MAE 545: Lecture 4 (9/29) Statistical mechanics of proteins

Size: px

Start display at page:

Download "MAE 545: Lecture 4 (9/29) Statistical mechanics of proteins"

Blanche Joseph
5 years ago
Views:

1 MAE 545: Lecture 4 (9/29) Statistical mechanics of proteins

2 Ideal freely jointed chain N identical unstretchable links (Kuhn segments) of length a with freely rotating joints ~r 2 a ~r ~R ~r N In first part of the lecture we will ignore interactions between different segments: e.g. steric interactions, van der Waals interactions, etc. 2

3 Ideal freely jointed chain N identical unstretchable links (Kuhn segments) of length a with freely rotating joints ~r 2 a ~r ~R ~r N What are statistical properties of the end to end vector ~R? partition function (sum over all possible chain configurations) Statistical mechanics Z = X c expected value of hoi = X e E c/ O c observables Z c For ideal chain all configurations have zero energy cost and contribute equally! e E energy of a given c/ E c configuration 3 T k B temperature Boltzmann constant k B = JK

4 Ideal freely jointed chain N identical unstretchable links of length a with freely rotating joints ~r 2 a ~r ~R ~r N Each ideal chain configuration is a realization of random walk. individual links h~r i i = ~r 2 i = a 2 NX D E D E ~R = ~R 2 ~R = ~r = Na 2 i end to end distance i= For large N the probability distribution for p ~R (2 Na 2 /3) e R ~ 2 /(2Na 2 /3) 3/2 4 ~R approaches Note: not accurate in tails of distribution where ~ R Na

5 Ideal freely jointed chain ~r 2 a ~r ~R ~r N D ~R m E = Z Statistical mechanics D ~R m E = X c d 3 ~ R ~ R m ~R m c e E c/ Z (2 Na 2 /3) e R ~ 2 /(2Na 2 /3) 3/2 } proportional to the number of random walks with end to end distance ~R. 5

6 Stretching of ideal freely jointed chain ~r 2 a ~rn fixed end ~r ~R Each chain configuration is a realization of biased random walk. Work due to external force W = ~ F ~R Statistical mechanics ~F partition function Z = X c e (E c W c )/ = X c e (E c ~ F ~ Rc )/ average end to end distance D ~R E = X c ~R c e (E c ~ F ~ Rc )/ Z ln ~ F 6

7 Bias for a single chain link ẑ ŷ ˆx ~r ~F x /a partition function bias in direction of force Z = hx i = Z Z hx i = a d(cos ) 2 d(cos ) 2 coth e Facos /kbt = sinh (F a/k BT ) F a/ a cos e+facos /k BT apple Fa Fa Z small force hx i Fa2 3 large force Langevin function Fa Fa F a/ 7 hx i a F

8 Stretching of ideal freely jointed chain ẑ ŷ ˆx ~r 2 a ~rn ~r Exact result for stretching of ideal chain Z = fixed end hxi = N hx i = Na Z coth ~R apple Fa Fa small force hxi NFa2 3 F entropic spring k constant Gaussian approximation d 3 R ~ [2 Na 2 /3] e R ~ 2 /(2Na 2 /3) F e ~ ~R/ = e NF2 a 2 /6k 2 3/2 B T 2 hxi ln = NFa2 3 Gaussian approximation is only valid for small forces! 8 ~F k = 3k BT Na 2

9 Experimental results for stretching of DNA L = 32.8µm T e - c ideal chain Ideal chain fails to explain experimental data at large forces! c? LU -! L / C L - hxi /L =.5 r J / P -3, /2 F /2 [kbt/nm] 3 hxi hxi [µm] 25 extension z (p) 35 i I Ideal chain predicts hxi L Fa Experiments suggest hxi L C pf -~ -2 lo-' loo ' F force [/nm] f (kt/nm) J.F. Marko and E.D. Siggia, /nm 4pN 9 Macromolecules 28, (995) igure Fit numerical exact solution force

10 Worm-like chain Coordinate along the chain s 2 [,L] R ~r ~t Position vector ~r(s) Unit tanget vector ~t(s) = d~r(s) Radius of curvature R(s) = d ~t(s) = d2 ~r(s) 2 Unstretchable chain of length L, but energy cost of bending is included. E bend = apple 2 R(s) 2 = apple 2 d ~t(s) 2 bending modulus apple Example: energy cost for bending of chain to semicircle 2R =2L/ E bend = apple 2 L R 2 = 2 2 apple L Thermal fluctuations can easily bend chains that are longer than L `p = apple E bend

11 Worm-like chain ~r ~t Distribution of end to end distances ~r(l) = ~r(l) 2 * = ~t(s ) ~r(l) 2 =2`pL apple `p L 2 ~t(s 2 ) ~t(s) h~r(l)i = + = e L/`p Persistence length ~ t(s) ~t(s + `) = e ` /`p `p apple Example: DNA at room temperature 2 ~ t(s ) ~t(s 2 ) = L `p `p 5nm 2 e s ~r(l) 2 =2`pL = 2Lapple s 2 /`p L = Na ~r(l) 2 = Na 2 This is equivalent to ideal chain for a =2`p N = L/(2`p) length of Kuhn segment number of Kuhn segments Polymers shrink, when temperature is increases!

12 Stretching of worm-like chains ẑ ŷ ˆx ~r ~F ~t In order to calculate the response of worm-like chains to external force, we need to evaluate partition function Z = X c e E bend(c)/ hxi ln This is very hard to do analytically for the whole range of forces, but we can treat asymptotic cases of small and large forces. 2

13 Stretching of worm-like chains at small forces ẑ ŷ ˆx ~r F `p ~F fixed end At small forces Gaussian approximation works well hxi NFa2 3 = 2FL`p 3 = 2FLapple 3k 2 B T 2 F k entropic spring constant k = 3k BT = 3k2 B T 2 2L`p 2Lapple 3

14 Stretching of worm-like chains at large forces ẑ ŷ ˆx fixed end ~t At large forces chains are nearly straight and oriented along x direction. q ~t(s) = A y (s) 2 A z (s) 2 ˆx + A y (s)ŷ + A z (s)ẑ ~F F `p A y (s),a z (s) Bending energy to the lowest order in Ay and Az. E bend = apple 2 d~t(s) apple 2 " day 2 # 2 (s) daz (s) + Work due to external forces to the lowest order in Ay and Az. W = ~ F ~r(l) = ~ F ~t(s) FL F 2 Ay (s) 2 + A z (s) 2 4

15 Stretching of worm-like chains at large forces ẑ ŷ ˆx fixed end ~t ~F F `p E = E bend W FL+ 2 " day 2 (s) apple + Free energy to the lowest order in Ay and Az. # 2 daz (s) + F A y (s) 2 + A z (s) 2! A y,z (s) = X q Rewrite free energy in terms of Fourier modes e iqs Ã y,z (q) Ã y,z (q) = E FL+ 2 X q From equipartition theorem we can find average fluctuations of Ay and Az D Ãy(q) 2E = L(F + appleq 2 ) D Ãz(q) 2E L e iqs A y,z (s) h Ãy(q) 2 + Ãz(q) 2i L(F + appleq 2 ) q =2 m/l m =, 2, 5

16 Stretching of worm-like chains at large forces ẑ ŷ ˆx fixed end ~t ~F F `p At large forces chains are nearly straight and oriented along x direction. ~t(s) apple A y (s) 2 + A z (s) 2 2 ˆx + A y (s)ŷ + A z (s)ẑ From equipartition theorem we can find average fluctuations of Ay and Az D Ãy(q) 2E = D Ãz(q) 2E L(F + appleq 2 ) hxi * " hxi L Stretching due to large force is Ay (s) 2 + A z (s) 2 #+ apple 2 2 p F apple = L 6 = L L 2 " X q hd Ãy(q) 2E + s 4F `p # D Ãz(q) 2Ei

17 Stretching of worm-like chains ẑ ŷ ˆx ~F ~F Small force F `p hxi L 2F `p 3 hxi L Large force F `p " s 4F `p # Approximate expression that interpolates between both regimes F `p = 2 hxi 4 L 4 + hxi L 7 J.F. Marko and E.D. Siggia, Macromolecules 28, (995)

Stretching of worm-like chains --T I ' ""I I 8 L -exact t - -interpolation 3 6 - - - variational hxi /L L 2-.

18 Stretching of worm-like chains --T I ' ""I I 8 L -exact t - -interpolation variational hxi /L L 2-./ exact (numerics) Interpolation formula F `p = 4 hxi L hxi L t A & lo-z lo-' ' o2 force F `p/kfa/k,t B T 8 J.F. Marko and E.D. Siggia, Macromolecules 28, (995)

Experimental results for stretching of DNA L = 32.8µm T e c? LU -! - c L / C L - hxi /L ideal chain =.5 r J / P -3, 3 hxi hxi [µm] 25 extension z (p) 35 i -~ -2 lo-' loo ' F force [/nm] f (kt/nm) J.F. Marko and E.

19 Experimental results for stretching of DNA L = 32.8µm T e c? LU -! - c L / C L - hxi /L ideal chain =.5 r J / P -3, 3 hxi hxi [µm] 25 extension z (p) 35 i -~ -2 lo-' loo ' F force [/nm] f (kt/nm) J.F. Marko and E.D. Siggia, /nm 4pN 9 Macromolecules 28, (995) igure Fit numerical exact solution force /2 F /2 [kbt/nm] worm-like chain I Stretching of the DNA backbone " s # hxi L + FL 4F `p For DNA `p = 5nm 5/nm 2nN Improved interpolation formula F `p = 4 + hxi L hxi L + F F 2 4

20 Steric interactions So far we ignored interactions between different chain segments, but in reality the chain cannot pass through itself due to steric interactions. Example of forbidden configuration in 2D Polymer chains are realizations of self-avoiding random walks! ~F Steric interactions are important for long polymers in the absence of pulling forces 2 Steric interactions are not important in the presence of pulling forces.

21 Paul Flory Mean field estimate for the radius of self-avoiding polymers R Approximate partition function: estimate number of self-avoiding random walks of N steps of size a that are restricted to a sphere of radius R. Z(R, N) g N e 3R2 /2Na 2 total number of random walks ln C ev = NX k= ln [2 Na 2 /3] 3/2 2 apple reduction in entropy when constrained to sphere of radius R ka 3 R 3 a 3 R 3 Excluded volume effect ln Z(R, N) N ln g NX k= 2a 3 R 3 (N )a 3 } ka 3 R 3 k 2 a 6 2R 6 Approximate partition function 3 2 ln(2 Na2 /3) C ev reduction in entropy due to excluded volume N 2 a 3 N 3 a 6 2R 3 6R 6 3R 2 N 2 a 3 N 3 a 6 2Na 2 2R 3 6R 6 R 3

Paul Flory Mean field estimate for the radius of self-avoiding polymers R ln Z(R, N) N ln g Approximate partition function 3 2 ln(2 Na2 /3) 3R 2 N 2 a 3 N 3 a 6 2Na 2 2R 3 6R 6 Estimate polymer

22 Paul Flory Mean field estimate for the radius of self-avoiding polymers R ln Z(R, N) N ln g Approximate partition function 3 2 ln(2 Na2 /3) 3R 2 N 2 a 3 N 3 a 6 2Na 2 2R 3 6R 6 Estimate polymer radius by maximizing the partition function N ln g ln ln Z(R, 3R Na R an `p Flory exponent L N 2 a 3 R 4 + N 3 a 6 R 7 = `p =3/5 (higher order term can be ignored) R Exact result from more sophisticated metho non-avoiding random walks =/2

23 R Paul Flory Self-avoiding walks in d dimensions Z(R, N) g N e dr2 /2Na 2 Approximate partition function [2 Na 2 /d] d/2 apple a d R d 2a d R d (N )a d R d ln Z(R, N) N ln g Estimate R by maximizing the partition function R an = 3 d +2 For d 4 Flory exponent is apple /2, but for non-avoiding walk =/2. For d 4 excluded volume is irrelevant! ln Z(R, d 2 ln(2 Na2 /d) /4 3/5 /2 23 dr Na 2 + d 2 dr 2 2Na 2 N 2 a d = Rd+ N 2 a d 2R d Note: except for d=3 these exponents are exact!

Models for Single Molecule Experiments

Models for Single Molecule Experiments Binding Unbinding Folding - Unfolding -> Kramers Bell theory 1 Polymer elasticity: - Hooke s law (?) - FJC (Freely joined chain) - WLC (Worm-like chain) 3 Molecular