Parameter-Free Prediction of DNA Conformations in Elongational Flow by Successive Fine Graining

Parameter-Free Prediction of DNA Conformations in Elongational Flowby Successive Fine Graining

P. Sunthar and J. Ravi Prakash*Department of Chemical Engineering, Monash University, Melbourne, VIC 3800, AustraliaReceived December 18, 2003; Revised Manuscript Received October 21, 2004

ABSTRACT: Brownian dynamics simulations are used to predict the evolution of DNA conformations inelongational flow. The DNA molecule is represented by a bead-spring chain model, and excluded-volumeand hydrodynamic interactions between the beads are taken into account. Two distinct types of behaviorare examined, corresponding to infinite and finite chains. In the former case, Hookean springs are used,and simulations data accumulated for chains with increasing numbers of beads, N, are extrapolated tothe limit N f ∞. In this nondraining limit, universal DNA stretch vs strain curves are obtained as afunction of the solvent quality parameter z. In the case of finite chains with Nk Kuhn steps, finitelyextensible springs are used, and the extrapolation of finite chain data is restricted to the limit (N - 1)f Nk. It is shown that in the presence of finitely extensible springs both the excluded-volume andhydrodynamic interaction parameters need to be redefined appropriately. The role of finite size effectsand sensitivity to the choice of parameter values is examined by comparing the finite chain stretch vsstrain curves with the universal curves. Theoretical predictions are also shown to compare favorablywith experimental observations of DNA stretch.

1. Introduction

The measurement and prediction of the properties ofdilute polymer solutions, both at equilibrium and in thepresence of an induced flow, has been a fundamentalconcern of polymer physics. Until relatively recently, thevalidation of statistical theories of polymer solutiondynamics has largely been confined to comparing pre-dictions of macroscopic observables, such as viscoelasticproperties, with experimental observations. However,recent experiments on dilute solutions of DNA mole-cules,1-4 in which a protocol for the measurement ofcertain microscopic quantities was introduced, haveraised the issue of theoretical validation to a new level.Smith and Chu3 observed, under a microscope, theevolution of λ-phage DNA molecules in an elongationalfield in a tiny cross slot cell and measured the totalextent of the molecules, as they were stretched from acoillike state to the fully elongated state. As a result,they were able to obtain one of the first direct measure-ments of a microscopic quantity that is characteristicof a dilute polymer solution undergoing flow. Whilerecent advances in the theoretical description of polymersolution dynamics have shown that it is essential to takeinto account the existence of certain nonlinear micro-scopic phenomena (which are neglected in the Rousetheory5,6), to obtain accurate predictions of macroscopicvariables, the experimental results of Chu and co-workers provide an opportunity, for the first time, toexamine the importance of these phenomena in thecontext of microscopic variables.

The most important development in molecular theo-ries for the equilibrium and linear viscoelastic behaviorof dilute polymer solutions, since the introduction of theRouse theory (which represents the polymer moleculeby a chain of N beads connected together by (N - 1)linear Hookean springs), has been the recognition thatthe incorporation of excluded-volume (EV) effects isnecessary in order to explain the scaling behavior of

static properties (such as the radius of gyration) in goodsolvents, while the incorporation of hydrodynamic in-teractions (HI) is required to explain the scaling ofdynamic properties (such as the diffusivity and intrinsicviscosity). In addition, far from equilibrium, it has beenfound necessary to replace the Hookean springs of Rousetheory, with finitely extensible springs, to capturecertain observed features, such as the shear thinningexhibited by the shear viscosity, and the boundednessof the elongational viscosity,6 that are a consequence ofthe finite contour length of the polymer molecule.Several theoretical papers,7-9 published after the ex-periments of Smith and Chu, have used some or all ofthese phenomena in their models and have shown thatobtaining quantitative comparison with the DNA stretchmeasurements requires the inclusion of all these threephenomena.

The incorporation of each of these phenomena leadsto the introduction of at least one new parameter intothe theory. For instance, the pairwise strength ofexcluded-volume interactions between any two beads inthe bead-spring chain is typically represented by thenondimensional parameter z*, which is proportional to(1 - Tθ/T), where T is the solution temperature and Tθis the θ-temperature of the polymer-solvent system.The strength of hydrodynamic interactions, on the otherhand, which arise due to the hydrodynamic drag expe-rienced by the beads, is usually measured in terms ofthe magnitude of the parameter h*, the nondimensionalbead radius. Finally, the inclusion of finitely extensiblesprings introduces the parameter b, which is a nondi-mensional measure of the fully stretched length of aspring. Not surprisingly, for any particular choice of thenumber of beads N, a proper choice of the values of theseparameters turns out to have a crucial bearing on theaccuracy of the predictions of the various theories. Allthe previously mentioned papers7-9 determine the cor-rect values of the parameters to use in their simulations,by essentially finding a best fit to experimental data,in some limiting regime of behavior. For instance,Jendrejack et al. simultaneously obtain all three pa-

* Author to whom correspondence should be adressed. E-mail:[email protected].

617Macromolecules 2005, 38, 617-640

10.1021/ma035941l CCC: $30.25 © 2005 American Chemical SocietyPublished on Web 12/15/2004

rameters, z*, h*, and b, by fitting model predictions toexperimentally reported observations of equilibriumλ-phage DNA stretch, relaxation time, and diffusivity.8Larson and co-workers, on the other hand, exploit theexpectation that a polymer coil will unravel into a fullystretched state at high enough strains and consequentlybehave like a long slender cylindrical rod to estimateh*. They neglect the presence of excluded-volume in-teractions (which corresponds to setting z* ) 0) andobtain the parameter b from reported estimates of thenumber of Kuhn steps, Nk, for λ-phage DNA. Thoughthe procedure of Jendrejack et al. is a reasonable schemefor parameter estimation, it is not clear that the chosenset of values is unique and that an alternative set ofparameter values will not yield equally accurate predic-tions of either equilibrium properties or the extent ofstretch in extensional flow. Furthermore, a simulta-neous fit of all the parameters to experimental resultsdoes not recognize the fact that a few of the parametershave no influence on some of the experimental variables.For instance, the parameter h* has no influence on theequilibrium stretch of λ-phage DNA. The parameterestimation procedure of Larson and co-workers leads toaccurate predictions of the stretch at moderate to highvalues of the Hencky strain. However, it leads to a poorprediction of the stretch at low strains.

The aim of the present work, as in the case of theearlier theoretical efforts cited above, is to examine therelative importance of various nonlinear effects indetermining the evolution of DNA stretch in extensionalflow. The problem of parameter estimation is ap-proached here, however, within the broader context ofthe universal behavior exhibited by dilute polymersolutions, both at and close to equilibrium. In particular,an attempt is made to bring the significant insight thathas been gained in recent years into the proper treat-ment of hydrodynamic and excluded-volume interactioneffects in dilute polymer solutions to bear on theproblem of DNA conformational evolution prediction.

To elucidate the current understanding of the role ofsolvent mediated interactions (HI and EV) in determin-ing the dynamics of polymer solutions, it is necessaryto distinguish between the behavior of θ-solvents andgood solvents and further between experimental obser-vations and theoretical results.

For any observed dynamic property φexpθ in a θ-solu-

tion, one can construct a nondimensional ratio

where Rg,expθ is the experimentally measured radius of

gyration in a θ-solvent and g is a function of Rg,expθ ,

such that it has the same dimensions and scaling withmolecular weight, M, as φexp

θ . Experimental observa-tions indicate that for a very large number of polymer-solvent systems the ratio Uφ,exp

θ attains a universalvalue, independent of molecular weight and chemistry,by relatively small values of M (of order 10 000). Anexample of such a universal ratio is the well-knownFlory constant, which is a nondimensional function ofthe intrinsic viscosity and the radius of gyration. Thus,the magnitude of any dynamic property φexp

θ , for alinear flexible polymer molecule with radius of gyrationRg,exp

θ , can be simply obtained from eq 1, provided oneknows the universal value Uφ,exp

θ .

The important role played by hydrodynamic interac-tions in θ-solutions is made evident by the fact thatuniversal ratios involving dynamic properties, as dis-cussed above, can be shown to exist theoretically onlywhen HI effects are incorporated. Both approximateanalytical theories, and more recent exact Browniandynamics simulations, have shown that the strength ofHI for a polymer-solvent system at equilibrium, or closeto equilibrium, is measured by the magnitude of thedraining parameter, h ≡ h*xN, and various nondi-mensional ratios of equilibrium and linear viscoelasticproperties attain constant universal values in the non-draining limit h f ∞.10 The experimental observationthat universal values are obtained by relatively smallvalues of M indicates that most systems reach thenondraining limit for relatively small values of h (sinceh is proportional to the square root of molecular weight).Recently, Kroger et al.11 have established the methodol-ogy for obtaining exact predictions of universal ratiosin the nondraining limit (within simulation error bounds)from Brownian dynamics simulations. By accumulatingdata for finite values of N, at various constant valuesof h*, they show that the values of a host of universalratios, obtained by extrapolating finite chain data to thelimit N f ∞, are unique, independent of the particularchoice of h* and reasonably close to experimentallymeasured values. Thus, provided one is in the nond-raining limit, the precise value of h* used to get thereis immaterial.

Experimental observations of the static behavior ofpolymer solutions under good solvent conditions haveshown that while both the deviation of the solutiontemperature T, from the θ-temperature Tθ, and themolecular weight M determine the “goodness” of thesolvent, they can be combined into an unique quantitycalled the solvent quality, z ) v0(1 - T/Tθ)xM, wherev0 is a constant, such that master plots of the depen-dence of material properties on solvent quality, for avariety of polymer-solvent systems, can be constructedwhen plotted in terms of z. Thus, for any observedproperty φexp, the ratio Rφ ) φexp/φexp

θ has been shown tobe a universal function of z, Rφ ) Rφ(z). A well-knownexample of such a function is the swelling of the polymerchain Rg, defined as the ratio of the radius of gyrationRg in a good solvent to the radius of gyration in aθ-solvent, which can be shown to be a universal functionof z.12 Thus, provided the universal properties Uφ,exp

θ

and Rφ(z) are known, the magnitude of any materialproperty φexp, for a polymer-solvent system under goodsolvent conditions, can be determined, given Rg,exp

θ andz.

Approximate analytical theories for excluded-volumeeffects in dilute polymer solutions10,13 have shown thatthe correct theoretical estimate of solvent quality isgiven by the expression z ) z*xN. Furthermore, theo-retical predictions can be mapped onto the experimen-tally observed universal dependence of material prop-erties on z by making a suitable choice of the constantv0. Recently, Kumar and Prakash14 have shown that byusing a narrow-Gaussian potential, and adopting afinite chain data extrapolation procedure similar to thatof Kroger et al., accurate predictions of universalcrossover scaling functions, free of any approximations,can be obtained by carrying out Brownian dynamicssimulations.

Uφ,expθ )

φexpθ

g(Rg,expθ )

(1)

618 Sunthar and Prakash Macromolecules, Vol. 38, No. 2, 2005

Since Kumar and Prakash neglect hydrodynamicinteractions, they obtain predictions of only staticproperties. However, by combining the approach ofKroger et al. and Kumar and Prakash, the coupledinfluence of HI and EV effects on universal propertiescan now be explored. It is worthwhile to note that boththe procedures of Kroger et al. and Kumar and Prakashrequire the extrapolation of finite chain data to the longchain limit, N f ∞. The limiting process is, however,carried out by simultaneously letting both the lengthof a single spring and z* tend to zero such that Rg

θ andz remain constant. Interestingly, while results obtainedin the limit N f ∞ imply that the contour length of thepolymer chain is infinite, they can still be used todescribe polymer-solvent systems with a finite molec-ular weight since both the radius of gyration Rg

θ andthe solvent quality z are kept finite. In brief, the centralconclusion of the discussion of HI and EV above is that(i) most polymer-solvent systems at equilibrium, orclose to equilibrium, approach the nondraining limit forrelatively small values of h and (ii) the quality of thesolvent can be characterized by a single variable z.Furthermore, accurate predictions in the universalregime, of all static and dynamic properties of a dilutepolymer solution, characterized by specific values of Rg

θ

and z, can be obtained, independent of model param-eters, by extrapolating appropriate results of bead-spring chain models to the limit N f ∞.

Recent investigations have shown that the generalmethodology of extrapolating finite bead-spring chaindata to the long chain limit, to obtain predictions of theuniversal behavior of dilute polymer solutions, can alsobe extended to shear flows far from equilibrium.15-17 Ithas been found that in this case master plots areobtained when the dependence on the shear rate γ isinterpreted in terms of the characteristic shear rate, â) [η]0Mηsγ/RT, where ηs is the solvent viscosity, [η]0 isthe zero shear rate intrinsic viscosity, and R is the gasconstant. As in the equilibrium and linear viscoelasticregimes, predictions in the asymptotic limit are obtainedwhile keeping z, Rg

θ, and â constant.The universal behavior that is predicted in each of

these situations is now well understood to have itsorigins in the self-similar nature of long chain flexiblemacromolecules in the coiled state. As more and moresprings are included in the bead-spring chain, localdetails of the chain are masked from the flow, leadingto parameter-free predictions. The use of infinitelyextensible Hookean springs in the model does notimpose any constraint on the contour length of thechain, which, as mentioned earlier, tends to infinity.

It is natural to attempt to extend these ideas to thecase of extensional flows. However, polymer moleculesdo not always remain coiled in an extensional flow. Asis well-known, in terms of the nondimensional Weis-senberg number Wi (defined by Wi ) λε, where λ is atime constant characteristic of the large scale relaxationof the polymer and ε is the extensional rate), for valuesof Wi J Wic, where Wic is a critical Weissenberg number(whose numerical value depends on the choice of λ), thepolymer molecule is unravelled from a coillike state toa fully stretched rodlike state, as the Hencky strain, ε) εt, increases. The coil-stretch transition occursbecause of the finite length of the molecule. In thestretched state, it is clear that universal behavior willnot be observed since the local details of the polymerare completely exposed to the flow. On the other hand,

for Wi < Wic and even in the case where Wi J Wic, forsome range of strain values, universal behavior may beexpected for sufficiently long polymer chains. The aimof this work is to carefully examine this question, in thecontext of the conformational evolution of λ-phage DNA,and determine the regime of behavior where the ideasof universality may be applied to extensional flows.

This task immediately raises a fundamental issue. Itis well-known that Hookean springs cannot be used todescribe extensional flows because they lead to un-bounded steady-state predictions for Wi J Wic. Asmentioned earlier, previous theoretical attempts haveshown that finitely extensible springs, with the ap-propriate force law, can be used successfully to describethe entire range of behavior. The question that naturallyarises in this context is, how should the procedure forobtaining predictions in the presence of HI and EV,discussed above for models with Hookean springs, berevised for a model with finitely extensible springs? Inother words, (i) what measure of HI should be heldconstant as N is increased so as to approach thenondraining limit, and (ii) how can a constant solventquality be maintained in the course of this procedure?The resolution of these issues forms the foundation forthe present work. We show that, even in the presenceof finitely extensible springs, it is possible to make useof the insight gained earlier with Hookean springs. Oncethe revised procedure is established, we show that theregime of applicability of the concepts of universalitycan be examined unambiguously.

After reviewing the necessary formalism for the modeland its parameters in section 2, we present a fairlydetailed discussion of the existing approaches to pa-rameter estimation and show how it is possible toexploit the existence of universal behavior to addressthe problem of parameter estimation in section 3. Thesystematic procedure that is developed as a conse-quence, for the treatment of finite chains, which istermed here as successive fine graining (SFG), isdiscussed in section 4. The universal stretch experiencedby a polymer chain in uniaxial extensional flow and thedeparture from universal behavior by finite chains arediscussed in section 5. The estimation of parameterswithin this framework, for λ-phage DNA, using avail-able experimental measurements, is given in section 6.1.Finally, the theoretical prediction of the conformationsof DNA molecules in elongational flow and a comparisonof predictions with experiments are given in section 6.2.

2. Bead-Spring Chain ModelIn this work, as mentioned earlier, a dilute solution

of DNA molecules is modeled as an ensemble of nonin-teracting bead-spring chains, each of which has Nbeads (which act as centers of hydrodynamic resistance)connected by massless springs representing an entropicforce between two points on the polymer chain. Ifψ(r1,...,rN), is the configurational distribution functionof the bead positions rµ, then the starting point in anystatistical theory of bead-spring chains is the “diffu-sion” equation or Fokker-Planck equation for theevolution of ψ:5,6,18

∂ψ

∂t) -∑

µ)1

N ∂

∂rµ

· [K·rµ +1

ú∑ν)1

N

Γµν·(Fνs + Fν

int)]ψ +

kT

ú∑

µ,ν)1

N ∂

∂rµ

·Γµν ·∂ψ

∂rν

(2)

Macromolecules, Vol. 38, No. 2, 2005 DNA in Elongational Flow by SFG 619

Here, K is a time-dependent, homogeneous, velocitygradient tensor of the surrounding fluid motion, ú is thehydrodynamic friction (drag) coefficient associated withthe bead, k is Boltzmann’s constant, and Γµν is thehydrodynamic interaction tensor, representing the effectof the motion of a bead µ on another bead ν throughthe disturbances carried by the surrounding fluid. Fν

s

is the entropic spring force on bead ν due to adjacentbeads, Fν

s ) Fc(Qν) - Fc(Qν-1), where Fc(Qν-1) is theforce between the beads ν - 1 and ν, acting in thedirection of the connector vector between the two beadsQν ) rν - rν-1. The quantity Fµ

int is the sum total of theremaining interaction forces on bead µ due to all otherbeads, Fµ

int ) ∑ν)1N Fe(rµν), where Fe is the binary force

acting along the vector rµν ) rν - rµ connecting beads µand ν. In the case of DNA we assume that this force isderived from a purely repulsive potential representingexcluded-volume effects.

Specifying the exact nature of the forces and thehydrodynamic interaction tensor completes the model.The simplest realization of this model is the Rouse chainwith a Hookean (linear) spring force and no other forcesarising due to excluded-volume or hydrodynamic inter-actions. However, as discussed earlier, it is essential toinclude a nonlinear spring force, hydrodynamic interac-tions, and excluded-volume effects in order to capturemany of the features of dilute solution behavior. ForDNA a commonly employed spring model is a wormlikechain (WLC) force “law” which is an interpolatoryformula between the limits of the Hookean (linear)regime and the fully stretched singular regime19

where H is the linear spring constant and q ) Q/Q0 isthe ratio of the magnitude of the connector vector Q andthe fully stretched length Q0. With regard to the EVeffects, recent simulations14 with Hookean springs haveshown that with the help of a narrow-Gaussian excluded-volume potential the universal equilibrium crossoverbehavior of dilute polymer systems can be accuratelypredicted. The advantages of using this potential are(a) it reduces to the δ-function potential in an appropri-ate limit, (b) it permits a simple calculation of terms ina perturbation expansion, and (c) it is also computa-tionally easy to handle. It is written as a regularizationof a the δ-function potential:20,21

where v is the excluded-volume parameter and d is aparameter that specifies the range of interaction. Theexcluded-volume force between beads separated by adistance r is hence given by

In eq 2, the hydrodynamic interaction tensor Γµν iswritten in general as

where Ωµν, for µ * ν, represents the actual interactionmatrix, and δ and δµν represent a unit tensor and aKronecker delta, respectively. For theoretical analyses,it is usually convenient to use the Oseen-Burgersfunction for the tensor Ωµν ) Ω(rµν), where the func-tional form, Ω(r), is given by18

In the simulations employed later in this paper, how-ever, we have used the regularized form given by theRotne-Prager-Yamakawa (RPY) interaction tensor.22,23

The expression for the RPY tensor is given in a nondi-mensional form shortly below.

It is convenient to render the equations dimension-less. For a Hookean spring xkT/H is its characteristiclength at equilibrium, and (ú/4H) is the characteristicrelaxation time of a single spring (in other words therelaxation time for a dumbbell), in the absence ofhydrodynamic interactions. Choosing these to be thereference length scale lH ≡ xkT/H and time scale λH ≡ú/4H, the dimensionless form of the various governingequations are given below.

The nondimensional spring force is given by

where q* ) Q*/xb and b ) Q02/lH

2. The excluded-volume force becomes

where the nondimensional parameters are z* ) v/(2πlH

2)3/2 and d* ) d/lH. The hydrodynamic interactionfunction given by the Rotne-Prager-Yamakawa regu-larization of the Oseen function in terms of nondimen-sional variables is

where for r* g 2xπh*

and for 0 < r* e 2xπh*

The quantity h* has been introduced previously as thenondimensional radius of the bead. It is also known asthe hydrodynamic interaction parameter and is definedby

The above expressions reduce to the correspondingexpressions for the Oseen tensor in the limit of r* .

FWLCc (Q) ) HQ 1

6q(4q + 1(1 - q)2

- 1) (3)

φe(r) ) kTv

(2πd2)3/2e-r2/2d2

(4)

Fe(r) ) - ∂φe

∂r

) kT v(2π)3/2d5

e-r2/2d2r (5)

Γµν ) δµνδ + úΩµν (6)

Ω(r) ) 18πηsr(δ + rr

r2) (7)

FWLCc (Q*) ) Q* 1

6q*(4q* + 1(1 - q*)2

- 1) (8)

Fe(r*) ) z*d*5

e-r*2/2d*2r* (9)

Ω(r*) ) [Ω1δ + Ω2r*r*r*2 ] (10)

Ω1 ) 3xπ4

h*r*(1 + 2π

3h*2

r*2),Ω2 ) 3xπ

4h*r*(1 - 2π h*2

r*2) (11)

Ω1 ) 1 - 932

r*

h*xπ, Ω2 ) 3

32r*

h*xπ(12)

h* ≡ axπlH

(13)


h*. In deriving the above expressions, the Stokes dragformula ú ) 6πηsa for a spherical bead of radius a hasbeen used. With these redefined expressions, the non-dimensional diffusion equation can be written as

where the parameter dependence for the functions hasbeen retained to show their explicit occurrence, and theforces are to be understood as having been nondimen-sionalized by xHkT.

The exact solution of the time evolution of variousaverages carried out with the distribution function ψ,without any further approximations, can be obtainedby writing a stochastic differential equation (SDE)equivalent to eq 14 and solving it with the help of aBrownian dynamics algorithm.18 In the presence offluctuating HI, the problem of the computational inten-sity of calculating the Brownian term has been attenu-ated significantly recently23 by the use of a Chebyshevpolynomial representation24 for the Brownian term. Wehave adopted this strategy, and the details of the exactalgorithm followed here are given in ref 25.

3. Model ParametersOne of the main aims of this paper is to examine the

region in Weissenberg number and Hencky strain space,in which a parameter-free prediction of the conforma-tional evolution of λ-DNA can be made. The generalprocedure is to determine the model parameters suchthat the model correctly reproduces known equilibriumproperties of the chain. Once this is done, predictionsin elongational flow at any deformation rate are thenobtained. The procedure followed in ref 8 is similar inspirit to this approach. However, as mentioned earlier,the procedure given in ref 9, though novel, adopts adifferent strategy which does not require conformationto equilibrium properties. The emphasis in the presentwork is to determine the parameters that yield thecorrect equilibrium properties, while at the same timebeing consistent with our current understanding of therole of HI and EV effects. In this spirit, a detaileddiscussion of the model parameters and their influenceon equilibrium properties is taken up below. It isworthwhile to note that the arguments are not specificto DNA alone and might be applied to any long-chainflexible macromolecule.

The theoretical prediction of any dimensional physicalproperty φ of the polymer solution is obtained bycarrying out a suitable configurational average with thedistribution function ψ, which satisfies the diffusionequation, eq 14. The property can then be representedas a function of nondimensional model parameters, inthe general form

where S is the corresponding function that provides thedimensions. Since lH and λH are the reference lengthand time scales for the nondimensional model, they donot alter the model variable φ*. However, unlike manyproblems in physics where a measurable quantity isused for the purposes of nondimensionalization, lH and

λH are quantities that are completely intrinsic to thebead-spring model and are not experimentally observ-able. Therefore, to make a dimensional comparison ofthe model predictions with experimental measurements,it is essential to fix lH and λH by matching dimensionalmodel variables with some suitable choice of experi-mental quantities, as discussed in greater detail below.

3.1. Parameter Determination at Equilibrium. Inthis section we discuss some possible procedures, alongthe lines previously suggested by Ottinger,26 that mightbe adopted for a systematic determination of the pa-rameters for any polymer-solvent system, includingthat of λ-phage DNA. As will be seen shortly, this willturn out to be quite a difficult task and will serve as amotivation both for the procedures that are currentlyadopted and for the procedure suggested in our work.

3.1.1. Free Draining Hookean Springs at θ-Con-ditions To illustrate a simple case of parameter deter-mination, we consider the Rouse model, which hasHookean springs (b f ∞), is free-draining (h* ) 0), andis under θ-conditions (z*) 0, d*) 0). Therefore, the onlyparameters are lH, λH, N, in which N is arbitrary,while the remaining two can be determined given twoexperimental measurements. The general procedure isto match some characteristic experimental size and timeconstant to the corresponding values obtained from themodel. If the experimentally observed radius of gyrationRg,exp

θ is known, then for any given N the dimensionalradius of gyration of the model is Rg,mod

θ ) Rg/(N)lH,

where Rg/(N) is the nondimensional model value for the

given value of N. For Hookean bead-spring chains of agiven N, Rg

/2 ) (N2 - 1)/(2N); thus, lH is determined bymatching Rg,mod

θ with Rg,expθ

Similarly, let λexp be some characteristic macroscopictime constant measured experimentally (such as thelongest relaxation time or a time derived from theintrinsic viscosity or the diffusivity). For example, in thecase of the relaxation time constructed from the intrinsicviscosity [η]0, we have5

where NA is the Avogadro number. Equating the relax-ation time from the model λmod ) λ*(N) λH to theexperimental relaxation time λexp, the reference time λHcan be fixed. In the case of a Rouse model, λ*(N) isstraightforward to calculate.

3.1.2. Hookean Springs with HI at θ-Conditions.In the case of a model with HI, the scheme for thedetermination of the length scale remains the same,since inclusion of hydrodynamic interaction does notalter static properties. However, the characteristic timefrom the model now has an additional dependence onthe parameter h* expressed as λ*(N,h*), and the deter-mination of h* and λH is coupled due to the relationship

It is to be noted that it is not required to actuallymeasure λexp, since it is known that for a polymer with

∂ψ

∂t*) ∑

µ,ν)1

N ∂ψ

∂r*µ

‚ Γµν(h*) ‚∂ψ

∂r*ν

- ∑µ)1

N ∂ψ

∂r*µ

·

[K*·r*µ + ∑ν)1

N

Γµν(h*)·(Fνs(b) + Fν

int(z*,d*))]ψ (14)

φ ) φ*(N, h*, b, z*, d*) S(lH, λH) (15)

lH )Rg,exp

θ

Rg/

(16)

λexp ) λη ≡ [η]0MNA

ηs

kT(17)

λH )6πηsa

4H)

ηs

kT32

π3/2lH3h* (18)


a sufficiently large molecular weight the time constantis related to the macroscopic size by an universalconstant. For example, a relaxation time can be relatedto the radius of gyration by

where UλRexp is a known universal constant. For in-

stance, the universal number corresponding to λη,denoted here by UηR (which is equivalent to the Flory-Fox parameter Φ), is given by

based on the reported27 value of Φ ) 2.5 × 1023.Consequently, all quantities on the rhs of eq 19 areexperimentally measurable, and the universal numberis also known for a given choice of λ.

Using eqs 18 and 19 and the fact that lH has alreadybeen determined by matching with Rg,exp

θ , we obtain

which is a nonlinear algebraic equation in the unknownsN and h*. For a given choice of N, eq 21 needs to besolved for the value of h*, and subsequently eq 18 canbe used to determine λH. There is no closed form solutionto this equation for a model which includes fluctuatingHI. However, it is well-known that with the preaver-aging approximation for HI it is possible to analyticallyobtain the solution to λ*(N,h*) by solving an eigenvalueproblem.28 It is also known that a value of h* ≈ 0.24provides a uniformly good approximation to the solutionof eq 21 for N . 1.29 It will be shown later that thisapproximation is not always true when finitely exten-sible springs are used instead of Hookean springs.

3.1.3. Combined Effects Including Finite Exten-sibility The results obtained so far are valid only for achain of Hookean springs. As mentioned earlier, such achain will not be a useful approximation in most strongflow situations, and the finite length of the polymer hasto be incorporated (i.e., the parameter b, defined beloweq 8, will have a finite value). In this case the determi-nation of the length scale lH and the parameter b arecoupled and require an additional experimental mea-surement, namely, the fully stretched length of thepolymer denoted by L0,exp. The fully stretched length ofthe bead spring chain is given by

Note that the static properties at equilibrium are alsofunctions of b. For instance, the radius of gyration isgiven by

where ø(b) is a known function of b for a given springforce model (see Appendix A for further details). Thisleads to two simultaneous equations in the unknownslH and b, from which the solution for b can be formallywritten as

where it is to be understood that the correspondingexperimental ratio has been used in place of the modelratio. In eq 24, we have also used the definition

where Nk is the number of Kuhn statistical segments(or Kuhn steps) in an equivalent bead-rod model.6 Onceb is determined, and therefore the value of ø, lH can bedetermined as before, but in this case by using eq 23for the radius of gyration obtained with finitely exten-sible springs.

The problem of fixing λH and h*, however, becomesconsiderably more difficult, since for finite N even thepreaveraging approximation is difficult to solve; i.e., anequation similar to eq 21 must be solved with a functionλ*(N,h*,b). Hsieh et al.9 have suggested a novel proce-dure to estimate the value of h*, namely, by matchingthe drag from a fully extended spring to that of a slenderrod. However, this does not ensure that the relaxationtime of the model matches the experimental value. InAppendix A we show that by using a preaveragingapproximation it is possible to obtain an approximatesolution of this equation, similar to that obtained in ref29 for Hookean springs.

With the need to incorporate good solvent effectsthrough the excluded-volume potential, two more pa-rameters, z* and d*, enter both the size Rg

/ and thetime constant λ*(N,h*,b,z*,d*). This problem is difficulteven when considering Hookean springs. There is nosystematic procedure to obtain these parameters, andbecause of this difficulty, it is not clear how many moreexperimental measurements are required to determinethe parameters. It is not surprising, therefore, thatwithin this framework of parameter estimation the bestoption might be to simply adopt a parameter fittingprocedure to all the available experimental data forDNA, as has been pursued in ref 8.

The difficulties associated with the approach sug-gested above for parameter estimation can be sum-marized as (a) the arbitrariness in the choice of numberof beads N, (b) the failure to isolate the effects of HIand EV, and (c) the failure to incorporate the existenceof a unique relationship between the size (static) andtime (dynamic) constants of long-chain polymers, therebyrequiring several additional experimental measure-ments to determine the model length scales, time scales,and parameters. In the following, we show how thesedrawbacks can be overcome by exploiting the universalbehavior of bead-spring chain models to obtain themodel parameters with minimal experimental measure-ments.

3.2. Parameter-Free Predictions for a Chain ofHookean Springs at Equilibrium. Insofar as purelyequilibrium properties are concerned, where the finitelength of the molecule is not a concern, we can use achain of Hookean springs to represent the long chainpolymer molecule, as is commonly done in most theo-retical treatments.5 This will provide a good illustrationof the method to obtain parameter-free predictions,which can then be carried over to a chain with finitelyextensible springs. The general principle behind the

λexp )ηs

kT4π3

(Rg,expθ )3UλR

exp (19)

UηR ) 9x62π

ΦNA

) 1.45 (20)

9xπ8

h*λ*(N, h*)

Rg/3(N)

) UλRexp (21)

L0,mod ) (N - 1)xblH (22)

(Rg,modθ )2 ) ø2(b)N

2 - 12N

lH2 (23)

bø2(b)

)L0,exp

2

(Rg,expθ )2

N + 12N(N - 1)

≡ Nk3(N + 1)N(N - 1)

(24)

Nk ≡ L0,exp2

6(Rg,expθ )2

(25)


following analysis is to exploit the universal behaviorof long chain polymers, which will remove the arbitrari-ness in the choice of N. We first reconsider the case ofa solution under θ-conditions, with the idea being toisolate the parameter estimation procedure for HI andEV effects.

3.2.1. Polymer Solution under θ-Conditions. SinceHookean springs are used, the procedure for determin-ing lH is still the same as discussed in section 3.1.1, i.e.,by matching Rg,mod

θ to Rg,expθ . However, the matching of

time constants is done by a different procedure. Asmentioned earlier, the unique behavior exhibited bysolutions of long chain polymers is the universalityobserved in experiments, which relates static propertiesto dynamic ones through a universal ratio.30 This isrepresented in general by eq 1 and illustrated in aparticular case by eq 19. For a chain of Hookean springs,we can define a similar ratio with model variables

Since Uφ,modθ is nondimensional, we can write it

in terms of the model parameters as Uφ,modθ ) Uφ,mod

θ

(N,h*). As discussed previously, the value of this ratioin the nondraining limit approaches a constant value,independent of h*. This has recently been demonstratedfor several ratios using simulations with fluctuatingHI,11 where independence from the value chosen for h*was observed as N f ∞. Both approximate analyticaltheories and simulations results have shown that theasymptotic behavior of this ratio can be expressed in ageneral form as

where Uφ∞ is the asymptotic value and cφ is a constant.

The leading order correction to the limiting value wasshown to be O(1/xN) in the Zimm theory29 as well asin simulations.11 The quantity hf

/ is also a constant,which is typically called the fixed point. It has a specialsignificance, as is evident from eq 27: For the choiceh* ) hf

/, the leading order correction to the limitingvalue changes from being of O(1/xN) to O(1/N). Thus,the asymptotic value Uφ

∞, which is attained in thenondraining limit h f ∞, is reached for smaller valuesof N.

Combining eqs 1 and 27, we have for any dynamicproperty obtained from the model, in the limit of N f∞

Since for a given value of N, Rg,modθ is matched to the

experimental value Rg,expθ , this equation implies that

the model prediction for any dynamic property reachesan asymptotic value φ∞

θ , irrespective of h*. Moreover, ithas been shown11 that the simulation values for theuniversal constants (Uφ

∞) agree very closely with theexperimental value (Uφ,exp

θ ) for several dynamic prop-erties. Therefore, in the limit of N f ∞ the value φ∞

θ

automatically agrees with the experiment value φexpθ ,

for any value of the parameter h*. The importantdifference between this approach and the finite Nmethod discussed earlier is that it is no longer necessaryto solve an equation like eq 21 to obtain h*; anyreasonable value will suffice. For this choice of h*, thetime scale λH is determined from eq 18.

In practice, the asymptotic value φ∞θ , for models with

fluctuating HI, is typically obtained using the followingprocedure. The ratio Uφ,mod

θ (N,h*) is calculated by car-rying out simulations, with h* kept constant, for in-creasing values of N. The limiting value Uφ

∞ is thenobtained by plotting the finite chain simulation valuesvs 1/xN and extrapolating to the limit N f ∞. Finally,φ∞

θ is obtained by using the expression φ∞θ )

Uφ∞g(Rg,mod

θ ). While physically meaningful values of h*lie in the range 0 < h* < 0.5,26 it has been shown thata value in the range 0.20 < h* < 0.25 provides betterconvergence even at small values of N, since hf

/ ≈0.24.11

3.2.2. Polymer under Good-Solvent ConditionsNow we consider a chain which includes excluded-volume effects to model a polymer dissolved in a goodsolvent at equilibrium. As mentioned earlier, it has beenexperimentally observed that the temperature andmolecular weight dependence of any macroscopic prop-erty φ under good solvent conditions (for various poly-mer-solvent systems, molecular weights, and temper-atures) can be collapsed on to a master curve,12 whichcan be expressed as

where φexpθ is the value of the same property under

θ-conditions and z is the solvent quality defined by z )v0 (1 - T/Tθ)xM, where v0 is a chemistry-dependentconstant used to shift the curves on to the master curve,Rφ(z). Since it has also been observed that all macro-scopic properties under θ-conditions can be related tothe Rg,exp

θ , we have from eq 1

which is a statement of the two-parameter theory,13

which hypothesizes that all macroscopic properties ofsolutions of long chain flexible macromolecules aredependent only on two parameters, Rg,exp

θ and thesolvent quality z. The function Rφ(z) is referred to as thecrossover function (or the swelling ratio) for the propertyφ due to excluded-volume effects.

For the purposes of identifying the parameters for asimulation, this equation clearly separates out HI andEV effects. While HI enters through the universal ratiounder θ-conditions, the EV effect is purely determinedby the parameter z. We have established elsewhereusing BDS that a narrow-Gaussian potential reproducesthe crossover behavior of static properties.14 In particu-lar, for z defined in terms of the model parameters, z )z*xN, we have shown that the experimental data canbe made to lie on simulation predictions by a suitablechoice of the constant v0. Thus, to determine the valueof z under the given experimental conditions, a mea-surement of only a single property φ is required, andthe corresponding theoretically predicted function Rφ(z)can be used from eq 29 to perform an inversion.Therefore, on the whole, only two experimental mea-

φ ) φexpθ Rφ(z) (29)

φ ) Uφ,expθ g(Rg,exp

θ )Rφ(z) (30)

Uφ,modθ ≡ φmod

θ

g(Rg,modθ )

(26)

Uφ,modθ (N,h*) ) Uφ

∞ + cφ( 1h*

- 1hf/) 1

xN+ O(1

N) (27)

φ∞θ ≡ lim

Nf∞φmod

θ (N, h*, lH, λH) ) φexpθ Uφ

∞

Uφ,expθ

g(Rg,modθ )

g(Rg,expθ )

(28)


surements are required. For example, if the radius ofgyration under θ-conditions, Rg,exp

θ , and at the givenconditions, Rg,exp, are known, then z is uniquely givenby

The function Rg(z) is a known function of z, which canbe determined by carrying out simulations as describedin ref 14. Once z is known, all other properties of thesystem are also known and given by their correspondingfunctions Rφ(z). Very importantly, no additional mea-surements are required to characterize the system.Though it would seem that z is a model parameter, it isin fact appropriate to consider it as an macroscopicexperimental quantity (like the radius of gyration)defined by the transfer function in eq 31. We note thatwe have not used any model specific definitions to arriveat the above equation. Equation 31 is an experimentalreality, as it is in the two-parameter theory and in theBDS results obtained here. The only difference is asingle arbitrary multiplicative factor relating experi-mental results to either approximate two-parametertheory or to exact Brownian dynamics simulations.

As discussed in ref 14 in detail, in the case of BDS,the Rφ(z) curve is obtained by keeping z ) z*xNconstant, while approaching the simultaneous limits ofN f ∞ and z* f 0. In this limit the parameter d*becomes irrelevant, and as a consequence, the modelpredictions are insensitive to the exact value of d* thatis chosen. Further, it was shown that the corrections tothe asymptotic value of Rφ scale as 1/xN, and anextrapolation can carried be carried out similar to thatfor Uφ,mod

θ , discussed above. Therefore, with just twopolymer specific macroscopic measurements at equilib-rium, all the equilibrium predictions of the model, whichare expected to be in agreement with the experimentalvalues, are obtained when we consider a bead-springchain model and take the limit N f ∞. In particular,there is no need to fit other experimental measurementssuch as diffusivity, intrinsic viscosity, or the longestrelaxation time to obtain the model parameters. Theirrelevant parameters h* and d* drop out of the modeland can therefore be assigned any arbitrary value forcomputational convenience.

3.3. Parameter-Free Predictions for a Chain ofFinitely Extensible Springs at Equilibrium. Asdiscussed previously in section 1, to accurately predictproperties in nonequilibrium situations, it becomesnecessary to account for the finiteness of the chain, anda commonly adopted method for doing so is to use abead-spring chain model with finitely extensible springs(FES). It is not quite straightforward, however, to applythe equilibrium parameter estimation procedure, givenabove for a chain with Hookean springs, to the case ofa chain with FES. A study of the long chain limitingbehavior of a chain of FES, along the lines of previousworks,11,14 is necessary in order to understand theapproach to universal behavior of such systemssthiswould then make it possible to obtain parameter-freepredictions at equilibrium, or close to equilibrium, evenwhen using a chain of FES. Though such a study isnovel and important in itself, we have thought itappropriate to give the details in Appendix A to main-tain continuity in our discussion of the DNA problem.

Only the salient conclusions of the examination of theuniversal behavior of chains with FES are summarizedbelow.

We first consider a solution under θ-conditions, mod-eled as an ensemble of noninteracting chains with FES.The starting point is an equation similar to eq 27,derived in Appendix A:

While this equation is similar in form to eq 27, thereare some significant differences. The first difference isthe appearance of the variable ø, which is proportionalto the nondimensional length of a single spring and isa function of the finite extensibility parameter b. Asdiscussed in greater detail in Appendix A, the form ofthe function ø(b) depends only on the spring force model,which in the present case of DNA is a wormlike spring.The second difference is the modification of the fixedpoint value hf

/, which is now a function of ø. (Thedependence of the fixed point value hf

/ on ø is discussedin greater detail in Appendix A.) Recall that theparameter b depends on the number of Kuhn steps inthe chain Nk and the choice of the number of beads N(see eq 24). For Hookean springs, b f ∞ and ø f 1. Inthis limit, as we would expect, eq 32 becomes identicalto eq 27. However, it can be shown that as the stiffnessof the springs increases, the magnitude of b decreasesand ø f 0. A value of ø < 1, therefore, represents a FESwith a certain degree of stiffness. In this case, both thedefinition of the hydrodynamic interaction parameterand the value of the fixed point are modified. It is shownin Appendix A that in the presence of FES h* is rescaledto h*, where

Consequently, the strength of HI is measured by thedraining parameter h ) h*xN, and eq 32 can berewritten as

As is evident from eq 34, the most important result ofthis analysis is that, even in the presence of finitelyextensible springs (with a prescribed value of ø), thelimiting universal ratios under θ-conditions are unal-tered, i.e, limNf∞ Uφ,mod

θ (N,h*,b) ) Uφ∞. However, the

asymptotic value Uφ∞ is now attained in the nondrain-

ing limit, h ) h*xN f ∞.The various points discussed above are nicely il-

lustrated by considering the universal ratio URDh con-structed from the Kirkwood diffusivity Dh θ and theradius of gyration

where the second equation is in terms of nondimen-sional model variables and Dh θ is defined by eq 63. Theapproach of URDh to its value in the long chain limit is

z ) Rg-1(Rg,exp

Rg,expθ ) (31)

Uφ,modθ (N, h*, b) ) Uφ

∞ + cφ(ø(b)h*

- 1hf/(ø)) 1

xN+ O(1

N)(32)

h* ) h*ø

(33)

Uφ,modθ (N, h*, b) ) Uφ

∞ + cφ( 1h*

- 1hf/(ø)) 1

xN+ O(1

N)(34)

URDh ≡ 6π Dh θ ηs

kTRg

θ ) 4xπh*

Rgθ*Dh θ* (35)


shown in Appendix A to obey the expression

The asymptotic value is consequently URDh∞ ) 8/3xπ ≈

1.505. URDh can be evaluated numerically by simulatingan ensemble of chains with FES at equilibrium, sincethe connector-vector distribution function for eachspring is known from the assumed force law. We haveused a simple acceptance-rejection method with auniform distribution majorizing function to generate thechains. Results obtained by carrying out these simula-tions, at various constant values of ø, while keeping therescaled parameter h* fixed at a value of 0.24, aredisplayed in Figure 1a. Symbols on each of the linescorrespond to simulations carried out for a particularvalue of ø at various values of N. The finite chain dataare then extrapolated to the limit N f ∞.

It is very clear from the figure, where it can be seenthat the various lines approach a common extrapolatedvalue, that the asymptotic value of URDh is independentof ø. In Figure 1b, finite chain data, accumulated forthree different values of h*, at a fixed value of ø, areshown to extrapolate to the same asymptotic value ofURDh

∞ , indicating the independence of the universal ratio

from the particular choice made for h*, in the nond-raining limit.

It is important to bear in mind that the limit N f ∞also corresponds to the limit Nk f ∞ (since ø is heldconstant on a trajectory). Thus, while the springs arefinitely extensible, the chain itself can be considered tobe infinitely long. This is because we are interested inequilibrium properties in this section, where the finite-ness of the chain is not a concern. However, theequilibrium analysis has revealed what the measure ofhydrodynamic interaction is in the case FES chains, andthis will prove to be of some significance when wediscuss the treatment of finite chains with FES (whichbecomes essential in the presence of flow) in the nextsection.

In the case of an ensemble of noninteracting FESchains in a good solvent, it is shown in Appendix A thatthe excluded-volume parameter z* is rescaled to z*,where

and the strength of excluded-volume interactions ismeasured by the solvent quality parameter z, given inthis case by

rather than by z*xN. We recall that the crossoverbehavior in the solvent quality is obtained in thesimultaneous limit of N f ∞ and the interactionparameter z* going to zero. Therefore, in the case ofFES, to keep the solvent quality z at a constant value,the rescaled interaction parameter z* f 0 is the correctlimit. The swelling ratios are then obtained at this valueof z by extrapolating finite chain data to the limit of Nf ∞. (More details on the extrapolation procedure canbe found in ref 14.) In Figure 2, the crossover functionfor the swelling of the radius of gyration, Rg(z), obtainedusing Hookean springs and that obtained using a chainof wormlike springs with various values of ø are shown.They can be seen to coincide with each other. Thisconfirms that eq 38 is indeed the correct form for thesolvent quality z. Further, the universal good solventcrossover behavior, when described in terms of z,remains identical to that obtained with Hookean springs

Figure 1. Scaling of the universal ratio URDh . Finite N datahave been obtained by carrying out simulations using a FENEmodel. (a) The rescaled HI parameter h* is maintainedconstant in all simulations, while ø is kept constant on eachtrajectory. (b) The parameter ø is kept constant in all simula-tions, while the rescaled HI parameter h* is kept constant oneach trajectory.

URDh ) 8

3xπ+ 1

x2πN( 1h*

- 1hf/(ø)) + O(1

N) (36)

Figure 2. Swelling ratio for the radius of gyration, obtainedfor a chain of wormlike springs, with two different values ofø. The rescaled solvent quality parameter z is defined by eq38. The continuous line corresponds to predictions obtainedwith Hookean springs.

z* ) z*ø3

(37)

z ≡ z*xN ) z*ø3

xN (38)


and therefore to the experiments as demonstrated inref 14. Thus, the functions Rφ(z), discussed in theprevious section, are identical for FES and Hookeansprings.

The essential principle underlying the existence ofuniversal behavior at equilibrium or close to equilibri-um, for chains with FES, demonstrated in Figures 1 and2, is that when there are a large number of modes inthe model, the macroscopic properties become indepen-dent of microscopic details, such as the force law.

As in the earlier discussion of HI effects, the discus-sion above of EV effects at equilibrium has been in thecontext of models where the springs have a finite length,but the properties of the chains themselves have beenobtained in the long chain limit, N f ∞. However, aswill become clear from the discussion in the sectionbelow, the key result of this equilibrium analysis,namely, that the solvent quality is measured in termsof the variable z ) z*xN, will prove to be of centralrelevance to the treatment of EV effects in finite chainswith FES.

4. Method of Successive Fine Graining (SFG)

In the presence of flow, it is clear that it in additionto using a model with finitely extensible springs thenumber of springs must also be finite, so that theconsequences of the finite size of the molecule may beaccurately represented. It is generally accepted that themost accurate representation of a linear, flexible, poly-mer molecule, of fixed length, is a bead-rod model withNk rods of fixed length. However, the computationalintensity of simulating bead-rod chains for the valuesof Nk that are typically encountered is the principalreason for the use of more coarse-grained bead-springchain models with finitely extensible springs. As is well-known, each spring in a bead-spring model is consid-ered to represent a set of Kuhn segments of the“underlying” polymer chain. In general, the springs ina coarse-grained FES model are chosen such that theforce-extension response of a spring mimics the re-sponse of a corresponding set of segments in theunderlying chain. For instance, while the FENE forcelaw (which is actually an approximation for the inverseLangevin force law6) describes the force-extensionbehavior of a set of segments of a completely flexiblechain, the WLC force law is an approximate representa-tion of the force-extension behavior of a set of segmentsof a wormlike chain.

Simulations of bead-spring chain models with FESin extensional flows have shown that, in general, topredict the behavior accurately across a wide range ofregimes it is necessary to use models with a largenumber of springs.9 In principle, the maximum numberof springs, (N - 1), that one can use is equal to thenumber of Kuhn steps in the underlying chain. Inpractice, with the force models currently used, FENEor WLC, it is not possible to reach (N - 1) f Nk. Thisis because of the limitations imposed by the nature ofø(b) in eq 24. It is straightforward to see, for instance,in the case of FENE springs, where ø(b) ) xb/(b+5),that b becomes negative near N ) 3Nk/5. A similarbehavior can be seen in Figure 6 for the WLC forcemodel.

The central hypothesis of the present work is that thebehavior of an underlying chain with Nk Kuhn stepsmay be predicted by accumulating data for a bead-spring chain model with FES, at various values of N

where the force law is valid, and extrapolating theaccumulated results to the limit (N - 1) f Nk. Sinceincreasing values of N represent more fine-grainedversions of the underlying chain, the procedure is called“successive fine graining”.

For a polymer with fixed contour length (or equiva-lently, a fixed value of Nk), successive fine grainingimplies that the parameter b becomes a function of N,given by eq 24. Furthermore, during the SFG procedure,the value of h* is held constant as (N - 1) f Nk.Physically, this is equivalent to assuming that thehydrodynamic interaction parameter at each level of thefine-graining process is constant and equal to that ofthe underlying chain. (Note that the conventional HIparameter h*) ø h* changes as b changes, according toeq 33.) Before discussing the treatment of solventquality in the SFG procedure, an example of this schemefor a polymer solution at equilibrium under θ-conditionsis demonstrated in Figure 3.

Figure 3 displays the dependence of the ratio URDh onN, obtained by the SFG procedure, for a bead-springchain with FENE springs, as described above. The opensymbols are results of simulations carried out keepingthe ratios Nk ) 200 (squares) and Nk ) 2000 (triangles)constant and calculating the value of URDh as the chainis fine grained, at two different fixed values of h*. Eachsymbol, consequently, represents a coarse-grained bead-spring chain with the same equilibrium size and contourlength as an underlying bead-rod chain, with thenumber of rods equal to the specified value of Nk andwith a hydrodynamic interaction parameter equal to thespecified value of h*. The filled symbols are exact nu-merical values of URDh for the bead-rod model, obtainedas described in ref 31. The SFG hypothesis is describedby the dashed and dot-dashed curves through the opensymbols, representing an extrapolation of the bead-spring chain results to the limit (N - 1) f Nk. As canbe seen, the extrapolated limit approaches the exactbead-rod result. It is also worthwhile to note that thebead-rod results for the different values of Nk can beextrapolated to a unique limit URDh

∞ , as Nk f ∞, which isidentical to that obtained in the previous section forchains with FES. This is in line with our expectationthat universal ratios are model-independent predictions.

Figure 3. Illustration of the SFG procedure at equilibrium.The open symbols represent values of the nondimensional ratioURDh for a FENE model, obtained at various values of N, bykeeping the ratios Nk ) 200 (square) and Nk ) 2000 (triangle)constant. While the open symbols above the horizontal linedrawn at URDh ) URDh

∞ ) 1.505 correspond to h* ) 0.17, thosebelow the horizontal line correspond to h* ) 0.30. The filledsymbols represent values obtained in the bead-rod model,31

with the corresponding values of Nk and h*.


With regard to solvent quality, we have seen earlierin the context of chains with FES that in order to keepz at a constant value the rescaled interaction parameter,z* ) z/ø3, should be chosen according to z* ) z/xN. Inthe SFG procedure, this rule is applied at each level ofthe fine-graining process.

It is straightforward to see that the SFG procedure(i.e., (N - 1) f Nk), carried out for chains with finitelyextensible springs, is identical in the limit of Nk f ∞ tothe schemes developed earlier by Kroger et al.11 andKumar and Prakash14,17 for obtaining the universalbehavior of models with Hookean springs, both atequilibrium and in the presence of a flow field (wherethe limit N f ∞ is used.) For Nk f ∞ and ø = 1 at allthe levels of fine graining which are accessible compu-tationally, and as a result, h* = h* and z* = z*. Indeed,the universal behavior that was observed for modelswith Hookean springs, with results independent of thechoice of both h* and d*, was a consequence of theinfinite length of the chain. For chains with finite valuesof Nk, the SFG procedure implies that in the limit (N -1) f Nk the draining parameter has a finite value h )h*xNk and z* ) z/xNk * 0. For instance, for λ-phageDNA, with an estimated value of Nk ) 200 (as discussedin greater detail subsequently), a choice of h* ) 0.2 leadsto h = 3. We might anticipate therefore that, in thiscase, the extrapolated results might not be universal,since the nondraining limit may not be attained, andthere might even be a dependence on the choice of d*.The existence of universal behavior clearly dependsquite crucially on the value of Nk and on the interactionof the flow with the polymer, determined by the mag-nitudes of the Weissenberg number Wi and the Henckystrain ε.

In the next section, we carefully examine this ques-tion, using all the procedures that have been describedso far. First, universal results in extensional flow areobtained for bead-spring chain models with Hookeansprings, both at θ-conditions and under good-solventconditions. Since such models correspond to infinitelylong polymers, they do not exhibit a coil-stretch transi-tion, and model predictions grow without bound withincreasing values of strain. Nevertheless, as will be clearfrom the discussion below, they serve as a basis withwhich the behavior of finite chains may be compared.

5. Universality and Finite Size EffectsIn the experiments reported in ref 3, λ-DNA was

subjected to a sudden elongational flow starting froman equilibrated state and observed under a microscopeas it evolved from a coillike state to the fully stretchedstate, at various elongational rates. The size of afluorescently dyed DNA coil, measured by this method,is a projected extent in the flow direction, commonlycalled the “extension” or “stretch”. In terms of a bead-spring model this is defined as xmax ≡ maxµ,ν|rµ

x - rνx|,

where rµx is the x-component of vector rµ, with x being

the flow direction. The statistical properties of thisquantity, unlike Rg or end-to-end distance, are not easyto calculate analytically.5 On the other hand, since it isstraightforward to evaluate the stretch from a givenconfiguration of bead positions, a knowledge of thedistribution function obeyed by the configurations im-plies that the average stretch can be obtained numeri-cally from an ensemble of configurations. Here, theaverage along the direction of intended elongation isdenoted as xj ≡ ⟨xmax⟩. In particular, while the equilib-

rium stretch is denoted by xj, the stretch at any nonzerofinite value of the strain is denoted by xj+, in keepingwith the accepted rheological convention for transientproperties. It is worthwhile to note that at equilibriumthe average stretch of a long flexible chain is not equalto exactly twice the radius of gyration Rg, as assumedin ref 3. For instance, in the limit of long chains obeyinga Gaussian distribution (θ-conditions), we can numeri-cally evaluate this to be

In the first part of this section, the universal stretchexperienced by a polymer chain in uniaxial extensionalflow is predicted by examining the behavior of a bead-spring chain model, with Hookean springs, in the longchain limit. While experimental data on the stretch aretypically reported in terms of the nondimensional ratioxj+/L, where L is the contour length of the DNA molecule,this is clearly not usable in the case of Hookean springs,where L f ∞. Consequently, universal behavior isexamined here in terms of the expansion factor E,defined by

which is the ratio of the transient stretch at a givensolvent quality to its value at equilibrium at the samesolvent quality. This ratio has the correct scalingbehavior for Hookean spring chains in the limit N f ∞.

As mentioned earlier, a crucial parameter that arisesin flow is the Weissenberg number, Wi. In this work, itis taken to be the same as defined in experiments ofSmith and Chu,3 namely

where λ1 is the longest relaxation time. Smith and Chuevaluated this characteristic time constant by fitting anexponential to the tail end of the relaxation of thefunction xj2(t), where xj(t) is the average instantaneousstretch, when a fully stretched DNA is left to relax toequilibrium. We have employed a similar procedure todetermine the longest relaxation time, as described ingreater detail below.

It has been established in section 3.2 that macroscopicpredictions of bead-spring chains with Hookean springs,in the limit N f ∞, do not depend on the microscopicparameters of the model, h*, z*, and, d*. However, sincethe simulations are carried out with finite N, it isnecessary to assign values to these variables, consistentwith known equilibrium data, but arbitrarily chosen forcomputational convenience. The stepwise procedure forparameter determination is outlined below.

1. The number of beads N: An arbitrary number ofbeads N is chosen, and all the following parameters aredependent on this choice of N.

2. The interaction parameters for HI (h*) and EV (z*,d*): Since any arbitrary value of h* can be chosen, wehave chosen h* ) 0.19 and 0.3, which bracket the fixedpoint value, hf

/) 0.24 (with preaveraged HI). Note thatfor Hookean springs, as mentioned earlier, h* ) h*,since ø ) 1.

For a given value of the solvent quality z, the EVparameters used in the narrow-Gaussian potential are

xjθ

2Rgθ

) 1.132 ( 0.001 (39)

E ) xj+

xj(40)

Wi ) ελ1 (41)


determined from

where K is an arbitrary constant. Clearly, for a fixedvalue of z, since z* f 0 as N f ∞, this choice of d*implies that the asymptotic limit is reached alongtrajectories in the (z*, d*) parameter space that convergeto the origin. Kumar and Prakash14,17 have shownpreviously that the asymptotic results are independentof the particular choice of K and that choosing d* valuesaccording to eq 43 permits the use of larger step sizesin the numerical integration scheme.

3. The relaxation time λ1/: Under good solvent con-

ditions, the longest relaxation time is obtained here byfitting a single exponential to the decay of three differ-ent size measures: (a) the mean-square end-to-enddistance, (b) the mean-square radius of gyration, and(c) the mean-square stretch, after initially extending achain by stretching each spring in the chain to roughlytwice its equilibrium length. The curve is fitted using

to determine the three parameters λ1/, s0

/, and s∞/ , where

s* is one of the variables listed above. The range of thefit was adjusted over 2 decades in the variable (s* -s∞/ )/s∞

/ starting from a value of 1. The difference in therelaxation times for a given N, obtained from the decayof the three different size measures listed above, wasfound to be within 3%.

Under θ-conditions, the well-known Thurston’s ex-pression,6 which predicts the longest relaxation time forany given choice of h* and N, has been used. It wasfound, by rigorous comparison for a few cases with theexponential curve-fitting procedure described above,that Thurston’s expression was valid within about 6%.

4. Elongation rate ε*: The experiments were per-formed at various Weissenberg numbers Wi, rangingfrom 2 to 55. The nondimensional elongational raterequired for the simulations is given by ε* ) Wi/λ1

/.Once data are accumulated for various finite values

of chain length N, it can be extrapolated to the longchain limit. In Figure 4, the expansion factor E, obtainedby following the procedure above, is plotted against 1/xN, at a constant strain ε ) 4 and constant Weissen-berg number Wi ) 2 for various cases. In Figure 4a,b,the HI parameter, h* ) h*, is held constant at twodifferent values during the fine-graining procedure,while in Figure 4c, the parameter K is held constant attwo arbitrarily chosen values. Figure 4a corresponds toa θ-solution, while Figure 4b,c corresponds to the caseof a good solvent with z ) 1. The independence of theextrapolated result in the long chain limit, in each ofthese cases, from the particular choice made for theseparameter values, is clearly seen in these figures. As aresult, the asymptotic value of E, obtained in thismanner, is the universal value of this property at ε ) 4and Wi ) 2.

By carrying out an extrapolation procedure, similarto that displayed in Figure 4, at various values of theHencky strain, the universal transient stretch experi-enced by an infinitely long polymer chain, in thenondraining limit, may be predicted. Figure 5 displays

the results of such an exercise, for Wi ) 2, under bothθ-conditions and for a good solvent with z ) 1.

As is expected for a bead-spring chain model withHookean springs, the curves increase without bound forincreasing values of ε. As a result, the stretch neverreaches a steady-state value. Physically, the infinitepolymer coil is being unravelled continuously but neverundergoes a coil-stretch transition because of theexistence of infinite length scales. The role played bythe presence of excluded-volume interactions is alsoclearly discerniblesthe initially expanded coil structureis unravelled more rapidly by the flow. The influence

Figure 4. Independence of the asymptotic value of theexpansion factor, E, from the choice of h* and d*, at Henckystrain ε ) 4 and Weissenberg number Wi ) 2. (a) θ-conditions,effect of h*, (b) z ) 1, effect of h* with K ) 1, and (c) z ) 1,effect of d* with h* ) 0.3. d* is varied according to eq 43, withtwo different values of K.

z* ) zxN

(42)

d* ) Kz*1/5 (43)

s* ) s∞/ + (s0

/ - s∞/ )e-t*/λ1

/

(44)


of EV effects in infinite chains, relative to its influencein finite chains, is considered in greater detail later inthis section (see discussion related to Figure 11).

While the data for the two values of h*, at z ) 1, aresuperimposed on each other for all the strain valuesdisplayed in Figure 5, the data at high values of strain,under θ-conditions, appear to be coming apart. Thismerely indicates that the presently accumulated finitechain data are not adequate to obtain accurate andunique extrapolated values (as seen, for instance, inFigures 4a) at these values of strain. However, inprinciple, this is easily remedied by carrying out similarsimulations at even larger values of N.

It is worthwhile to note that the results displayed inFigure 5 are exact universal predictions, with fluctuat-ing hydrodynamic and excluded-volume interactions,obtained here for the first time by exploiting theextrapolation ansatz for Hookean springs developed byKroger et al.11 and Kumar and Prakash.14

We turn now to predictions of the expansion factorE, for a chain of finite length, using the method ofsuccessive fine graining, outlined in section 4. Aspointed out in that section, the two key differences fromthe case of a chain with infinite length considered aboveare the use of finitely extensible springs and therestriction of the extrapolation process to the limit (N- 1) f Nk. These differences imply a change in some ofthe steps of the finite chain parameter determinationprocedure, described above for Hookean springs. Basi-cally, an additional parameter b is introduced throughthe use of FES, with the concomitant appearance of thefunction ø and the need to rescale the HI and EVinteraction parameters. For the sake of clarity, the stepsthat are changed are enumerated below.

1. The finite extensibility parameter b: For givenvalues of Nk and N, the corresponding value of b can befound by solving the nonlinear expression, eq 24. Thesolution is not available in closed form for a wormlike-spring force law, since ø(b) itself has to be evaluatednumerically, as given in Appendix B. However, anapproximate procedure can be developed which yieldsthe value b to a high degree of accuracy. First, note thatthe function y(N) ≡ b/3ø2(b) can be readily evaluatedgiven Nk and N, since from eqs 24

By numerically evaluating ø(b) for various values of b,the relationship between b and y can be obtained for agiven force law and plotted once and for all, as shownfor WLC springs in Figure 6. As a result, the value ofb, corresponding to given values of Nk and N (and,therefore, y), can be simply read off from the figure. Itturns out that this procedure can be further simplifiedsince it was found that the numerical data could bedescribed very accurately with the following seriesrepresentation

where the parameters c1 ) -0.63, c2 ) -1.0, c3 ) -0.70,and c4 ) 1.47 have been obtained by a systematicnonlinear least-squares regression fit of the numericaldata in the b vs b/3ø2 space. Thus, using the value of y,b is determined from eq 44. Subsequently, ø(b) can beobtained, as given in Appendix B. Note that the serieshas the correct asymptotic form for y f ∞, correspondingto the Hookean limit. The accuracy of the approximationis evident from Figure 6, where it can be seen thatvalues obtained from the series representation virtuallycoincide with the numerical data. We have taken Nk )200 in the calculations to follow.

2. The rescaled interaction parameters for HI (h*) andEV (z*, d*): As described in section 4, any arbitraryvalue of h* can be chosen. The corresponding value ofh* is determined from h* ) ø(b)h*. We have chosen thevalues h* ) 0.19 and 0.3 to carry out comparisons withthe results obtained earlier for Hookean springs. For agiven value of the solvent quality z, the strength of theexcluded-volume interaction, at each level of the fine-graining process, is determined from z* ) ø3(b)z/xN,while d* is obtained from eq 43, as in the case ofHookean springs.

3. The reference scales lH and λH: In a situation whereit is desired to compare the model predictions to adimensional experimental measurement, it is necessaryto obtain the values of the reference scales of length lHand time λH at any given level of discretization, N. Asdescribed in section 3.1, lH can be found from eqs 16and 23, while λH can be obtained from eq 18.

4. The relaxation time λ1/: The procedure for deter-

mining the relaxation time is identical to the case ofHookean springs, with the minor difference that the

Figure 5. Universal behavior in extensional flow. Expansionfactor E as a function of Hencky strain, at Wi ) 2, underθ-conditions and at z ) 1, obtained using Hookean springs inthe limit N f ∞.

y(N) ) NkN + 1

N(N - 1)(45)

Figure 6. Determination of the parameter b for a wormlikespring, for given values of N and Nk, using the series ap-proximation, eq 46.

b ) 3y[1 +c1

xy+

c2

y+

c3

y2+

c4

y3] (46)


stretch relaxation was carried out after first extendingchains to 85% of their maximum length.

Table 1 lists the key parameter values and theresultant nondimensional relaxation time λ1

/, obtainedby the above procedure, at various levels of discretiza-tion, as an illustration of typical magnitudes encoun-tered in the simulations. It is important to note herethat for a given vale of N the nondimensional relaxationtime is relatively insensitive to the value of Nk when itwas chosen in the range 150-300. This will prove to bea crucial point when comparisons are made with ex-perimental observations on DNA in section 6. As willbe discussed in greater detail below, it was found thatfor λ-DNA a maximum of N ) 33 was sufficient to obtainreliable extrapolations of the average stretch evolutionin elongational flow in the limit (N - 1) f Nk. The SFGprocedure is tedious to some extent since for each valueof N, we need to first determine the corresponding λ1

/,as given in Table 1. Note, however, that for a givenvalue of N the choice of h* and the d* is still arbitrary.We have used 104 polymer chains in an equilibriumensemble to determine the relaxation time and about105 polymer chains for a nonequilibrium simulation.

Values of the expansion factor E, for various valuesof 1/xN, obtained for a model with WLC springs, usingNk ) 200, and two different values of h*, at fixed valuesof ε ) 4 and z ) 1 (with the parameter values fordifferent choices of N determined as described above),are displayed in Figure 7. While Figure 7a correspondsto the Weissenberg number Wi ) 2, Figure 7b corre-sponds to Wi ) 55. The lines through the symbolsrepresent extrapolations of the finite chain data to thelimit (N - 1) f Nk. It is clear from Figure 7a that atWi ) 2 the extrapolated values at N ) Nk are insensitiveto the choice of h*. On the other hand, at a highWeissenberg number, such as Wi ) 55, the extrapolatedvalues for the two different values of h* are distinct.This is clearly dependent on the value of strain, as canbe seen from Figure 8, where the results of carrying outa similar extrapolation procedure, at various values ofthe Hencky strain, are displayed.

At the relatively low value of Weissenberg number,Wi ) 2, the expansion factor is clearly insensitive to thechoice of h* at all values of the strain. On the otherhand, at Wi ) 55, the different values chosen for h* get“revealed” as the strain increases. This is entirelyconsistent with our previous discussion, where it wasargued that the coillike structure of the chain atequilibrium, and close to equilibrium, masks the localdetails of the chain (such as the nondimensional beadradius) from the flow, while at higher Weissenbergnumbers, and large values of strain, the polymer chainis exposed to the flow field as it unravels to a stretchedstate. In the case of bead-spring chain models withFENE springs, the SFG procedure reveals a similarinsensitivity to the value of h* at all values of the strainfor Wi ) 2, while a slight separation of the curves forlarge values of strain occurs at Wi ) 55 (see also Figure10).

In Figure 4c, the independence of the infinite chainlength results, from the value chosen for the narrow-Gaussian potential parameter, K, was clearly demon-strated. As discussed earlier, when eq 41 is used, d* f0 as N f ∞ for fixed values of z. Since the parameterd* represents the range of excluded-volume interactionsbetween the beads, this limit clearly corresponds to a

Table 1. Typical Parameter Values Encountered in theSuccessive Fine-Graining Procedure for a Wormlike

Chain with Nk ) 200, L ) 22 µm, z ) 1, ηs ) 43.3 cP, T )296 K, h* ) 0.19, and K ) 1

N b ø(b) lH (× 10-5 cm) λH (s) λ1/

11 54.00 0.910 2.99 0.41 16.5 ( 0.119 25.71 0.858 2.41 0.20 36.5 ( 0.427 15.56 0.808 2.14 0.13 60.0 ( 0.5

Figure 7. Illustration of the SFG procedure for a model withWLC springs, at a fixed value of Hencky strain, ε ) 4 and atWeissenberg numbers (a) Wi ) 2 and (b) Wi ) 5. Parametervalues are Nk ) 200, z ) 1, and K ) 1. Symbols for N < Nkhave been obtained by carrying out simulations at two differ-ent values of h*. Lines through the symbols represent anextrapolation to the limit (N - 1) f Nk.

Figure 8. Expansion factor E, in extensional flow, for a modelwith WLC springs at two different Weissenberg numbers andtwo different values of h*. Symbols are extrapolated values,obtained by the SFG procedure. Parameter values are thesame as in Figure 7.


δ-function excluded-volume potential. In the case offinite chains, however, we have seen earlier that thelimit (N - 1) f Nk implies z* ) z/xNk * 0. As a result,different choices for the parameter K imply differentlimiting values for the range of excluded-volume inter-actions.

In Figure 9, the expansion factor for two differentvalues of K, at Wi ) 2 and Wi ) 55, obtained with theSFG procedure for a chain with WLC springs, isdisplayed. Differences between the extrapolated predic-tions for the two values of K lie within the error bars ofthe simulations, indicating the relative insensitivity ofthe expansion factor to this parameter. Since, for Wi )2, we have previously observed the independence of theexpansion factor from the particular choice of h*, thedisplayed curve for Wi ) 2 in Figure 9 is in some sense“parameter-free” and dependent only on the number ofKuhn steps in the chain and the spring force law.

In both Figures 8 and 9, the expansion factor at largestrains is seen to level off to an asymptotic steady statevalue, clearly highlighting the central difference be-tween finite and infinite chains. At a threshold valueof strain that depends on the Weissenberg number, thefinite character of the chain leads to the existence of apoint of inflection on the curves, where the slope stopsincreasing and starts to decrease. Since the finitenessof the chains is incorporated in the model by the use offinitely extensible springs, it may be anticipated thatnature of the deviation of the finite chain curves, fromthe universal Hookean curve, depends on the choice ofthe spring force law.

Differences in predictions by the various force lawsare highlighted in Figure 10, where the expansion factorobtained with Hookean, WLC, and FENE force laws,using the SFG procedure, at two different Weissenbergnumbers and two different values of h*, are compared.While the expansion factor predicted by the Hookeanforce law is unbounded, the incorporation of FESensures a bounded value of the expansion factor at highvalues of the strain. The expansion factor predicted byall three force laws is identical at small values of strain.However, at a value of strain ε = 1.5, the FES force lawpredictions depart from the Hookean predictions. Theagreement at low values of strain is not surprising sincethe polymer chain is predominantly in the coiled state.In this case, as we have seen earlier, results becomerelatively independent of local details such as the spring

force laws. For Wi ) 2, both the WLC and the FENEforce law predictions are identical until ε = 2.5, beyondwhich a difference in the predictions can be discerned.Interestingly, at Wi ) 55, the separation of the curvesfor the two values of h*, and the separation of the FEScurves from the Hookean curves, occur at roughly thesame value of strain, ε = 1.5. (It is perhaps appropriatehere to note that maintaining a constant value of h*for both the WLC and FENE models actually impliesusing different values of h* at each stage of the fine-graining process, since the function ø(b) is different forthe two force laws.)

It is clear from Figure 10 that the universal (Hookean)stretch vs strain curve acts as an envelope for allpossible finite chain, stretch vs strain, curves. Es-sentially, for a given value of the Weissenberg number,the strain at which a finite chain curve departs fromthe Hookean curve, would depend on the value of Nk.For increasing values of Nk, the point of departure wouldoccur at increasing values of strain.

The role of excluded-volume interactions in modelswith Hookean springs was elucidated earlier in Figure5. Before we examine their role in models with FES, itis necessary to introduce some new notation. Consistentwith the definition of equilibrium swelling ratios, theswelling of the equilibrium stretch is denoted by

In a transient extensional flow, we introduce thetransient swelling ratio Rx

+ to represent the ratio of thestretch in extensional flow under good solvent conditionsto its value under θ-conditions, evaluated at the samevalue of strain:

The relationship between the transient swelling ratioRx

+ and the expansion factor E defined earlier can bederived from

Figure 9. Sensitivity of the expansion factor to the range ofexcluded-volume interactions. Symbols are extrapolated valuesfor two different values of K, obtained by the SFG procedure.Parameter values are Nk ) 200, h* ) 0.19, and z ) 1.

Figure 10. Expansion factor for Hookean, WLC, and FENEforce laws, at two different Weissenberg numbers and twodifferent values of h*, obtained by the SFG procedure. At Wi) 2, data for the two values of h* agree to within simulationerror bars (see also Figure 8). Parameter values are the sameas in Figure 7.

Rx ) xjxjθ

(47)

Rx+ ) xj+

xj+θ(48)

Rx+ ) xj+

xjxjθ

xj+θxjxjθ

(49)


where Eθ ) xj+θ/xjθ. Note that the equilibrium expansionfactor depends on z through its crossover function, Rx) Rx(z).

In Figure 11, the dependences of Rx+ on strain, for

models with Hookean springs and WLC springs, arecompared. The data for Hookean springs can be ob-tained from that displayed earlier in Figure 5 and eq50. The curves for WLC springs have been obtained ina similar manner by applying the SFG procedure. It isclear that Rx

+ for Hookean springs continues to in-crease with increasing strain, indicating that excluded-volume interactions are always present. This is aconsequence of the fact that the chain is infinitely long,and at any value of flow strength, there are alwayslength scales at which beads exclude each other. On theother hand, in the case of FES chains, once the coilbecomes stretched, the excluded-volume interactions getswitched off as the beads are drawn increasingly furtherapart. Though at equilibrium (ε ) 0) the swelling of achain with z ) 3 is greater than that with z ) 1, Rx

+

decreases in both cases ultimately to a common steady-state valuesalbeit after an initial range of strain valuesin which it increases slightly.

It would be worthwhile to confirm these predictionswith careful experimental investigations. At the mo-ment, the available experimental data on λ-phage DNAare insufficient to enable a precise estimation of thesolvent quality, as will be seen from the detaileddiscussion below. However, we can obtain a reasonablevalidation of the ideas presented so far by comparingthe predictions of the present simulations, obtained withthe SFG procedure, with the experimental data of Smithand Chu.3

6. Application of SFG to DNA

Before we use the method of successive fine graining,we first justify applying the general approach formodeling flexible macromolecules to λ-DNA. DNA is apolyelectrolyte, and when in solution the electrostaticeffects can lead to spatially long-range interactions. Thisprecludes the use of the scaling arguments used forflexible macromolecules. In the presence of excess salt,however, the electrostatic interactions become screened32

and become local (the range of interactions falling belowthe persistence length of the chain) and can in effect betreated as an excluded-volume interaction betweendistant portions of the chain, similar to the excluded-volume effects seen in good solvents. It is reasonable,therefore, to accord DNA a treatment similar to aflexible molecule in a good solvent.

6.1. Estimation of Parameters for λ-DNA atEquilibrium. As discussed in the previous section, tocapture the behavior of a finite-length polymer moleculeunder flow, we require, in the successive fine-grainingmethod, only three equilibrium properties of the poly-mer, namely, the radius of gyration under θ-conditionsRg

θ, the solvent quality z, and the contour length L.Further, when we are interested in making nondimen-sional predictions shown in the previous section, onlythe number of Kuhn steps Nk (which is related to ratioof L to Rg

θ through eq 24) and z are required.In the context of estimating these parameters for the

purpose of comparing predictions with the experimental

observations of Smith and Chu,3 there are certain issuesthat need to be addressed with regard to the particularexperimental systems considered in their work. Thecontour length for YOYO-1 stained λ-DNA is known tobe L = 22 µm.3 To the best of our knowledge, a directmeasurement of the radius of gyration of λ-DNA underthe given solvent conditions considered has not beenreported. Additionally, various experimental quantitiesrequired for the elongational flow experiment wereobtained in different solvent media (the buffer solutionwas changed to alter the viscosity), and this could beresponsible for changes in solvent quality. For instance,the diffusivity was measured in a 0.95 cP solvent,2 andthe longest relaxation time was determined in a 41 cPsolvent,33 while the elongational flow experiments wereperformed in solvent viscosities ranging from 43.3 toabout 180 cP3 to obtain a control on the strain raterelative to the longest relaxation time of the molecule.Therefore, it is difficult to ascertain the two parametersRg

θ (or Nk) and z accurately.An estimate of Nk can be obtained from the force

extension curves,34 which suggest a persistence lengthλp ) 66 nm for YOYO stained DNA.35 From this we haveNk ) L/(2λp) ≈ 167 or equivalently Rg

θ ) 0.69 µm. Whilethis interpretation is widely used, there is also analternative suggestion, based on other experiments, thatthe persistence length is much lower36saround λp ∼ 28.5nm at high salt concentrations, which corresponds to∼38 nm for stained DNA. This would then imply Nk ∼289 and Rg

θ ) 0.53 µm.It is also possible, in principle, to obtain an estimate

for Nk and z using other measurements of λ-DNA, whichare consistent with the universal behavior discussed insection 1. For instance, the equilibrium stretch, diffu-sivity, and relaxation time can be used to constructuniversal ratios, from which Rg

θ and z can be estimatedusing the universal crossover behavior. However, suchan attempt is presently infeasible for two reasons: (i)an absence of careful studies of the swelling ratios forDNA under various solvent conditions, to ensure thatDNA solutions exhibit universal behavior, and (ii) thepresence of a large uncertainty in the measured stretchof λ-DNA close to equilibrium. (It has been reported thatthe observed coil size is exaggerated due to a “bloomingeffect” in the video camera, and a constant factor of 1

Rx+ ) E

EθRx (50)

Figure 11. Dependence of the transient swelling ratio on thestrain and solvent quality, for models with Hookean springs(filled circles) and WLC springs (open symbols). For the latter,Nk ) 200. In both cases, Wi ) 2. Hookean results have beenobtained in the limit N f ∞, while WLC results have beenobtained by the SFG procedure. At this value of Wi, the latterare independent of the choice of K and h*.


µm has been suggested as the offset correction to besubtracted from the observed size. The magnitude of thesuggested correction is based on an approximate esti-mate of the actual equilibrium stretch, obtained from aknowledge of the radius of gyration. However, the radiusof gyration is in turn only calculated from the Zimmmodel under good-solvent conditions, using diffusivitymeasurements.2)

It is clear, therefore, that there are several uncertain-ties in assigning values to Rg

θ and z. The main empha-sis in the present paper, however, is on developing anappropriate framework for the treatment of solventquality effects, under nonequilibrium conditions. It isconsequently perhaps not so crucial that at present itis not possible to obtain accurate estimates of thesequantities.

The arguments above suggest estimates of Nk in therange 166-289. To make comparisons with the experi-ments, this does not pose a serious problem if werepresent all the quantities in a nondimensional form.This is because (a) the relaxation times λ1

/ are insensi-tive to the value of Nk in the range 150-300 (see section5) and (b) extrapolated values of E are indistinguishablewithin error bars when (N - 1) f Nk, for values of Nkin this range (as can be seen in Figure 7). Therefore,for the purpose of comparing with experimental obser-vations below, we take Nk ) 200 as a representativevalue.

A similar argument cannot, however, be made for thesolvent quality, as its effect in elongational flow isclearly evident from Figure 11. In the absence ofaccurate estimates, we assume z ) 1. In combinationwith Nk ) 200, as will be seen shortly below, this valueof z leads to reasonably accurate estimates of both theequilibrium stretch and the stretch during extension.

With Nk ) 200 and L ) 22 µm, we can compute Rgθ )

0.63 µm using eq 25 and xjθ ) 1.43 µm using eq 39. Wehave determined the universal crossover of the swellingof the equilibrium stretch and found that at z ) 1, Rx )1.25. This gives the equilibrium stretch in good-solventconditions to be xj ) 1.79 µm. The measured stretch(extension) in Figure 2A in ref 3, when extrapolated tozero strain, is 2.55 ( 0.02 µm. The difference of ∼0.8µm can therefore be attributed to blooming. It shouldbe noted that the available data have been used only toobtain a rough estimate of the various parameters, andin the light of more accurate measurements in thefuture, some of these calculations would need to berepeated.

6.2. DNA in Elongational Flow. With the specifica-tion of Rg

θ, z, and the contour length, the equilibriumspecifications required to carry out the successive finegraining is complete. The only other relevant parameterentering the flow situations is the Weissenberg numberWi. We use the same definition of Wi (or the Deborahnumber De) as considered in ref 3: Wi ) ελ1. In all thesimulation results discussed in this work, we considertwo cases, namely, the smallest and largest Wi reportedin ref 3, Wi ) 2 and 55, to explore model predictions.

First, we superimpose the results displayed earlierin Figure 10, which has been computed with Nk ) 200and z ) 1, on the experimental observations of stretchvs strain, as shown in Figure 12. For the experiments,we have subtracted a blooming correction of 0.8 µm,found above, from the measured stretch uniformly atall strains, and divided it by the stretch at zero strain.Figure 12 clearly shows the universal nature of the

unravelling of λ-DNA (when compared with Hookeancurve), up to about strains = 2 units, after which finitesize effects begin to play a role. As we know from thediscussions in the earlier sections, the existence ofuniversality implies that the expansion factor E isindependent of the parameters h* and d* and dependsonly on the solvent quality z.

At values of strains beyond ε ) 2, Figure 12 showsthat, within experimental errors, (a) the curves agreewith experiments, for the values chosen for h* in Figure10, namely, h* ) 0.19 and h* ) 0.3, and (b) both theFENE and WLC model predictions (for these values ofh*) agree with the experimental measurements. Clearly,only data obtained at higher strains and more accuratemeasurements can resolve the differences seen in thetheory.

In Figure 13, we examine the effect of solvent qualityusing only the WLC force model for the springs. Herewe choose to compare the absolute values of the stretchxj instead of using the expansion ratio E. Ideally, acomparison of the transient swelling ratio (displayed inFigure 11), with experimental observations, would haveclearly revealed the differences that solvent quality caninduce in the unravelling rate. In the absence of suchdata, we use the absolute value to highlight the differ-

Figure 12. Comparison of the expansion factor predicted forHookean, WLC, and FENE force laws, at two different Weis-senberg numbers, with the experimental observations of Simthand Chu.3 Parameter values are the same as in Figure 7.

Figure 13. Effect of solvent quality on the absolute stretchof λ-DNA in elongational flow. Theoretical predictions underθ-conditions and two different solvent qualities, z ) 1 and z )3, are compared with the experimental results of Smith andChu.3 The maximum contour length of the chain, L ) 22 µm,is also displayed. Parameter values are Nk ) 200, h* ) 0.19,and Wi ) 2.


ences. It is clear from Figure 13 that the quality of thesolvent not only influences the equilibrium swelling (atε ) 0) but also increases the rate of chain unravellingwhen compared to the z ) 0 or (θ state) elongation. Thecurves in Figure 13 were obtained with h* ) 0.19 andWi ) 2. It was seen earlier that at Wi ) 2 and z ) 1 thestretch curves are not very sensitive to the choice of h*(Figure 8). Similar behavior was also observed for theother two solvent qualities displayed in Figure 13.

It is perhaps appropriate to conclude this section byreiterating that the parameter estimates L ) 22 µm,Nk ) 200, and z ) 1, used to obtain theoreticalpredictions, are not adjustable parameters in the modelbut can be determined independently by experimentalmeasurements. This combination has been shown tolead to a reasonable prediction of the experimental dataof Smith and Chu. However, because of the uncertaintyin the extent of blooming and the experimental errorbars on xj+, it is possible that other combinations of theseparameters could also represent the data well.

7. Conclusion

A framework has been introduced for describing thebehavior of dilute solutions of flexible macromolecules,which enables a consistent representation of the impor-tant physical phenomena of hydrodynamic interactions(HI), excluded-volume effects (EV), and finite extensibil-ity. Incorporation of all these three nonlinear phenom-ena has been found to be essential in order to quanti-tatively explain the behavior of these solutions. The coreproblem of estimating the magnitude of parametersgoverning these effects has been addressed separatelyin the two regimes of behavior encountered in exten-sional flows, namely, (i) the universal regime where thechain is predominantly coiled and (ii) the nonuniversalregime where the chain unravels into a stretched statedue to its finite size.

In the universal regime, two recently introducedmethodologies for using Brownian dynamics simulationsof bead-spring chain models with Hookean springs,namely, the procedure suggested by Kroger et al.11 forexamining the influence of hydrodynamic interactionsin the nondraining limit and the procedure suggestedby Kumar and Prakash14 for obtaining the asymptoticbehavior due to excluded-volume effects, have beencombined to obtain parameter-free predictions in thepresence of both HI and EV effects. In particular, thedependence of the expansion ratio E on the Henckystrain ε has been shown to become independent, in thelong chain limit, of the hydrodynamic interaction pa-rameter h* and the range of excluded-volume interac-tions d* (Figure 4). In this regime, the only variablesrequired to describe the behavior of the solution are theradius of gyration under θ-conditions, Rg

θ, the solventquality, z, and the Weissenberg number, Wi (Figure 5).

The modification of the definitions of the hydrody-namic interaction parameter, and the excluded-volumeinteractions parameter, as a consequence of usingfinitely extensible springs, has been examined by con-sidering the approach of chains with finitely extensiblesprings to the long chain limit. While the rescaled HIparameter is h* ) h*/ø, the rescaled EV parameter isz* ) (z*/ø3), where ø is proportional to the nondimen-sional length of a single spring. As a result, though thepredictions of universal ratios at equilibrium underθ-conditions are identical to those obtained with Hookeansprings, universal behavior is attained in the nondrain-

ing limit h ) h* xN f ∞ (Figure 1). Similarly, whilethe crossover dependence of the swelling of the polymerchain is identical to the prediction obtained withHookean springs, the solvent quality parameter z isredefined as z ) z*xN (Figure 2).

A description of finite size effects in elongational flow,for polymer chains with Nk Kuhn steps, has beenobtained by introducing the successive fine-graining(SFG) procedure. In this two-step method, first, predic-tions of material properties are obtained with bead-spring chains consisting of progressively higher numberof finitely extensible springs N, while keeping thecontour length L, the radius of gyration Rg

θ, the solventquality z, and the Weissenberg number Wi constant. Inthe second step, the accumulated data are extrapolatedto the limit (N - 1) f Nk. Though the results obtainedwith the SFG procedure are not strictly parameter-free,they were found to be relatively insensitive to the choiceof h* in the range 0.19-0.30 (Figure 8) and to d* (Figure9) for Nk ) 200. Interestingly, predictions of the expan-sion ratio for springs obeying two different force laws,namely, FENE and WLC, were found to differ from eachother significantly only at large strain units (Figure 10).While the influence of excluded-volume effects is presentat all values of strain in the case of infinite chains(Figure 5), EV effects die out as polymer molecules offinite length unravel to a stretched state (Figure 11).

Because of uncertainties in the persistence length ofDNA and in the extent of video-camera blooming in theobservations of stretch reported in ref 3, the values Rg

θ

) 0.63 µm (corresponding to Nk ) 200) and z ) 1 wereused in this work in order to compare theoreticalpredictions with experimental results. Reasonable agree-ment between theory and experiment was found at boththe Weissenberg numbers, Wi ) 2 and Wi ) 55, thatwere examined here (Figure 12).

The SFG procedure introduced here has opened upthe exciting possibility of systematically studying theinfluence of HI, EV, and finite extensibility effects onthe behavior of dilute polymer solutions far from equi-librium. This procedure, which enables an accuratemodeling of the dynamics of DNA in high salt solutionswith bead-spring chain models, could also prove to bea useful tool for computational biophysicists, since it issignificantly less computationally expensive than usingbead-rod chain models with bending potentials.37

Acknowledgment. This work has been supportedby a grant from the Australian Research Council underthe Discovery-Projects program. We thank Mr. R. Prab-hakar for help with the development of the BDS codesand for very fruitful discussions. We are also gratefulto Prof. Tam Sridhar for his suggestions and comments.

Appendix A. Rescaling of InteractionParameters for FES at Equilibrium

The need to incorporate the finite extensibility of thespring arises only in flow situations where the Hookean(or linear) spring force model results in unphysicalbehavior of the predicted material functions of a dilutesolution.6 However, to be able to do so, it is essential tofirst understand the effects of finite extensibility atequilibrium. In this appendix, the influence of usingfinitely extensible springs on the scaling of equilibriumstatic and dynamic properties of a bead-spring chainmodel is analyzed using theoretical approximations andconfirmed by the use of BDS.


The finite extensibility of the spring is incorporatedby considering a model in which the spring forcebecomes singular at some finite extension. This intro-duces a parameter Q0, which is the fully stretchedlength. Some of the commonly used spring force modelsare the finitely extensible nonlinear elastic (FENE) law,the inverse Langevin chain (ILC), and the wormlikechain (WLC). They may be represented in a generalform consisting of a Hookean part and a nonlinear partFc(Q) ) HQ f (Q/Q0), where the non-Hookean part f (Q/Q0) is given for the various cases by

where q ) Q/Q0. These expressions are statisticalapproximations to the equilibrium force extension be-havior of a chain of discrete elements (segments or rods)and smoothly go from a linear regime at small exten-sions to singular behavior when fully stretched. TheFENE and the ILC approximate a chain of freely jointedrods, and both these models have the same asymptoticbehavior as q f 1, the ILC being a Pade approximationto the actual inverse Langevin function.10 The wormlikechain, however, has a distinctly different asymptoticbehavior, and the above WLC expression19 is a simpleinterpolation formula between the two limiting behav-iors for the normalized extension as q f 0 and q f 1.

Considering a case when the only contribution to thepotential energy of the chain comes from the spring force(i.e., a chain under θ-conditions) and using the forcelaws, it is straightforward to calculate equilibriumaverages6 of various moments of the distribution func-tion ψ. In particular, we can find the covariance betweentwo connector vectors to be ⟨QµQν⟩ ) ⟨Q2⟩δµνδ, where ⟨Q2⟩is the average evaluated using the equilibrium distribu-tion for a single spring. In three dimensions this can bewritten in a general form as

Here we introduce the function ø(b) which can beevaluated with the knowledge of the force law for thefinitely extensible spring. For a FENE spring this canbe written explicitly as6

In the asymptotic limit of b f ∞, which corresponds toa linear spring force by construction of the nonlinearpart f, we recover the Hookean result for ⟨Q2⟩, in whichcase, ø(b) f 1. The function ø(b) is therefore boundedbetween zero and unity.

We now obtain some general exact results for a chainof connected springs in equilibrium. The static proper-ties can be written down explicitly under θ-conditions.The end-to-end vector for the chain is Re ) ∑µ

N-1Qµ,and the average mean-square end-to-end distance is⟨Re

2⟩ ) (N - 1)⟨Q2⟩. As a result, using eq 52, we obtainat equilibrium

It can be shown that the mean-square gyration radiuswith an arbitrary spring force law at equilibrium is6

These results for the static properties are exact in Nand are applicable for any spring force law underθ-conditions. While similar closed form expressionscannot be obtained under good-solvent conditions andfor dynamic properties even under θ-conditions, theycan be obtained by a numerical method (such as BDS),as will be discussed later in this section. Nevertheless,the asymptotic behavior of these properties can beobtained by some analytical approximations, which aregiven for each of the cases below.

A.1. Rescaling of the HI Parameter. In the pres-ence of HI between the beads, as discussed in section2, the HI tensor is often approximated using the Oseentensor given in eq 7. It is known that this approximationhas a limitation for small values of rµν and requires acorrection.22 However, this correction can be ignored inthe long chain limit of N f ∞, for an analyticaltreatment. In the equilibrium averaged approximation,the tensor, which is a function of the bead positions, isreplaced by an isotropic constant tensor Ωµν f ⟨Ωµν⟩eq,which is given by6

The ensemble average appearing in the above equationcan be carried out using the equilibrium distributionfunction of the vector between any two bead positionsrµν. For a bead-spring chain with Hookean springs, inwhich the distribution function for the vector betweenadjacent beads is Gaussian by definition, it can beshown that the probability density P(rµν) for any twobeads is also a Gaussian6 and is given by

where σH2 ) |µ - ν|(kT/H). It is then straightforward to

evaluate

In the case of a finitely extensible spring the distributionfunction between adjacent beads is not Gaussian, andtherefore the probability density P(rµν) cannot be re-duced to a Gaussian, as in the case of Hookean springsabove. However, we can make an approximation when|µ - ν| . 1. Since adjacent springs are “freely jointed”(independent) at the bead center, we know from thecentral limit theorem that the distribution function forthe vector rµν will be Gaussian for |µ - ν| . 1 and willbe given by

fFENE ) 11 - q2

fILC ) 3 - q2

3(1 - q2)

fWLC ) 16q(4q + 1

(1 - q)2- 1) (51)

⟨Q2⟩ ) 3ø2(b)kTH

(52)

ø2(b) ) bb + 5

(for FENE) (53)

⟨Re2⟩ ) 3(N - 1)ø2(b) kT

H(54)

⟨Rg2⟩ ) ⟨Re

2⟩N + 16N

(55)

⟨Rg2⟩ ) ø2(b)N

2 - 12N

kTH

(56)

⟨Ωµν⟩eq )1 - δµν

6πηs⟨ 1rµν

⟩eq

δ (57)

P(rµν) ) ( 12πσH

2)3/2e-rµν

2/2σH2

(58)

⟨ 1rµν

⟩eq

) x 2πσH

2) x 2H

πkTx 1|µ - ν| (59)


where σF2 ) |µ - ν|l 2 and l is the characteristic length

of a spring. For an arbitrary force law this is given byl 2 ) ⟨Q2⟩/3 and along with eq 52 we can write for anyforce law

The average occurring in the hydrodynamic interactiontensor can then be written as

The first case in the above expression represents thesituation where the average is evaluated with a Gauss-ian distribution, corresponding to beads for which |µ -ν| > |µ - ν|min. In the second case, the function f (ø, |µ- ν|) represents the value of the analytically unknownaverage that arises due to the distribution functionP(rµν) being non-Gaussian for beads with |µ - ν| < |µ -ν|min. An estimate of the bounds on |µ - ν|min can beprovided by the following argument. For a freely jointedbead-rod chain, |µ - ν|min ≈ 10 leads to a Gaussiandistribution for the end-to-end vector. On the otherhand, we know from eq 58 that for a chain of Hookeansprings even the vector between adjacent beads obeysa Gaussian distribution, implying |µ - ν|min ) 1.Therefore, we can anticipate that for a finitely extensiblespring, with a stiffness parameter b varying frominfinity (Hookean) to zero (rodlike), that 1 < |µ - ν|minj 10. The function f (ø, |µ - ν|) must also clearly satisfythe requirement that f (ø, |µ - ν|) f 0, as ø f 1.

We now obtain the scaling of the diffusivity for a chainof finitely extensible springs and make some generaliza-tions based on this. The translational diffusion coef-ficient under θ-conditions in the preaveraged approxi-mation is given by6

Substituting eq 62, converting sums to integrals, andtaking into account the correction due to the singularityat µ ) ν,29,38 we obtain in the limit N . 1

where g(ø,N) ) ∑µ,ν;|µ-ν|<|µ-ν|minf (ø, |µ - ν|)/x|µ-ν|

represents non-Gaussian contributions to the sum in eq63, and hZf

/ ≈ 12/35x2 ) 0.2424. (A further smallrefinement to this calculation can be obtained,39 which

gives hZf/ ) 0.242 16.) The only difference between the

Zimm result (for Hookean springs) and eq 64 is thepresence of the functions ø and g(ø,N). It is instructiveto examine the behavior of universal ratios at thisjuncture. The ratio of radius of gyration and thehydrodynamic radius can be written as

In the limit of large N we can write eq 56 as Rg ) øxN/2 xkT/H, and with eq 64 we have

Since the long chain limit must remain unchangedregardless of the force law used to describe the springs,we can write the expansion

This implies that, correct to leading order, the universalratio can be written as

where

Comparing eq 68 with the Zimm expression for Hookeansprings

it can be concluded that (i) in the presence of finitelyextensible springs a rescaled hydrodynamic interactionparameter can be defined by absorbing the factor 1/øinto the definition of h*

and (ii) the fixed point is modified and becomes afunction of ø, as can be seen from eq 69. In general, anyuniversal ratio in the presence of finitely extensiblesprings can consequently be written in the form

A close parallel can be drawn with the resultsobtained for a chain with consistently averaged HI andfinite extensibility.26 In this approximation the equilib-rium distribution function is replaced by a Gaussiandistribution as in the FENE-P model. In such anapproximation for FENE springs, one obtains ø2(b) )b/(b + 3), instead of eq 53, and the whole treatment can

P(rµν) ≈ ( 12πσF

2)3/2e-rµν

2/2σF2

(60)

σF2 ) |µ - ν|ø2(kT/H) (61)

⟨ 1rµν

⟩eq

)

x 2πσF

2)x2

π1lH

1ø x 1

|µ - ν|,for|µ-ν|>|µ - ν|min

x2π

1lH

1ø x 1

|µ-ν|(1+f(ø,|µ-ν|)) otherwise

(62)

Dh θ )kT

Nú(1 +h*xπ

N xkT

H∑µ,ν

µ*ν

⟨ 1

rµν⟩

eq) (63)

Dh θ ) kTNú[1 + h*

ø (83x2N - 1hZf/ )] + O(N-3/2) +

kTNú

x2h*Nø

g(ø,N) (64)

URDh ≡ Rg

RH≡ 6πηs

kTDh θRg ) 1

xπh*Rgx H

kTDh θúkT

(65)

URDh ) 8

3xπ+ 1

x2πN( øh*

- 1hZf/ ) + O(N-1) +

1xπN3/2

g(ø,N) (66)

g(ø,N) ) g0(ø)N + g1(ø)xN + ... (67)

URDh ) 8

3xπ+ 1

x2πN( øh*

- 1hf/(ø)) + O(N-1) (68)

1hf/(ø)

) 1hZf/

- x2g0(ø) (69)

URDh ) 8

3xπ+ 1

x2πN( 1h*

- 1hZf/ ) + O(N-1) (70)

h* ) h*ø(b)

(71)

Uφ ) Uφ∞ + cφ( 1

h*- 1

hf/(ø)) 1

xN+ O(1

N) (72)


be carried out with the rescaled interaction parameterh* in eq 71, as was shown in ref 26.

A.2. Rescaling of EV Interactions. In this work,excluded-volume effects have been incorporated in abead-spring chain model by using a narrow-Gaussianpotential, given in eq 4. Kumar and Prakash14 haverecently established that, in the long chain limit, thispotential correctly captures both the solvent qualitycrossover and the excluded-volume limit behavior ofHookean chains. It was shown that in this limit the onlyrelevant parameter is the solvent quality z, expressedin terms of the model variables as z ) z*xN. However,this interpretation of solvent quality in terms of themodel variables cannot be directly carried over to thecase of a chain of finitely extensible springs, and areinterpretation of z in terms of model parameters isnecessary.

As pointed out earlier, it is difficult to obtain exactexpressions for the static properties with excluded-volume interactions. However, a perturbation expansionin the strength of interactions can be carried out.5,13

Here we present some essential steps required to obtainthe perturbation parameter when carrying out anexpansion in small excluded-volume effects using thenarrow-Gaussian potential. The derivation is based onan expansion obtained for a chain of Hookean springs,40

which we extend to the case of finitely extensiblesprings. The easiest starting point is the calculation ofthe swelling of a static property, such as the root-mean-square end-to-end distance

which requires the evaluation of the covariance atequilibrium

where the normalizing constant

and φ is the potential energy of the chain given by

where φks is the entropic spring potential for a spring k,

obtained from the nonlinear model for the force chosen,and φe is the narrow-Gaussian potential in eq 4. In thefollowing, we omit explicit limits on sums for ease ofnotation, and the limits given above should be takenappropriately. The weighting function in the momentequation (eq 74) can be expanded to first order in EVeffects

Before we evaluate the covariance, we need to find thenormalization constant correct to first order. It can beshown that

where

for the leading order potential, and ⟨φe⟩θ is the ensembleaverage evaluated with the distribution at θ-conditions

To evaluate ⟨φe⟩θ, we recognize that, similar to thecase of HI, EV is a “long-range” effect; i.e., the interac-tions are between parts of the chain that are farseparated along the chain length are responsible for themacroscopic properties.5 Therefore, we can make thesame approximation as we did before and use eq 60(which is a Gaussian distribution) to evaluate theequilibrium average ⟨φe⟩θ. Since φe is also a Gaussian,we can show

where σz2 ) d2 + σF

2. Substituting this in eq 78 andexpanding 1/N to first order in v, which is required forevaluating the moments, we have

which also defines N. We now find ⟨QiQj⟩, correct to firstorder in v, to be

Therefore, to evaluate the second moment to first order,we only have to evaluate averages with Gaussiandistributions, a convenience brought about due to theassumption of a narrow-Gaussian potential for EVeffects. Following the approach given in ref 41 todecompose complex moments of Gaussian distributionfor a multidimensional Gaussian distribution in severalvariables, we can show that

Finally, it follows from eq 84 that

⟨Re2⟩ ) ∑

i,j

N-1

⟨Qi‚Qj⟩, (73)

⟨QiQj⟩ ) 1N∫dQN-1 e-φ/kTQiQj (74)

N ) ∫dQN-1 e-φ/kT (75)

φ ) ∑k)1

N-1

φks +

1

2∑

µ,ν)1µ*ν

N

φe(rµν) (76)

e-φ/kT ) ∏k

e-φsk/kT[1 -

1

2kT∑µ*ν

φe(rµν)] (77)

N ) Nθ[1 -1

2kT∑µ*ν

⟨φe(rµν)⟩θ] (78)

Nθ ) ∏k∫dQN-1 e-φs

k/kT (79)

⟨φe⟩θ )1

Nθ∏

k∫dQN-1 e-φs

k/kT(φe) (80)

⟨φe(rµν)⟩θ ) vkT(2πσz

2)3/2(81)

1

N)

1

Nθ[1 + ∑µ*ν

v

2(2πσz2)3/2] ≡ 1

Nθ(1 +1

N) (82)

⟨QiQj⟩ )Nθ

N⟨QiQj⟩θ -

1

2kT

Nθ

N∑µ*ν

⟨QiQj φe(rµν)⟩θ (83)

⟨QiQj⟩ ) ⟨QiQj⟩θ(1 +1

N) -1

2kT∑µ*ν

⟨QiQjφe(rµν)⟩θ (84)

⟨QiQjφe(rµν)⟩θ )

⟨φe(rµν)⟩θ × [⟨QiQj⟩θ - 1σz

2⟨QiQjφ

e(rµν)⟩θ · ⟨rµνQj⟩θ](85)

⟨QiQj⟩ ) ⟨QiQj⟩θ + ∑µ*ν

v

2(2πσz2)3/2

1

σz2⟨Qirµν⟩θ · ⟨rµνQj⟩θ

(86)


Using the following identities, which are again simpleGaussian averages

and observing that rµν occurs as an even multiple,implying the indices µ and ν can be swapped to obey µ< ν, we can show that

With the following identities and simplifications (usingeq 61)

we can write the final expression for the covariance fromeq 89 as

where we have used the convenient notation introducedin ref 41

which picks only those µ, ν that span a range across iand j. Substituting the expression for the covariance ineq 73, converting the sums to integrals in the limit Nf ∞, as given in ref 40, the actual perturbationparameter for excluded-volume effects can be shown tobe equal to

This provides the definition of the solvent quality interms of the model parameters. With this redefinition,the swelling ratio of the radius of gyration can beobtained using a chain of finitely extensible springs.Just as the universal ratios do not depend on the forcelaw, we expect the crossover functions to be universalfunctions of z and consequently the same irrespective

of the nature of the springs used to obtain the crossover.Using BDS, we have demonstrated that this is indeedthe case.

A.3. Physical Interpretation of Rescaling. Whilethe discussion of the rescaling of HI and EV interactionparameters above has been carried out on formalgrounds, it is instructive to provide a physical interpre-tation of the rescaling. First, to understand the meaningof ø, we turn to eq 52: for a Hookean force law, whereø ) 1, we can interpret xkT/H to be the characteristiclength of a single spring. In standard treatments,18 thishas been considered as the characteristic length lH ≡xkT/H used to nondimensionalize the governing equa-tions. For any finitely extensible spring, it is clear fromeq 52 that ø is the characteristic length of the spring inthe units of xkT/H. The HI parameter h*, which hasbeen considered to represent a nondimensional effectivehydrodynamic radius, eq 13, for the Hookean springs,must be reinterpreted in a more general way as the ratioof the bead radius to the characteristic length of thespring, so that the important nondimensional numbercharacterizing the strength of HI should be

as has appeared naturally in eq 68.Similarly, in the case of EV, the local parameter is a

volume term v and the important nondimensionalparameter characterizing excluded-volume interactionscan be written down from eq 93 as

This reinterpretation of HI and EV parameters isconsistent with Hookean results with ø ) 1.

Appendix B. Evaluation of ø(b) for a WLCThe factor ø(b) is defined by the nondimensional

second moment in eq 52. The distribution function withwhich the moment is evaluated is simply that of theconnector vector of a single spring (or a dumbbell)

with φ given by Fc ) ∂φ/∂Q. Nondimensionalizing lengthby xkT/H, inserting the WLC force model expressionfrom eqs 51, and performing the integration in sphericalcoordinates, the expression for ø can be written as

where φ* ) (b/6)[2q2 + 1/(1 - q) - q]. This integral isnot known in closed form; however, it might be evalu-ated by using a numerical quadrature. We present some

⟨QiQj⟩θ ) ø2 kTH

δδij (87)

⟨Qirµν⟩θ ) ø2 kT

Hδ ∑

k)µ

ν-1

δik for µ < ν (88)

⟨QiQj⟩ )

ø2 kT

Hδ × [δij + ∑

µ*ν

1

2

v

(2πσz2)3/2

ø2 kT

H

1

σz2

∑k,l)µ

ν-1

δikδjl](89)

σz2 ) ø2 kT

Hσz/2 (90)

σz/2 ≡ [d*2

ø2+ |µ - ν|] (91)

v(2πσz

2)3/2) z*

(ø2σz/2)3/2

(92)

z* ≡ v(2πkT/H)3/2

(93)

⟨QiQj⟩ ) ø2 kT

Hδ[δij +

z*

2ø3∑µ*ν

1

σz/5

θ(µ,i,j,ν)] (94)

θ(µ,i,j,ν) ) 1 if (µ e i, j < ν) or (ν e i, j < µ)0 otherwise ) (95)

z ) z*ø3

xN (96)

h* ) bead radiuscharacteristic length of a spring

(97)

h* ) 1xπ

a

øxkT/H) h*

ø(98)

z* ) excluded volumecube of the characteristic length of a spring

(99)

z* ) 1(2π)3/2

v

(øxkT/H)3) z*

ø3(100)

ψeq (Q) ) e-φ/kT

∫dQ e-φ/kT(101)

ø2(b) ) 13

∫0

1dq q4e-φ*

∫0

1dq q2e-φ*

(102)


useful approximations to evaluate this integral numeri-cally. Consider the integral in general

where n is any number. The problem of evaluating thisintegral numerically using a generic formula arisesbecause, for very large b, the contributions beyond acertain value of q become negligible, and to obtain anaccurate value of the integral, several quadrature pointsneed to be placed in the whole interval 0 < q < 1, sothat the dominant interval is well represented. Toovercome this problem, we perform a rescaling of theintegral and obtain its value in an asymptotic limit. Thisis done by evaluating the integral only up to a certainvalue of q. The upper bound of this integration is foundin the following manner. We first note that for large bcontributions come from q , 1, in which limit we canapproximate φ* ≈ b/6 + bq2/2. Therefore, we have forlarge b

The contributions to this integral for q > x2M/b can beneglected, where M is a large number. This providesan upper bound so that we can write eq 103 as

With this rescaling, the integral can be evaluated with

a small number of Gauss quadrature points: what hasbeen essentially achieved is that the quadrature pointsare efficiently spaced in the dominant interval.

The evaluation of the integral with the WLC forcemodel has an additional complication. This arises fromthe small q behavior of φ* where φ* ∼ b/6 + bq2/2. Notethat there is an additional constant apart from theHookean factor. Though this does not have any conse-quence for the physics of the problem, since in the forceexpression (used in the BDS) this term is zero upondifferentiation, it does lead to numerical problems inevaluation of the integral for large b. For large b thisleads to (a) floating point overflow, in both the numera-tor and the denominator in eq 102 because of thepresence of the constant e-b/6 as well as (b) a precisionerror when one small number is divided by another. Toovercome this, we multiply the numerator and denomi-nator of eq 102 by eb/6, which will not alter the integraland which also leads to the correct Hookean limit for q, 1.

Using these approximations, ø can be evaluated for agiven value of b. The behavior of ø as a function of Mfor various values of b is shown in Figure 14. The valueof ø approaches its asymptotic value to within four digitsof accuracy even for M ) 18. The function ø(b) soobtained is shown in Figure 15. We note that thisprocedure can be uniformly applied to any value of band is also common to all finitely extensible force lawswhere no closed form solutions are available for ø(b).

Note Added after ASAP Publication. The correctreference 4 was inadvertantly omitted from the versionposted ASAP December 15, 2004; the corrected versionwas posted December 29, 2004.

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MA035941L


Parameter-Free Prediction of DNA Conformations in Elongational Flow by Successive Fine Graining

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