Polymer Chains. Ju Li. GEM4 Summer School 2006 Cell and Molecular Mechanics in BioMedicine August 7 18, 2006, MIT, Cambridge, MA, USA

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1 Polymer Chans Ju L GEM4 Summer School 006 Cell and Molecular Mechancs n BoMedcne August 7 18, 006, MIT, Cambrdge, MA, USA

2 Outlne Freely Jonted Chan Worm-Lke Chan Persstence Length

3 Freely Jonted Chan (FJC) x 0 b 1 x 4 b 4 x 3 b 3 degrees of freedom: {x }, =0..N b 5 b x 1 x x 5 under constrants: x x -1 = b = b b s called the Kuhn length x N b N x N-1 no energy penalty otherwse In partcular, the Kuhn segments can penetrate rght through each other. 3

4 When t s not pulled Probablty dstrbuton of b : 3 b δ dp = ρ( b ) d = ( b b) db 4 b = 0, var[ b ] b = dω b π b and b +1 are uncorrelated. Defne end-to-end dstance: D b 1 + b + + b N. 4

5 We can now apply central lmt theorem: D = 0, var[ D] D = Nb Furthermore, we know dstrbuton of D must approach Gaussan dp D exp d π Nb Nb 3 D The radus of gyraton R g <D D> 1/ characterzes the spatal extent of the molecule N 1/ b, whch s much smaller than the contour length L=Nb. 5

6 Suppose x 0 s fxed, and x N s pulled wth external force f: Gbbs free energy of the system: G = F f D= E TS f D= TS f D F k Tln Z = const B N 3 b 3 b 3 b b B 1 N = 1 ktln d d d exp( β h( )) where h( b ) δ ( b b) express the constrant condton. 6

7 G = k Tln Ze = k Tln Ze βfd βf ( b1+ b+ + bn ) B B N = kt B d 1d d N β h = 1 const ln b b b exp( ( b )) where h ( b ) δ ( b b) f b The above s equvalent to N based segments, stll ndependent of each other. Each based segment s controlled by z d e πe dcosθ snh( β fb) β fb ln z 1 cosθ = = coth( β fb). ( β fb) β fb β fbcosθ cos 4 β fb θ π Ω = = 7

8 ˆ 1 X = D f = N bcosθ = Nb coth( β fb) β fb X fb = Λ L kbt Langevn functon Λ( x) coth( x) x 1 ~1-x Λ(x) ~x/3 x 8

9 Worm-Lke Chan (WLC) l=l l=0 contnuously bendable but unstretchable dl also called Kratky-Porod model L κc Bendng energy penalty: E[ x( l)] = dl curvature d x, Y π C = = κ = r for sold cylnder R dl 4 9

10 dx dt L κd d, = 1, C =, E[ ( l)] = t t t t t dl dl 0 dl over sold angle 4π Z dt( L) dt( L)... dt( L)exp β κ t t L The above path ntegral can be shown to be equvalent to dffuson of a free partcle constraned on a sphercal shell (quantum propagator n magnary tme), wth dffusvty dentfed as 1/βκ ( tme s l). t(l) t(0) orentaton space 10

11 l So, t( l) t(0) when l s small. βκ l l t() l t(0) 1 exp( ) βκ βκ The exponental decay form turns out to be vald n both short- and long-l lmt. Let us defne persstence length p l t( l) t(0) exp( ) p = βκ = κ kt The above orentaton correlaton functon s smlar to velocty auto-correlaton functon n Brownan moton. B 11

12 Two segments on the same polymer chan can be consdered to be strongly correlated n orentaton f they are separated by less than the persstence length p. 1 But ths correlaton des rather quckly for segments separated by chan length > p. Consder a quarter-crcle: t R t f The ntal tangent t and the fnal tangent t f can be consdered an example of orentaton decorrelaton. But, ths quarter-crcle could be very expensve to make f R s too small.

13 What s the rght prce? π R κ κ κ kt kt R R p B B R R kbt You can afford to make a loop of R > p. The force-extenson behavor of WLC-Kratky-Porod model does not admt smple closed-form soluton lke the Langevn functon. The problem can be mapped to a charged quantum partcle confned on a unt sphere, based by an electrc feld a Sturm-Louvlle egenfuncton problem seekng the lowest egenvalue. Requres dagonalzaton of a small matrx, after convertng to sphercal harmonc bass. 13

14 Interpolaton form: fp kt B X 1 = + L 4 1 X 4 1 L Good thng about ths nterpolaton form s that both lmts are asymptotcally correct. Marko and Sgga, Macromolecules 8 (1995)

15 The concept of persstence length s also applcable to Freely Jonted Chan model, f we make the followng calbraton. D D = dlt() l dl t( l ) 0 L L 0 0 = dldl t() l t( l ) L g() l dl We know g(0) = 1. If we clam g( l) s "equvalent" to exp( l/ p), then D D = Lp. L However, we know n FJC model, D D = Nb = Lb. b So the equvalent persstence length n FJC model s p =. 15

16 How well does t actually work? In the low-stretch regme: 3kTX B f (WLC) p L 3kTX b L B = (FJC) Marko and Sgga, Macromolecules 8 (1995)

17 17 Self-Avodng Random Walk (SAW) Experments show that R g grows more quckly than N 1/. Flory suggested ths s due to excluded volume of the polymer chan. He predcted that: R g ~ N 3/(+d) = N 0.6 (n 3 dmensons) = N 0.75 (n dmensons) Self-avodng random walker moves on a lattce and never re-vsts a ste f t has been vsted before. The more precse result n 3D s R g ~ N 0.588±0.001

18 18 Readng Lst Do, Introducton to Polymer Physcs (Oxford Unversty Press, Oxford, 1996). Do and Edwards, The Theory of Polymer Dynamcs (Clarendon Press, New York, 1986). Boal, Mechancs of the Cell (Cambrdge Unversty Press, New York, 00).