Statistical of Branched Polymers Physical Virology ICTP Trieste, July 20, 2017

Alexander Grosberg Center for Soft Matter Research Department of , New York University My question to an anonymous referee: why do you think this is a perpetual 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 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

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!