Statistical Mechanics of Proteins Ideal Freely Jointed Chain

MAE 545: Lecture 4 (9/29)

Statistical mechanics of proteins Ideal freely jointed chain

N identical unstretchable links (Kuhn segments) of length a with freely rotating joints

~r2 a ~r ~r1 N R~

In ﬁrst part of the lecture we will ignore interactions between different segments: e.g. steric interactions, van der Waals interactions, etc.

2 Ideal freely jointed chain N identical unstretchable links (Kuhn segments) of length a with freely rotating joints

~r2 a ~r ~r1 N R~ What are statistical properties of the end to end vector R~ ? Statistical mechanics partition function energy of a given E /k T Ec (sum over all possible Z = e c B configuration chain configurations) c X T temperature E /k T e c B Boltzmann expected value of = kB hOi Oc Z constant observables c X 23 1 For ideal chain all configurations have kB =1.38 10 JK ⇥ zero energy cost and contribute equally! 3 Ideal freely jointed chain N identical unstretchable links of length a with freely rotating joints

~r2 a ~r ~r1 N R~

Each ideal chain conﬁguration is a realization of random walk. individual links ~r =0 ~r 2 = a2 h ii i N ⌦ ↵ end to end ~ ~ ~ 2 2 distance R = ~ri R =0 R = Na i=1 D E D E For large N the probabilityX distribution for R ~ approaches

1 R~ 2/(2Na2/3) Note: not accurate in tails p R~ e ⇡ (2⇡Na2/3)3/2 ⇥ of distribution where ⇣ ⌘ R~ Na | | ⇠ 4 Ideal freely jointed chain

~r2 a ~r ~r1 N R~

Statistical mechanics e Ec/kB T R~ m = R~ m c Z c D E X m 3 m 1 R~ 2/(2Na2/3) R~ = d R~ R~ e ⇥ (2⇡Na2/3)3/2 ⇥ D E Z }

proportional to the number of random walks with end to end distance R ~ .

5 Stretching of ideal freely jointed chain

~r2 a ~r ~r1 N ~ ﬁxed R~ F end Each chain conﬁguration is a realization of biased random walk.

Work due to external force W = F~ R~ · Statistical mechanics

(E W )/k T (E F~ R~ )/k T partition function Z = e c c B = e c · c B c c X X

(E F~ R~ )/k T average end to e c · c B @ ln Z R~ = R~ =+k T end distance c Z B ~ c @F D E X

6 Bias for a single chain link

zˆ ~r1 yˆ xˆ ✓ F~

1 d(cos ✓) Facos ✓/kB T sinh (F a/kBT ) partition function Z1 = e = 1 2 F a/kBT Z 1 d(cos ✓) e+Facos ✓/kB T bias in direction x1 = a cos ✓ h i 1 2 ⇥ ⇥ Z1 of force Z Fa k T x = a coth B Langevin function h 1i k T Fa ✓  B ◆ 1

small force Fa kBT 0.8 ⌧ Fa2 x1 0.6 h i⇡3kBT /a ⟩ 1 x large force Fa kBT ⟨ 0.4 k T 0.2 x a B h 1i⇡ F 0 0 1 2 3 4 5 6 7 8 9 10 F a/kBT 7 Stretching of ideal freely jointed chain

zˆ yˆ ~r2 a xˆ ~r ~r1 N ~ ﬁxed R~ F end Exact result for stretching of ideal chain Fa k T x = N x = Na coth B h i h 1i k T Fa ✓  B ◆ NFa2 F entropic spring 3k T small force x k = B h i⇡ 3kBT ⌘ k constant Na2 Gaussian approximation

3 1 R~ 2/(2Na2/3) F~ R/k~ T NF2a2/6k2 T 2 Z = d R~ e e · B = e B 2 3/2 ⇥ Z [2⇡Na /3] @ ln Z NFa2 x = kBT = h i @F 3kBT Gaussian approximation is only valid for small forces!

8 Experimental results for stretching of DNA 8760 Marko and Siggia Macromolecules, Vol. 28, No. 26, 1995 L = 32.8µm A long enough linear DNA is a flexible polymer with 10- ideal Idealrandom-walk chain fails statisticsto with end-to-end mean-squared distance Ro = (bLY2,where b is the Kuhn statistical chain explain experimental T c monomer size (excluded volume effects can be ignored e data inat mostlarge of forces! the experimental data considered in this LU c? ! paper;* see section 1II.C). The bending costs an energy - Idealper chain length predicts of ~BTAK~/~,where K = laS2rlis the curvature / L (the reciprocal of the bending radius) and where A is

C 2 L -3 / x kBT ,1 the characteristic length over which a bend can be made

h i 1 - /L r L ⇡ Fa i 0.5 with energy cost kBT. This inextensible polymer model nm] 1= x h is variously called the wormlike chain (WLC), the J T/ / B ExperimentsKratky-Porod suggest model, and the persistent chain model. k [

2 = = / For the WLC, Ro2 2AL,and thus b 2Ae7

1 x C h iSince1 A is also the characteristic distance along the P F LWLC⇡ over pwhichF the tangent vector correlations die 30 25 extension z (p) it is called the persistence length. For DNA in vivo x x[µm] 35 h hi i iI (where there is about 150 mM Na+ plus other ions), one 10-~ 10-2 lo-' loo 10' should keep in mind a value A x 50 nm or 150 b~,~ Fforce[kB T/f (kT/nm)nm] although at low ionic strengths electrostatic stiffening J.F. Markocan and causeE.D. Siggia, A to appear as large as 350 nm. Throughout Figure 2. Fit of numerical1k T/ exactnm 4pN solution of WLC force- B 9 Macromolecules 28, 8759-8770 (1995) extension curve to experimental data⇡ of Smith et a1.l (97004 this paper, L >> A is always assumed. bp DNA,10 mM Na+). The best parameters for a global least- Like any flexible polymer, separation of the ends of a squares fit are L = 32.8 pm and A = 53 nm. The FJC result DNA by an amount z << L costs free energy F = 3k~Tz~/ for b = 2A = 100 nm (dashed curve) approximates the data (2R02) and therefore requires a force f = aF/az = 3k~Tz/ well in the linear low-fregime but scales incorrectly at large (W).Below the characteristic force of kBTIA, the f and provides a qualitatively poorer fit. Inset: f1l2 vs z for z the highest forces; the exact WLC result (solid line) is in this extension is small compared to L and this linear force plot a straight line extrapolating to L = 32.8 pm from which law is valid. Since 1 kBT/nm = 4.1 piconewtons (pN), the experimental points begin to diverge above z = 31 pm; for A = 50 nm, kBTIA = 0.08 pN: the forces needed to including intrinsic elasticity (eq 19 with y = 500 kBT/nm, extend DNAs are very small compared to the piconew- dotted curve) improves the fit. tons needed to fully extend conventional polymers (e.g., polystyrene) with Kuhn length b < 1 nm. also make plausible a crossover from an entropic For forces beyond kBTIA, the nonlinear entropic elasticity regime to an intrinsic stretching elasticity elasticity11J2 of the WLC model with fixed total contour regime (where the DNA contour length slightly in- length determines the force-distance behavior. Only creases), recently suggested by Odijk.16 In the same for forces of order the base-stackinglpairing energies/ section, we describe why one can largely ignore effects length 10kBT/nm = 500k~T/Awill the constraint of of excluded volume and spontaneous bends that may fixed arc length cease to be a good approximation, thus occur along DNA because of its heterogeneous base-pair rendering inapplicable the WLC model (see section V). sequence. The effective energy of a stretched WLC is11,20 Section IV discusses experiments that stretch teth- ered DNAs with one free end (Figure lb) with an electric field (again relying on the polyelectrolyte character of DNA) or with hydrodynamic Because of the complexities of dealing with a nonuniform and self- consistently determined tension, these kinds of experi- where the force f appears as a Lagrange multiplier to ments furnish less stringent tests of elastic theory but fix the end-to-end extension z = &*[r(L)- r(0)I. Below are closer to the kinds ofways DNAs and other polymers we will compute the equilibrium extension using the get stretched in the natural world. Finally, section V Boltzmann distribution e-E1kBT. In the remainder of this discusses recent experiments17 showing that strong paper, forces and extensions are taken to be along the forces cause the double-helical "secondary structure" of z axis, and when forces appear with inverse-length B-DNA to abruptly lengthen by a factor of about 1.85. dimensions, a factor of kBT has been suppressed. Although the precise nature of the new DNA state is at A. Simple Calculation of WLC StrongStretch- this time unclear (perhaps it is an extended flat ribbon ing Behavior. When large forces are applied to a or separated random-coil-like single strands), the ge- WLC, the extension approaches the total length L, and ometry of the lengthening is consistent with straighten- the tangent vector fluctuates only slightly around 2.12 ing of the double helix, and the force scale is consistent From the constraint It1 = 1, we see that if t, and ty are with what is necessary to overcome the cohesive free taken as independent components, the t, fluctuations energy binding the DNA strands together. are quadratic in the two-vector tl= [t,,t,], namely, t, = 11. Entropic Elasticity of the Wormlike Chain 1 - tL2/2+ 0(t14). To quadratic order, K~ = (a,tl)2,and we obtain the Gaussian approximation to (1):12 Double-helical B-DNA is a stiff-rod polymer. At length scales comparable to the double-helix repeat of 3.5 nm or the diameter of 2.1 nm, the pairing and stacking enthalpy of the bases makes the polymer very rigid, with a well-defined contour length that may be measured either in nanometers or in base pairs (1 bp where we have expressed the extension in (1)as z = = 0.34 nm).4 DNA conformations may therefore be Jds t, and where terms of higher than quadratic order described by a space curve r(s) of fixed total lengthAL, in tl have been dropped. where s is arc length and where the tangent vector t = Fourier transforms ($L(q) Jds eiqstl(s))decouple the a,r is a unit vector.l8Jg energy into normal modes: Worm-like chain Coordinate along the chain s [0,L] 2 Position vector R ~r(s) Unit tanget vector d~r(s) ~t(s)= ds ~r Radius of curvature ~ t d~t(s) d2~r(s) R(s)= = ds ds2 Unstretchable chain of length L, but energy cost of bending is included. 2  L 1  L d~t(s) bending modulus E = ds = ds bend 2 R(s)2 2 ds  Z0 Z0 Example: energy cost for  L ⇡2  E = = bending of chain to semicircle bend 2 ⇥ R2 2 L Thermal ﬂuctuations can easily bend chains that are longer than  L `p = Ebend kBT kBT ⌧ 2R =2L/⇡ 10 Worm-like chain Persistence length ` /` ~r ~t(s) ~t(s + `) = e| | p ·  ⌦ `p ↵ ⌘ kBT Example: DNA at room temperature ~ t ` 50nm p ⇡ Distribution of end to end distances L ~r(L)= ds ~t(s) ~r(L) =0 0 h i L L Z L L L L 2 s s /` ~r(L) = ds ~t(s ) ds ~t(s ) = ds ds ~t(s ) ~t(s ) = ds ds e| 1 2| p 1 1 · 2 2 1 2 1 · 2 1 2 *Z0 Z0 + Z0 Z0 Z0 Z0 ⌦ ↵ ⌦ ↵ ` 2 p L/`p 2 2L ~r(L) =2`pL 1 1 e ~r(L) =2`pL = L L `p k T  B ⌦ ↵ ⇣ ⌘ ⌦ ↵ This is equivalent to ideal chain for

L = Na a =2`p length of Kuhn segment Polymers shrink, when temperature 2 2 ~r(L) = Na N = L/(2`p) number of Kuhn segments is increases!

⌦ ↵ 11 Stretching of worm-like chains zˆ ~r yˆ ~ xˆ F

In order to calculate the response of worm-like chains to external force, we need to evaluate partition function

E (c)/k T Z = e bend B c X @ ln Z x = k T h i B @F This is very hard to do analytically for the whole range of forces, but we can treat asymptotic cases of small and large forces.

12 Stretching of worm-like chains at small forces

F `p kBT zˆ ⌧ yˆ xˆ ~r F~

ﬁxed end

At small forces Gaussian approximation works well

2 NFa 2FL`p 2FL F x = = 2 2 h i⇡ 3kBT 3kBT 3kBT ⌘ k entropic spring constant 3k T 3k2 T 2 k = B = B 2L`p 2L

13 Stretching of worm-like chains at large forces F ` k T zˆ ~t p B yˆ xˆ ~ ﬁxed F end At large forces chains are nearly straight and oriented along x direction.

~t(s)= 1 A (s)2 A (s)2 xˆ + A (s)yˆ + A (s)zˆ y z y z q A (s),A (s) 1 y z ⌧ Bending energy to the lowest order in Ay and Az.  L d~t(s)  L dA (s) 2 dA (s) 2 E = ds ds y + z bend 2 ds ⇡ 2 ds ds Z0 Z0 "✓ ◆ ✓ ◆ # Work due to external forces to the lowest order in Ay and Az. L L F W = F~ ~r(L)= ds F~ ~t(s) FL ds A (s)2 + A (s)2 · · ⇡ 2 y z Z0 Z0 ⇥ ⇤

14 Stretching of worm-like chains at large forces F ` k T zˆ ~t p B yˆ xˆ ~ ﬁxed F end

Free energy to the lowest order in Ay and Az. 1 L dA (s) 2 dA (s) 2 E = E W FL+ ds  y + z + F A (s)2 + A (s)2 bend ⇡ 2 ds ds y z Z0 "✓ ◆ ✓ ◆ # ! ⇥ ⇤ Rewrite free energy in terms of Fourier modes L iqs ds iqs q =2⇡m/L Ay,z(s)= e A˜y,z(q) A˜ (q)= e A (s) y,z y,z m =1, 2, q 0 L X Z ··· 1 E FL+ L(F + q2) A˜ (q) 2 + A˜ (q) 2 ⇡ 2 | y | | z | q X h i From equipartition theorem we can ﬁnd average ﬂuctuations of Ay and Az

k T A˜ (q) 2 = A˜ (q) 2 B | y | | z | ⇡ L(F + q2) D E D E 15 Stretching of worm-like chains at large forces F ` k T zˆ ~t p B yˆ xˆ ~ fixed F end At large forces chains are nearly straight and oriented along x direction. A (s)2 + A (s)2 ~t(s) 1 y z xˆ + A (s)yˆ + A (s)zˆ ⇡ 2 y z  From equipartition theorem we can find average fluctuations of Ay and Az k T A˜ (q) 2 = A˜ (q) 2 B | y | | z | ⇡ L(F + q2) D E D E Stretching due to large force is L 2 2 Ay(s) + Az(s) L x ds 1 = L A˜ (q) 2 + A˜ (q) 2 h i⇡ 2 2 | y | | z | * 0 " ⇥ ⇤#+ q Z X hD E D Ei k T k T x L 1 B = L 1 B h i⇡ p 4F `  2 F  " s p # 16 Stretching of worm-like chains zˆ yˆ xˆ F~ F~

Small force Large force

F ` k T F `p kBT p ⌧ B 2F `p k T x L x L 1 B h i⇡ 3kBT h i⇡ " s4F `p # Approximate expression that interpolates between both regimes 2 F ` 1 x 1 x p = 1 h i + h i k T 4 L 4 L B ✓ ◆ J.F. Marko and E.D. Siggia, 17 Macromolecules 28, 8759-8770 (1995) Stretching of worm-like chains 8762 Marko and Siggia Macromolecules, Vol. 28, No. 26, 1995

1 0 --T I ' ""I I

0 8 L -exactexact

(numerics)- -interpolation t 0 6 - - - variational 3 L

/L Interpolation formula i x h 2 F ` 1 x 1 x p = 1 h i + h i k T 4 L 4 L B ✓ ◆

02- __1./ a I tA& 00 0' ' 11111'1' ' ' '~"l ' ".'"' ' ' "" lo-z lo-' 100 10' 1 o2 10-~ lo-' 10-1 loo 10' force fA/k,T force f (kT/nm) F `p/kBT Figure 3. Comparison of three different calculations of the Figure 4. Apparent persistence lengths as defined in (15) for WLC extension zlL as a function of fXlk~T.The solid line J.F. highMarko and and low E.D. forces, Siggia, extracted from the 10 mM experimental shows the numerical exact result (Appendix);the 18 dashed lineMacromolecules data of 28,Smith 8759-8770 et al.' in(1995) Figure 2 using L,tf = 33.7 pm. The shows the variational solution (13- 14); the dot-dashed line curves are the two theoretical fits of sections I11 (solid curves shows the interpolation formula (7). All three expressions are show L = 32.8 pm, A0 = 50.8 nm, A = 53 nm, and y = 500 asymptotically equal for large and small fMkBT. kBT/nm; dashed curves show L = 33.7 pm, A0 = 15 nm, A = 52 nm, and y infinite) for z vs f transformed in an identical way matches the result of a battery of other, less direct fashion. measurements.516 The inset of Figure 2 shows that f1I2 goes to zero 2 however, L,ff = 33.4 pm would make Aeff in Figure 4 approximately linearly in z for z - L (the solid line nearly f-independent, thereby establishing that a single shows the asymptotic WLC result zlL = 1 - (kBT/ persistence length can describe both limits. However, [4Af1)1/2for A = 53 nm and L = 32.8 pm). However, for this would still leave the deviations shown in the inset the highest forces, the measured extensions exceed L of Figure 2 pointing to a failure of the naive WLC model. = 32.8 pm, and f1I2 is not quite linear in z (note that The scatter in A,ff for intermediate forces is in any event this effect is very hard to see in the main part of Figure independent of the choice of LeE within the range L,ff = 2 where zlL is plotted versus f). These effects point to 33.4-33.7 pm. The next section shows how the gentle a failure of the pure WLC model at high forces. A decrease of A,a for 10 mM Na+ (Figure 4) may be possible explanation is that the B-DNA is slightly explained by electrostatic self-repulsion, either with or stretched by the highest tensions (e.g., consider that a without intrinsic stretching elasticity. slight untwisting of the double helix will slightly lengthen it; the linear twisting elasticity of B-DNA is 111. Electrostatic Stiffening well characterized5Q Odijk has recently noted that Figures 5a and 5b show A,R, (151, for the experimental relaxing the constraint of fixed length in the WLC model data of Smith et a1.l for 97 004 bp DNAs with L,ff = via the addition of linear stretching elasticity yields the 33.7 pm for 1 and 0.1 mM Na+ (Figures 6a-c show asymptotic stretching16 zlL = 1 - (kBT/[4Af1)112i- fly, extension z versus force f for 10, 1, and 0.1 mM Na'). where L is the total contour length of the unstressed The limiting low-force persistence length in these cases chain and where y is an elastic constant with dimen- greatly exceeds its high-force limit, with most of the sions of force (when y-l - 0, the inextensible WLC variation occurring for zlL > 0.5. There is compara- result is obtained). The estimate y = 16k~TAID2 can tively little variation in A,R measured for high salt (10 be made by supposing DNA to be a homogeneous elastic mM, Figure 4). rod of radius D = 1.2 nm;16takingA = 53 nm, we guess For the low ionic strengths of 0.1 and 1 mM, the y % 600 kBT/nm. Turning to the data shown in the inset electrostatic screening length AD falls within the range of Figure 2 and keeping the previously fit L = 32.8 pm of the WLC elastic correlation length, 6 = (A/flU2.(For and A = 53 nm, we find that y = 500 kBT/nm can the Na2HP04 solution used,l we expect complete 2: 1 account for the slight deviation from WLC behavior at Na-HP04 dissociation at 300 K,23which gives AD % high forces (inset of Figure 2, dotted line). 0.25M-u2 nm,24where M is the Na+ molarity; M is used A more revealing way to plot the data and directly to label the data sets). At low forces, electrostatic self- test the WLC model is to define an effective persistence repulsion increases the effective persistence length; for length, Ace, computed separately in the high- and low- high enough forces that << AD, only the "intrinsic" force limits from the simple exact analytic formulas elastic persistence length is observed. Although the relating z to f: electrostatic contribution to the low-force persistence length was discussed long ago by Odijk, Skolnick, and Fixman13 (OSF), Barrat and Joanny14 (BJ) only recently emphasized that the OSF results indicate scale depen- dence of the persistence length. We now incorporate these electrostatic effects into the force-extension Figure 4 shows the 10 mM data of Figure 2 transformed calculations of the previous section to explain the low using L,ff = 33.7 pm in (15). This value of L,R was ionic strength results of Smith et ale1and also examine chosen for consistency with the low-salt data discussed how the resulting fits change when intrinsic stretching in the following section. Given only the data in Figure elasticity (see section 1I.C) is included. Experimental results for stretching of DNA 8760 Marko and Siggia Macromolecules, Vol. 28, No. 26, 1995 L = 32.8µm A long enough linear DNA is a flexible polymer with 10- ideal Stretchingrandom-walk of the statistics with end-to-end mean-squared distance Ro = (bLY2,where b is the Kuhn statistical chain DNA backbone T c monomer size (excluded volume effects can be ignored e in most of the experimental data considered in this LU k T FL c? worm-like x L paper;*1 seeB section+ 1II.C). The bending costs an energy ! chain h i⇡ 4F `p - "per lengths of# ~BTAK~/~,where K = laS2rlis the curvature / L (the reciprocal of the bending radius) and where A is

C 2 L -3 For DNA / ,1 the characteristic length over which a bend can be made

/L ` = 50nm - r p i 0.5 with energy cost kBT. This inextensible polymer model nm] 1= x 500k T/nm 2nN h is Bvariously called the wormlike chain (WLC), the J T/ ⇡ ⇡ / B Kratky-Porod model, and the persistent chain model. k

[ Improved interpolation formula

2 = = / For the WLC, Ro2 2AL,and thus b 2Ae7

1 2 F `p 1 Sincex A Fis also 1the characteristic distance along the

F = 1 h i + k T 4 L 4 P B WLC✓ over which◆ the tangent vector correlations die 30 25 extension z (p) itx is calledF the persistence length. For DNA in vivo x x[µm] 35 +h i h hi i iI (whereL there is about 150 mM Na+ plus other ions), one 10-~ 10-2 lo-' loo 10' should keep in mind a value A x 50 nm or 150 b~,~ Fforce[kB T/f (kT/nm)nm] although at low ionic strengths electrostatic stiffening J.F. Markocan and causeE.D. Siggia, A to appear as large as 350 nm. Throughout Figure 2. Fit of numerical exact solution of WLC force- 1kBT/nm 4pN 19 Macromolecules 28, 8759-8770 (1995) extension curve to experimental data⇡ of Smith et a1.l (97004 this paper, L >> A is always assumed. bp DNA,10 mM Na+). The best parameters for a global least- Like any flexible polymer, separation of the ends of a squares fit are L = 32.8 pm and A = 53 nm. The FJC result DNA by an amount z << L costs free energy F = 3k~Tz~/ for b = 2A = 100 nm (dashed curve) approximates the data (2R02) and therefore requires a force f = aF/az = 3k~Tz/ well in the linear low-fregime but scales incorrectly at large (W).Below the characteristic force of kBTIA, the f and provides a qualitatively poorer fit. Inset: f1l2 vs z for z the highest forces; the exact WLC result (solid line) is in this extension is small compared to L and this linear force plot a straight line extrapolating to L = 32.8 pm from which law is valid. Since 1 kBT/nm = 4.1 piconewtons (pN), the experimental points begin to diverge above z = 31 pm; for A = 50 nm, kBTIA = 0.08 pN: the forces needed to including intrinsic elasticity (eq 19 with y = 500 kBT/nm, extend DNAs are very small compared to the piconew- dotted curve) improves the fit. tons needed to fully extend conventional polymers (e.g., polystyrene) with Kuhn length b < 1 nm. also make plausible a crossover from an entropic For forces beyond kBTIA, the nonlinear entropic elasticity regime to an intrinsic stretching elasticity elasticity11J2 of the WLC model with fixed total contour regime (where the DNA contour length slightly in- length determines the force-distance behavior. Only creases), recently suggested by Odijk.16 In the same for forces of order the base-stackinglpairing energies/ section, we describe why one can largely ignore effects length 10kBT/nm = 500k~T/Awill the constraint of of excluded volume and spontaneous bends that may fixed arc length cease to be a good approximation, thus occur along DNA because of its heterogeneous base-pair rendering inapplicable the WLC model (see section V). sequence. The effective energy of a stretched WLC is11,20 Section IV discusses experiments that stretch teth- ered DNAs with one free end (Figure lb) with an electric field (again relying on the polyelectrolyte character of DNA) or with hydrodynamic Because of the complexities of dealing with a nonuniform and self- consistently determined tension, these kinds of experi- where the force f appears as a Lagrange multiplier to ments furnish less stringent tests of elastic theory but fix the end-to-end extension z = &*[r(L)- r(0)I. Below are closer to the kinds ofways DNAs and other polymers we will compute the equilibrium extension using the get stretched in the natural world. Finally, section V Boltzmann distribution e-E1kBT. In the remainder of this discusses recent experiments17 showing that strong paper, forces and extensions are taken to be along the forces cause the double-helical "secondary structure" of z axis, and when forces appear with inverse-length B-DNA to abruptly lengthen by a factor of about 1.85. dimensions, a factor of kBT has been suppressed. Although the precise nature of the new DNA state is at A. Simple Calculation of WLC StrongStretch- this time unclear (perhaps it is an extended flat ribbon ing Behavior. When large forces are applied to a or separated random-coil-like single strands), the ge- WLC, the extension approaches the total length L, and ometry of the lengthening is consistent with straighten- the tangent vector fluctuates only slightly around 2.12 ing of the double helix, and the force scale is consistent From the constraint It1 = 1, we see that if t, and ty are with what is necessary to overcome the cohesive free taken as independent components, the t, fluctuations energy binding the DNA strands together. are quadratic in the two-vector tl= [t,,t,], namely, t, = 11. Entropic Elasticity of the Wormlike Chain 1 - tL2/2+ 0(t14). To quadratic order, K~ = (a,tl)2,and we obtain the Gaussian approximation to (1):12 Double-helical B-DNA is a stiff-rod polymer. At length scales comparable to the double-helix repeat of 3.5 nm or the diameter of 2.1 nm, the pairing and stacking enthalpy of the bases makes the polymer very rigid, with a well-defined contour length that may be measured either in nanometers or in base pairs (1 bp where we have expressed the extension in (1)as z = = 0.34 nm).4 DNA conformations may therefore be Jds t, and where terms of higher than quadratic order described by a space curve r(s) of fixed total lengthAL, in tl have been dropped. where s is arc length and where the tangent vector t = Fourier transforms ($L(q) Jds eiqstl(s))decouple the a,r is a unit vector.l8Jg energy into normal modes: 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!

Steric interactions are Steric interactions are not important for long polymers in important in the presence

the absence of pulling forces 20 of pulling forces. Mean ﬁeld estimate for the radius Paul Flory of self-avoiding polymers

Approximate partition function: estimate number of R self-avoiding random walks of N steps of size a that are restricted to a sphere of radius R.

3R2/2Na2 3 3 3 e a 2a (N 1)a Z(R, N) gN 1 1 1 1 ⇡ ⇥ [2⇡Na2/3]3/2 ⇥ · R3 · R3 ··· R3  ✓ } ◆ ✓ ◆ ✓ ◆

Cev total number of reduction in entropy random walks reduction in when constrained to entropy due to sphere of radius R excluded volume Excluded volume effect N 1 N 1 ka3 ka3 k2a6 N 2a3 N 3a6 ln C = ln 1 ev R3 ⇡ R3 2R6 ··· ⇡ 2R3 6R6 kX=1 ✓ ◆ kX=1 ✓ ◆ Approximate partition function 3 3R2 N 2a3 N 3a6 ln Z(R, N) N ln g ln(2⇡Na2/3) ⇡ 2 2Na2 2R3 6R6

21 Mean ﬁeld estimate for the radius Paul Flory of self-avoiding polymers

Approximate partition function R 3 3R2 N 2a3 N 3a6 ln Z(R, N) N ln g ln(2⇡Na2/3) ⇡ 2 2Na2 2R3 6R6 Estimate polymer radius by maximizing the partition function @ ln Z(R, N) 3R 3 N 2a3 N 3a6 + + =0 ln Z @R ⇡Na2 2 R4 R7 (higher order N ln g L ⌫ term can be R aN ⌫ ` ignored) ⇠ ⇠ p ` ✓ p ◆ Flory exponent ⌫ =3/5

Exact result from more non-avoiding sophisticated methods random walks ⌫ 0.591 ⌫ =1/2 R ⇡ 22 Paul Self-avoiding walks in d dimensions Flory Approximate partition function

dR2/2Na2 d d d e a 2a (N 1)a R Z(R, N) gN 1 1 1 1 ⇡ ⇥ 2 d/2 ⇥ · Rd · Rd ··· Rd [2⇡Na /d]  ✓ ◆ ✓ ◆ ✓ ◆ d dR2 N 2ad ln Z(R, N) N ln g ln(2⇡Na2/d) ⇡ 2 2Na2 2Rd Estimate R by maximizing the partition function @ ln Z(R, N) dR d N 2ad + =0 @R ⇡Na2 2 Rd+1 3 R aN ⌫ ⌫ = ⇠ d +2

For d 4 Flory exponent is ⌫ 1 / 2 , but for non-avoiding walk ⌫ =1 / 2 .  For d 4 excluded volume is irrelevant! d 1 2 3 4 Note: except for d=3 these ⌫ 1 3/4 3/5 1/2 exponents are exact!