Semiflexible chains under tension

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1 Semiflexible chain under tenion B.-Y. Ha and D. Thirumalai Intitute for Phyical Science and Technology, Univerity of Maryland, College Park, Maryland 74 Received 16 September 1996; accepted 5 December 1996 A functional integral formalim i ued to derive the extenion of a tiff chain ubject to an external force. The force veru extenion curve are calculated uing a mean-field approach in which the hard contraint u ()1 i replaced by a global contraint u ()1 where u() i the tangent vector decribing the chain and i the arclength. The theory quantitatively reproduce the experimental reult for DNA that i ubject to a contant force. We alo treat the problem of emiflexible chain in a nematic field. In the limit of weak nematic field trength our treatment reproduce the exact reult for chain expanion parallel to the director. When the trength of the nematic field i large, a ituation in which there are two equivalent minima in the free energy, the intrinically mean-field approach yield incorrect reult for the dependence of the peritence length on the nematic field American Intitute of Phyic. S I. INTRODUCTION Inpired by the elegant experiment by Smith et al. 1 on the repone of DNA to tetching by a contant force, few theoretical paper 5 have conidered the effect of external field on emiflexible chain. Since DNA i tiff it can be decribed, at leat approximately, a a worm-like chain. The eariet theoretical paper dealing with thi problem wa initiated by Fixman and Kovac. Thee author ued a modified verion of the Gauian model for tiff chain and provided expreion for the tretching a a function of applied force. Their treatment i only valid when the applied force i mall and ignificant deviation from thee prediction are oberved at ufficiently large value of the external force. Marko and Siggia 3 have calculated the extenion a a function of force for worm-like chain and found that their reult fit the experimental data very well. Some apect of thi theory have alo been conidered by Odijk 5 who alo dicue the competition between entropically dominated effect and elaticity effect. In thi paper we reviit the problem of a emiflexible chain ubject to tenion. A functional integral formalim together with a mean-field treatment allow u to et up a general way of tackling problem involving emiflexible chain. When our theory i applied to the cae of tretching of DNA by a contant force, we obtain excellent agreement with experimental reult for the force veru extenion. II. EASTIC RESPONSE OF A SEMIFEXIBE CHAIN A. Mean-field theory The implet model for a emiflexible chain SC i obtained by taking into account the peritence in the tangential direction. A SC can be decribed by a pace curve r() of total contour length with being the arclength. The unit tangent vector i u()r()/ with the contraint that u ()1 for all. Since the molecule i tiff it cot energy to bend and the bending energy per egment length i proportional to (u()/). The difficulty with calculation involving thi formulation of SC i that one ha to invoke the contraint that u ()1. Thi make even the free-chain problem nonlinear. There ha been extenive treatment of model of SC in the literature Now conider applying an external field that tretche the chain. The effective free-energy can be written a where with Fk B T ln Z, Z Duu 1e H/k B T H k B T l p u d f ud. 1a 1b 1c In Eq. 1c l p i the peritence length of the emiflexible chain which in the experiment of Smith et al. 1 on DNA i etimated to be 53 nm. For generality we have aumed that the external field f() depend on the arclength. The cae of uniform f()f i appropriate for the experiment of Smith et al. and i the one treated in the previou theoretical paper. In particular Marko and Siggia 3 have ued an eigenfunction expanion to obtain the extenion veru f by exploiting the analogy between thi problem and the quantum problem of a dipolar rotor in an external electric field. Here we ue a mean-field approach 6,9 that effectively replace the local contraint u ()1 by a global one u ()1. Such a theory ha been hown to give an exact expreion for the mean end-to-end ditance of SC when f(). 6 The baic idea i to enforce the contraint u ()1 uing an auxillary field variable () and evaluate the reulting integral over () by the tationary phae approximation. The free energy cf. Eq. 1 can be rewritten a J. Chem. Phy. 16 (1), 8 March /97/16(1)/443/5/$ American Intitute of Phyic 443 Downloaded 3 May 4 to Reditribution ubject to AIP licene or copyright, ee

2 444 B.-Y. Ha and D. Thirumalai: Semiflexible chain under tenion i D D uu,f expf/k B T i exp d cont i D expf,fcont, i a where u,fexp 1 uq,u u fd and F,fln D uu,f d with Q, l p. b c d The contant in Eq. a come from normaliation aociated with D () and D u(). Thee contant will be omitted from now on. The function F (),f() i a generating functional which can be ued to calculate variou correlation function. For example the connected correlation function u u c u uu u 3a f F 3b f 3Q 1,. Similarly R R R can be written a R d d f F f. 3c By performing the functional integration with repect to u() the free-energy F (),f() can be written a F,u 3 ln Q 1 d df Q 1,f d. The variable (), that enforce the contraint u ()1, i evaluated by tationary phae approach. 6,9 The tationary phae condition i obtained by requiring that F /() bean extremum. Thi lead to the equation Q, d dfq 1, Q 1,f1. The value of () that make F,f() tationary depend on the precie form of f(). We now pecialie to the condition that f()fcont, independent of. Thi i the ituation that ha been treated in the literature and i aumed to be relevant to the experiment of Smith et al. 1 For contant f the tationarity condition Eq. 6 give a uniform value for. More preciely Eq. 6 reduce to 3 1 l p // f , In order to evaluate the firt term in Eq. 7 in a tranparant manner we ue the boundary condition uu() and u()/u()/. Thee condition were implicitly aumed in Eq. b. The eaiet way to compute the firt term in Eq. 7 i in term of an eigenfunction expanion. et denote the eigentate with the curvilinear pace label and let n be the tate that are Fourier conjugate to in uch a way 6 n i n n. 8 The tate n are eigentate of momentum uch that n1/ exp(in/). In term of thee eigentate the firt term in Eq. 7 become 3 1 l p //, 3 1 l p // 3 1 l p // 3 n If we ue the identity n1 n 1 l p /n /. nn co nx n coh x inh 1. the tationarity condition for become 3 4 l p coth 1 f a 1b with /l p. For large Eq. 1b implifie to l p f 4. 1c A f thi tationarity condition coincide with the one derived previouly. J. Chem. Phy., Vol. 16, No. 1, 8 March 1997 Downloaded 3 May 4 to Reditribution ubject to AIP licene or copyright, ee

3 B.-Y. Ha and D. Thirumalai: Semiflexible chain under tenion 445 B. Correlation function and mean-quare internal ditance In term of the tationarity olution of, the variou correlation function can be computed. For example u u 3Q 1, f 3 4 l p 4 coh/ inh/ f For the above equation reduce exactly to the tationarity condition for cf. Eq. 1b leading to u ()1. Thu our approach enure that the contraint u ()1 i atified globally. For Eq. 11 become u u l p e f 4. 1 The mean-quared internal ditance of the emiflexible chain under tenion can be obtained a rr d1 d u 1 u. Subtituting Eq. 1 in Eq. 13 yield rr u1 u d 1 d 3 4 l p 1e 13 f The above equation naturally ugget a length cale, l f, below which entropy factor dominate and above which the mechanical energy aociated with the orienting field dominate. For mall value of the contour length l the field-dependent term become negligible wherea for long l the third term in Eq. 14 dominate. A cale l f i obtained by balancing the firt and lat term of Eq. 14 and i given by 3 f l f f. 15 When fl f 1 then the tationary phae condition can be eaily olved and one get ( f )1/l p and conequently Eq. 15 become l f 1/f l p. Under thee condition one can think of the emiflexible chain to break up into a equence of blob of effective egment length given by f f The above reult coincide with the blob ie predicted by flexible chain under tenion. 16 The above blob length i called tenile creening length. 16 For Eq. 16 to be valid it i neceary that within the length f there hould be a large number of egment each of length l p. Since the effect f within f i negligible the chain on thi length behave effectively a a Gauian chain containing l f /l p unit. It i for thi reaon f in the law f limit coincide with the Pincu blob length. In the limit of l f f, Eq. 15 give l f 1/f and hence the chain conformation i dictated by coupling to the mechanical energy. In thi limit we expect the chain to be aligned with the external field with almot complete uppreion of fluctuation. The effect of f i felt at all length cale, i.e., l f i effectively ero. C. Average chain extenion The quantity of experimental interet i the average extenion of the chain parallel to the external field. The average elongation i computed uing f u 17 f. The tatitical average in Eq. 17 can be conveniently expreed in term of the free-energy functional F. When f i uniform we get f f F f f l p 1 f d f f. 18 In Eq. 18 we have hown the argument of to emphaie the dependence of on f. The above equation and the aociated tationarity condition cf. Eq. 1c determine the average chain extenion. The chain extenion can be eaily calculated for the cae of l p f and for l p f. When l p f weget f R 3, 19 where R l (/3l p ) i the ie of the correponding ideal chain with l /3l p the mean-field peritence length. 6 The above reult i alo obtained for a Gauian chain under tenion. The leading-order correction to Eq. 19 can be obtained for l p f 1 by expanding Eq. 18 in power of f. The tationarity condition for ( f )uptoof become f f, where i the tationary phae condition for f. 6 Thu the average elongation become fl f l. 1 In the mall f limit the entropy conideration dominate the effect due to the orienting field. The dependence of on f i quite different in the oppoite limit. If fl p 1 then we expect the external field to be relevant at all length cale. Thi would uppre the chain fluctuation on cale greater than l p. When l p f the ta- J. Chem. Phy., Vol. 16, No. 1, 8 March 1997 Downloaded 3 May 4 to Reditribution ubject to AIP licene or copyright, ee

4 446 B.-Y. Ha and D. Thirumalai: Semiflexible chain under tenion tionary phae condition for ha a olution 1 reulting in 1/f cf. Eq. 1c. In thi limit / 1 and the chain aume a rod conformation. In the limit of l p f 1 wecan approximately olve Eq. 1c to get f 1 f l p f The chain extenion in the limit of large f become l p f Thi reult ha already been noted in the literature. 5 The f 1/ behavior for emiflexible chain i in contrat to the Gauian cae. Apart from a numerical factor, the reult in Eq. 3, valid for l p f 1, coincide with that dicued recently by Odijk 5 for emiflexible chain near the rod limit. The prefactor in front of 1/l p f obtained by Odijk i 1/ which i lightly maller than our reult. The reaon may lie in the fact that our treatment utilie a tationary phae method to enforce the hard contraint u ()1 globally. In the limit of l p f 1 the chain fluctuation are relatively mall and can be expanded in term of f ( x, y ) where f i the angle between the tangent vector and the external field. The average extenion i related to f () a 1 1 f. 4 For the uniform field we expect f () to be independent of hence f () f. Comparing Eq. 3 we get an etimation of the mean fluctuation f l p f which alo coincide with the reult of Odijk 5 apart from the numerical factor. In addition to yielding the reult reported in the literature our theory offer a imple etimate of the extenion / a a function of f for arbitrary value of f. Thi i achieved by imultaneouly olving Eq. 1c and 18. A a tet of the utility of the tationary phae approach we compare our theory with the experiment of Smith et al. 1 on DNA. Marko and Siggia 3 have already hown that their numerically exact force extenion for the emiflexible chain yield excellent agreement with experiment. In Fig. 1 we plot veru f uing the parameter appropriate for DNA. For comparion a few point from the experiment, a preented in Fig. 1 of Ref. 3, are alo hown. The mean-field approach reproduce the data quantitatively. III. CONCUSIONS In thi article we have conidered the problem of emiflexible chain ubject to an external field uing a functional integral formalim. The crux of our method hinge on replacing the local contraint that u ()1 by a global contraint FIG. 1. The olid line i the force-extenion curve obtained by imultaneouly olving Eq. 1c and 18. The parameter have been choen from the fit of force-extenion curve for DNA reported by Smith et al. Ref. 1. The value of 3.9 m and l 53 nm. The quare repreent the experimental reult of Smith et al. a reported in Ref. 1 and 3. u ()1 where u() i the tangent vector. The intrinically mean-field approach ha been hown to be ucceful in producing the configurational propertie of emiflexible chain in the abence of the external field. 6,7,9 Here we have hown that the tationary phae method of enforcing the global contraint yield excellent reult even when the chain i ubject to tenion. The very good agreement between the theory and the experiment on DNA ubject to force i a confirmation of thi aertion. The approach we have decribed here i ytematic but i not without limitation. Thee limitation become evident by conidering the behavior of a emiflexible chain in a nematic environment, a problem that ha received coniderable attention in the literature. 17 A mean-field Hamiltonian of a tiff tet chain in a matrix of other chain in a nematic tate can be written a 19a H 1 d duq,ug u d, 6 where u()[u (),u ()] and g with the dimenion of length i the trength of the nematic potential. By following the procedure outlined in the previou ection the tationary equation one that extremie the free energy with Eq. 6 a the Hamiltonian become 1 1 l p 1 4 l p g. The above equation enure that 1u u u 7 intead of u ()1 for all. The mean extention of the chain parallel and perpendicular to the director axi can be calculated a J. Chem. Phy., Vol. 16, No. 1, 8 March 1997 Downloaded 3 May 4 to Reditribution ubject to AIP licene or copyright, ee

5 B.-Y. Ha and D. Thirumalai: Semiflexible chain under tenion 447 R u u d d 8a R u u d d. 8b Conider the weak nematic limit, i.e., g. For mall g the tationary condition can be olved and the reult can be ued to get R and R. Thee lead to R /3l gl, 9a R 1/3l gl, 9b where l /3l p. The peritence length along the nematic field i increaed by a factor of 11/3gl which compare well with the exact reult in the limit of of Warner et al. 17 A imilar analyi for g ugget that chain fluctuation perpendicular to the director are totally uppreed wherea the peritence length along the director i predicted to increae by a factor of. Thi reult i in contrat with the analyi of everal author 17,19a,1 who have hown that the effective peritence length grow exponentially a (gl p ) 1/. The nematic potential, gu, ha two deep minima at large g at u 1. Thu in the large g limit the chain configuration i dominated by tunneling between the two minima by an intanton. Mathematically the partition function i dominated by intanton contribution which our mean-field theory fail to capture. It i clear that if there i ymmetry breaking in the problem then the replacement of the hard contraint by a global one can lead to incorrect reult. ACKNOWEDGMENT Thi work wa upported in part by a grant from the National Science Foundation through Grant No. NSF CHE S. B. Smith,. Fini, and C. Butamante, Science 58, M. Fixman and J. Kovac, J. Chem. Phy. 58, J. F. Marko and E. D. Siggia, Macromolecule 8, ; Science 65, A. Kholondenko and T. Vilgi, Phy. Rev. E 5, T. Odijk, Macromolecule 8, B.-Y. Ha and D. Thirumalai, J. Chem. Phy. 13, J. B. agowki, J. Noolandi, and B. Nickel, J. Chem. Phy. 95, R. G. Winkler, P. Reineker, and. Harnau, J. Chem. Phy. 11, M. Otto, J. Eckert, and T. A. Vilgi, Macromol. Theory Simul. 3, A. M. Gupta and S. F. Edward, J. Chem. Phy. 98, K. F. Freed, J. Chem. Phy. 54, M. G. Bawendi and K. F. Freed, J. Chem. Phy. 83, K. Soda, J. Phy. Soc. Jpn. 35, N. Saito, J. Takahahi, and Y. Yunoki, J. Phy. Soc. Jpn., R. A. Harri and J. E. Heart, J. Chem. Phy. 44, P. Pincu, Macromolecule 9, M. Warner, J. M. F. Gunn, and A. B. Baumgurtner, J. Phy. A 18, Z.-Y. Chen, Phy. Rev. ett. 71, a P. G. de Genne, Polymer iquid Crytal, edited by A. Ciferri, W. R. Kringhaum, and R. B. Meyer Academic, New York, 198; b See alo Polymeric iquid Crytal, edited by A. Blumtein Plenum, New York, A. N. Semenov and A. R. Khokhlov, Sov. Phy. Up. 31, G. J. Vroege and T. Odijk, Macromolecule 1, ; T. Odijk, ibid. 19, K. D. Kamien, P.. Doual, and D. R. Nelon, Phy. Rev. A 45, J. Chem. Phy., Vol. 16, No. 1, 8 March 1997 Downloaded 3 May 4 to Reditribution ubject to AIP licene or copyright, ee