This journal is©The Royal Society of Chemistry 2019 Soft Matter, 2019, 15, 1785--1792 | 1785

Cite this: SoftMatter, 2019,

15, 1785

The shape of pinned forced polymer loops

Wenwen Huang a and Vasily Zaburdaev *ab

Loop geometry is a frequent encounter in synthetic and biological polymers. Here we provide

an analytical theory to characterize the shapes of polymer loops subjected to an external force field.

We show how to calculate the polymer density, gyration radius and its distribution. Interestingly, the

distribution of the gyration radius shows a non-monotonic behavior as a function of the external force.

Furthermore, we analyzed the gyration tensor of the polymer loop characterizing its overall shape. Two

parameters called asphericity and the nature of asphericity derived from the gyration tensor, along with

the gyration radius, can be used to quantify the shape of polymer loops in theory and experiments.

1 Introduction

Looping is ubiquitous among natural and synthetic polymers.1–6

However, compared to linear polymer chains, the theory ofpolymer loops is less developed.6–8 Recently, the movement ofchromosomes during meiosis in fission yeast was analyzed ingreat detail.8–11 It was shown that yeast chromosomes formloops and are actively pulled inside the cell in an oscillatorymanner.10,12 As a result, the overall shape of the chromosomes,as seen in the experiments, varies from rounded to stretched.It was suggested that these oscillations help to align thechromosomes for recombination.11 That promoted us todevelop a theory capable of quantifying the shapes of forcedchromosomal loops. Previously, we used an approach ofrandom walks to demonstrate how the pulling force maysuppress fluctuations of the polymer loop, but the problemof polymer shape remained unexplored. In this paper, we showhow to quantify the shape of pulled polymer loops in asystematic manner. Using a coordinate transformation, thepulled polymer loop setup is mapped to a model of the pinnedpolymer loop in an external field. We study how the shape ofthe polymer loop changes under different strengths of theexternal force field.

Unlike solid objects, the shape of polymers in a fluctuatingenvironment is less visual. The typical size of a polymer can bequantified by its gyration radius.13,14 A number of previousstudies was devoted to the description of polymer shapes.15–19

Dating back to 1962, Fixman calculated the gyration radiusdistribution of polymer chains.20,21 Solc et al. used the gyration

tensor to characterize the shape of a random flight chain.22,23

The shape of pinned polymers was investigated both theore-tically24,25 and experimentally26–28 with the development ofsingle molecule imaging techniques. The looping constraintsignificantly affects the polymer shape. Alim et al. studied theshape of semi-flexible polymer loops without pinning and theexternal force field.17 Crumpled or elliptical shapes weredescribed depending on whether the polymer was flexible orstiff.17 To the best or our knowledge, no previous work wasfocused on pinned polymer loops under an external force field.

The paper is organized as follows. In the next section, weintroduce the model of pulled polymer loops. We then discussthe segment density function in Section 3. Section 4 is devotedto the gyration tensor and the gyration tensor based shapedescriptors, such as gyration radius, asphericity and the natureof asphericity. We discuss our results in the concludingSection 5.

2 Model

To describe the polymer loop, we employed the freely jointedbead-rod model. We use the bead-rod instead of the morecommon bead-spring model as it provides a simpler way totake into account the finite extensibility of the polymer, whichmay be important, for example, to model chromosomes.We will, however, discuss the differences between bead-rodand bead-spring in the Appendix. We consider one polymerloop which is pulled by a single monomer with constantvelocity through a viscous medium at a certain temperature.By transferring the coordinate system to the frame co-movingwith that bead, we arrive at the setup where one single bead ispinned and a constant velocity of the surrounding fluid withthe direction opposite to the pulling direction is imposed on all

a Max Planck Institute for the Physics of Complex Systems, Nothnitzer Str. 38,

D-01187 Dresden, Germany. E-mail: vzaburd@pks.mpg.deb Friedrich-Alexander Universitat Erlangen-Nurnberg, Cauerstr. 11,

91058 Erlangen, Germany

Received 20th November 2018,Accepted 30th January 2019

DOI: 10.1039/c8sm02357c

rsc.li/soft-matter-journal

Soft Matter

PAPER

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1786 | Soft Matter, 2019, 15, 1785--1792 This journal is©The Royal Society of Chemistry 2019

other beads. We neglect the hydrodynamic interactions14,29

of the beads with each other and model the effect of flowby assuming the Stokes’ law. Thus the external force applied toevery bead can be written as F = �xv. Here, v is the pullingvelocity and x is the friction coefficient. We can approximatethe effect of different pulling velocities by varying the magni-tude of the applied force field.

An illustration of a pinned polymer loop is shown in Fig. 1.Without loss of generality, we assume the first bead, whoseposition is denoted by r0, to be pinned at the origin point, r0 = 0.The looping constraint is thus r0 = rN = 0 where N is the totalnumber of beads. The rod length is denoted by a and thedirection of the external force field is chosen along the zdirection. The orientation of the jth rod is denoted by the unitvector uj = (rj � rj � 1)/a. With this we can rewrite the loopingcondition as

XNj¼1

uj ¼ 0: (1)

By construction, we have ri ¼ aPij¼1

uj . Thus the potential energy

of the pinned polymer loop configuration can be written as

E ¼ �XNi¼1

F � ri ¼ �XNi¼1

F � aXij¼1

uj

!¼ aF �

XNj¼1

juj : (2)

The polymer is immersed in the solution which leads tothermal fluctuations. We characterize the strength of these fluc-tuations by temperature T. The excluded volume effect will only bediscussed briefly in the concluding section. Other interactionssuch as Coulomb interactions and bending energy are alsonot considered here.13,14 With the help of this simple setup,we calculate several shape descriptors in the following sections.

3 Segment density function

The configuration of the bead-rod polymer can be consideredas a realization of a random walk trajectory which starts from r0

= 0 and comes back to r0 after N steps. According to the centrallimit theorem,32 the distribution of the ith bead position is wellapproximated by the Gaussian distribution. We can write

pðxiÞ ¼1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2p var xi½ �p exp �ðxi � xih iÞ2

2var xi½ �

� �; (3a)

pðziÞ ¼1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2p var zi½ �p exp � zi � zih ið Þ2

2var zi½ �

!: (3b)

where, xi and zi are the coordinate components of the ith beadposition. According to the rotational symmetry along the z-axis,the distribution of p(yi) is identical to p(xi).

The mean and variance of every bead position can be writtenas the following expressions:10,32

rih i ¼ aXij¼1

uj� �

; (4a)

var ri½ � ¼ a2

Pij¼0

var uj� � PN

j¼N�ivar uj� �

PNj¼0

var uj� � (4b)

Interestingly, the variance is not given by a simple sum, buta more elaborate construction, which is a consequence of theBrownian Bridge condition. The random walk trajectory has toreturn to the origin, which makes the variance to be symmetricwith respect to the middle of the loop (for details see ref. 10).The mean and variance of every rod orientation uj can beobtained by using the grand canonical ensemble approxi-mation

Zj ¼ðduj exp �ð j � mÞaF � uj

kBT

� �¼ sinh w

w; (5)

where Zj is the partition function, m is the chemical potential,which can be fixed later by the averaged looping conditionPNj¼1

uj� �¼ 0 and w = ( j � m)Fa/kBT is a dimensionless parameter.

It is now straightforward to calculate the mean and variance ofthe rod orientation:

uj;z� �

¼ kBT

Fa@m ln Zj ¼ cosh wþ 1=w; (6a)

var uj;z� �

¼ kBT

Fa

� �2

@m2 ln Zj ¼ 1

w2 � csc h2w: (6b)

Using eqn (6a), the chemical potential can be readily calcu-lated as m = (N + 1)/2. The x and y components of rodorientations can be obtained by symmetry argument and thefact that uj

2 = 1. We thus have

huj,xi = huj,yi = 0, (7a)

var[uj,x] = var[uj,y] = (1 � huj,z2i)/2. (7b)

Here huj,z2i = var[uj,z] + huj,zi2. With the results above, the

distribution of the ith bead position eqn (3) can be calculated ina straightforward manner.10 The segment density function asthe sum over individual segment distributions, is:

PðxÞ ¼ 1

N

XNi¼1

pðxiÞ; (8a)

Fig. 1 Sketch of a bead-rod polymer loop with the first bead pinned.r0 = 0. The force F is pointing in the z-direction. Every two neighboringbeads are joint by a rigid rod of length a: |rj+1 � rj| = a.

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PðzÞ ¼ 1

N

XNi¼1

pðziÞ: (8b)

We show in Fig. 2 the segment density distribution of a 3Dpinned polymer loop projected on the 2D x–z plane. The upperpanel shows the marginal distribution of density along theforce direction, while the right panel shows the marginaldistribution of density perpendicular to the force direction.The red lines represent our theoretical results above andmatch very well with the Monte Carlo simulation data shownby blue histograms. Fig. 2(a) shows the case of the pinnedpolymer loop in a strong external force field. We can see thatthe polymer shape is fully stretched with most of fluctuationshappening at the free end, which corresponds to the middlepart of the loop. Fig. 2(b) shows the case of the pinnedpolymer loop in a moderate force field. The difference

between the free end and pinned ends are clearly demon-strated here, which make the shape of the polymer loop lookmore like a pendulum. The case of the weak external forcefield is shown in Fig. 2(c), where a droplet-like shape isobserved. Having an intuition of the overall shape of pinnedpolymer loop in an external field, we discuss the gyrationtensor in the next section.

4 The gyration tensor

In order to quantify the shape in a more precise manner, wecalculate the three-dimensional gyration tensor of the polymer,which is defined as follows:

Q ¼

Qxx Qxy Qxz

Qyx Qyy Qyz

Qzx Qzy Qzz

26664

37775!

lx2 0 0

0 ly2 0

0 0 lz2

26664

37775 (9)

where the elements Qab can be written as

Qab ¼1

N

XNi¼1

ri;a � rc;a �

ri;b � rc;b �

; (10)

with a, b A {x, y, z} and rc is the center of mass position

averaged over the position of all beads rc ¼1

N

PNj¼1

rj . The right

arrow in eqn (9) indicates the diagonalization, and lx2, ly

2, lz2

represent the three non-negative eigenvalues of the gyrationtensor. Physically, the intuition of the three eigenvalues can beinterpreted as the magnitude of three orthogonal axes of thefitted ellipsoid enclosing the polymer loop. For convenience,we sort them by lx

2 r ly2 r lz

2.Based on the gyration tensor, the first shape indicator to be

discussed is the gyration radius of the polymer, which isdefined and can be calculated as follows:

Rg2 :¼ 1

N

XNi¼1

ri � rcð Þ2 ¼ lx2 þ ly2 þ lz2: (11)

We show how the mean gyration radius hRg2i of the polymer

loop varies with the strength of the external force field in Fig. 3.The gyration radius increases monotonically with the externalforce field. Note that the exact results of the two extremes can beeasily obtained as hRg

2i = Na2/12 (F = 0) and hRg2i = N2a2/48

(F -N). In the zero external force limit, the polymer is a randomcoil while it is fully stretched in the strong external force limit. Forthe case in between, we can calculate the mean gyration radius as

Rg2

� �¼ 1

2N2

Xi;j

ðri � rjÞ2� �

; (12)

which can be expanded as

Rg2

� �¼ 1

2N2

Xi;j

rih i � rj� � �2 þ var ri½ � þ var rj

� �

� 2Cov ri; rj �

:

(13)

Fig. 2 The segment density distribution of pinned polymer shapeindicated by the color map for (a) strong external force field with F = 1,(b) moderate external force field with F = 0.1, (c) weak external forcefield with F = 0.01. The unit of the force is rescaled to be kBT/a.r indicated by the colormap is the segment density. The panels aboveand on the right of each density plot are the marginal distributions alongthe z- and x-axis respectively. The solid red lines are showingthe analytical expressions eqn (8). Other parameters are set as followskBT/a = 1, N = 100.

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The mean and variance of ri can be obtained from eqn (4)and the covariance can be obtained in a way similar to thevariance

Covðri; rjÞ ¼ a2

Pik¼0

var uk½ �PNk¼j

var uk½ �

PNk¼0

var uk½ �: (14)

Taken together we arrive at the result shown by the solid line inFig. 3. To further investigate the fluctuations of the gyration radius,in Fig. 4, we show the normalized distributions of the gyrationradius for several strengths of the external force field. Interestingly,the height of the distribution varies non-monotonically with thestrength of the external force. In other words, the gyration radiusfluctuates most under a moderate force field.

To explain this non-monotonic behavior, we use the conceptof accessible volume,33,34 which can be estimated by n = lxlylz.

We show, in the inset of Fig. 4, the variance of gyration radius,which characterizes the width of the distributions, as a functionof the external force field. On top of it, we also plot the accessiblevolume n. As we can see the trends of these two curves arevery similar, which indicates that the accessible volume can becorrelated with the non-monotonic behavior observed in Fig. 4.The discrepancy of these two curves after the maximal point isbecause of the symmetry break due to the pinning: as we can seein Fig. 2(b), the overall shape of the polymer is an asymmetricrather than a symmetric ellipsoid.

In the weak external force regime, the behavior of thebead-rod system is well approximated by the bead-spring modelwhere the equilibrated spring length equals to the rod length.We have calculated characteristics of the correspondingbead-spring loop in the Appendix. An important fact for thebead-spring model is that the variance of the bead positiondoes not depend on the external force field. Consequently, theradius of the fitted ellipsoid in the direction perpendicular tothe external force direction is unchanged in different externalforce fields, while the radius along the external force directiongrows with the external force field. Thus, a larger accessiblevolume is obtained, and we observe the initial increase in theinset of Fig. 4. However, the finite extensibility of rods becomesimportant when we further increase the strength of the externalforce field. The radius of the fitted ellipsoid perpendicular tothe force direction is narrowed down and the accessible volumealso decreases, which corresponds to the decay part in the insetof Fig. 4.

We next study the asphericity and the nature of asphericity,two other descriptors that commonly used to quantify theshape of polymers. Their definition is based on the gyrationtensor.17 The asphericity is defined as

D ¼ 3

2

TrQ2

ðTrQÞ2; (15)

where Qij = Qij � dijTrQ/3; and the nature of asphericity isgiven by

S ¼ 4 det Q

2

3TrQ2

� �3=2: (16)

Intuitively, the asphericity D is a parameter related to the

variance of the three eigenvaluesP3i¼1ðli � �lÞ2; where li A {lx, ly, lz}

and �l = (lx + ly + ly)/3. The nature of asphericity S is aparameter related to the product of the three eigenvaluesQ3i¼1ðli � �lÞ. For visualization and comparison purposes we use

the rescaled version of these two measures such that 2ffiffiffiffiDp2 ½0; 2�

and arccosS/3 A [0,p/3].17 The geometrical interpretation ofthese parameters is illustrated in Fig. 5(b). The limit of D = 0corresponds to a fully spherical object while D = 1 describes theshape of the rod-like object, whereas S measures whether theobject is prolate or oblate.

Fig. 3 The mean of gyration radius square as a function of the strength ofthe external force field. N = 100, kBT/a = 1.

Fig. 4 The distribution of gyration radius square of the pinned polymerloop under different strength of external force field. The inset showsthe width of these distribution (blue curve) and the accessible volume(red curve) as a function of the external force. N = 100, kBT/a = 1.

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In Fig. 5, we plot the distribution of the polymer shapes onthe phase diagram of asphericity and the nature of asphericity.We observe that the shapes are distributed in a smaller regionof the parameter space with the increasing force. In addition,the prolate and elongated shapes are preferred largely inde-pendent of the external force. The diagrams we plotted here aresimilar to those obtained for semi-flexible unpinned polymerloops.17

5 Conclusion and discussion

To summarize, we have studied the shape of one pinned andforced polymer loop. Using the position distribution functionof each bead, we calculated the segment density function of thewhole polymer loop. We found the overall shape of the polymeris rod-like when the external force field is strong, pendulum-like when the external force field is moderate and droplet-likewhen the external force field is weak. To quantify the shapemore precisely, we calculated the three-dimensional gyrationtensor of the pinned polymer loop. A detailed study of thegyration radius shows that the mean of gyration radius squaredincreases monotonically with the strength of the external forcefield while its variance does not. The gyration radius has amaximal fluctuation in a moderate external force regime. Wecan explain this observation by using the concept of accessiblevolume and the Rouse theory of bead-spring polymer approxi-mation in the weak force regime. Finally, we calculated theshape distribution of one pinned polymer loop on the two-dimensional diagram of asphericity and the nature of asphericity,which further highlights the shape of the polymer to be morerod-like even when the external force field is very weak.

Here we focused on one freely jointed bead-rod loop. We didnot take into account the excluded volume effect in our model.

Our simulation results show that adding the excluded volume doesnot change the results presented so far qualitatively but leads to theincrease in the system size by 5–10 percent (results not shown).

There are several interesting aspects we can further explorein the future. One example is the shape of multiple pinnedpolymer loops. Our preliminary simulation results show thatthe non-monotonic behavior in Fig. 4 disappears when thereare two or more pinned loops in the system. Note that in fissionyeast, there are three pairs of chromosomes pinned at the samepoint, namely six loops.12 Another topic to explore is the changeof polymer shape in the time-dependent force field. For instance,the turning process of the chromosomes during oscillationsin fission yeast is an important shape change that carries a bio-logical function and requires further investigations.

Conflicts of interest

There are no conflicts to declare.

AppendixA Rouse theory of the pinned polymer loop

We mainly discussed the bead-rod polymer loop model in themain text. However, when the external force field is weak, thebehavior of the bead-rod system can be approximated by abead-spring model. For the bead-spring loop, we can use Rousetheory to calculate the stationary and dynamical properties ofthe polymer. We demonstrate here the calculation of a pinnedbead-spring loop in an external force field.

Rouse theory is a theoretical framework to calculate thedynamics of a bead-spring polymer.14,30 We consider a pinnedpolymer loop modeled by beads and connecting springs. As inour previous discussion of the bead-rod model, the beadlabeled by 0 is assumed to be pinned at the origin and thereare N beads in total in the loop. Again, the periodic indexing isused. We can write the pinned condition as r0 = rL = 0.

The dynamical equation, which is a Langevin equation inthe overdamped regime, can be written as:

xdri

dt¼ �kH

Xk

Aikrk þ fei þ fbi ; (17)

where x is the friction coefficient of a bead in a fluid, ri is theposition of the ith bead, kH is the spring constant of a linearHookean spring. f e

i is the external force exerted on beads, f bi is a

typical Brownian force satisfying h f bi,ai = 0 and h f b

i,a(t) f bj,b(t0)i =

2xkBTdijdabd(t � t0). For a pinned loop, A is a (N � 1) � (N � 1)connectivity matrix and has the following form

A ¼

2 �1 0 � � �

�1 2 �1 � � �

..

. . .. . .

. ...

� � � �1 2 �1

� � � 0 �1 2

266666666664

377777777775: (18)

Fig. 5 The shape of pinned polymer loop distributed over the phasediagram of asphericity and the nature of asphericity. (a) F = 10�3,(b) F = 0.02, (c) F = 0.1, (d) F = 1. ~r (indicated by the color) is the normalizedrelative frequency of the polymer shape. Other parameters are set askBT/a = 1, N = 100.

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As in the main text, we do not take into account hydro-dynamic interactions, bending energy and excluded volumeeffect in this simple model. For convenience, we use the vectornotation and rewrite eqn (17) as:

xd

dtR ¼ �kHARþ Fe þ Fb; (19)

where R = [r1,r2,. . .,rN�1]T. Similar vector notation is alsoapplied for Fe, Fb. In order to solve this set of dynamicalequations, we first notice that the connectivity matrix A is avery special type of matrix called tridiagonal Toeplitz matrix,31

and can be diagonalized exactly. To do this, let us introduce asimilarity transform such that

[O�1AO]jk = Djk = lkdjk, (20)

here O is normalized to be a unitary matrix, and lk are theeigenvalues of matrix A. The eigenvalues and eigenvectors canbe written as follows

lk ¼ 4 sin2kp2N

� �; k ¼ 1; 2; . . . ;N � 1; (21a)

Ojk ¼ Okj ¼ ½O�1�jk ¼ ½O�1�kj ¼ffiffiffiffi2

N

rsin

jkpN

� �: (21b)

Multiplying both sides of eqn (19) by O�1, we arrive at

xdðO�1RÞ

dt¼ �kHO�1AOO�1Rþ O�1Fe þ O�1Fb: (22)

Here O�1AO = D. Using the notation such that R = O�1R, we getthe set of decoupled dynamical equations

xd~rj

dt¼ �kHlj~rj þ ~f

e

j þ ~fb

j : (23)

Eqn (23) can be solved easily by standard methods. Thegeneral solution can be written as:

~rjðtÞ ¼ ~rjð0Þe�kHljx t

þ 1

x

ðt0

~fe

j e�kHljx t�t 0ð Þ

dt 0�

þðt0

~fb

j e�kHljx t�t 0ð Þ

dt 0�:

(24)

Here the transformed Brownian force f bi also fulfills h f b

i,ai = 0and h f b

i,a(t), f bj,b(t)i = 2xkBTdijdabd(t � t0). Given the solution of

eqn (24), the position of each bead can be obtained by theinverse transformation R = OR. In the simple case of constantexternal force field, f e

j = f eez, eqn (24) can be rewritten as

~rjðtÞ ¼ ~rjð0Þe�kHljx t þ

~f eezkHlj

1� e�kHljx t

� �

þ 1

x

ðt0

~fb

j e�kHljx ðt�t

0Þdt 0:

(25)

Finally, the bead position can be obtained by the inversetransformation:

riðtÞ ¼Xj

Oij~rjðtÞ (26)

Now the stationary statistics of the polymer, such as themean and variance of each bead position can be calculatedeasily. Substituting rj (t) from eqn (25) to eqn (26) and taking thelimit t - N, we get

r1i� �

¼Xj

Oij

~f eezkHlj

¼ f eez

2NkH

Xj;k

sinijpN

� �sin

jkpN

� �

sin2jp2L

� � : (27)

The summation over j can be calculated explicitly to get

r1i� �

¼ f eez

kH

XNþ12

k¼1

sinið2k� 1Þp

N

� �

ð2k� 1Þp sin2 ð2k� 1Þp2L

� �: (28)

One can clearly see from eqn (28) that hrNi i = hrNN�ii asexpected. In addition, the components of mean positionperpendicular to the force field direction are vanished.

In order to calculate the variance of the bead position, it isnontrivial to first calculate the two-time correlation function ofnormal coordinate position as

~rmðtÞ~rnðt 0Þh i ¼ ~rmð0Þ~rnð0Þh ie�khlmx t�khlmx t 0

þ ð~f eÞ2kH2lmln

1� e�kHlmx t

� �1� e

�kHlnx t 0� �

þ 3kBT

kHlme�kHlmx tdmn:

(29)

Then the second moment of the bead position can becalculated as:

ri2ðtÞ

� �¼Xm;n

OimOin ~rmðtÞ~rnðtÞh i: (30)

Finally, taking the limit t - N, we get the equilibriumvariance of the bead position

var r1i� �

¼ ðr1i Þ2� �

� ðr1i Þ� �

2

¼ 3kBT

2NkH

XN�1k¼1

sinikpN

� �

sinkp2L

� �2664

37752

’ 3kBT

kHNiðN � iÞ:

(31)

It is worth mentioning here we also have the symmetry thatvar[rNi ] = var[rNN�i]. Moreover, we want to point out that thevariance does not depend on the external force. It means thatthe statistical distance between two beads does not depend onthe external force field. This is essentially because infiniteextensible Hookean springs are used in the Rouse theory.Furthermore, we also want to remark that the result ofeqn (31) is identical to the Brownian bridge result withoutexternal force.

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This journal is©The Royal Society of Chemistry 2019 Soft Matter, 2019, 15, 1785--1792 | 1791

Now we try to fit the Rouse theory to the pinned bead-rodloop. To do this, we take the spring in the bead-spring modelas an entropic spring and then relate the spring constant tothe length of the rod. If the spring in the polymer is the three-dimensional entropic spring, then the spring constant can befound by the equipartition theorem as

1

2kHa

2 ¼ 3

2kBT ; (32)

where a can be interpreted as the equilibrium length of thespring and is set to the length of a rod for the comparison.

We obtain kH ¼3kBT

a2. Plugging it into eqn (28) and (31) we

arrive at

r1i� �

¼ 1

3 ~TaezXNþ12

k¼1

sinið2k� 1Þp

N

� �

ð2k� 1Þp sin2ð2k� 1Þp

2L

� �; (33a)

var r1i� �

¼ a2iðN � iÞ

N; (33b)

where T = kBT/DE = kBT/f ea. Now we can compare these resultswith the results of the bead-rod model.10 From Fig. 6, we noticethat unlike the bead-rod model, the variance of bead positiondoes not depend on the external force field here. This is afundamental difference between the bead-rod and bead-springmodel. As discussed in the main text, eqn (33) leads to amonotonically increasing accessible volume with the increasingof force. Thus the non-monotonic behavior found in bead-rodmodel (shown in Fig. 4) is not observed in the harmonic bead-spring model. Actually, it is possible to analytically calculatethe distribution of gyration radius for the bead-spring model,which we will show in our future work. The main results showthat the distribution is broadening as the external forceincreases.

Acknowledgements

We would like to acknowledge stimulating discussions withYen Ting Lin, Hui-Shun Kuan, Lennart Hilbert, Jaeoh Shin andDaniela Froemberg. Open Access funding provided by the MaxPlanck Society.

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Fig. 6 The equilibrium mean and variance of bead position for the bead-spring model (dots: simulation results, solid lines: theory), compared with thebead-rod model (dashed lines). (a) Mean bead position of the z component. (b) The variance of bead position. Different colors denote differentdimensionless temperature T which is indicated in the legend. The black dash line in both (a) and (b) shows the limit of no external force field.

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