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Statistical Mechanics of Branched Polymers Physical Virology ICTP Trieste, July 20, 2017 Statistical Mechanics of Branched Polymers Physical Virology ICTP Trieste, July 20, 2017 Alexander Grosberg Center for Soft Matter Research Department of Physics, New York University My question to an anonymous referee: why do you think this is a perpetual motion machine? First part based on this image never published… From RNA to branched polymer Drawing by Joshua Kelly and Robijn Bruinsma To view RNA as a branched polymer is an imperfect approximation, with limited applicability … but let us explore it. Drawing by Angelo Rosa What is known about branched polymers? Classics 1: Main result: gyration radius 1/4 Rg ~ N Excluded volume becomes very important, because Bruno Zimm N/R 3~N1/4 grows with N Walter Stockmayer 1920-2005 g 1914-2004 “Chemical diameter” other people call it MLD Zimm-Stockmayer result: “typical” chemical diameter ~N1/2 Then its size in space is (N1/2)1/2=N1/4 Interesting: similar measures used to characterize dendritic arbors as well as vasculature in a human body or in a plant leave… Two ways to swell Always May or may not possible be possible Quenched Annealed Flory theory: old work and recent review Flory theory Highly Even less nontrivial trivial Quenched Annealed Kramers matrix comment Already a prediction J.Kelly developed algorithm to compute the right L Kramers theorem J.Kelly developed algorithm to compute the right L Quenched Qkm = ±K(k)M(m) 2 Trace(Qkm) = Rg (see Ben Shaul lecture yesterday) Maximal eigenvalue of Qkm gives the right L Hendrik (Hans) Kramers (1894-1952) Prediction works reasonably well Other predictions are also OK Green – single macromolecule in a good solvent; Magenta – single macromolecule in Q-solvent; Blue – one macromolecule among similar ones in a melt. ρ L~Nρ R~Nν~Lνpath path path ν ν ν ν ν Flory theory can be useful Observation: viral RNA seem to be “compact” Viral RNA (Qβ, ground state found by Vienna algorithm) Randomly reshuffled RNA (typical representative) Question: how does the work of compression depend on the sequence? Tricky question, because RNA has to be much compressed (~1bp/nm3), so all interactions are hugely important A.M.Yoffe, …, A.Ben-Shaul “Predicting the sizes of large RNA molecules”, PNAS, v. 105, 16153-16158, 2008 Work of compression Our main idea: Strongly Does NOT dependent on significantly the sequence depend on the sequence For interactions part, we can use, any of the classical interpolation formulas, e.g., van der Waals or Flory-Huggins Work of compression Randomly shuffled Dashed lines Flory-Huggins, Qβ solid lines van der Waals Differential work of compression: simplified formula Maximal eigenvalue of Kramers matrix Can we do better? Hundreds of states within 1kBT of the ground state … Let us go to the opposite limit of annealed branching pattern Confinement work: de Gennes style scaling argument 1/4 Gyration radius without confinement Rg~aN 1/4 x Confinement entropy ∆S=f(R/Rg) -> (R/aN ) Should be extensive, proportional to N -> ∆S~N(a/R)4 Different from uncertainty principle A primer on Schrödinger equation method for a linear chain Uncertainty principle: To confine particle within R you need to provide kinetic energy (ħ/R)2/m To confine polymer within R you need to provide entropy (a/R)2 Interesting: another version of quantum mechanics Oscar Klein 1894-1977 What is known about branched polymers? classics 2: P.-G. de Gennes 1932-2007 One step more after de Gennes Schrodinger How side eq., treats branches affect “diameter” as the “diameter” a linear chain Recent paper in an unusual journal Ilya M. Lifshitz (1917-1982) turns 100 this year Simplest problem: confinement Maps on classical mechanics Maps on quantum mechanics Main results for the confinement problem Confinement free energy is consistent with scaling 1D 3D “Territories” in influenza virus (genome consists of 8 RNAs) Acknowledgements and Conclusions • Robijn Bruinsma • Joshua Kelly • Ralf Everaers, Angelo Rosa, Michael Rubinstein • Alexander Gutin, Eugene Shakhnovich • Roya Zandi Thank you!.
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