10/15/15 Phys 175/275 - ! Biological Physics! Olga Dudko UCSD Physics Dna and RNA looping!A shorter scale phenomenon of widespread significance:...
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The relation between force and extension for a random walk “macromolecule” in three


(a) In class, we have derived the relation between force and extension for a one-dimensional random walk “macromolecule”. Derive the analogous relation between force and extension for a three-dimensional random walk “macromolecule”, in which each monomer is permitted to point in one of six directions. Make a plot of the resulting function. Hint: find the average length of one segment in the direction of the force, <L0>, and then the average length for the entire molecule is N times as large, N<L0>.

(b) In the small-force limit, the force-extension curve is linear; that is, in this regime, the polymer behaves like an ideal Hookean spring with a stiffness constant k ∝kBT/Ltota

Demonstrate this claim and deduce the numerical factor that replaces the proportionality with a strict equality.

10/15/15 1 Phys 175/275 – Biological Physics Olga Dudko UCSD Physics Dna and RNA looping A shorter scale phenomenon of widespread significance: formation of loops in genomic DNA and RNA. Loops in a variety of different settings: Bubble formation in a dsDNA helix Propensity of RNA to form hairpin loops. Example: riboswitch. Long distance DNA looping of chromosomal DNA: a precursor to genetic recombination event L7 DNA looping due to a TF, part of the process of gene regulation
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10/15/15 2 Dna looping: gene activation at a distance Bacterial gene activator protein NtrC contacts RNA polymerase directly even though the two proteins are bound several hundred nucleotide pairs apart The interaction btw NtrC and RNA polymerase, with the intervening DNA looped out, seen in the electron microscope. Entropic cost of Loop formation: 1D RW Model for DNA loop formation by the Lac repressor (DNA-protein interface: X-ray) In order for a protein (e.g., Lac repressor) to spontaneously form a loop in the DNA, the DNA and protein must together suffer a fluctuation that brings all the pieces into physical proximity Cost: elastic energy for the DNA to bend + entropy contribution to the free energy of looping 1D RW: Probability that the two ends of the chain are within a small distance of one another, : The cyclization probability: ! 1 => => the cyclization probability of long DNA strands will decay with polymer length to the power of –1/2 1D !
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