Soft Matter

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aBiophysics Program, Stanford University

lenafabr@stanford.edu; ajspakow@stanfordbChemical Engineering Department, Stanfor

Cite this: Soft Matter, 2013, 9, 7016

Received 30th January 2013Accepted 9th April 2013

DOI: 10.1039/c3sm50311a

www.rsc.org/softmatter

7016 | Soft Matter, 2013, 9, 7016–70

Discretizing elastic chains for coarse-grained polymermodels

Elena F. Koslover*a and Andrew J. Spakowitz*ab

Studying the statistical and dynamic behavior of semiflexible polymers under complex conditions generally

requires discretizing the polymer into a sequence of beads for purposes of simulation. We present a novel

approach for generating coarse-grained, discretized polymer models designed to reproduce the polymer

statistics at intermediate to long lengths. Our versatile model allows for an arbitrary discretization

length and is accurate over a larger range of length scales than the traditional bead–rod and bead–

spring models. In its generality, the discrete, stretchable, shearable wormlike chain (dssWLC) model

incorporates the anisotropic elasticity inherent in a semielastic chain on intermediate length scales. We

demonstrate quantitatively the statistical accuracy of this model at different discretizations, thereby

allowing for efficient selection of the number of segments to be simulated. The approach presented in

this work provides a systematic procedure for generating coarse-grained discrete models to probe

physical properties of a semielastic polymer at arbitrary length scales.

I Introduction

Many of the polymers commonly encountered in the biologicalcontext and elsewhere exhibit semiexible properties. That is,they are effectively rigid, with a linear bending elasticity onsome short length scale, begin to look increasingly exible atlonger lengths, and can be approximated as fully exiblethreads at the longest length scales. Such polymers are oenmodeled as “wormlike chains” (WLC),1,2 continuous chains withan energy density that is quadratic in the curvature and char-acterized by the persistence length l p – the length scale at whichthe polymer transitions from rigid to exible behavior. TheWLC model has been shown to be an effective description of avariety of locally stiff polymers, including double-strandedDNA3–5 and RNA,6 actin laments,7 unstructured polypeptides,8

and carbon nanotubes.9

The statistical mechanics of the pure WLC model have beenstudied extensively,2,10,11 with the partition function analyticallycalculated under various end constraints.12–14 However, theincorporation of more complicated effects for realistic poly-mers, such as sterics, hydrodynamics, connement, or inter-actions with packaging proteins, generally requires numericalsimulation of the chain properties. For purposes of simulation,the wormlike chain is generally represented by a linearsequence of beads connected by an effective potential. Suchdiscrete chains constitute a coarse-grained model of the poly-mer in question, as each bead generally represents a segment of

, Stanford, CA 94305, USA. E-mail:

.edu

d University, Stanford, CA 94305, USA

27

the original detailed polymer containing many atoms. Simula-tions based on theWLCmodel have been employed to study thepacking of DNA into chromatin bers,15,16 the packaging of viralgenomes,17 and themechanical properties of actin networks.18,19

Two common approaches are employed for generatingdiscrete versions of the wormlike chain model. The “bead–rod”models connect beads along the polymer contour with stiff rodsand incorporate a bending potential between subsequentrods.20 A closely related set of models allows for elasticstretching of the connected segments, with the stretch modulusset equal to that of the polymer being modelled.15,19 Whenapplied to modeling biopolymers such as DNA or actin la-ments, the stretch modulus is usually quite high, approxi-mating the bead–rod model. This potential is oen harmonic inthe bend angle, and such models can accurately reproduce thecontinuous wormlike chain provided that the segment length issignicantly shorter than the persistence length. However, suchan approach becomes computationally prohibitive when thetotal chain length is much longer than the length-scale of thepolymer stiffness. For instance, the bead–rod model isimpractical for modeling genomic DNA which can range from104 to 107 persistence lengths. Another class of models repre-sents the chain as beads connected by isotropic harmonicsprings.3,20,21 These models are essentially equivalent tomapping the polymer onto an effective Gaussian chain,22 andcan accurately represent chain behavior only when each beadrepresents a contour length signicantly larger than thepersistence length. Finer discretization can be achieved bymodied bead–spring models which make use of alternative,non-harmonic spring force laws extracted from the force–extension behavior of the wormlike chain.23,24

This journal is ª The Royal Society of Chemistry 2013

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The appropriate choice of discretization length (D) for apolymer chain arises from a balance between the availablecomputational power and the length-scales relevant to thephysics of interest. Increasing values of D result in a chain withfewer degrees of freedom, leading to more efficient simulations.However the minimal length-scale of accuracy will also increaseas the discretization length increases. Currently, there is nostandard approach to generating discrete chain models in a waythat can incorporate both ne-grained and coarse-grained dis-cretization within a single unied framework.

The process of generating a discrete model to represent acontinuous chain can be thought of as a coarse-grainingprocedure. One seeks a representation of the chain that isaccurate at length scales above the discretization length whileforgoing accuracy at shorter length scales. In a previous work,we presented a systematic procedure for coarse-graining poly-mer models by mapping onto a family of continuous chainstermed the stretchable, shearable wormlike chain (ssWLC)model.25 The ssWLC is dened by both a position and anorientation vector at each point along the contour, with theenergy for a given path containing quadratic penalties forbending of the orientation, stretching of the position along theorientation vector, shearing away from the orientation vector,and quadratic coupling between bend and shear deviations. Weshowed that such a chain, which incorporates both the classicKratky–Porod wormlike chain and the Gaussian chain as speciallimits, can be used reproduce the intermediate to long-lengthstatistics of any polymer dened entirely by localized interac-tions. The ssWLC provided a systematic approach to describingthe behavior of complicated polymer chains by mapping to ananalytically tractable continuous model.

In this work we describe a novel polymer model – thediscrete, stretchable, shearable wormlike chain (dssWLC) –

which serves as a discrete analog to the continuous ssWLC. Thenew model can be used to systematically generate discreterepresentations of the continuous wormlike chain at anydesired length-scale of discretization. We provide a quantitativeestimate of the length scale of accuracy for this model and showthat the dssWLC can span length scales where neither bead–rodnor Gaussian spring models are accurate. Our discretizationprocedure is based on selecting effective parameters for themodel in such a way as to match the intermediate to long-lengthstatistics of the original polymer model being studied. Althoughwe focus here on the discretization of the continuous wormlikechain, the methodology described could be employed todevelop coarse-grained discrete models for an arbitrary polymerwith semielastic properties, provided the physics dening itslocal behavior are known.

II Discrete models for the WLC

One of the most common approaches for creating a discretizedversion of the continuous WLC polymer model involves simplydiscretizing the Hamiltonian. For a chain of length L and dis-cretization length D, the polymer is split into beads numberedfrom 0 to N ¼ L/D, separated by segments of xed length. Thepolymer conguration is given by the bead positions {~ri} or

This journal is ª The Royal Society of Chemistry 2013

equivalently by the orientations of the segments connectingadjacent beads~ui¼ (~ri�~ri�1)/D. The free energy associated witha particular conguration is given by,

E��~ui�� ¼XN

i¼1

3b

2D

���~ui �~ui�1

���2; (1)

where 3b is commonly set to equal to persistence length (l p) ofthe original wormlike chain.

Although such a “bead–rod” model is accurate for very ne-grained discretizations, the long-chain statistics of this discretemodel diverge from those of the continuous WLC as thesegment length D increases. In particular, the lowest-ordermoment of the end-to-end distance for the bead–rod chain canbe calculated as,

hR2i ¼ ND2 1þ x

1� xþ 2D2x

xN � 1

ð1� xÞ2 ;

x ¼ �~ui$~ui�1

� ¼ coth3bD

� D

3b:

(2)

The long-chain statistics of the polymer are encompassed inthe term of this lowest order moment that is linear in the chainlength (here denoted as C(1)

2 ). This term is also known as theeffective Kuhn length,22 and nding C(1)

2 for a particular model isequivalent to mapping onto an effective Gaussian chain. InFig. 1a we demonstrate how this term diverges from the valueassociated with a continuous wormlike chain as D increases. Toproperly reproduce the long-chain behavior, one must adjustthe bending modulus 3b in order to match the effective Kuhnlength of the continuous chain,

Cð1Þ2 ¼ D

�1þ x

1� x

�¼ 2l p: (3)

The appropriate bending modulus obtained by solvingeqn (3) is plotted in Fig. 1a. We note that this bending modulusdeparts substantially from the continuous persistence lengthnear D ¼ 0.5l p, and decreases to zero as D / 2l p. Equivalently,the WLC has the same effective Gaussian behavior at longlength as the freely jointed chain with segment length 2l p. Forsegment lengths longer than 2l p, the bead–rod model cannotaccurately reproduce long length-scale behaviour of thecontinuous WLC. The signicance of adjusting the bendingmodulus as a function of the discretization length is shown inFig. 1b, which compares the end-to-end distributions fromdiscretized bead–rod chains using Monte Carlo simulations.The discrete model with the adjusted value of the bendingmodulus is notably more accurate at reproducing the statisticsof the continuous WLC at low to intermediate chain extensions(corresponding to the longest length-scale effects).

We note that the adjustment of 3b to reproduce the effectiveKuhn length has been previously described.26,27We reiterate thisanalysis here because a large number of studies in recentliterature continue to use the approximation 3b ¼ l p regardlessof the discretization length. Furthermore the selection of thebending modulus to match long-length behavior provides asimple analogue for the more detailed discretization methodthat is the focus of the current work. All subsequent results in

Soft Matter, 2013, 9, 7016–7027 | 7017

Fig. 1 Statistics of the bead–rod model. (a) Black: the linear term (C(1)2 ) of themean square end to end distance, for different discretization lengths using theconvention 3b ¼ l p. Red: the appropriate value of 3b to match the effective Kuhnlength to that of the continuous WLC. All units are in terms of l p. (b) Distributionof the end-to-end distance for the bead–rod model with conventional (green)versus adjusted (red) values of the bending modulus, compared to the continuousWLC (blue), from Monte Carlo simulations of a chain of length 4l p with segmentlength D ¼ 0.5l p.

Fig. 2 Statistics for the discrete stretchable WLC model. (a) Effective parametersfor discretizing a continuous WLC with discretization length D. All length units arein terms of l p. (b) Distribution of the end-to-end distance for a wormlike chain oflength 6l p with a discretization length of D ¼ 2l p. The bead–rod model (green),bead–spring Gaussian model (red), and discrete stretchableWLCmodel (cyan) arecompared to the analytic results for the continuous chain (blue).

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this work pertaining to the bead–rod model use the adjustedbending modulus obtained by solving eqn (3).

In order to more accurately reproduce the WLC statistics atintermediate length scales, it is necessary to match both thelinear (C(1)

2 ) and constant (C(0)2 ) terms of hRz

2i, as well as thelinear term (C(1)

4 ) of the next highest order moment, hRz4i. If

these three terms match between two chain models, then allmoments of the end-to-end distance will match out to an errorthat scales as O (1/L2) as a fraction of the long-length limit.25 Inorder to develop a discrete chain model that can match thesethree constants, we require two additional free parameters inthe model.

A natural choice is to consider a stretchable discrete WLC,where the segments between beads are allowed to stretch awayfrom their preferred length, with a quadratic penalty. Such amodel could reproduce more accurately the physics of thecontinuous WLC by incorporating the tendency of a chainsegment of contour length D to be compressed, on average, as aresult of thermal wiggles. Again, the chain conguration isgiven by a set of beads at positions {~ri}, i¼ 0,.,N. We dene thefree energy function for the discrete stretchable WLC as follows,

7018 | Soft Matter, 2013, 9, 7016–7027

En

~Ri

o¼XNi¼1

3b

2D

��~ui �~ui�1

��2þ 3k2D

�~Ri

�� ��� DgÞ2; (4)

where ~Ri ¼~ri �~ri�1 and~ui ¼ ~Ri/|~Ri|. This model includes threeparameters—the bending modulus (3b), the stretch modulus 3k,and the ground-state segment compression (g). The low-ordermoments of the end-to-end distance for such a model can becalculated as described in Appendix A.

The effective parameters of the discrete stretchable WLC areobtained by matching the terms C(1)

2 ¼ 2/3l p, C(0)2 ¼ �2/3l p

2,C(1)4 ¼ �208/45l p

3 of the continuous WLC (see Fig. 2a). We seethe ground state length g of the segments decreases withincreasing D as longer contour lengths are on average morecompressed under thermal uctuations. Interestingly, thebending modulus 3b increases with D, though we note that theoverall bending stiffness between adjacent segments (3b/D) doesdecrease for longer segments, as expected. This relative stiff-ening is a manifestation of the additional correlations beyondthe Gaussian chain model (which has C(0)

2 ¼ C(1)4 ¼ 0) that are

imposed by matching these additional terms to those of thecontinuous WLC. As the segment length increases, thesegments also become more exible, with decreasing stretch

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modulus 3k. The improvement of the stretchable WLC over thestandard bead–rod model or the isotropic bead–spring modelobtained by mapping to a Gaussian chain is illustrated by thedistribution of chain extensions obtained from Monte Carlosimulations (Fig. 2b).

We note that with this discretization procedure the stretchmodulus becomes innite as the segment length decreasesdown to D ¼ 0.79l p. At this particular discretization length, theinextensible bead–rod model accurately reproduces themoments of the continuous chain out to the second highestorder in chain length. The discrete stretchable WLC cannot beused to model the continuous WLC for segment lengths belowthis critical value, and a more complicated polymer model withadditional parameters is required. In the subsequent section,we develop a discrete stretchable, shearable wormlike chain(dssWLC) model, which serves as a discretized version of theuniversal continuous elastic model used for coarse-graining ofpolymer chains in a previous work.25 We show that this modelallows us to reproduce the intermediate to long length behaviorof the continuous WLC at arbitrary discretization length D.

III Discrete SSWLC model

The dssWLC model chain consists of a sequence of beadsnumbered 0 through N, with the polymer conguration fullydescribed by the positions (~ri) of each bead and the unit vectors(~ui) attached to each bead. For convenience, we dene thevectors connecting adjacent beads as~Ri¼~ri�~ri�1. We note thatthe orientation vectors ~ui constitute additional degrees offreedom in this model, and are not equal to the tangent vectorsconnecting the beads. The free energy function for a particularconguration of the polymer chain is given by,

En

~Ri;~ui

o¼XNi¼1

3b

2D

���~ui �~ui�1 � h~Rt

i

���2þ 3k2D

~Ri$~ui�1 � Dg

2þ 3t

2D

���~Rt

i

���2�(5)

where~Rti ¼~Ri� (~Ri$~ui�1)~ui�1 is the component of the interbead

vector perpendicular to the orientation vector~ui�1. This model,illustrated in Fig. 3, discretizes the continuous-chain Hamilto-nian of the previously described ssWLC model.25 Its sixparameters include the bending modulus (3b), stretch modulus

Fig. 3 Schematic of the discrete stretchable, shearable WLC model. The chain concharacterized by a position and an attached orientation vector~ui. The free energy fuas well as coupling between the bend and shear degrees of freedom.

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(3k), shear modulus (3t), bend–shear coupling (h), andfractional ground-state segment length (g), which serve analo-gous roles to the parameters in the continuous model. Anadditional parameter (D) corresponds to the discretizationlength—the continuous contour length encompassed by eachsegment of the discrete model. We note that when 3b¼ l p, g¼ 1,3k, 3t / N, the dssWLC reduces to the standard bead–rodmodel used to discretize a wormlike chain with persistencelength l p.

The propagator for a single segment of the discrete chain canbe dened as,

G1

�~u; ~R|~u0

� ¼ 1

Nexp�� E1

�~u; ~R;~u0

��(6)

where N is a normalization constant and E1 corresponds to thefree energy for a single segment (the summand in eqn (5)). Thepartition function for the entire chain of N segments can thenbe found by a convolution of N copies of G1,

GN

�~u; ~R|~u0

� ¼ ð d�~ri�d�~ui�YNi¼1

G1

�~ui;~ri �~ri�1|~ui�1

�(7)

To simplify the convolution structure we perform a Fouriertransform~R/~k. For simplicity, we assume that the wavevectoris oriented along the z canonical axis (~k ¼ k z). The transformedsingle-segment propagator is given by,

G1

�~u ; k|~u0

� ¼ 1

Nexp

"l pD~u$~u0 þ 3b

2h2

2D3t

h1� �~u$~u0�2i

þ ik

(Dgz$~u0 þ h3b

3t

�~u$z� �~u$~u0��z$~u0��

)

� k2

(D

2

1

3k� 1

3t

! �z$~u0

�2þ D

23t

)#(8)

where 3t ¼ 3t + h23b.Furthermore, if we expand the propagator in terms of the

spherical harmonics,28

G1

�~u; k|~u0

� ¼ Xl 0 ;l ; j

g jl ;l 0

Yjl

�~u�Yj*

l 0

�~u0�

(9)

then the convolution in eqn (7) turns into a matrix multiplica-tion and the overall partition function for the chain can beexpressed as

tour is discretized into beads separated by segments of length D, with each beadnction incorporates quadratic penalties for stretch, shear, and bending deviations,

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GN

�~u; k|~u0

� ¼Xh�gðjÞ�Ni

l ;l 0Yj

l

�~u�Yj*

l 0

�~u0�

(10)

where g ( j ) is the matrix of coefficients for the Fourier, spherical-harmonic transformed propagator. Note that throughout thismanuscript we use boldfaced symbols for matrices. If we areinterested specically in the distribution of the N th bead posi-

tion (given by�ðgð0ÞÞN�0;0), then we may assume j ¼ 0, and this

index is dropped for all subsequent calculations. This formu-lation is analogous to previously developed approaches forcalculating the partition function of a chain consisting of linkedsegments by convolution over the degrees of freedom for thejoints.25,29 The dening physics of the specic discrete chainmodel used here is encompassed by the matrix g (k). Details onthe calculation of the gl ,l 0(k) coefficients for the dssWLC areprovided in Appendix B.

Moments of the end-to-end displacement for the discretechain are found, as in the continuous case, by taking derivativesof the coefficient gN00 with respect to the wave-vector magnitudek (see Appendix C). The discrete chain structure factor is denedas the Fourier-transformed density correlation function,

SdssWLCðkÞ ¼ 2

N2

XNn1¼0

Xn1n2¼0

�exp�ik�~rn2 �~rn1

�$z��

¼ 2

N2

XNn1¼0

Xn1n2¼0

�gðn2�n1Þ�

0;0

¼h�

NI þ gðNþ1Þ � ðN þ 1Þg�$ðg � IÞ�2i0;0

(11)

where I is the identity matrix.

Fig. 4 Error in the structure factor (eqn (13)) for the bead–rod (dashed) anddssWLC models (solid), compared to the continuous WLC. Discretization lengthsare D ¼ 0.1, 0.5, 1, 2 (from blue to red). A constant value of a ¼ 1 is used in thesecalculations.

IV Discretizing with the DSSWLC model

In a previous work,25 we described how to coarse-grain a poly-mer chain by mapping to the continuous ssWLC model, usingthe effective persistence length l p as a sliding parameter to setthe degree of coarse-graining. Here, we proceed in an analogousfashion, by nding the effective parameters 3b, 3k, 3t, h, g of thedssWLC in such a way as to ensure that all moments of the end-to-end distribution match up to the second highest order inchain length. For simplicity, we non-dimensionalize lengthunits such that the continuous chain has a persistence length of1. We note that the effective persistence length (l effp ) of thediscrete chain can be dened as

�~ui$~ui�1

� ¼ g11ðk ¼ 0Þ ¼ exp � D

l effp

!: (12)

Continuing the analogy to the continuous ssWLC, we denea dimensionless parameter a ¼ h23b/3t, so that the effectivepersistence length depends only on 3b, a. For xed values of l effp ,a, the components of the low-order moments can be expressedas polynomial functions (of order 8 or lower) in the remainingthree parameters, g, 3k

�1, 3t�1 (see Appendix D). We solve these

coupled polynomial equations to nd the effective parametersof the dssWLC model that will reproduce the continuous chainat intermediate to long lengths. Since l effp served as an indica-tion of the coarse-graining scale for the continuous chain

7020 | Soft Matter, 2013, 9, 7016–7027

mapping, we proceed to select the lowest possible value ofl effp such that a solution exists with g > 0, 3k > 0, 3t > 0.

In order to quantify the length-scale of accuracy for thedssWLC model, we examine the structure factor as compared tothat of the continuous WLC. The discretized structure factor forthe continuous WLC (Scont(k)) is dened as in eqn (11), with theFourier, spherical-harmonic transformed propagator g taken tobe the previously characterized11 propagator for a continuouschain of length D. Fig. 4 shows the fractional error in thestructure factor,

EmodelðkÞ ¼ SmodelðkÞ � ScontðkÞScontðkÞ ; (13)

as a function of the wavevector magnitude k, for both thedssWLC and the bead–rod models. Small values of k correspondto long length-scale behavior, while larger values dial in statis-tics for progressively shorter length-scales. The dssWLCoutperforms the bead–rod model, with the difference betweenthe two increasing signicantly as the discretization length D isincreased. We dene the length-scale of accuracy for eachmodel as lmodel ¼ 1/k* where Emodel < 10�4 for all k < k*. That is,we pick a cutoff for accuracy in the structure factor and nd thelength-scale such that the model exceeds that accuracy at alllonger lengths.

Up to this point, in selecting the effective parameters of thedssWLCmodel, we have set the parameter a to an arbitrary xedconstant. Using the above denition of the length-scale ofaccuracy, we can now allow a to vary for each discretizationlength in such a way as to minimize ldssWLC. The resultingeffective parameters of the discrete model are plotted in Fig. 5.We note that for D $ 1.8, the shear modulus becomes inniteand the appropriate dssWLC model reduces to one where theorientation vectors ~ui fully align with the bead-to-bead vectors.In this limit, the dssWLC model constitutes a modied versionof the discrete stretchable WLC described in Section II. InAppendix E we describe a procedure for efficiently representing

This journal is ª The Royal Society of Chemistry 2013

Fig. 5 Parameters for the dssWLC model for discretizing a continuous WLC at different discretization lengths D. The dimensionless parameter a is adjusted in such away as to minimize the length scale of accuracy. (a) Bending modulus (blue), effective persistence length for the discrete chain (black), and ground state segmentcompression (red). (b) Inverse stretch (green) and shear (magenta) modulus. 3t

�1 ¼ 0 corresponds to a modified discrete stretchable WLC model as described inAppendix E. (c) Coupling coefficient (cyan) and a parameter (yellow).

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the dssWLC with 3t /N in simulations without keeping trackof the ~u orientation vectors.

For small values of the discretization length D / 0, thestretch and shear moduli become innite, with the ground statecompression g/ 1, as expected for increasingly short segments

Fig. 6 Length scale of accuracy (l ¼ 1/k*) for the bead–rod and the dssWLCmodel (with parameters given in Fig. 5) as a function of the discretization length.The line l ¼ D/2 is also shown for comparison.

Fig. 7 Distributions of the end-to-end displacement for a chain of intermediate lenthe continuous WLC are shown in blue. Monte Carlo simulations for the bead–rod musing different discretization lengths.

This journal is ª The Royal Society of Chemistry 2013

of a semiexible chain. The effective persistence length alsoapproaches the original value for the continuousWLC l effp / 1 asthe auxiliary vectors ~ui attached to each bead become identicalwith the discrete tangent vectors connecting consecutive beads.

Finally, in Fig. 6 we plot the effective length-scale of accuracyfor the dssWLC model with optimized values of a, as comparedto the bead–rod model. We see that the minimal length-scale ofaccuracy for the bead–rod model rises sharply with increasingdiscretization length D, so that models with longer segmentscan only be efficiently applied for very long chains. The dssWLCmodel on the other hand, allows for discretization at anysegment length D while remaining accurate at signicantlyshorter length-scales. The length-scale of accuracy for ourmodel can be approximated as l � D/2 over the range D < 1.8.The transition to a higher scaling with D occurs at the pointwhere the shear modulus becomes innite and the coarse-grained model becomes a modied version of the discretestretchable chain (see Appendix E).

We further illustrate the utility of this model by comparingthe distribution of end-to-end extensions for chains of inter-mediate length (Fig. 7). Although both models can reproducethis distribution at a ne-grained discretization of D ¼ 0.1, thedssWLC model allows the discretization length to increase by a

gth (L¼ 4, in terms of the continuous chain persistence length). Analytic results forodel (green) and bead–spring model (red) and dssWLC model (cyan) are shown,

Soft Matter, 2013, 9, 7016–7027 | 7021

Fig. 8 Average extension along direction of applied force, for the continuouswormlike chain and simulated discrete models, using a discretization length D ¼2l p and total chain contour length L ¼ 4l p. Error bars are standard error of themean from Monte Carlo simulations.

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factor of 10–20 while still remaining relatively accurate. We notethat the dssWLC model deviates from the continuous chainbehavior at larger values of the end-to-end extension becausethis model does not include any nite length constraint pre-venting the segments from stretching beyond their contourlength. The statistics of the chain near full extension aredominated by short length-scale wiggles, explaining the loss ofaccuracy in this regime.

As a further application of the discretization proceduredescribed here, we consider the simulated force–extensioncurves for the dssWLC chain, the bead–rod chain, and thecontinuous wormlike chain (Fig. 8). Such curves are commonlyused for interpreting single-molecule experiments that aim toextract mechanical properties of polymers such as DNA, RNA, orproteins by pulling on them via optical or magnetic traps.3,5,6,30,31

Simulations of polymers under force have been employed formodeling the unpeeling of nucleosomes,15 stretching andunfolding of chromatin,32,33 and passage of DNA and otherpolymers through nanopores.34 Our results show that thedssWLC model can accurately reproduce the low-force exten-sion of the polymer at relatively large discretization lengths (D¼2). Again, high force behavior is dominated by short length-scale uctuations, so that coarse-grained models are inappro-priate for studying this regime. In the low-force regime we seesignicantly improved agreement with the continuous chainstatistics as compared to the bead–rod model.

V Concluding remarks

The results described in this manuscript demonstrate how thedssWLCmodel canbeused todiscretize a continuous semiexiblechain to an arbitrary segment length, with signicantly improvedaccuracy in the chain statistics when compared to the conven-tional bead–rod model. We note that for a given discretizationlength, the dssWLC has two extra degrees of freedom per beaddening the additional orientation vector~ui. However, while thetotal number of degrees of freedom in the simulation is less thandoubled, the length scale of accuracy, as denedby comparisonof

7022 | Soft Matter, 2013, 9, 7016–7027

the structure factor deviation from the continuous WLC,decreases by asmuch as 50-fold for segment lengths between oneand two persistence lengths. Thus, the modest increase incomplexity of the model is balanced by signicant gains in accu-racy. Overall, the dssWLCmodel allows for accurate simulation ofthe long-length behavior of the wormlike chain model at asignicantly higher degree of coarse-graining and thus with asignicant savings in computational time as compared to thestandard approach of discretizing the chain into rigid segments.

The discretization method presented here has a wide rangeof possible applications in the eld of polymer simulations. Theappropriate discretization length D for any given applicationwill depend on the length-scale of the relevant physics ofinterest. For instance, when simulating polymer networks ormeshes, the discretization length needs to be smaller than thecontour length separating interaction points or cross-links. Oneof the main strengths of the model described here is itsuniversality, in that it can be applied to nd an effective modelat any given discretization length. By incorporating both ne-grained and coarse-grained discretizations within a singleformalism, the dssWLC model opens up the possibility ofadjusting the discretization between simulation runs withdifferent parameter sets or even within an individual simulationas the relevant length-scale varies between different portions ofthe chain or different simulation time points.

A number of unansweredquestions remain regarding efficientways to incorporate non-local chain effects into simulations thatmake use of the dssWLC model with coarse discretization. Forinstance, the appropriate formulation of steric interactions forchains where each segment represents a relatively long contourlength of the polymer of interest remains a topic for future study.We note that the same difficulties regarding steric interactions,topological constraints, and hydrodynamics plague the conven-tional discrete elastic models. One possibility that arises with thedssWLC is that of using ner discretization lengths in areaswhere the polymer chain becomes crowded or self-intersecting,while longer discretizations can be maintained in isolatedregions of the polymer, with the discretization level adjusteddynamically throughout the simulation. Another approachwouldbe to make use of the WLC propagator function to determine theshape of the chain density between xed beads of the discretemodel, using the result to create effective steric shapes for thesegments connecting the beads. The dynamic behavior of thedssWLC is also an enticing topic for future study. A dynamicsimulation making use of this model would require an effectivefriction coefficient for rotation of the orientation vectors~ui.

The dssWLC model as presented here could, in principle, beexpanded to include the twist of the polymer as well as itscontour path. To this end the vectors ~u would need to beexpanded to orthogonal triads representing a full coordinateorientation in space. The bending and shear moduli could thenbe made anisotropic with respect to this coordinate orientation.In addition to matching the moments of the end-to-enddistance at long lengths, the additional parameters shouldmake it possible to match moments of the coordinate orienta-tion, so that the statistics of the polymer twist would also bereproduced at long length-scales. While such an expansion of

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the model is beyond the scope of this manuscript, the use of thedssWLC to model polymers with twist, including global twist-writhe constraints remains a ripe topic for further study.

In a previous work we demonstrated how detailed polymerchains can be coarse-grained by mapping onto a continuouselastic polymer model with analytically tractable statistics.25

Here we have presented an alternate, though closely related,approach that allows for mapping a continuous polymer modelonto a discrete elastic chain suitable for simulations. ThedssWLC allows the discretization, and thus the computationalcost, of the simulation to be systematically adjusted to theminimal requirements necessary given the length-scale ofinterest. The discrete polymer model described here thussignicantly expands our ability to model semiexible polymerchains with the use of coarse-grained simulations.

�s4� ¼ 66

gD2

3kþ 28D3g4 þ 2D43kg

5 þffiffiffiffiffiffiD3

33k

s 15þ 45D3kg

2 þ 15D23kg4 þ D333kg

6X

23kgþ ffiffiffiffiffiffiffiffiffiffi3k=D

p �1þ g23kD

�X

(S7)

VI AppendixA Low order moments of the discrete stretchable WLC

In this section we calculate the lowest-order moments hRz2i and

hRz4i of the discrete stretchable WLC model, whose free energy

function is dened by eqn (4). For convenience, we introducethe notation si ¼ |~Ri| for the length of segment i and ri ¼ (~Ri$z)/|~Ri| for the component of the segment orientation in thecanonical z direction. Because the energy function has nocoupling between the segment lengths and their orientations,the values of si and ri are uncorrelated with each other. The low-order moments can then be expressed as

�Rz

2� ¼

* XNi¼1

siri

!2+¼ 2hsi2

Xi. j

�rirj�þN

�s2��ri

2�

(S1)

�Rz

4� ¼

*�XNi¼1

siri

�4+

¼ 24X

i\j\k\h

hsi4�rirjrkrh�þ36

Xi\k\h

�s2�hsi2�ri2rkrh�þ 6

Xi\k

�s2�2�

ri2rk

2�

þXi\h

�s3�hsi�ri3rh�þN

�s4��r4�

(S2)

Moments of the segment length can be found by directintegration,

hsni ¼

ðN0

snþ2exph� 3k2D

ðs� DgÞ2ids

ðN0

s2exph� 3k2D

ðs� DgÞ2ids

; (S3)

yielding the following expressions:

This journal is ª The Royal Society of Chemistry 2013

hsi ¼ 4þ 23kg2Dþ gffiffiffiffiffiffiffiffi3kD

p �3þ g23kD

�X

23kgþ ffiffiffiffiffiffiffiffiffiffi3k=D

p �1þ g23kD

�X

(S4)

�s2� ¼ 10gDþ 2D23kg3 þ ffiffiffiffiffiffiffiffiffiffi

D=3kp �

3þ 6D3kg2 þ D232kg4�X

23kgþ ffiffiffiffiffiffiffiffiffiffi3k=D

p �1þ g23kD

�X

(S5)

�s3� ¼

16D

3kþ 18D2g2 þ 2D33kg

4 þ g

ffiffiffiffiffiffiD3

3k

s 15þ 10D3kg

2 þ D232kg4X

23kgþ ffiffiffiffiffiffiffiffiffiffi3k=D

p �1þ g23kD

�X

(S6)

X ¼ffiffiffiffiffiffi2p

pexp�g2D3k=2

� 1þ erf

"g

ffiffiffiffiffiffiffiffiD3k2

r #!(S8)

To calculate the fourth order correlations in the segmentorientations, we expand the propagator G(~uk|~uk�1) for the kth

segment orientation ~uk given the orientation of the previoussegment ~uk�1 in terms of the spherical harmonics,

G~uk |~uk�1

¼ 1

Nexph3bD

~uk$~uk�1

i¼XNl¼0

Xl

m¼�l

xlYm*l

~uk�1

Ym

l

~uk

; (S9)

where xl ¼ilð3b=DÞi0ð3b=DÞ and il is the l

th modied spherical harmonic

Bessel function of the rst kind. Eqn (9) combines the usualplane wave expansion in terms of the relative angle between theorientations together with the spherical harmonic additiontheorem.28 The distribution function of the kth segment orien-tation given the orientation of the 0th segment can be found as aconvolution over all intermediate segments. Using the ortho-normality of the spherical harmonics this gives,

Gð~uk |~u0Þ ¼ð d~u1.d~uk�1X

l1.lk

m1.mk

xl1Ym1*l1 ð~u0ÞYm1

l1 ð~u1Þxl2Ym2*l2 ð~u1ÞYm2

l2 ð~u2Þ.

.xlkYmk*lk ð~uk�1ÞYmk

lk ð~ukÞ

¼XNl¼0

Xl

m¼�l

_xl

kYm*

l ð~u0ÞYml ð~ukÞ

(S10)

We nd the correlation between segment orientations, withi # j as,

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�rirj� ¼ ð d~uid~ujrirjX

l;m

xj�il Ym*

l ð~uiÞYml ð~ujÞ ¼

1

3xj�i1 : (S11)

Using the property rYml (r)¼ aml+1Y

ml+1(r) + a

ml Y

ml�1(r), with aml ¼

[(l � k)(l + k)/(4l2 � 1)]1/2, we can also calculate the fourth-ordercorrelation function of segment orientations, for i # j # k # h,�

rirjrkrh� ¼ ðd~uid~ujd~ukd~uhrirjrkrhX

xj�il1 Ym1*

l1 ð~uiÞYm1l1 ð~ujÞxk�j

l2 Ym2*l2 ð~ujÞYm2

l2 ð~ukÞ

� xh�kl3 Ym3*

l3 ð~ukÞYm3l3 ð~uhÞ

¼ 1

9xj�iþh�k1

�4

5xk�j2 þ 1

�(S12)

Plugging into eqn (S1) and carrying out the summations overindices i, j, k, h yields the two lowest order moments for a chainwith N segments in the form,�

RZ2� ¼ C

ð1Þ2 DN þ C

ð0Þ2 þ.�

RZ4� ¼ C

ð2Þ4 D2N2 þ C

ð1Þ4 DN þ C

ð0Þ4 þ.

(S13)

where (.) contains terms that decay exponentially with N. Todiscretize a continuous wormlike chain with persistence lengthl p as an effective discrete stretchable WLC, we numerically solvethe following equations to nd the effective parameters 3b, g, 3k:

Cð1Þ2 ¼ 2

3

hsi2x1ð1� x1Þ

þ 1

3

�s2� ¼ 2

3l p (S14)

Cð0Þ2 ¼ �2

3

hsi2x1ð1� x1Þ2

¼ �2

3l p2 (S15)

Cð1Þ4 ¼ 4hsi4x12ð � 25þ 5x1 þ 33x2 � 13x1x2Þ

15ð1� x1Þ3ð1� x2Þ

þ 4hs2ihsi2x1ð15� 23x2 � 14x1 þ 22x1x2Þ15ðx1 � 1Þ2ðx2 � 1Þ

þ hs2i2ð5� 13x2Þ15ðx2 � 1Þ þ 8hs3ihsi

5ð1� x1Þþ hs4i

5¼ �208

45l p3

(S16)

B Projected propagator coefficients for the DSSWLC

In this section we calculate the matrix of coefficients obtainedby projecting the Fourier-transformed propagator G1(~u, k|~u0) ofthe dssWLC model onto the spherical harmonics (eqn (9)),focusing specically on the case with j¼ 0. We begin by deningthe relative orientation vector u, such that if R is a three-dimensional rotation that places~u0 on the z axis, then u ¼ R$~u.Additionally, we dene u0 ¼ R$z. This yields the followingidentities in terms of the spherical harmonic functions,

~u0$z ¼ffiffiffiffiffiffi4p

3

rY 0

1 ðu0Þ ¼ffiffiffiffiffiffi4p

3

rY 0

1

�u0�

~u$~u0 ¼ffiffiffiffiffiffi4p

3

rY 0

1

�u�

~u$z ¼ffiffiffiffiffiffi4p

3

rY 0

1 ðuÞ ¼4p

3

X1j¼�1

Yj*1

�u0�Y

j1

�u�

(S17)

7024 | Soft Matter, 2013, 9, 7016–7027

where the last equation is an instance of the spherical harmonicaddition theorem.28

The propagator can then be expressed as,

G1 ¼ F�u�exphH�k; u0; u

�i

F�u� ¼ 1

Nexp

(3b

D

�u$z� þ h23b

2

2D3t

1� �u$z�2�

)

H�k; u0; u

� ¼ ik

(Dg

ffiffiffiffiffiffi4p

3

rY 0

1

�u0�

þ 4ph3b33t

hY 1

1

�u�Y 1*

1

�u0�þ Y�1

1

�u�Y�1*

1

�u0�i)

� k2

(D

6

1

3k� 1

3t

! " ffiffiffiffiffiffiffiffiffi16p

5

rY 0

2

�u0�þ 1

#þ D

23t

)

(S18)

In order tond thematrix of coefficients for projecting G1 ontothe spherical harmonics, we assume that this matrix will be trun-cated at the lmax level and proceed to expand the quantity exp[H(k,~u0, u)] out to the lmax order in the wave-vector k. We note that theexpansion of a product of two spherical harmonics is given by,35

Yj1l1Y

j2l2¼ ð�1Þj1þj2ffiffiffiffiffiffi

4pp

Xl1þl2

l3¼jl1�l2 j

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið2l1 þ 1Þð2l2 þ 1Þð2l3 þ 1Þ

p

��l1 l2 l30 0 0

��l1 l2 l3j1 j2 �j1 � j2

�Y

j1þj2l3

(S19)

where the parentheses enclose Wigner 3j symbols.35 The matrixelement gl,l0 contains terms that are of order at least max(l,l0) ink. Thus, truncating the matrix of coefficients is equivalent tosetting a length scale cutoff below which the partition functionbecomes inaccurate.

The quantity H is a linear combination of terms of the formYja(~u0)Y

jb(u). Expanding out the exponent yields

exp�H�k; u0; u

�� ¼Xa;b;j

hja;bY

ja

�u0�Y

jb

�u�

(S20)

where a $ 0, b $ 0, |j| # min(a,b) and the coefficients hja,b areobtained by applying eqn (S19) repeatedly.

We then perform another spherical harmonic expansion togive

F�u� ¼X

l

xlY0l

�u�: (S21)

The coefficients xl ¼ ÐF(u)Y0l(u)du are found recursively

using the usual formula for the Legendre polynomials:28

lþ 1ffiffiffiffiffiffiffiffiffiffiffiffiffi2lþ 3

p Y 0lþ1 ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffi2lþ 1

prY 0

l � lffiffiffiffiffiffiffiffiffiffiffiffiffi2l� 1

p Y 0l�1; (S22)

together with the additional recursion,

In ¼ð1�1

rnexp�ar� br2

�dr

�2b

nþ 1Inþ2 ¼ 1

nþ 1

hea�b � ð�1Þnþ1

e�a�bi

� a

nþ 1Inþ1 � In;

(S23)

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where in this case r ¼ u$z. All the coefficients are normalizedsuch that x0 ¼ 1

The coefficients in the spherical harmonic projection of thepropagator (eqn (9)) are given by,

gl;l0 ¼ðY 0

l0

�~u0�G1

�~u; k��~u0���Y 0

l

�~u�d~ud~u0 (S24)

For convenience we dene the tensor,

Ma;b;jl;l0

¼ðd~ud~u0Y

0l0ð~u0ÞYj*

a

�u0�Y

jb

�u�Y 0

l ð~uÞF�u�

gl;l0 ¼Xa;b; j

hja;bM

a;b;jl;l0

(S25)

As our expansion of the propagator is expressed in terms ofthe relative orientation vector u rather than ~u, we make use ofthe spherical harmonics addition theorem to interconvertbetween them:28

Y 0l

�~u� ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffi

4p

2l þ 1

r Xm

Yml

�u0�Ym*

l

�u�

(S26)

Consequently, we have

Ma;b;jl;l0

¼ffiffiffiffiffiffiffiffiffiffiffiffiffi4p

2l þ 1

r Xm

( ð du0Y

0l0

�u0�Ym

l

�u0�Yj*

a

�u0��

�"X

l

xl

ð duY

jb

�u�Y 0

l

�u�Ym*

l

�u�#)

¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið2aþ 1Þð2bþ 1Þð2l0 þ 1Þð2l þ 1Þ

4p

r Xlþb

l¼jl�bjxl

ffiffiffiffiffiffiffiffiffiffiffiffiffi2lþ 1

p

�

l l0 a

0 0 0

! l l0 a

0 j �j

! b l l

0 0 0

! b l l

j 0 �j

!

(S27)

where the coefficients xl are found using eqn (S21)–(S23).

C Moments of the end-to-end displacement

Moments of the end-to-end distribution for a chain of length Nsegments are found via,

�R2n

z

� ¼ ðiÞn vngN00vkn

: (S28)

In a previous work,25 we showed that such moments can beexpressed as an expansion in terms of the chain length,�

R2z

� ¼ Cð1Þ2 DN þ C

ð0Þ2 þ.�

R4z

� ¼ Cð2Þ4 ðDNÞ2 þ C

ð1Þ4 DN þ C

ð0Þ4 þ.�

Rð2nÞz

� ¼ CðnÞ2n ðDNÞn þ C

ðn�1Þ2n ðDNÞn�1 þ.C

ð0Þ2n þ.;

(S29)

where all terms that decrease exponentially with chain lengthare dropped. Furthermore, if the three coefficients C(1)

2 , C(0)2 ,

C(1)4 are matched for two chain models, then all moments of the

end-to-end distance will match out to an error that scales as O (1/N2) as a fraction of the long-length limit. In the appendix of thisprevious work25 we employed stone-fence diagrams to calculatethe lowest-order moments for the case of a kinked WLC,another form of discrete polymer. The current case of the

This journal is ª The Royal Society of Chemistry 2013

discrete shearable WLC model yields analogous results, whereeach block matrix Bl0,l in the propagator for the kinked chain isreplaced with a single element gl,l0 for the dssWLC propagator.For completeness, we give the nal expressions here, in terms ofthe derivatives of gl,l0 at k ¼ 0:

DCð1Þ2 ¼ 2

x1 � 1 g001g010 � g0000

Cð0Þ2 ¼ 2

x1 � 1 2 g001g010

DCð1Þ4 ¼ �

12 x1 � 5 x1 � 1 3 g0201g

0210 þ

12 x1 � 3 x1 � 1 2 g0000g

001g

010

�3g00200 �24

x1 � 1 2

x2 � 1 g001g012g021g010

þ 12

x1 � 1 2 g001g0011g010 � 6

x2 � 1g0002g

0020

þ 12

x1 � 1

x2 � 1 �g0002g021g010 þ g001g

012g

0020

�

þ 4

1� x1

�g00001g

010 þ g001g

00010

�þ g000000 ;

(S30)

where we dene xl ¼ xl=ffiffiffiffiffiffiffiffiffiffiffiffiffiffi2lþ 1

p.

D Finding effective DSSWLC parameters

For purposes of discretizing a continuous WLC, the parameters3b, g, 3k, 3t, a for the effective dssWLC model with discretiza-tion length D are extracted by matching the coefficients C(1)

2 ¼ 2/3, C(0)

2 ¼ �2/3, C(1)4 ¼ �208/45 to those of the continuous chain.

We note that the coefficients xl depend only on 3b and a. Theeffective persistence length of the discrete chain is given by,

g11(k ¼ 0) ¼ x1 ¼ exp(�D/l effp ) (S31)

All the derivatives of matrix elements that enter into eqn(S30) are polynomial functions of g, 3k

�1, 3t�1, as well as

depending on the xl. We nd the effective parameters for thedssWLC according to the following scheme,

1 Fix a value of a.2 Fix a value of l effp .3 Solve for the appropriate value of 3b.4 Calculate coefficients xl for 1 # l # 4.5 Match the coefficient C(0)

2 to express g3t�1 as a quadratic

polynomial in g.6 Match the coefficient C(1)

2 to express g23k�1 as a fourth-

order polynomial in g.7 Match the coefficient C(1)

4 , solving an eighth-order poly-nomial to nd an appropriate value of g.

8 Repeat steps 2–7 to nd the lowest value of l effp such that apositive real solution exists for all parameters.

The xed parameter a can then be adjusted to minimize thelength-scale of accuracy based on the comparison between thediscrete chain structure factor and that of the continuous WLC.

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E Non-shearable limit of the DSSWLC

In the limit where the shear modulus goes to innity, thedssWLC model described in this manuscript reduces to amodied version of the discrete stretchable WLC (dened byeqn (4)). The shear component of the single segment propagatorG1 behaves in this limit as,

exph� 3t

2D

�~Ri �

�~Ri$~ui�1

�~ui�1

�2i �����!3t/N

1

jRj2�d�~Ri$~ui�1 � j~Rij

�þ d�~Ri$~ui�1 þ j~Rij

��;

(S32)

where d is the Dirac delta function. The overall density for asequence of segments ~R1,.,~RN, integrated over the auxiliaryorientation vectors ~ui is then given by,

P�~R1.~RN

� ¼ 1

N

X1m1¼0

.X1mN¼0

expn� 3k2D

½R1 � ð�1Þm1gD�2.

� 3k2D

½RN � ð�1ÞmNgD�2 þ ð�1Þm1þm23b

Dð~n1$~n2Þ.

þð�1ÞmN�1þmN3b

Dð~nN�1$~nNÞ

o:

(S33)

Here we dene the normalized bead-to-bead vectors~ni ¼ ~Ri/|~Ri|. This summation can be expressed in terms of transfermatrices,

P�~R1.~RN

� ¼ exp

(� 3k2D

XNi¼1

�j~Rij � gD�2 þ 3b

D

XNi¼2

~ni$~ni�1

)

�~AT

1 B1;2A2B2;3.An�1Bn�1;n~An

�QNi¼1

j~Rij2

Ai ¼ 1 0

0 exp�� 2j~Rijg3k

�!; ~Ai ¼

1

exp�� 2j~Rijg3k

�!

Bi;j ¼

0BBB@

1 exp

� 23b

D~ni$~nj

�

exp

� 23b

D~ni$~nj

�1

1CCCA

(S34)

The limiting behavior of this chain can then be described bythe free energy function,

En

~Ri

o¼

PNi¼1

h� 3b

D~ni$~ni�1 þ 3k

2D

�j~Rij � Dg�2 þ log

�j~Rij2�i

�log~AT

1 B1;2A2B2;3.An�1Bn�1;n~An

:

(S35)

There are thus two additional terms in the energy comparedto the usual formulation of the discrete stretchable WLC (eqn(4)). These terms arise because the energy associated withaltering the orientation of the segments in the dssWLCmodel iscontrolled by the unit vectors ~ui rather than the inter-segmentvectors directly. The nal term results because the innite shearlimit does not prevent a segment from being oppositely

7026 | Soft Matter, 2013, 9, 7016–7027

oriented to its orientation vector (~Ri$~ui�1 ¼ �|~Ri|). We note thatthis last term becomes exponentially small in the limit of stiff,relatively inextensible segments (large 3k). For very stiff springs(3k / N) the additional energy terms vanish (or becomeconstant) and themodel reduces to the bead–rodmodel denedby eqn (1).

The energy function given by eqn (S35) can be implementedin a straight-forward manner for simulating the thermody-namic behavior of the dssWLC model in the 3t / N limitusing Monte Carlo simulations. This formulation eliminatesthe need to keep track of the auxiliary orientation vectors ~ui,halving the degrees of freedom, and thus resulting in consid-erable computational savings.

Acknowledgements

We are grateful for helpful discussions with Ekaterina Pilyuginaand Jay Schieber. Funding for this project was provided by theFannie and John Hertz Foundation and the NSF CAREER award.

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