RNA-directed Viral Assembly
I) Self-Assembly and Free Energy Minimization.
II) Fundamental Interactions.
III) Self-Assembly Empty Capsids.
IV) Condensation of RNA genome molecules.
V) Free Energy Landscape Viral Assembly.
I)
(1995, Scientific American)
“Mark I” Self-Assembly
Self-assembling Monolayer
“Thermodynamic Assembly”: assembled and disassembled components in thermal equilibrium.
Variational Principle:
Gibbs Free Energy
G = U – TS – m N
dG = 0
Amphihilic Molecules
“Hydrophilic”
“Hydrophobic”
Limited Complexity
Synthetic Chemistry:
J-M Lehn, D.Cram
circular helical
* weak, non-covalent bonds
* water soluble
“host-guest”
“Mark II” Coded assembly.
•
DNA encoded assembly program -> protein synthesis-> assembly
• Constant free energy consumption. dG ≠ 0
• Complexity: unlimited.
• Is viral assembly Mark I or Mark II ? Free energy ?
II ) Fundamental Interactions.
Cowpea Chlorotic Mottle Virus: CCMV
Genome:
(J. Johnson et al.)
• In-vitro Self-Assembly
T=3 Capsid
180 identical proteins
Layer capsid proteins
A) Capsid Proteins: Amphiphilic
Water
Hydrophobic
Water
N-terminal tail
•
Layer: Spontaneous Curvature
• Expect reversible, thermodynamic assembly
B)
• Strength attractive interactions increases with acidity.
C) Electrostatic Interactions
• Water-accessible equipotential surfaces. Blue positive; Red negative.
• Inside-Outside Voltage Difference
(McCammon et al.)
Electrical Charges CCMV Dimers
U EL
QC(pH=7) = -28
“ core” charges
(physiological)
+
(
Qc2
pH = 7 =
» 390 k BT
eD
)
Very large !
+
QT=+20 “tail” charges/dimer
Some “just so” questions about CCMV electrostatics
RNA has a total negative charge ≈ -3,000
Positive tail charge ≈ 90 x 20 = + 1,800
• Neutralization promotes viral assembly.
1) Why neutralize only a fraction of RNA charge ?
Outer layer charge ≈ - 28 x 90 = -2,520
2) What’s the role of the large negative protein charge?
• Prevents aggregation of viruses.
• Prevents RNA from sticking to capsids.
III)
Self-Assembly Empty Capsids
Electrostatic Repulsion vs. Hydrophobic Attraction
+
+
• Treat viral assembly as a chemical reaction:
Assembled T=3 capsid
90 Free CP Dimers (“subunits”)
Thermal Equilibrium
dG =
å
Concentrations
mi dN i = 0
components i
“Law of Mass Action”
Dimer Concentration
• DG = assembly energy/dimer
“Signature” of Thermodynamic Self-Assembly
Empty capsid assembly experiments
* Acidic environment (low pH)
Chromatography
dG = 0
• DG ≈ 30 kBT/dimer.
• Capsid assembly is irreversible!?
reversible
irreversible
Capsid Van der Waals/Landau Free Energy
Ns adsorbed proteins
R
D
rs = [Ns / R2] area density
Entropic Free Energy 2D ideal solution “order parameter”
QC=28
FS (N s , R) / N s » kBT ln rs D2 - e + vs rs + ws rs2 +...
e:
vs:
Adsorption energy proteins on sphere.
Thermal equilibrium:
Second “virial coefficient”
ws : Third virial coefficient ≈
kBT
D4
¶F
=m
¶N s
Chemical potential proteins
Second Virial Coefficient
• Electrostatics vs Hydrophobicity
vS = vDH - J
-
Qc=-28
- - -
Capsid Proteins
- - Qc=-28
- 2D
“Bjerrum Length” ≈ nm
( )
vS y / k BT =
“Debye Parameter”
≈ 1/nm
lB Q
2
C
k
R
( )
ψ
- JS y
y = 2D / R
Angle-dependent
hydrophobic attraction
Optimal Angle/ Radius
CCMV (pH=5): QC = 20
vDH /kB T = 400 nm2
(
)
(
)
vS y = yc / k BT » -70 nm2
J S y = yc / k BT » -470 nm
2
yc = 2D / Rc
Measured for empty shells
CP-CP Hydrophobic Attraction
Capsid Radius
Debye-Hückel Theory of Aqueous Electrostatics
Macro ion Charge Density (CPs/RNA)
Dielectric Constant Water
Electrical Potential
Electrostatic Free Energy
Debye
parameter
k 2 = 4p e 2 [Salt] / e kBT
Bjerrum length
-
e2
= k BT
e lB
Sheet of charges
FDH
æ l Q2 ö N 2
/ k BT = çç B C ÷÷ 2S
è k øR
Summary
• Delicate balance between large repulsive interactions and large attractive interactions
• Second virial coefficient depends on the sphere radius R.
æ l Q2
F2 / N s = k BT çç B C - J S Rc / D
è k
(
)
ö
÷÷ r s
ø
VS ≈ VDH
RC
R
R*
Free energy “landscape”
F(R,Ns)
F(R,Ns)
R
vs> 0
R*
vs= 0
Rc
vs< 0
Ns
Nc =90
¶F
=m
¶N s
“Common-tangent construction”
Phase-coexistence: nearly closed shells and nearly bare spheres
IV) Condensation RNA genome molecules
• Highly branched, highly charged “polyelectrolyte”
Q ≈ - 3,000
L = 300 nm
Paired stretches l ≈ 5 bp
≈ 100 nm
Neutron scattering R ≈ 11 nm
(no “condensing agents”)
• Highly compactified
( Knobler et al. )
CCMV RNA 1
RNA Condensation
Condensing
Agent
Free energy F = U - TS
“Intermediary”
“Native”
dG = 0
“Folded
Fraction”
# condensing
agents per RNA θ
Gibbs Free Energy
G = F – m( [agent]) θ
Chemical potential
condensing agent
Condensing agent concentration
(polyvalent counterions)
Koculi E, Lee NK, Thirumalai D, Woodson SA.
J Mol Biol. 2004, 341(1):27-36.
• Highly cooperative, first-order phase transition.
N state: folded
• Ribozyme (Tetrahymena) RNAse
(Cech)
Tertiary contacts
• RNA inside T=3 virus:
What are the condensing agents ?
Highly condensed
CCMV Capsid protein
ss RNA
PO4
-
Disordered N-Terminal Tail:
+ 10 charges
RNA Condensing Agent
Numerical Simulation: e (tail/RNA)≈ 10-15 kB T
Zhang et al.
Biopolymers. 2004 November; 75(4): 325–337)
Remove Protein Cores
CCMV Dimers
QT=+20 tail charges/dimer
Condensation of CCMV RNA
Good Solvent: “fractal”
R(N) ≈ N 1/2
•
Flory-Landau mean-field theory
R
l =5bp
# segments
N=300
segmentsN
Entropic Elasticity
U
R2
N2
N3
FF (R, N ) = k BT 2 +V q 3 +W 6 + ..
R0
R
R
()
R0 (N ) = l N 1/4
Radius gyration of an “ideal” Flory-Stockmayer branched polymer
Linear polymers: much larger
• V(q): Second Virial Coefficient.
• W: Third Virial Coefficient ≈ kB T l 6
R0 (N ) = l N 1/2
q: # tails / segment
Condensed Globule
Swollen Fractal
V
æV ö
R(N ) / l » ç ÷ N 1/3
çW ÷
è ø
1/3
V=0
“Theta Solvent”
( )
R(N ) » V l
2
1/3
N 1/2
CCMV RNA genome free in solution
R ≈ 11 nm
* No phase transition
• l = 0.5 nm
• V(q=0)/ kB T = 1-10 nm3
Second Virial Coefficient
Segment charge
Tail fraction
Bjerrum Length
V (q ) / k BT =
ql = - 10
3lB
k
2
(-q + Q q )
l
T
Maximum concentration
2
-VRNA
Non-electrostatic
q max = ql / QT » 0.5
Polyvalent Counterion Charge
Neutralization
Debye parameter
V (q = 0) / kBT » 100 nm3 -VRNA » 1-10 nm3
V (q = ql / QT ) / kBT = -VRNA » -100 nm3
(free RNA in solution)
RNA/tail association:
“unveils” strong RNA self-attraction
QT = + 10
(CCMV)
RNA Globule
()
V q
Voltmeter
DV
()
eDVD q / k BT =
“Donnan Potential”
(
lB -ql + QTq
k 2 R3
15-20 milli Volt
)
qM
Charge
neutral
q
RNA/tail affinity
( )
( )
FRNA R,q = FF R,q + kBT Nq lnq - (m + e )Nq
* Minimize with respect to R
Swollen,
charged
Chemical potential tails
V(q) > 0
V(q) < 0
Condensed,
neutralized
q
qm= ql /QT
• Common-tangent Construction: Phase Coexistence
Large, reversible first-order phase transition
Gel swelling/shrinking
V) Free Energy Landscape
Combine:
( )
( )
(
F R,q = FRNA R,q + FS R, Nq
)
# Surface-Adsorbed CPs = # Tails
( )
F R,q
Charged
Tailneutralized
Radius R
Micro-segregation
60 excess CP dimers
# proteins/segment
Is this processes thermodynamic reversible self-assembly?
Step 1
Protein-RNA assembly
Reversible
Same CP chemical potentials
Irreversible
Step 2
Micro-segregation
Step 3
Protein expulsion
“Michaelis-Menten like”
Lowered CP chemical potential
Enhanced RNA self-attraction
+
æ l Q2
k BT çç B C - J S Rc / D
è k
60
Lowered CP chemical potential
(
ö
)÷÷ r
ø
Irreversible
e - DVDQC
Donnan Potential + Protein Self-repulsion
s
How are excess proteins expelled?
Brownian Ratchet:
-
-
-
-
-
_
_
+
+
Capsid Proteins
_
+
Tails
RNA
How good is mean-field theory?
Protein-Protein binding sites
Toy T=1 Virus
Flexible linear polymer genome
genome binding sites”
* Genome molecule: no branching.
* Assembled state: # binding sites = chain length
Elrad and Hagen
Protein-genome affinity e > Protein-protein affinity J
time
* RNA/Protein pre-assembly condensate
A
B
C
D
Problem: Optimal angles
visible in A-C
E
C
D
• Local correlations.
E
A
Genome-protein affinity e weaker than protein-protein affinity J
RNA “glues” capsomers together one-by-one
* Heterogeneous nucleation of a shell on a flexible RNA scaffold
“Down the funnel”
Partial shells
Many possible assembly pathways
“Antenna-Assembly”
(Hu and Shklovskii)
“Hamiltonian Cycle”
• Graph-theoretic problem
(R.Twarock)
Conclusions
1) Assembly of small ss RNA viruses can be viewed as the combination of
reversible RNA condensation + quasi-reversible shell formation.
2) Combination of two simple thermodynamic assembly processes produces
a more complex free energy landscape with different possible multi-step
Irreversible pathways.
3) Viral assembly appears intermediate between Mark I and Mark II assembly.