Systematic Coarse-Graining of Microscale Polymer Models asEffective Elastic ChainsElena F. Koslover*,† and Andrew J. Spakowitz*,†,‡

†Biophysics Program, Stanford University, Stanford, California 94305, United States‡Chemical Engineering Department, Stanford University, Stanford, California 94305, United States

ABSTRACT: One of the key goals of macromolecular modeling is to elucidate howmacroscale physical properties arise from the microscale behavior of the polymer constituents.For many biological and industrial applications, a direct simulation approach is impractical dueto to the wide range of length scales that must be spanned by the model, necessitatingphysically sound and practically relevant procedures for coarse-graining polymer systems. Wepresent a highly general systematic coarse-graining procedure that maps any detailed polymermodel onto effective elastic-chain models at intermediate and large length scales. Our approachdefines a continuous flow of coarse-grained models starting from the detailed microscale model, proceeding through anintermediate-scale stretchable, shearable wormlike chain, and finally resolving to a Gaussian chain at the longest lengths. Wedemonstrate the utility of this procedure by coarse-graining a wormlike chain with periodic rigid kinks, a model relevant forstudying DNA−protein arrays and genome packing. The methodology described in this work provides a physically soundfoundation for coarse-grained simulation of polymer systems, as well as analytical expressions for chain statistics at intermediateand long length scales.

1. INTRODUCTION

Polymer models are crucial to the study of a wide variety ofmaterials ranging from DNA1−4 and cytoskeletal fibers5−7 inthe biological context to nanotubes,8 gels,9,10 and plastics11 fortechnological applications. One of the key challenges inmodeling polymeric materials lies in bridging the gap betweenthe length scales of the underlying molecular mechanics and themuch larger length scales relevant to the macroscaleapplications of interest. For instance, polyethylene can bemodeled as a freely rotating chain with angstrom-lengthsegments,12 but this level of detail makes it impractical tosimulate macroscale properties of a piece of plastic. Anotherexample is the DNA polymer, whose molecular structure givesrise to an elastic rigidity at the nanometer length scale,1 whereascontour lengths up to several meters must be packaged,processed, and accessed by the cell.13 A meaningful approach tocoarse-graining polymer models is thus essential for modeling abroad range of systems of practical relevance.A variety of general approaches have been used for

developing coarse-grained models of polymer systems. Oneclass of models consists of united-atom formulations, wheregroups of atoms in a specific polymer structure are representedas coarse beads and detailed atomistic simulations are used toextract appropriate parameters for effective coarse-grainedinteractions.14−17 These methods can incorporate complicatednonlocal interaction potentials while reproducing the behaviorof realistic polymer systems.18,19 However, the reliance onsimulations makes such coarse-graining procedures computa-tionally expensive to implement, limits them to individualspecific polymer systems, and restricts the coarsening scale tomesoscopic models. An alternate approach makes use ofidealized elastic models comprising either bead−spring

assemblies20,21 or continuous chains23,23 to address highlycoarse-grained macroscale properties of general polymericsystems, including rheology, chain size, and response toexternal force. In this work, we focus on the latter approachof treating polymers as elastic threads. A common procedurefor modeling elastic polymers involves representing the chain asa sequence of discrete rigid links with an appropriate effectivebending potential.24−26 This procedure is accurate for shortdiscretization lengths but is unable to resolve intermediate-length behavior as the discretization is increased. An alternateapproach for large-scale coarse-graining is to treat the polymeras an effective Gaussian spring. A number of works haveaddressed the coarse-graining of specific polymer models, suchas the wormlike chain and the freely jointed chain, by mappingto bead−spring models that are valid when the contour lengthencompassed by each spring is quite long.1,21,27 Higheraccuracy for more finely discretized models can be achievedby using alternative spring force laws, with the functional formextracted from the force−extension behavior of the poly-mer.28,29 The desired degree of coarse-graining arises from abalance between the relevant length scales dictating the physicsof interest and the limitations of computational power requiredto encompass the overall size of the system. A generalprocedure that can bridge from finely detailed to roughlycoarse-grained polymer models within a single frameworkremains imperative for modeling multiscale phenomena.We present a systematic analytical procedure for generating

increasingly coarse-grained models for a locally defined

Received: October 2, 2012Revised: December 10, 2012

Article

pubs.acs.org/Macromolecules

© XXXX American Chemical Society A dx.doi.org/10.1021/ma302056v | Macromolecules XXXX, XXX, XXX−XXX

polymer, while maintaining the correct statistics in the long-chain limit. The configuration statistics of any polymer chaincan be fully defined by calculating all moments of the end-to-end distance ⟨Rz

2n⟩ as a function of contour length L. Atsufficiently long lengths, any polymer can be mapped to aneffective Gaussian chain where all such moments match up toan error that scales as 1/L (as a fraction of the long-chain limit).While such a mapping is used routinely to approximate a longpolymer as a random walk, it fails to account for correlations inthe polymer contour that become increasingly important onintermediate length scales. Our novel coarse-graining proceduremaps a detailed polymer model to an effective elastic chain thatmatches all moments out to an error scaling as 1/L2.In section 2, we focus on developing a continuous polymer

model that serves as a logical extension beyond the Gaussianchain, capturing behavior at intermediate length scales. Weshow that a stretchable, shearable wormlike chain (ssWLC)with bend-shear coupling is a universally appropriate model forcoarse-graining across a range of length scales. Exact analyticalexpressions for low-order moments of the end-to-end distance,as well as the full structure factor, are provided for this model.We describe a procedure for extracting the model parameterswhen coarse-graining a detailed polymer. We then demonstratethat our procedure provides a continuous flow of increasinglycoarse-grained ssWLC models from the classic Kratky−Porodinextensible wormlike chain22 at short length scales to theGaussian chain at long lengths.In section 3, we provide an example of a more complicated

polymer model that can be coarse-grained using this procedure.Specifically, we show how a wormlike chain with periodic kinkscan be mapped to an effective ssWLC at intermediate and longlengths. This example is motivated by its applicability tostudying DNA−protein arrays. We show that, on length scalesmuch shorter than those of a Gaussian-chain random walk, thisprocedure can reproduce the statistics of such kinked arrayswith a simple, easy to simulate, and analytically tractablecontinuous polymer model.

2. SYSTEMATIC POLYMER COARSE-GRAININGPROCEDURE

We describe a general approach for modeling the statisticalbehavior of a polymer chain based on the most localized energycontributions. Our procedure provides the lowest ordercorrection beyond the Gaussian chain at long lengths, therebyencompassing intermediate length-scale properties. The modelthat emerges from this approach incorporates a local chainorientation, while simultaneously allowing stretching andshearing of the chain path relative to this orientation. Previouswork by Wiggins and Nelson26 has shown that a discretepolymer with rigid links and an arbitrary bending potential canbe renormalized to a wormlike chain at long lengths if thedistribution of consecutive segment orientations is narrowlydistributed around the straight angle. A recent study by Wolfeet al.30 compared coarse-grained models of double-strandedDNA, showing how more detailed models converge to ananisotropic elastic filament at long length scales. Here wedevelop a more generalized result, demonstrating that anylocally defined polymer chain can be mapped to a stretchable,shearable worm-like chain (ssWLC) model. When applied to astiff chain, the ssWLC encompasses both the classic wormlikechain model at short length scales and the Gaussian chain atlong lengths. The model incorporates a sliding parameter thatsets the degree of coarse-graining, enabling expansion or

contraction of the length scale of accuracy to suit the relevantphysics of interest.

Continuous Stretchable, Shearable Wormlike Chain.The path of a continuous polymer chain is defined by a spacecurve r(s), where 0 ≤ s ≤ L is a coordinate marking positionalong the contour. The energy E of a particular configuration ofthe polymer can be expressed as

∫ = E r s s r r r[ ( )] d [ , , , ...]L

s ss sss0 (1)

where rs = (∂r/∂s), rss = (∂2r/∂s2), and so forth. The increasingderivatives of r correspond to increasing nonlocal contributionsof chain deformation to the energy.We note that this class of chains includes many commonly

used models in polymer physics, including wormlike chains,freely jointed chains, and rotational isomeric state models.31

More general polymer models such as as the twisted helicalwormlike chain23 or the birod chain30 can include additionallocal degrees of freedom (e.g., twist density) that allow foranisotropic flexibility and intrinsic curvature. However, if we areinterested specifically in the spatial distribution of the chaincontour, then the formulation given by eq 1 remains fullyapplicable, with the Hamiltonian expressed as a function of rs(s)by integrating over any additional degrees of freedom.In this manuscript, we focus specifically on “locally defined”

polymer chainsones where the correlations and interactionsbetween different segments along the chain decay on somefinite length scale. When viewed on a length scale that is muchlarger than the microscopic correlation length scales, all suchpolymers have an energy function that is effectively localized.For the general continuous chain defined by eq 1, if we zoomout far enough then all nonlocal physical effects shoulddisappear, and to lowest order, the Hamiltonian can beapproximated as = | |r A r[ ]s s

2. This approximation results inthe Gaussian chain model,26 a single-parameter model to whichall polymer models flow at sufficiently long lengths.On shorter length scales, less localized contributions to the

Hamiltonian will begin to matter, and a more general function r r[ , ]s ss is required. Here, we develop a model that expands

this function out to second order, keeping only those termspermitted by symmetry considerations. For the subsequentanalyses, we require the Hamiltonian to be expressed as afunction of general coordinates and their first derivatives. Wethus define two auxiliary variables σ(s) and u (s), where u is aunit vector and the ground-state polymer configuration musthave rs(s) = σ(s)u(s). Essentially, the auxiliary vector u tracksthe chain orientation. We then have rss = σsu + σsu . The firstpart σsu is a second-order term that cannot enter linearly intothe Hamiltonian due to symmetry with respect to reversal ofthe chain direction. As we are expanding only to second-order,this term does not contribute. Consequently, we can forego theauxiliary variable σ entirely and construct the Hamiltonian as alinear combination of (rs·u ),(rs·u )2,(rs·rs),(u s·us), and (u s·rs),where we have kept only those terms of first or second orderthat are not disallowed by symmetry arguments. We refer tothis continuous chain as the stretchable, shearable wormlikechain model (ssWLC) and for greater physical clarity expressthe Hamiltonian as

εη

εγ

ε= | − | + · − + | |⊥ ⊥ ⊥u r r u r

2 2( )

2b

s s s s2 2 2

(2)

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where rs⊥ = rs−(rs·u )u . We note that the ssWLC model as

presented here constitutes a modified version of previouslysolved elastic chain models, including the anisotropic elasticchain23,32 and the extensible shearable wormlike chain,33 in thatit includes an additional coupling term between the bend andshear degrees of freedom.The model described by this Hamiltonian can be understood

intuitively by considering the coarse-grained behavior of anarbitrarily long polymer chain, where each point along thecontour has both a position r(s) and an orientation u (s). Wecan then look at the distribution of these quantities at position s+ Δs relative to position s, where Δs encompasses anintermediate length of the polymer compared to the lengthscale at which correlations in the chain contour die off. Such anintermediate length polymer would resist bending (Figure 1a),

corresponding to the (εb/2)|us|2 term. Its natural end-to-end

length would be a fraction γ of its contour length, due tocontraction induced by entropic wiggles of the chain segment.The chain would resist stretching or compression away fromthis natural length with stretch modulus ε∥ (Figure 1b). Theend-to-end vector would tend to be aligned along theorientation u, and resist shearing away from this orientationwith modulus ε⊥ (Figure 1c). Finally, we would expect acoupling term modulated by η that accounts for the tendency ofthe polymer orientation to bend in the same direction as theshear (Figure 1d).We note that this model encompasses both the standard

Kratky−Porod wormlike chain model22 in the limit {ε∥ → ∞,ε⊥ →∞, γ = 1} and the Gaussian chain in the limit {εb = γ = 0}or {γ = 0, ε⊥ = ε∥}. The statistics of the ssWLC without bend-shear coupling have been previously studied,33 and we utilizesimilar approaches for calculations for our more general model.In the subsequent sections, we develop expressions for thestructure factor and low-order moments for the ssWLC withbend-shear coupling. We then show how to set the parametersof the ssWLC model in such a way as to match the moments ofthe end-to-end separation for a detailed polymer chain at anadjustable level of coarse-graining.

Distribution Function for the ssWLC. To calculate thestatistics of the ssWLC model defined by eq 2, we make use ofthe path integral formulation of the Green’s function for thechain distribution,34 defined by

∫

∫

| =

−

= =

= = G R u u L r s u s

s r s u s

( , ; ) [ ( )] [ ( )]

exp{ d [ ( ), ( )]}

u u r

u L u r L R

L

0(0) , (0) 0

( ) , ( )

0

0

(3)

where the Hamiltonian is made dimensionless by kBT. ThisGreen’s function gives the probability density for a chain oflength L starting at the origin with orientation u 0 to end atposition R with orientation u . It is normalized according to[∫ G(R, u |u 0; L)dR du = 1] . Following an approach analogousto one developed for helical wormlike chains,23 we find thegoverning “Schrodinger″ equation by drawing an analogy to thequantum-mechanical propagator for a coupled symmetric topand point particle with a velocity-dependent potential energy.We define the angular momentum operator Λ for the chainorienation, where each component Λμ corresponds to adifferential rotation around the μ axis. The Green’s functionthen obeys

δ δ δ∂∂

+ | = − ⎡⎣⎢

⎤⎦⎥L

G R u u L L R u u( , ; ) ( ) ( ) ( )0 0 (4)

where

ηε ε

εγ η

ε

= − Λ + Λ − − ·∇

− ·∇ + ·∇ − × ∇ ·Λ

⊥ ⊥

⊥

I uu

u u u

12 2

( )1

2[( ) ]

12

( ) ( )

pu R

R R R

22

2 2

2

(5)

and lp = εb/(1 + η2εb/ε⊥). We proceed to perform a Fouriertransform from position R to wavevector k, a Laplace transformfrom L to Laplace variable p, and a spherical harmonictransform in the orientations u and u0. Namely, we expand theGreen’s function as

∑ | = *G k u u p g k p Y u Y u( , ; ) ( , ) ( ) ( )j l l

l lj

lj

lj

0, ,

, 0

ff f

00 0

(5)

and project eq 4 onto the spherical harmonics. Equation 4 thenbecomes

∑ δ+ ==

∞

pg g Al lj

nl nj

nlj

l l,0

, ,0 0 0

(6)

where the matrix A is defined by the action of the operator Hon the spherical harmonics:

∫= *A Y u Y u u( ) ( ) dnlj

lj

nj

(7)

To simplify subsequent calculations, reorient the universeframe of reference such that k = kz. The elements of A are thengiven by

Figure 1. Schematic of the ssWLC model, indicating the componentsof the energy function corresponding to the different deformations of acoarse-grained polymer chain away from its ground-state structure,including (a) bending of the chain orientation, (b) stretching orcompression along the contour, (c) shearing perpendicular to thechain orientation, and (d) coupling of the bend and sheardeformations.

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ηε ε ε

γ ηε

γ ηε

ε ε

ε ε

= + − + − +

= − +

= − − +

= − +

= − +

⊥ ⊥

+ +⊥

+

−⊥

+⊥

+ +

−⊥

Al l j k

ck

c

A i kaik

la

A i kaik

l a

Ak

bk

b

Ak

bk

b

( 1)2 2 2

(1 )2

( 1)

2 2

2 2

l lj

plj

lj

l lj

lj

lj

l lj

lj

lj

l lj

lj

lj

l lj

lj

lj

,

2 2 2 2

, 1 1 1

, 1

, 2

2

2

2

2

, 2

2 2

(8)

where {alj = [(l − j)(l + j)/(4l2 − 1)]1/2}, bl

j = aljal−1j , and cl

j =(al

j)2+(al+1j )2.

Finally, the coefficients for the spherical harmonic expansionof the Green’s function can be found as

= + −pg I A( ) 1(9)

We note that eq 4−9 closely parallel those developed for thehelical wormlike chain in the classic treatment by Yamakawa,23

although the form of our Hamiltonian, and thus the 5-diagonalstructure of our matrix A, differ to incorporate the stretching,shearing, and coupling terms of the ssWLC model.If we are interested only in the end distribution of the

polymer, then integrating eq 5 over the orientation u andaveraging over u 0 leaves G(k;p) = g00

0 (k) . Since there is nocoupling between the j indices of the coefficients, we set j = 0 inall subsequent calculations and drop the j index entirely. Thestructure factor of the chain is defined as

∫ ∫ = ⟨ · − ⟩

=−

⎛⎝⎜

⎞⎠⎟

S kL

s s ik r s r s

LG k p

p

( )1

d d exp[ ( ( ) ( ))]

2 ( ; )

L L

01

02 1 2

12

(10)

where the angled brackets indicate an average with respect tothe chain statistics, and −1 denotes the inverse Laplacetransform from p to the chain length L. The inverse Laplacetransform can be performed by summing over the poles of g00.Namely,

∑ λλ λ λ

= + +| + |Π −

λ− − −

≠ ′ ′

g

pL

eA A

l A( )

( )l

lL

l l l l l

1 002

1 1 2 02

l

(11)

where λl are eigenvalues of −A, and |...|0 denotes thedeterminant of the matrix with the first row and columnremoved. [i.e., the (0, 0) cofactor of the matrix]. While A is aninfinite matrix, it must be truncated at a finite value of l forpractical calculations. Truncation at level 2l yields exactlycorrect values of the moments up to ⟨Rz

2l⟩ and approximatelycorrect values of higher moments at large L. In all calculationsbelow, we truncate at l = 10.The full distribution of the end-to-end distances for the

ssWLC can be obtained by numerically inverting the Fourier−Laplace transform. That is, we find the Green’s function for assWLC with free ends as

∫

∫π

π

= − ·

=∞

−

G R L k ik R G k L

kk kR

Rg k p

( ; )1

(2 )d exp( ) ( ; )

2(2 )

dsin( )

[ ( , )]

3

2 0

100

(12)

where the Laplace inversion is performed by summing overpoles, analogously to eq 11.

Calculating low-order moments. To find moments ofthe end-to-end distribution, we use

⟨ ⟩ = −∂∂

R ig

k( )z

n nn

n00

(13)

We make use of the fact that Al,l′(k = 0) is a diagonal matrix,[(∂All′/∂k)|k=0] is nonzero only when l′ = l ± 1, and [(∂2 All′/∂k2)|k=0] is nonzero only when l′ = l or l′ = l ± 2. Consequently,the derivatives of the matrix inverse can be expressed with thehelp of stone-fence diagrams that have been employed forcalculating moments of plain wormlike chains,35 expanded toinclude the k2 terms arising from the shear and stretch

components (see Figure 2). For instance, the lowest nonzeromoment is given by the expression

⟨ ⟩ = − +∂∂

+∂∂

+

+ +∂∂

+

− − −

− −

R A pA

kA p

Ak

A p

A pAk

A p

2( ) ( ) ( )

( ) ( )

z2

001 01

111 10

001

001

2002 00

1

(14)

corresponding to the stone-fence diagrams in Figure 2a. Themoment ⟨Rz

2n⟩ is given by putting together all diagrams of order2n that start and end at the l = 0 level. The coefficientassociated with each diagram is given by (−1)(n−m2)(2n)!/2m2,where m2 is the number of second-order steps. It is clear fromthis diagrammatic notation that only terms up to the l = n levelcontribute to ⟨Rz

2n⟩.When inverting the Laplace transform in chain length (p →

L), each diagram contributes a term scaling as Lt−1 where t isthe number of times that the diagram touches the l = 0 level.Thus, for all moments of Rz, only the diagrams confined to thelowest levels contribute in the long length limit. We expand themoments at long length as

Figure 2. Diagrams for calculating the first two nonzero moments ofthe end-to-end distance: (a) ⟨Rz

2⟩, and (b) ⟨Rz4⟩. Each point on the

diagram corresponds to a term of (p + All)−1. Each straight line is a

first-order step corresponding to a first derivative All′′ , with l′ = l ± 1.Each line with two ticks is a second-order step corresponding to asecond derivative All′″ , with l′ = l, l ± 2. All derivatives and matrixelements are evaluated at k = 0. The moment ⟨Rz

2n⟩ includes alldiagrams of total order 2n starting and ending at l = 0.

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∑⟨ ⟩ = + Ε=

R C L L( )zn

k

n

nk k2

02( )

(15)

where E(L) includes all terms that decay exponentially withlength. The two highest order terms in this expansion are thengiven by

⟨ ⟩ = + +

⟨ ⟩ = + + +

⟨ ⟩ = !!

+ !! − !

+ !− !

+

−−

− −

⎡⎣⎢

⎤⎦⎥

R aL b

R a L ab c L

Rnn

a Lnn

a c

n nn

a b L

...

3 (12 ) ...

(2 )2

(2 )2 4 ( 2)

(2 )2 ( 1)

...,

z

z

zn

nn n

nn

nn n

2

4 2 2

22

2

1 1

(16)

where a and b are obtained from Laplace inversion of thediagrams in Figure 2a, and c from the inversion of the last fivediagrams in Figure 2b.We note that the development of the moment calculations

from eq 9 up to this point can be generalized to any polymermodel with an appropriate definition of the matrix A. In thecase where A contains higher-order terms in k, the stone-fencediagrams must be modified to include higher-order steps, as willbe seen in section 3 of this manuscript. The expression for thelong-length limit of the moment expansion given in eq 16 isfully general. Consequently, if two polymer models match inthe three expansion coefficients C2

(1), C2(0), C4

(1), then allsubsequent moments will also match up to the two highestorder terms in L. This matching procedure is analogous to thestandard mapping of a polymer to the Gaussian chain byfinding its Kuhn length, C2

(1).12 The Gaussian mapping returnsmoments that deviate from those of the original chain with anerror that scales as 1/L as a fraction of the long length limit.Our procedure extends beyond the Gaussian by including thenext term in the chain length expansion, yielding errors thatscale as 1/L2.For the continuous ssWLC model, we can plug the

expressions for matrix A defined in eq 8, into the diagramsillustrated in Figure 2, and invert the Laplace transform to get

γ γ ηε ε ε

γ γ ηε

γ γ ηε

γ ηε

γ ηε

γ γ ηε ε ε ε ε

ε εγ γ η

εγ η

ε

γ ηε

= + + +

= − +

= + − +

+ + + + −

+ − − + +

+

⊥ ⊥

⊥

⊥ ⊥ ⊥

⊥ ⊥ ⊥

⊥ ⊥ ⊥

⊥

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟⎛⎝⎜

⎞⎠⎟⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟⎛⎝⎜⎜

⎞⎠⎟⎟

⎛⎝⎜⎜

⎞⎠⎟⎟

⎛⎝⎜⎜

⎞⎠⎟⎟⎡⎣⎢⎢

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟⎤⎦⎥⎥

a l

b l

c l

l l

l

23

2 23

13

23

2

3245

2 3

42 3

52

58

451 1

16

451 1 2

3

p

p

p

p p

p

2

3

22

2

(17)

Finally, we note that the persistence length for the ssWLCcan easily be found using

· = = = + → −( )u u g k p0 1/( 1/ ) epL

0 1,1/ p

(18)

where the final step is inversion of the Laplace transform. Thus,the parameter ε η ε ε= + ⊥/(1 / )p b b

2 gives the persistencelength for the chain orientation.

Coarse-Graining with the ssWLC. To map an arbitrarypolymer model to an equivalent ssWLC that will exhibitmatching statistics at long lengths, we need to know the low-order moments of the polymer model. Specifically, we requirethe coefficients C2

(1),C2(0) corresponding to the linear and

constant terms of ⟨Rz2n⟩ and the coefficient C4

(1) for the linearterm of ⟨Rz

4⟩. For a continuous chain, these terms can be foundusing the procedure described in the previous section. For adiscrete chain, an analogous procedure is described in section 3of this manuscript. Furthermore, in the absence of ananalytically tractable form, these quantities can be found byrunning direct simulations of the detailed polymer in question.We want to find appropriate parameters for the ssWLC such

that these low-order moment terms match those of the detailedpolymer. Our model has 5 parameters (εb, ε⊥, ε∥, γ, η) withwhich to match the three terms (C2

(1), C2(0), C4

(1)). We removeone of the excess degrees of freedom by defining adimensionless parameter α = η2εb/ε⊥ and setting this equalto a constant. This allows us to decouple the persistence length(which measures correlations in the chain orientation) from theshear modulus. As a result, the chain orientation distributionnow depends entirely on εb (bending modulus) while thestretch and shear modulus, as well as γ influence only theposition of the chain ends. The α parameter can be set to anypositive constant, and could, in principle, be varied to minimizethe error in any additional statistical average term of thepolymer model (e.g., C4

(0) or C6(1)). However, there is no clear

choice of a single additional statistic that would improve theoverall accuracy of the coarse-grained model. For all resultspresented in this manuscript, we found a constant value of α =1 to be a reasonable choice.The second extra degree of freedom is used to adjust the

length scale of accuracy for the effective model. We recall thatthe development of the ssWLC as a general model for locallydefined polymer chains rested on the assumption that chaincorrelations decay on a sufficiently small length scale comparedto the overall chain length. Consequently, increasing thepersistence length p also raises the length scale above whichthe model is accurate. Thus, the parameter p functions as aslider to specify the requisite level of coarse-graining. Whensimulating the ssWLC, the chain should be discretized intosegments much shorter than the persistence length, soincreasing p allows for coarser discretization of the chaincontour and fewer degrees of freedom for the simulation. Wewill discuss a systematic procedure for selecting effectiveparameters for a discretized ssWLC in a subsequent manu-script.36

For fixed values of p and α, we solve for the remainingparameters to match C2

(1), C2(0) and C4

(1), yielding the solutions

εα

α α

α α

α α α

=−

+ + −

+ + − −

+ + − −

⊥− C C

C C C

C C

1(12 4)

{6 (3 1) 4( 1)

[144( ) 12 (15 6 5)

64 ( ) 10(1 2 3 ) ] }

pp

p p

p p

12 2

(0) 22(1)

2(0) 2

2(0)

2(1) 2

2 22(1) 2 2

4(1) 1/2

(19)

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ε ε= − + +−⊥−

⎛⎝⎜⎜

⎞⎠⎟⎟

CC2 3

p

1 1 2(0)

2(1)

(20)

γαε

ααε

α=

−

+−

+⊥ ⊥l C

l l

4 6

4 (1 ) (1 )p

p p

2(0)

2(21)

The above three equations constitute a simple, systematicprocedure for mapping a polymer model of interest onto aneffective ssWLC model, with p setting the length scale ofaccuracy. As this length scale becomes very large ( → ∞p ), wereach the limiting behavior γ→0, ε⊥

−1 → C2(1), ε∥

−1 → C2(1). In

this limit, the local trajectory of the chain position r becomesdecoupled from the orientation vector u. Path integration overall values of u (s) results in the position statistics dictated by theparameters (γ,ε⊥, ε∥) analogous to those of the Gaussian chain.We note here that the orientation vector u (s) is not identical tothe tangent vector of the original chain being coarse-grained.Instead the u provides an indication of the anisotropy inherentin the chain distribution under the ssWLC model. By forcing aconstant term in the lowest-order moment ⟨R2⟩ to match thatof the original chain, we maintain an initial rigidity in the chainat shorter lengths, in contrast to a purely Gaussian chain whichis flexible on all length scales. As the coarse-graining scaleincreases, these orientation vectors are rigidified (increasing p)in order to maintain this initial correlation. The length scale ofaccuracy increases concomitantly, so that the ssWLC providesan accurate description of the detailed polymer statistics onlyfor chain lengths much larger than p.As an example of this systematic zooming-out procedure, we

coarse-grain a plain, inextensible wormlike chain31 by mappingto a ssWLC at increasingly longer length scales. The low-ordermoments for a plain WLC with persistence length lp

(0) = 1 canbe found from eq 16−17 by setting γ = 1, ε⊥

−1 = ε∥−1 = η = 0.

In Figure 3, we plot the parameters of the effective ssWLC as a

function of the coarse-graining scale p. In the limit →p p(0),

the plain WLC chain limit is recovered. We note that the shearmodulus decreases toward the Gaussian chain value faster thanthe stretch modulus, implying that the shearing degree offreedom predominates in the short to intermediate length-scalebehavior of the WLC. This result is in agreement with analyticcalculations, which show spreading over the end-to-end vector

away from a fixed initial orientation at much shorter lengthsthan compression along the orientation vector (see Figure 5 inref 37).The coarse-graining procedure described here defines a

continuous flow from the plain WLC to the Gaussian chain. Todemonstrate this progression more explicitly, we calculate thestructure factor S(k) the Fourier transform of the densitycorrelations for segments along the chain. At small values of thewavevector k this quantity describes long length scale behavior,while larger values of k correspond to shorter length scales.Thus, the structure factor addresses polymer statistics acrossthe full range of length scales.Figure 4 compares the structure factor of the effective ssWLC

to those of the plain WLC and the Gaussian chain, illustrating

the transition between the two limits as the coarse-grainingparameter p is varied. At small values of k the models fall on auniversal curve, as all chains map to the Gaussian at sufficientlylong length-scales. Larger values of k show a divergencebetween the behavior of the WLC and the Gaussian, due to thedifferent local structures of these chains. The length scale ofaccuracy for the ssWLC model can be approximated byconsidering the maximal value of k at which the structure factorclosely matches that of the original WLC. As shown in Figure 4,this k value decreases with increasing p. Thus, mapping to assWLC with a larger value of p is accurate only at longer lengthscales, validating the use of this parameter as a slider todetermine the degree of coarse-graining.

3. COARSE-GRAINING A WLC WITH KINKSWe employ our coarse-graining procedure to explore thestatistics of a wormlike chain with twist and periodic fixedkinks. This polymer is of particular interest for studying DNA−protein arrays. Above length scales of about ten basepairs, DNAis known to behave as a wormlike chain.1 Many DNA-bindingproteins locally bend or kink the substrate DNA upon binding,and the elastic polymer properties of DNA have been shown toimpact the cooperative binding of such proteins.38 Theformation of long arrays consisting of many proteins bindingand kinking the DNA is particularly relevant to studyingchromatin condensation, where the first step in the packaginghierarchy consists of wrapping DNA around histone proteins to

Figure 3. Parameters from mapping a plain inextensible wormlike

chain with persistence length = 1p(0) to an effective ssWLC, plotted as

a function of the coarse-graining scale p. The effective bendingmodulus is given by ε = 2b p, and the bend-shear coupling by (η =(ε⊥/εb)

1/2).

Figure 4. Structure factor for increasingly coarse-grained polymermodels, ranging from the plain inextensible wormlike chain (top solidline), to the Gaussian chain (bottom dashed line). Intermediate curvescorrespond to effective ssWLC models with = 1.2, 2, 4, 10p , fromtop to bottom. Inset: value of the wavevector k at which the structurefactor matches that of the plain wormlike chain to within an error of10%.

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form periodically spaced nucleosomes.39 Previous work hasshown that the local geometry and linker spacing of the kinksinduced by nucleosomes play an important role in determiningthe structure and elastic properties of nucleosome arrays40,41

and compact chromatin fibers.42,43

The biological relevance of modeling locally perturbed DNAstructures has motivated a number of computational andtheoretical studies of the statistics of kinked DNA. Analyticalresults have been developed for a “kinkable WLC” comprisingan elastic polymer with spontaneously formed, randomlyinserted, fully flexible kinks.44 The mean square end-to-enddistance for a chain with rigid kinks held at a fixed dihedralangle without twist fluctuations has also been calculated.45

More recently, a general form for the end distribution of elasticchains with kinks of limited flexibility was derived by Zhou andChirikjian,46 though no explicit expressions for a twisted chainor multiple kinks were provided. In this section we follow asimilar procedure to find the low-order moments of a chainconsisting of rigid kinks linked together by a straight elasticpolymer with twist. The twist of the linker polymer is shown tohave a fundamental effect on the statistics of the kinked chain.We proceed to map the kinked WLC to an effective ssWLCand demonstrate the accuracy of this simple model inreproducing kinked chain statistics at intermediate to longlength scales. This example problem serves to validate thecoarse-graining procedure described in this manuscript byapplying it to a detailed polymer system of biological relevance.Kinked WLC Model. We define the kinked WLC of length

N as consisting of N repeating units, where each unit comprisesan inextensible twisted wormlike chain47 of length Lu followedby a rigid kink. The linker chain has bend persistence length

p(0), twist persistence length t , and ground-state twist τ. The

orientation of the chain at each point along the contour isdefined by an orthonormal triad of vectors that form a rotationmatrix Ω(s). The kink itself is defined by a rotation matrix Ωk

that maps the orientation of the incoming polymer onto theorientation of the outgoing chain (Figure 5a). The kink canthus incorporate both twist and bend perturbations to thechain.From symmetry considerations, the kinked WLC forms a

helical ground-state structure (Figure 5b), with the helix

parameters determined by the kink geometry and the spacingbetween kinks. Thermal fluctuations result in fluctuations awayfrom the ground-state helix (Figure 5c). If viewed on asufficiently large length scale, the local helical structure vanishesand the polymer can be approximated as a stretchable elasticchain. Here, we map this wormlike chain with periodicallyspaced rigid kinks (kinked WLC) onto an effective ssWLCmodel which can be used to study the statistics of such arrays atlong length scales.

Low-Order Moments for the Kinked WLC. To find aneffective ssWLC matching to the kinked WLC at long lengthscales, we need to calculate the terms C2

(1), C2(0), C4

(1) for thelow-order moments of the detailed polymer model. Tocalculate these quantities, we begin by finding the localpropagator [G1(R, Ω|Ω0)], corresponding to the distributionfunction for position and orientation at the end of onerepeating unit (R, Ω), relative to the position and orientation atthe start of the unit (R0 = 0, Ω0). As before, we carry out aFourier transform of the position (i.e., R → k) and expand thepropagator in terms of Wigner-functions l

mj, a generalizationof the spherical harmonics for the full three-angle rigid-bodyrotation.23 The transformed propagator can then be expressedas

∑ = Ω Ω*G kB ( ) ( ) ( )l l j j m

l lj j

lmj

lfmj

1, , , ,

,,

0

f f

f

f f

0 0

0

0

00

(22)

The elements of matrix B(k) in the above expansion havebeen derived in previous work by a convolution in position androtation space of the propagator for the elastic linker polymerand that of the rigid kink.32 For completeness and consistencyof notation, we include the expressions for B and its derivativesin the Appendix. This matrix has a structure analogous to thematrix A in eq 8, but where each element indexed by l,l′ isreplaced with a (2l + 1) × (2l′ + 1) matrix corresponding to thej indices. Thus, B(k = 0) is block diagonal, (∂B/∂k)|k=0 is block-tridiagonal, and so forth.The partition function GN for the entire kinked WLC,

consisting of N repeating units, is found from the convolutionin both position and orientation space of N copies of the localpropagator. These convolutions turn into multiplications underthe Fourier and Wigner-function transforms, giving

∑ = Ω Ω*G B[ ] ( ) ( )Nl l j j m

Nl lj j

lmj

lmj

, , , ,,,

0

f f

f

f

f

f

0 0

0

0

00

(23)

Calculation of the low order moments, ⟨Rz2n⟩, for the kinked

WLC thus reduces to finding derivatives of BN with respect tok. This can be accomplished using the same stone-fencediagrams described in the preceding section, with the additionof triple and quadruple steps. For instance, ⟨Rz

2⟩ is given by

∑

∑ ∑

⟨ ⟩ = −∂

∂

= − ″

+ ′ ′

=

−− −

=

−

=

− −− − −

Rk

B

B B B

B B B B B

[ ]

[

2 ]

z

N

n

Nn N n

n

N

n

N nn n N n n

22

002

0

11

0

2

0

22

00

1

1 1

1 2

11 2 1 2

(24)

where all the matrices are evaluated at k = 0. Here thesubmatrix Bll′ is nonzero only for l = l′, corresponding to pointsin the stone-fence diagrams. Bll′″ is nonzero for l = l′ ± 1

Figure 5. (a) Schematic of the kinked WLC model. Ω is the chainorientation at each point along the contour. Ωk is the rotation matrixdefining the kink. G1 is the propagator across a single chain unit. Kinksare separated by linker chains of length Lu. (b) Ground-state structureof the kinked WLC forms a helix. (c) Snapshot of a kinked WLC fromsimulations incorporating thermal fluctuations away from the ground-state structure.

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corresponding to single steps in the diagrams, and Bll′′ isnonzero for l = l′,l′ ± 2 corresponding to double steps. The twoterms in eq 24 thus correspond to the two diagrams in Figure2a.Performing the eigendecomposition of Bll yields

∑ λ==

+

B U[ ]lln

j

l

lj n

lj

1

2 1( ) ( )

(25)

where λl(j) are the eigenvalues and Ul

(j) is the dyadic product ofthe jth right and left eigenvectors.The discrete convolutions in eq 24 can then be carried out

analytically to extract the linear and constant terms of thelowest moment as

∑λ

= − ″ −−

′ ′=

L C UB B B2

1u

jj

j2(1)

001

3

1( ) 01 1

( )10

(26)

∑λ

=−

′ ′=

C UB B2

(1 )jj

j2(0)

1

3

1( ) 2 01 1

( )10

(27)

The analogous calculations for the next highest moment,⟨Rz

4⟩, can be found in the Appendix. C2(1) corresponds to 1/3 of

the Kuhn length that can be used to approximate the kinkedWLC as a Gaussian chain at very long lengths. The additionalcomponents C2

(0),C4(1) allow us to find an effective ssWLC

model as described in section 2.

■ RESULTSWe perform the coarse-graining procedure described in theprevious sections for the kinked WLC with kinks Ωk defined by

Euler angles (π/4, π/4, π/4) separated by varying lengths Lu ofa semiflexible polymer with all physical parameters set to matchthose of DNA. That is, the linker polymer chains have bend

persistence length = 50p(0) nm, twist persistence length

= 100t nm, and natural twist τ = 2π/(10.5bp ×0.34 nm/bp).1,48 Because of the nonzero twist τ, changing the linkerlength Lu alters the relative alignment of the kinks and results invery different ground-state structures for the polymer.42

Figure 6 illustrates the parameters of the effective ssWLC as afunction of the linker length, for a coarse-graining scale of

= 70p nm. As expected, kinked chains that are naturally more

extended have larger values of γ, the relaxed extension of theeffective shearable chain. The parameter γ is always less thanthe ground-state helix height h per contour length because thecoarse-grain model includes contraction of the overall polymersize due to thermal fluctuations.More extended chains have higher values of the stretch

modulus ε∥ and lower values of the shear modulus ε⊥. Althoughthe semiflexible polymer connecting the kinks is itselfinextensible, a compact helical spring created by the kinks(such as the 49 bp structure in Figure 6c) can be easilystretched by exploiting the bend and twist flexibilities of thepolymer. By contrast, more extended chains (such as the 45bpstructure in Figure 6c) are easier to shear by displacing the endsperpendicular to the helix axis via bending of the linkerpolymers. We note that for extremely compact kinked chainsthere is no solution for the effective ssWLC with the selectedvalue of the coarse-graining parameter p. While solutions exist

at higher coarse-graining levels, they require a negative value ofγ (or, equivalently, a negative value of η). Such points are left

Figure 6. Mapping of kinked wormlike chains with different linker lengths onto an effective ssWLC model. The coarse-grained ssWLC has fixedpersistence length = 70nmp and bending modulus εb = 140 nm. (a) Dashed line gives the height along the helix axis (h) per unit contour length ofthe ground-state helix. Solid line plots the natural stretch (γ) for the effective coarse-grained model. (b) Inverse shear (ε⊥

−1) and stretch (ε∥−1)

moduli for the effective shearable chain as a function of linker length. (c) Ground-state structures for three different kinked chains corresponding tothe marked points in parts a and b.

Figure 7. Distribution of the end-to-end distance from simulations of the kinked WLC, compared to effective coarse-grained models. Blue circles:distributions for a kinked WLC with 15 kinks, separated by contour lengths of Lu = 40,45,49 bp. Solid green lines: analytical results for distribution ofthe effective ssWLC, with = 70nmp and parameters γ, ε∥, ε⊥, η as labeled. Dashed red lines: Gaussian chain distributions, with effective Kuhnlength κ.

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out of Figure 6. We do not attempt to address these extremecases, since for such compact structures the statistics of thepolymer will be dominated by steric interactions betweendifferent coils of the chain, an effect that is not included in ourmodel.The accuracy of the coarse-grained model is illustrated in

Figure 7, which shows the distribution of the end-to-enddistances obtained by direct simulation of the kinked WLC,compared to the analytical results for the ssWLC withparameters given in Figure 6. Simulations were carried out bydiscretizing the chains into segments much shorter than boththe persistence length and the twist 1/τ and sampling from themultivariate normal distribution to obtain bend and twistperturbations for each segment in the kinked WLC. Thesimulated chains consisted of 15 kinks with linker lengths 40,45, and 49 bp (where 1 bp =0.34 nm). The ground-statestructures of the kinked chains are shown in Figure 6c. The

total chain lengths simulated (4 , 4.5 , 5p p p(0) (0) (0) for the three

different linker lengths) are in the intermediate regime whereGaussian statistics do not accurately reproduce the behavior ofthe detailed kinked chain. We see that in this regime theeffective ssWLC model provides a greatly improved approx-imation to the chain statistics.The greatest deviation from the kinked chain occurs for the

most extended kinked array, with 45bp linkers. Here, the finiteextensibility of the linker chain, which is not reproduced by thessWLC model, has the greatest effect on the end-to-enddistribution, as evidenced by the extended tail of the ssWLCdistribution at large |R|. A finite length constraint could, inprinciple be implemented in discretized simulations based onthe ssWLC model. Nonetheless, even in this special case theshearable chain provides a significantly better model of thekinked chain statistics than does the Gaussian.These results demonstrate that the coarse-graining procedure

described here accurately reproduces the end-to-end statisticsof the kinked WLC model at lengths much shorter than thosethat allow the polymer to be treated as a Gaussian chain. Wenote that although the kinked DNA model is based on a twistedpolymer, with the twist degrees of freedom significantlyimpacting the overall structure of the kinked chain, it isnonetheless possible to map this detailed polymer onto ourssWLC model, which lacks twist. This example highlights thegenerality of the mapping procedure described in thismanuscript. All locally defined polymer chains can beapproximated as a ssWLC on intermediate to long lengthscales.

4. CONCLUSIONS

In this work, we present a fully general analytical method forcoarse-graining polymer models by mapping to an effectiveshearable wormlike chain. Our approach allows for systematiczooming in on a polymer model by decreasing the coarse-graining parameter p. For large values of p, this procedure isequivalent to approximating the polymer as a Gaussian chain,whereas smaller values more accurately model the elasticproperties of the polymer at intermediate length scales,including its resistance to stretching, bending, and shearingaway from the structure of lowest free energy. For a continuoussemiflexible chain, this coarse-graining approach defines acontinuous flow from the original chain model to the Gaussianchain. For more complicated polymer models, the smallestcoarse-graining scale at which an effective mapping can be

found is limited to those values of p where the effective ε⊥, ε∥are positive real numbers.By mapping a locally complicated polymer model onto an

effective ssWLC, we gain both the ability to perform morecoarse-grained simulations and the ability to analyticallycalculate certain aspects of the chain statistics. Effective modelswith larger values of p can be discretized into correspondinglylarger segments, allowing for more efficient simulations. In asubsequent publication,36 we address the topic of how best tomap a polymer onto a discrete shearable wormlike chain forarbitrarily large discretization. The systematic reduction ofsimulation degrees of freedom is key to in silico studies of large-scale polymer systems such as genomic DNA, cytoskeletalfilament networks, or entangled polymer meshes.The structure factor and the end-to-end distance distribution

for the ssWLC can be calculated exactly using the methodsdescribed in section 2. The joint distribution of chain positionsand orientations is also analytically tractable via an analogousprocedure. Thus, aspects of the chain statistics such as thelooping probability with or without fixed end orientations canbe approximated for the polymer chain on intermediate to longlength scales.As an example of our coarse-graining procedure we map a

kinked WLC, consisting of an inextensible, twisted, semiflexiblepolymer with periodic kinks onto an effective shearablewormlike chain. We show that this mapping yields a significantimprovement over the Gaussian chain in reproducing chainstatistics at intermediate lengths. This example can be appliedto studying the role of DNA-bending proteins in modulatingthe large-scale structure of genomic DNA. Future work willinclude examining the looping behavior of DNA−proteinarrays, including the nucleosome arrays that form the lowestlevel of hierarchical chromatin packing, across a large span oflength scales.Several recent studies have raised concerns regarding the

applicability of elastic chain models in studying realistic DNAstructures at short lengths.49−51 However, we demonstrate inthis work the importance of the elastic chain as a general,unifying model for describing long-length polymer behavior.Local effects, such as the binding of kinking proteins, can besubsumed into the effective parameters of the coarse-grainedssWLC model. We therefore emphasize that the influx of newmeasurements regarding the behavior of short DNA segmentsshould not lead to complete abandonment of elastic chainmodels. Instead, information on local polymer structure can beincorporated into the framework of an elastic model fordescribing chain statistics at long lengths. While there aresituations where physical constraints are such that small lengthscale effects must be acknowledged, coarse-grained elasticchains such as the ssWLC remain critical to bridging fromlocally detailed models to large-scale behavior.We emphasize that the method presented here is highly

general and can be applied to modeling any locally definedpolymer on a sufficiently large length scale. However, nonlocalinteractions cannot be incorporated directly into the model.Steric exclusion effects, for instance, can be included in the localbut not the global sense. That is, steric interactions betweennearby segments along the chain contour can be incorporatedinto the effective elastic parameters of the model. Dependingon the structure of the underlying self-avoiding polymer, therecan then be an intermediate length scale where the ssWLC isapplicable but the inherent rigidity of the chain makes distal

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steric effects negligible. In some cases, such as the highlycompact helical structures seen for the kinked WLC, stericinteractions between distal segments will dominate thestructure and the ssWLC model described here cannot beused. Similarly, other nonlocal effects such as topologicalentanglements, global constraints on twist and writhe, andhydrodynamic interactions are not currently included in themodel. However, these global effects could be incorporated intolarge-scale simulations using the effective ssWLC as a startingpoint for the polymer model. In cases where the chain twistaround its axis as well as the overall contour path is of interest,the ssWLC model can be expanded to include the twist degreeof freedom and the concomitant coupling between twist andposition. The optimal procedure for handling these aspects ofcoarse-grain systems is an area deserving future study.The methodology described in this work is designed to serve

as a springboard for the development of coarse-grained polymermodels, an indispensable task for researchers interested instudying macromolecular properties on macroscopic lengthscales.

■ APPENDIX

Unit Propagator for Kinked WLCHere we derive the propatagor G1(R,Ω|Ω0) for the position andorientation at the end of one unit of the kinked WLC relative tothe position and orientation at the start of the unit. A unit ofthe kinked chain consists of a wormlike chain with twist47 oflength Lu followed by a rigid kink defined by the rotation matrixΩk. If we define Ω− as the orientation at the end of the linkerchain prior to the kink, then the Fourier-Laplace-transformedpropagator across the linker only is given by an expansion inWigner functions,

∑ Ω |Ω = Ω Ω− * −G k p g k( , ; ) ( ) ( ) ( )linkl l m j

l lj

lmj

lmj

0, , ,

, 0

ff f

00 0

(S1)

where the coefficients gl0,lfj can be expressed in matrix form as47

= + −pg A I( )j j 1(S2)

τ= + + − +⎛⎝⎜⎜

⎞⎠⎟⎟

l lj i jA

( 1)2

12

12

l lj

p t p, (0)

2

= −+ +ikaAl lj

lj

, 1 1

= −− ikaAl lj

lj

, 1

To get the propagator for the full unit of the chain, we notethat the orientation following the kink is given by Ω = Ω−

°Ωkand then make use of the Wigner function expansion for acomposition of rotations,23

∑πΩ =+

Ω Ω− *

l( )

82 1

( ) ( )lmj

nlmn

ljn

k

2

(S3)

We can define a kink matrix M that is block-diagonal in the lindices

π=+

Ω*

lM

82 1

( )ljn

ljn

k

2

(S4)

and treat the linker propagator matrix g as diagonal in the jindices and block-tridiagonal in the l indices. The final

expression for the unit propagator is then given by eq 22with the matrix of coefficients,

= ·B g M (S5)

The derivatives of g with respect to k can be found usingstone-fence diagrams with only single-order steps allowed. Thediagrams for the first 4 derivatives are shown in Figure 8. The

inverse Laplace transform p → Lu can be easily carried out foreach diagram by summing over the poles. We note that since Ais a symmetric matrix for this system, all g derivatives aresymmetric as well. The derivatives of B are then found by right-multiplying the derivatives of g by M.Low Order Moments for Kinked WLC with Many KinksThe first two nonzero moments for the end-to-end distance ofa kinked WLC with N kinks can be expressed as,

⟨ ⟩ = + +R C L N C( ) ...z u2

2(1)

2(0)

(S6)

⟨ ⟩ = + +R C L N C L N C3( ) ( ) ( ) ...z u u2

2(1) 2 2

4(1)

4(0)

(S7)

where ... refers to terms that decay exponentially with n. Theterms C2

(0),C2(1) are found from eq 24−27. We perform

analogous calculations to find the term C4(1), using the stone-

fence diagrams in Figure 2b, with additional diagramsincorporating third and fourth derivatives of the unitpropagator matrix B (see Figure 9). Each point on thediagrams corresponds to a term of Bll

n(k = 0), a (2l + 1) × (2l +1) submatrix block of B. Discrete convolutions are carried outover the powers n, with the total of the powers equal to the

Figure 8. Stone-fence diagrams for calculating k derivatives of thecoefficients gl0,lf

j in the Wigner-function decomposition of thepropagator for the twisted wormlike chain. Each point on the diagramrepresents a factor of (1/(p + All

j )), and each step represents a factor of(∂All,l+1

j /∂k), evaluated at k = 0.

Figure 9. Additional stone-fence diagrams for calculating ⟨Rz4⟩ for the

kinked WLC. Together with the diagrams in Figure 2b these representall the terms in eq S9.

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number of kinks N minus the number of steps in the diagram.Each step in the diagram corresponds to a term of ∂mBll′/∂k

m,with the order of the derivative equal to the order of the step.The coefficient for each diagram is 4!/(2!m23!m34!m4), where miis the number of ith-order steps. For example, the first diagramin Figure 2b contributes the following term:

∑ ∑ ∑ ∑ ′ ′ ′ ′

− − − − −

B B B B B B B B

B

24 [

]

n n n n

n n n n

N n n n n

00 01 11 10 00 01 11 10

004

1 2 3 4

1 2 3 4

1 2 3 4 (S8)

Using the eigendecomposition of B given by eq 25, togetherwith the normalization condition B00 = 1, we perform thediscrete convolutions directly to give

∑

∑

∑ ∑

∑

∑ ∑

∑ ∑

∑ ∑

∑

λ λ λ λλ λ

λλ

λ λ λ

λ λ

λ λ

λ λ

λ λ

λ

=− + +

− −

× ′ ′ ′ ′ +−−

× ′ ′ ″ − ″ ″

+′ ′ ′ ′

− − −

+′ ″ ′

− −

+″ ′ ′

− −

+′ ′ ″

− −

+″ ″

−+

‴ ′

−

+′ ‴

−+ ⁗

=

=

= =

=

= =

= =

= =

=

⎡⎣⎢⎢

⎤⎦⎥⎥

⎡⎣⎢⎢

⎤⎦⎥⎥

L C

B U B B U B

B U B B B B

B U B U B U B

B U B U B

B U B U B

B U B U B

B U B B U B

B U BB

245 3( )

2( 1) ( 1)

243

2( 1)

3

24(1 )(1 )(1 )

12( 1)( 1)

12( 1)( 1)

12( 1)( 1)

6(1 )

4(1 )

4(1 )

ui j

i j i j

i j

i j

j

j

j

j

i j k

i k j

i k j

i j

i j

i j

k j

k j

k j

k j

j k

k j

k

k

kj

j

j

j

j

j

4(1)

, 1

31( )

1( )

1( )

1( )

1( ) 2

1( ) 2

01 1( )

10 01 1( )

101

31( )

1( ) 2

01 1( )

10 00 00 00

, 1

3

1

501 1

( )12 2

( )21 1

( )10

1( )

2( )

1( )

, 1

301 1

( )11 1

( )10

1( )

1( )

1

5

1

302 2

( )21 1

( )10

2( )

1( )

1

5

1

301 1

( )2 2

( )20

2( )

1( )

1

502 2

( )20

2( )

1

301 1

( )10

1( )

1

301 1

( )10

1( ) 00

(S9)

This provides the final term necessary for mapping thekinked WLC to a coarse-grained ssWLC model.

■ AUTHOR INFORMATIONCorresponding Author*E-mail: lenafabr@stanford.edu (E.F.K.); ajspakow@stanford.edu (A.J.S.).NotesThe authors declare no competing financial interest.

■ ACKNOWLEDGMENTSWe thank Jay D. Schieber and Ekaterina Pilyugina for helpfuldiscussions over the course of this work. Funding was providedby the Fannie and John Hertz foundation and the NationalScience Foundation CAREER Award program.

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