Problem #2 Polymer entropic elasticity.

a) Please explain the difference between the physics assumptions for freely-jointed chain, extensible freely jointed chain, worm-like chain and extensible worm-like chain. For worm-like chain, what is the physical meaning of its persistence length lp?

b) Please explain the difference between the freely-jointed chain and the freely-rotating chain models, and cite your references. Which model yields a greater in general? And why?

c) For a 3D freely-jointed chain, with n rigid bonds at fixed length b, knowing that the distribution function of its end-to-end separation distance r follows Gaussian distribution:

where Please derive its mean-squared end-to-end distance , most probable end-to-end distance rmax (where P(r) at its maximum), and the entropic spring constant k at small extensions. Hints:

(1) By definition, the full integration of probability function equals 1:

(2) The free energy of the freely jointed chain is dominated by its conformational entropy:

(3) For small extension, the contribution of r^2 term in P(r) to the free energy is negligible.

d) The Excel file attached contains the data of extending one polymer molecule via AFM single molecule force spectroscopy. Assuming the persistence length lp = 0.55 nm, please estimate its contour length from worm-like chain model by using the nonlinear fit functions in Matlab.

Hints: Choose a value slightly higher than the maximum measured extension, e.g. 100 nm, as the start point, please check the documents for functions: fitoptions, fittype and fit.

P(r)=4r2(1/2)3exp(2r2) =3/(2na2) 0P(r)dr=1 S=kBlnP(r)+ const.. Problem \#2 - Polymer entropic elasticity a) Please explain the difference between the physics assumptions for freely-jointed chain, extensible freely jointed chain, worm-like chain and extensible worm-like chain. For worm-like chain, what is the physical meaning of its persistence length lp ? b) Please explain the difference between the freely-jointed chain and the freely-rotating chain models, and cite your references. Which model yields a greater r2 in general? And why? c) For a 3D freely-jointed chain, with n rigid bonds at fixed length b, knowing that the distribution function of its end-to-end separation distance r follows Gaussian distribution: P(r)=4r2(1/2)3exp(2r2) where =3/(2na2). Please derive its mean-squared end-to-end distance r2, most probable end-to-end distance rmax (where P(r) at its maximum), and the entropic spring constant k at small extensions. Hints: (1) By definition, the full integration of probability function equals 1 : 0P(r)dr=1. (2) The free energy of the freely jointed chain is dominated by its conformational entropy: S=kBlnP(r)+const.. (3) For small extension, the contribution of r2 term in P(r) to the free energy is negligible. d) The Excel file attached contains the data of extending one polymer molecule via AFM single molecule force spectroscopy. Assuming the persistence length lp=0.55nm, please estimate its contour length from worm-like chain model by using the nonlinear fit functions in Matlab. Hints: Choose a value slightly higher than the maximum measured extension, e.g. 100nm, as the start point, please check the documents for functions: fitoptions, fittype and fit