RNA-directed Viral Assembly

1. I) Self-Assembly and Free Energy Minimization. RNA-directed Viral Assembly II) Fundamental Interactions. III) Self-Assembly Empty Capsids. IV) Condensation of RNA genome molecules. V) Free Energy Landscape Viral Assembly.

2. I) (1995, Scientific American) “Mark I” Self-Assembly Self-assembling Monolayer

3. “Thermodynamic Assembly”: assembled and disassembled components in thermal equilibrium. Amphihilic Molecules Variational Principle: Gibbs Free Energy G = U – TS – m N dG = 0 “Hydrophilic” “Hydrophobic” Limited Complexity

4. Synthetic Chemistry: J-M Lehn, D.Cram circular helical * weak, non-covalent bonds * water soluble “host-guest”

5. “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 ?

6. Genome: II ) Fundamental Interactions. • Cowpea Chlorotic Mottle Virus: CCMV • (J. Johnson et al.) T=3 Capsid 180 identical proteins • In-vitro Self-Assembly

7. A) Capsid Proteins: Amphiphilic Layer capsid proteins N-terminal tail Water • Layer: Spontaneous Curvature Hydrophobic • Expect reversible, thermodynamic assembly Water

8. B) • Strength attractive interactions increases with acidity.

9. C) Electrostatic Interactions • Water-accessible equipotential surfaces. Blue positive; Red negative. • Inside-Outside Voltage Difference (McCammon et al.)

10. Electrical Charges CCMV Dimers QC(pH=7) = -28 “ core” charges (physiological) Very large ! + + D = 2 nm QT=+20 “tail” charges/dimer

11. 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.

12. III) Self-Assembly Empty Capsids Electrostatic Repulsion vs. Hydrophobic Attraction + +

13. Treat viral assembly as a chemical reaction: Assembled T=3 capsid 90 Free CP Dimers (“subunits”) Thermal Equilibrium Concentrations “Law of Mass Action” Dimer Concentration “Signature” of Thermodynamic Self-Assembly • DG = assembly energy/dimer

14. Empty capsid assembly experiments * Acidic environment (low pH) Chromatography dG = 0 • DG ≈ 30 kBT/dimer. • Capsid assembly is irreversible!? reversible irreversible

15. Capsid Van der Waals/Landau Free Energy Ns adsorbed proteins R D QC=28 rs = [Ns / R2] area density “order parameter” Entropic Free Energy 2D ideal solution e: Adsorption energy proteins on sphere. vs: Second “virial coefficient” ws: Third virial coefficient ≈ kBT D4 Thermal equilibrium: Chemical potential proteins

16. Second Virial Coefficient Qc=-28 • Electrostatics vs Hydrophobicity • vS = vDH - J Qc=-28 - - - - - - - 2D - Capsid Proteins “Bjerrum Length” ≈ nm R ψ “Debye Parameter” ≈ 1/nm Angle-dependent hydrophobic attraction Optimal Angle/ Radius CCMV (pH=5): QC = 20 vDH /kB T = 400 nm2 Measured for empty shells Capsid Radius CP-CP Hydrophobic Attraction

17. Debye-Hückel Theory of Aqueous Electrostatics Macro ion Charge Density (CPs/RNA) Dielectric Constant Water Debye parameter Electrical Potential - Bjerrum length Electrostatic Free Energy Sheet of charges

18. Summary • Delicate balance between large repulsive interactions and large attractive interactions • Second virial coefficient depends on the sphere radius R. VS ≈ VDH RC R R*

19. Free energy “landscape” F(R,Ns) F(R,Ns) R vs> 0 R* vs= 0 Rc vs< 0 Ns Nc =90 “Common-tangent construction” Phase-coexistence: nearly closed shells and nearly bare spheres

20. 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

21. Free energy F = U - TS RNA Condensation Condensing Agent “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) • Highly cooperative, first-order phase transition. Koculi E, Lee NK, Thirumalai D, Woodson SA. J Mol Biol. 2004, 341(1):27-36.

22. Tertiary contacts N state: folded • Ribozyme (Tetrahymena) RNAse • (Cech)

23. RNA inside T=3 virus: • Highly condensed What are the condensing agents ? 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)

24. CCMV Dimers Remove Protein Cores QT=+20 tail charges/dimer

25. Condensation of CCMV RNA R Good Solvent: “fractal” R(N) ≈ N 1/2 l =5bp • Flory-Landau mean-field theory # segments N N=300 segments Entropic Elasticity U Radius gyration of an “ideal” Flory-Stockmayer branched polymer Linear polymers: much larger • V(q): Second Virial Coefficient. q: # tails / segment • W: Third Virial Coefficient ≈ kB T l 6

26. Condensed Globule Swollen Fractal V V=0 “Theta Solvent” • CCMV RNA genome free in solution • R ≈ 11 nm • l = 0.5 nm • V(q=0)/ kB T = 1-10 nm3 * No phase transition

27. Second Virial Coefficient Segment charge Tail fraction Bjerrum Length ql = - 10 Maximum concentration Non-electrostatic QT = + 10 (CCMV) Polyvalent Counterion Charge Neutralization Debye parameter (free RNA in solution) RNA/tail association: “unveils” strong RNA self-attraction

28. RNA Globule 15-20 milli Volt Voltmeter DV qM q “Donnan Potential” Charge neutral

29. RNA/tail affinity * Minimize with respect to R Chemical potential tails V(q) > 0 Swollen, charged V(q) < 0 Condensed, neutralized q qm= ql /QT • Common-tangent Construction: Phase Coexistence Gel swelling/shrinking Large, reversible first-order phase transition

30. V) Free Energy Landscape Combine: # Surface-Adsorbed CPs = # Tails

31. Charged Tail- neutralized Radius R Micro-segregation 60 excess CP dimers # proteins/segment

32. Is this processes thermodynamic reversible self-assembly? Step 1 Reversible Protein-RNA assembly Same CP chemical potentials Irreversible Step 2 Micro-segregation Lowered CP chemical potential Enhanced RNA self-attraction Irreversible + 60 Step 3 Protein expulsion Lowered CP chemical potential Donnan Potential + Protein Self-repulsion “Michaelis-Menten like”

33. How are excess proteins expelled? - - Brownian Ratchet: - - - - + Capsid Proteins _ + _ + _ Tails RNA

34. 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

35. Protein-genome affinity e > Protein-protein affinity J time * RNA/Protein pre-assembly condensate

36. A B C D E Problem: Optimal angles visible in A-C A C D • Local correlations. E

37. 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

38. “Down the funnel” Partial shells Many possible assembly pathways

39. “Antenna-Assembly” (Hu and Shklovskii)

40. “Hamiltonian Cycle” • Graph-theoretic problem • (R.Twarock)

41. Conclusions Assembly of small ss RNA viruses can be viewed as the combination of reversible RNA condensation + quasi-reversible shell formation. 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.