulations
Statics
Dynamics
Context and Summary
Normal Mode Analysis, Principal Component Analysis
Diagonalize stiffness matrix to find low-frequency modes
Finds where the “hinges” are, but doesn’t give dynamics
(cf. Tirion 1996, Bahar et al 1998)
Relaxation Dynamics
Useful when we already know how to separate rigid domains
Diagonalize relaxation matrix R = Γ 1/2 KΓ 1/2
Breathing mode relaxation rate 4.5/ns
Relaxation times provide a check for simulation run times
Next: effects of point mutations, environmental conditions
Atomistic Simulations
Statics
Dynamics
Conclusions
Methods range from microscopic to whole-capsid descriptions
Relaxation dynamics applicable to many large systems?
MD simulations motivate choice of elastic parameters in
assembly model
Still no motivation for choice of growth rates
(simulating assembly from a melt of units might help)
Acknowledgments
D. Murray, V. M. Vogt, M. Widom, W. Sundquist, H. Weinstein,
D. Roundy, and CCMR Computing Facility
Supported by DOE Grant No. DE-FG02-89ER-45405