GEM4 Summer School OpenCourseWare

Transcription

1 GEM4 Summer School OpenCourseWare Lecture: Polymer Chains by Ju Li. Given August 16, 2006 during the GEM4 session at MIT in Cambridge, MA. Please use the following citation format: Li, Ju. Polymer Chains. Lecture, GEM4 session at MIT, Cambridge, MA, August 16, (accessed MM DD, YYYY). License: Creative Commons Attribution-Noncommercial-Share Alike. Note: Please use the actual date you accessed this material in your citation.

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

3 2 Outline Freely Jointed Chain Worm-Like Chain Persistence Length

4 Freely Jointed Chain (FJC) x 0 b 1 x 4 b 4 x 3 b 3 degrees of freedom: {x i }, i=0..n b 5 b 2 x 1 x 2 x 5 under constraints: x i x i-1 = b i = b b is called the Kuhn length x N b N x N-1 no energy penalty otherwise In particular, the Kuhn segments can penetrate right through each other. 3

5 When it is not pulled Probability distribution of b i : dp = ρ ( b )d 3 b = δ ( b b ) db dω i i i i 4π b i = 0, var[ b i ] b i 2 = b 2 b i and b i+1 are uncorrelated. Define end-to-end distance: D b 1 + b b N. 4

6 We can now apply central limit theorem: D = 0, var[ D ] D 2 = Nb 2 Furthermore, we know distribution of D must approach Gaussian 3 dp exp 3 D 2 d 3 D 2π Nb 2Nb 2 The radius of gyration R g <D D> 1/2 characterizes the spatial extent of the molecule N 1/2 b, which is much smaller than the contour length L=Nb. 5

7 Suppose x 0 is fixed, and x N is pulled with external force f: Gibbs free energy of the system: G = F f D= E T S f D= T S f D F k T B ln Z = const ktln d 3 b d 3 b 3 b exp( β h(b )) B 1 2 d N i i=1 N where h b ) δ ( b b) ( i i express the constraint condition. 6

8 G = k Tln Ze B fd = k Tln Ze β B βf ( b 1 + b 2 + +bn ) = const kt ln d 3 b d 3 3 b d b exp( β )) B 1 2 N h (bi i=1 where h (b i ) δ ( b i b ) f b i The above is equivalent to N biased segments, still independent of each other. Each biased segment is controlled by z i d eβ fbcos θ i = 2π eβ fbcos θ i d cos Ω i θ = 4π i sinh(β fb) β fb cos θ i = ln z i 1 = coth( β fb). (β fb) β fb 7 N

9 X = D f ˆ 1 = N b cos θ i = Nb coth( β fb ) β fb X fb = Λ L k T B Langevin function Λ( x ) coth( x ) x 1 ~1-x Λ(x) ~x/3 x 8

10 Worm-Like Chain (WLC) l=l l=0 continuously bendable but unstretchable dl also called Kratky-Porod model L κc 2 Bending energy penalty: E[ x ( l )] = dl d 2 x Yπ r 4 curvature C = =, κ = for solid cylinder 2 R dl 4 9

11 dx dt L κdt dt t, t = 1, C =, E[ t( l )] = dl dl 0 2dl over solid angle 4π Z dt( L) d t d ( )exp κ t t (2 L)... t L β 2 L The above path integral can be shown to be equivalent to diffusion of a free particle constrained on a spherical shell (quantum propagator in imaginary time), with diffusivity identified as 1/2βκ ( time is l). t(l) t(0) orientation space 10

12 2 2l So, t( l ) t(0) when l is small. βκ t( l ) t(0) 1 l l exp( ) βκ βκ The exponential decay form turns out to be valid in both short- and long-l limit. κ Let us define persistence length p = βκ = kt l t( l ) t(0) exp( ) p The above orientation correlation function is similar to velocity auto-correlation function in Brownian motion. B 11

13 Two segments on the same polymer chain can be considered to be strongly correlated in orientation if they are separated by less than the persistence length p. 12 But this correlation dies rather quickly for segments separated by chain length > p. Consider a quarter-circle: t i R t f The initial tangent t i and the final tangent t f can be considered an example of orientation decorrelation. But, this quarter-circle could be very expensive to make if R is too small.

14 What is the right price? π R κ κ κ kt kt R R p 2 2R 2 B B R k B T You can afford to make a loop of R > p. The force-extension behavior of WLC-Kratky-Porod model does not admit simple closed-form solution like the Langevin function. The problem can be mapped to a charged quantum particle confined on a unit sphere, biased by an electric field a Sturm-Liouville eigenfunction problem seeking the lowest eigenvalue. Requires diagonalization of a small matrix, after converting to spherical harmonic basis. 13

15 Extension z/l exact interpolation variational Interpolation form: fp X 1 = + kt L 4 B 1 2 X 4 1 L Force fa/k B T Comparison of three different calculations of the WLC extension z/l as a function of fa/k B T. The solid line shows the numerical exact result; the dashed line shows the variational solution; the dot-dashed line shows the interpolation formula. All three expressions are asymptotically equal for large and small fa/k B T. Good thing about this interpolation form is that both limits are asymptotically correct. Figure by MIT OpenCourseWare. After Marko and Siggia, Marko and Siggia, Macromolecules 28 (1995)

16 The concept of persistence length is also applicable to Freely Jointed Chain model, if we make the following calibration. b So the equivalent persistence length in FJC model is p = L D D = dlt( l 0 ) dl t( l ) 0 L = dldl t( l ) t( l ) 0 L g () l dl L We know g(0) = 1. If we claim g( l ) is "equivalent" to exp( l / p), then D D = 2Lp. However, we know in FJC model, D D = Nb 2 = Lb.

17 How well does it actually work? 1.0 Extension z/l for L-32.8 µ Force f (kt/nm) In the low-stretch regime: 3kB T X f (WLC) 2 p L 3kB T X = (FJC) b L Fit of numerical exact solution of WLC force-extension curve to experimental data of Smith et al. ( bp DNA, 10 mm Na + ). The best parameters for a global least squares fit are L = 32.8 m and A = 53 nm. The FJC result for b = 2A = 100 nm (dashed curve) approximates the data well in the linear low- f regime but scales incorrectly at large f and provides a qualitatively poorer fit. Figure by MIT OpenCourseWare. After Marko and Siggia, Marko and Siggia, Macromolecules 28 (1995)

18 Self-Avoiding Random Walk (SAW) Experiments show that R g grows more quickly than N 1/2. Flory suggested this is due to excluded volume of the polymer chain. He predicted that: R g ~ N 3/(2+d) = N 0.6 (in 3 dimensions) = N 0.75 (in 2 dimensions) Self-avoiding random walker moves on a lattice and never re-visits a site if it has been visited before. The more precise result in 3D is R g ~ N 0.588±

19 18 Reading List Doi, Introduction to Polymer Physics (Oxford University Press, Oxford, 1996). Doi and Edwards, The Theory of Polymer Dynamics (Clarendon Press, New York, 1986). Boal, Mechanics of the Cell (Cambridge University Press, New York, 2002).