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![Page 1: RNA-directed Viral Assembly I) Self-Assembly and Free Energy Minimization. III) Self-Assembly Empty Capsids. II) Fundamental Interactions. IV) Condensation.](https://reader030.fdocuments.in/reader030/viewer/2022032600/56649dc85503460f94abda39/html5/thumbnails/1.jpg)
RNA-directed Viral Assembly
I) Self-Assembly and Free Energy Minimization.
III) Self-Assembly Empty Capsids.
II) Fundamental Interactions.
IV) Condensation of RNA genome molecules.
V) Free Energy Landscape Viral Assembly.
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“Mark I” Self-Assembly
Self-assembling Monolayer
(1995, Scientific American)
I)
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“Thermodynamic Assembly”: assembled and disassembled components in thermal equilibrium.
Variational Principle: Gibbs Free Energy G = U – TS – m N dG = 0
“Hydrophobic”
“Hydrophilic”
Amphihilic Molecules
Limited Complexity
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“host-guest”
circular helical
* weak, non-covalent bonds* water soluble
Synthetic Chemistry:
J-M Lehn, D.Cram
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• DNA encoded assembly program -> protein synthesis-> assembly
• Constant free energy consumption. dG ≠ 0 • Complexity: unlimited.
“Mark II” Coded assembly.
• Is viral assembly Mark I or Mark II ? Free energy ?
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T=3 Capsid180 identical proteins
Genome:
(J. Johnson et al.)
Cowpea Chlorotic Mottle Virus: CCMV
• In-vitro Self-Assembly
II ) Fundamental Interactions.
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A) Capsid Proteins: Amphiphilic
N-terminal tail
Layer capsid proteins
Water
Water
Hydrophobic
• Layer: Spontaneous Curvature
• Expect reversible, thermodynamic assembly
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• Strength attractive interactions increases with acidity.
B)
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• Water-accessible equipotential surfaces. Blue positive; Red negative.• Inside-Outside Voltage Difference
C) Electrostatic Interactions
(McCammon et al.)
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QT=+20 “tail” charges/dimer
QC(pH=7) = -28 “ core” charges(physiological)
D = 2 nm
Electrical Charges CCMV Dimers
+ +
Very large !
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RNA has a total negative charge ≈ -3,000
Positive tail charge ≈ 90 x 20 = + 1,800
1) Why neutralize only a fraction of RNA charge ?
Some “just so” questions about CCMV electrostatics
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.
• Neutralization promotes viral assembly.
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+ +
III) Self-Assembly Empty Capsids
Electrostatic Repulsion vs. Hydrophobic Attraction
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90 Free CP Dimers (“subunits”) Assembled T=3 capsid
Concentrations
Dimer Concentration
• Treat viral assembly as a chemical reaction:
“Law of Mass Action”
• DG = assembly energy/dimer
Thermal Equilibrium
“Signature” of Thermodynamic Self-Assembly
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• Capsid assembly is irreversible!?
Chromatography
* Acidic environment (low pH)Empty capsid assembly experiments
• DG ≈ 30 kBT/dimer.
reversible
irreversible
dG = 0
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e: Adsorption energy proteins on sphere.
vs: Second “virial coefficient”
ws : Third virial coefficient ≈ kBT D4
Capsid Van der Waals/Landau Free Energy
Entropic Free Energy 2D ideal solution
R
QC=28
Ns adsorbed proteins
rs = [Ns / R2] area density“order parameter”
D
Chemical potential proteins
Thermal equilibrium:
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Second Virial Coefficient
CCMV (pH=5): QC = 20 vDH /kB T = 400 nm2
Angle-dependenthydrophobic attraction
Measured for empty shells
CP-CP Hydrophobic Attraction
ψ
- - --Qc=-28
--
--
Capsid Proteins
R
2DQc=-28
“Debye Parameter”≈ 1/nm
“Bjerrum Length” ≈ nm
Optimal Angle/ Radius
• Electrostatics vs Hydrophobicity vS = vDH - J
Capsid Radius
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Debye-Hückel Theory of Aqueous Electrostatics
Electrical Potential
Macro ion Charge Density (CPs/RNA)
Dielectric Constant Water
Electrostatic Free Energy
-Bjerrum length
Debyeparameter
Sheet of charges
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• Delicate balance between large repulsive interactions and large attractive interactions
Summary
• Second virial coefficient depends on the sphere radius R.
R
RC
VS ≈ VDH
R*
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R
Ns
R*Rc
vs> 0
vs= 0
Nc =90
F(R,Ns)
vs< 0
Phase-coexistence: nearly closed shells and nearly bare spheres
“Common-tangent construction”
Free energy “landscape” F(R,Ns)
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Q ≈ - 3,000L = 300 nm
Neutron scattering R ≈ 11 nm(no “condensing agents”)
( Knobler et al. )
IV) Condensation RNA genome molecules
• Highly branched, highly charged “polyelectrolyte”
≈ 100 nm
• Highly compactified
Paired stretches l ≈ 5 bp
CCMV RNA 1
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• Highly cooperative, first-order phase transition.
Free energy F = U - TS
“Intermediary”
“Native”
Koculi E, Lee NK, Thirumalai D, Woodson SA. J Mol Biol. 2004, 341(1):27-36.
Condensing agent concentration(polyvalent counterions)
“Folded Fraction”
RNA Condensation
G = F – ( m [agent]) θ
# condensing agents per RNA θ
Chemical potentialcondensing agent
dG = 0
Gibbs Free Energy
CondensingAgent
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N state: folded Tertiary contacts
• Ribozyme (Tetrahymena) RNAse(Cech)
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ss RNAPO4
-
Disordered N-Terminal Tail:+ 10 charges
CCMV Capsid protein
RNA Condensing Agent
Biopolymers. 2004 November; 75(4): 325–337)
Zhang et al. Numerical Simulation: e (tail/RNA)≈ 10-15 kB T
What are the condensing agents ? • RNA inside T=3 virus:
Highly condensed
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QT=+20 tail charges/dimer
CCMV DimersRemove Protein Cores
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Condensation of CCMV RNA
# segments NN=300 segments
R
Radius gyration of an “ideal” Flory-Stockmayer branched polymer
• V( ): q Second Virial Coefficient. q: # tails / segment
• Flory-Landau mean-field theory
Entropic Elasticity U
• W: Third Virial Coefficient ≈ kB T l 6
l =5bpGood Solvent: “fractal”R(N) ≈ N 1/2
Linear polymers: much larger
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VV=0
Swollen Fractal Condensed Globule
“Theta Solvent”
CCMV RNA genome free in solutionR ≈ 11 nm
• l = 0.5 nm• V(q=0)/ kB T = 1-10 nm3
* No phase transition
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ql = - 10
Polyvalent Counterion ChargeDebye parameter
Bjerrum Length
Segment charge
Neutralization
Non-electrostaticMaximum concentration
Tail fraction
QT = + 10 (CCMV)
(free RNA in solution)
RNA/tail association: “unveils” strong RNA self-attraction
Second Virial Coefficient
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VoltmeterDV
“Donnan Potential”
15-20 milli Volt
qM q
Chargeneutral
RNA Globule
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Large, reversible first-order phase transition
Chemical potential tails
• Common-tangent Construction: Phase Coexistence
* Minimize with respect to R
Gel swelling/shrinking
RNA/tail affinity
V( ) < q0
V( ) > q0
qm= ql /QTq
Condensed, neutralized
Swollen, charged
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Combine:
# Surface-Adsorbed CPs = # Tails
V) Free Energy Landscape
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# proteins/segment
Radius R
Micro-segregation60 excess CP dimers
Tail-neutralized
Charged
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Is this processes thermodynamic reversible self-assembly?
Step 1 Reversible
Same CP chemical potentials
Lowered CP chemical potentialEnhanced RNA self-attraction
IrreversibleStep 2
Lowered CP chemical potential
+ 60 IrreversibleStep 3
“Michaelis-Menten like”
Protein-RNA assembly
Micro-segregation
Protein expulsion
Donnan Potential + Protein Self-repulsion
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Capsid Proteins
RNA
Tails
Brownian Ratchet:
How are excess proteins expelled?
--
--
-
-
+
+
+_
__
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Toy T=1 Virus
genome binding sites”
Flexible linear polymer genome
Protein-Protein binding sites
Elrad and Hagen
* Genome molecule: no branching.* Assembled state: # binding sites = chain length
How good is mean-field theory?
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Protein-genome affinity e > Protein-protein affinity J
time
* RNA/Protein pre-assembly condensate
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A B C D E
A C
D
E
Problem: Optimal anglesvisible in A-C
• Local correlations.
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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
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Partial shells
“Down the funnel”
Many possible assembly pathways
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“Antenna-Assembly”
(Hu and Shklovskii)
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“Hamiltonian Cycle”
• Graph-theoretic problem (R.Twarock)
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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 producesa more complex free energy landscape with different possible multi-step Irreversible pathways.
3) Viral assembly appears intermediate between Mark I and Mark II assembly.