Eff ects of Shape of Crow ders on Dynam ics of a Polym er Chain Closure

Bai-cheng XiaDong-hua ZhangJia-jun WangWan-cheng Yu

CAS Key Laboratory of Soft Matter Chem istry,Department of Polymer Science and Engineering, University of Science and Technology ofChina,Hefei230026,China

Eff ects of Shape of Crow ders on Dynam ics of a Polym er Chain Closure

Bai-cheng Xia,Dong-hua Zhang,Jia-jun Wang,Wan-cheng Yu?

CAS Key Laboratory of Soft Matter Chem istry,Department of Polymer Science and Engineering, University of Science and Technology ofChina,Hefei230026,China

Using 3D Langevin dynam ics simu lations,we investigate the eff ects of the shape of crowders on the dynam ics of a polym er chain closure.The chain closure in spherical crowders is dom inated by the increasedmedium viscosity so that itgetsslowerw ith the increasing volume fraction of crowders.By contrast,the dynam ics of chain closure becomes very comp licated w ith increasing volum e fraction of crowders in spherocylindrical crowders.Notably,themean closure tim e is found to have a dramatic decrease at a range of volum e fraction of crowders 0.36?0.44.We then elucidate that an isotropic to nematic transition of spherocylindrical crowders at this range of volume fraction of crowders is responsible for the unexpected dram atic decrease in them ean closure tim e.

Loop formation,Crowding eff ects,Shape polydispersity

I.INTRODUCTION

The processwhen twom onom ersseparated by a large distance along the polymer chain come close enough to start interacting w ith each other is called loop formation.Loop formation of a polym er chain has been studied w idely by experiments,theory,and simulations due to itsgreat biological relevance[1–45].For instance,the loop formation of DNA in the cell nucleus induced by transcription factor proteinsm akes sites of DNA separated by severalμm on the geneticm ap be in molecu lar contact[1].Another exam ple is the contact formation ofpolypeptide chainswhich isconsidered asa basic step of protein folding[2].Apart from its prevalence in biological system s,loop formation also exists w idely in chem ical system s,such as in telechelic polymers[3]and in carbon nanotudes[4].

So far,many specific aspects affecting the dynamics of a chain closure have been discussed,including the chain stiff ness[5–7],the Cou lomb interactions[8], the confinem ent eff ects[9],the com p lex chain relaxation [10,11],and the excluded volume effects[12–17].Due to its close biological relevance,how loop formation occurs in realistic cellular environments is an intriguing issue.The cellular environment in living biological cells is highly crowded and fi lled w ith a plenty of biomacromolecules,such as proteins,ribosomes,lipids,and cytoskeleton fibers.The volum e fraction of these contents can be as large as 40%.It has been recognized in recent years that the crowded cellular environments could affectmany biological processes,including gene expres-sion and protein folding,etc.It is w ithout doubt that an investigation about the dynam ics of a polym er chain closure in crowded environments is of significant importance and meaning.Indeed,severaladvances in this aspect have been achieved recently[41–45].Toanet al.[43]have studied the looping kinetics of self-avoiding polymers in crowded media and found that looping is entropically aided by the dep letion effect while the increased friction im pedes the diff usive encounter of the chain ends.The interp lay of the dep letion eff ect and the increased friction makes the looping of short chains slower and that of long chains faster[43].Lately,Shinet al.[44]have reported how the crowder size aff ects the kinetics of polymer looping and showed that the loop formation gets slower in small crowderswhile it is accelerated in big crowders.M ore recently,they have investigated the polymer looping in crowded solutions of active particles and found that the presence of active particles yields a higher eff ective temperature of the bath so that the looping is facilitated[45].

It has been suggested by Kondratet al.[46]that the polydispersity of the size of crowders has a striking effect on the diff usivity ofmacromolecules.At the same volum e fraction of crowders,the chain diff usivity was shown to be slower w ith an increasing content of the small crowder in the composition[46].Meanwhile, Kanget al.[47]reported that the polydispersity of the shape of crowders aff ects the conform ational propertiesof sem iflexible chainssignificantly.Obviously,the chain diffusivity and the chain conformational properties are closely related to the dynam ics of a chain closure.However,up to now,how the polydispersity of the size and the shape of crowders affect the dynamics of a chain closure remains unclear.Therefore,by using three-dim ensional(3D)Langevin dynam ics simulations,we investigate in this work the dynam ics of achain closure in crowded environments where two different shapes of crowders,i.e.,the spherical and spherocylindrical crowders are introduced into the system. Note that the eff ects of the polydispersity of the size of crowders on the dynam ics of a chain closure is not the sub ject of the present work.

II.MODEL AND M ETHODS

The polymer in the simulations ismodeled asa beadspring chain[48].Each bead in the chain represents a segment.The finite extension nonlinear elastic(FENE) potential is applied between neighboring beads along the chain to achieve their connections described by the spring.Here,the FENE potential is given as

whereris the distance between consecutive beads,kis the spring constant andR0is themaximum allowed separation between connected beads.The repulsive nonbonded interactions between chain beads aremodeled by the truncated Lennard-Jones(LJ)potential,namely theWeeks-Chand ler-Andersen(WCA)potential[49]

Here,σ=1 is the diameter of a chain bead,ris the distance between beads,andε=1 is the interaction strength between beads.

In order to m im ic the crowding environments in realistic cells,crowders w ith different shapes are introduced into the cubic simulation box of a sizeL0=15σ. As shown in FIG.1,two diff erent shapes of crowders are considered in the present work,i.e.,spherical(S-type)crowders and spherocylindrical(SC-type)crowders.The diameter of a S-type crowder isσ.The SC-type crowders are formed by connecting 5 spherical crowders together through the above FENE potential and a bending potential between successive bonds is app lied

Here,θis the anglebetween adjacent bond vectorsw ith itsequilibrium valueθ0being set to beπ,andκ=1200 is the bending constant.Therefore,the SC-type crowder in our simulations can hard ly bend and behave like a rod.The interactions between chains beads and two kinds of crowders are purely repulsive,which can also be described by the aboveWCA potential.

In thesimulations,themotionsof chain beads,S-type crowders and beads in SC-type crowders are described by the Langevin equation[50]:

FIG.1 Schem atic illustration of the spherical and spherocylindrical crowders used in the simulations.The diameter of the spherical crow der isσ,and the spherocylindrical crowder is constructed by connecting 5 spherical crow ders.

Initially,a polymer chain of the lengthN=20 and a set number of crowdersNcare introduced into a cubic simulation box of a much larger sizeL=10L0.Ncis fi rst estimated according to a pre-specified volume fraction of crowders?0and then obtained by rounding.In this way,the p lacement of crowders becomes much easier.Note that the periodic boundary conditions are app lied in all directions.Then,the simulation box begins to contractgradually till itssizeequals toL0. During the contraction process,the thermal relaxation of the chain and crowders described by the Langevin thermostat proceedssimultaneously.As the contraction process comp letes,the thermal relaxation continues for 5×103tLJ.To ensure suffi cient equilibration of the system,we have calculated the autocorrelation function of the end-to-end vectorc(t),

FIG.2 The autocorrelation function of the end-to-end vector of a polym er chain in SC-type crowdersc(t)as a function of the timetunder three diff erent volume fractions of crow ders?=0,0.36,and 0.45.The solid lines are the linear fi ttings to the data.

Here,R(t)and R(0)is the end-to-end vector at timetand 0,respectively.c(t)is an exponential decay function of the timet.By fi ttingc(t)-tcurves at a range ofc(t)=[1,e?2]in a sem ilogarithm ic p lot,the autocorrelation tim eτacould be obtained from the negative recip rocals of the slopes of fi tting linear lines directly. Asshown in FIG.2,theautocorrelation timeτaat?=0, 0.36 and 0.45 is 37,439,and 690,respectively.Theseτaaremuch shorter than the thermal relaxation time 5×103tLJin the simulations so that the the system is actually equilibrated even for the highest volume fraction of crowders?=0.45 we have studied.W ith all of these done,the chain dynam ics ismonitored till its two end segments are w ithin a capture radiusa=2.5σ,i.e., a closure event com p letes.The closure tim etcis identified w ith the fi rst passage time of the searching process of the two end segments.Each datum point reported in thiswork is derived from averaging over 2000 independent runs so as to reduce statistical errors.Them ean closure timeτcis an average of 2000 closure timestc.

III.RESULTS AND DISCUSSION

The process of a polymer chain closure is a conform ational transition from the states that the chain w ith an end-to-end distanceRee>ato the stateRee=a.In realistic cells,the conformational transitionsofbiopolymers,such as proteins,take place in crowded environm ent.The presence of crowders leads to an increased m edium viscosity.M eanwhile,non-ignorable dep letion attractions between chain segmentsemerge.Obviously, the increased medium viscosity im pedes the contact of the two end beads of the chain as its conform ational transition could be considered as a diff usive process of barrier crossing according to the K ramers theory[53]. However,the dep letion attractions com press the con form ationalspace of the chain and thus facilitate the chain closure.The interplay of these two favorable and unfavorable factors for the chain closure process resulting from the crowding eff ects determ ines how the dynamics of chain closure depends on the volume fraction of crowders?.

FIG.3 Dependence of themean closure timeτcon the volum e fraction of crow ders?.Here,the crow ders have two diff erent shapes,i.e.,the S-type and SC-type crowders.

It hasbeen suggested by Shinet al.[44]that the size of crowders could determ ine the outcome of this interp lay.Specifically,the chain closure gets slower in small crowders as the increased medium viscosity dom inates; while it becomes faster in big crowders due to the prevailing con finem ent eff ects.We noted that the shape of crowders could aff ect the conformational transitions of sem iflexible polymer chains significantly[47].For flexible chains,this factor should also p lay an im portant role in the conform ational transitions of chains and the dynam ics of chain closure investigated here.Therefore, we have performed a set of simulation contrast tests to exam inehow theshapeof crowdersaffects thedynam ics of chain closure.

The core result of this work is presented in FIG.3. W hen the crowders in the system are spherical,the mean closure timeτcincreasesmonotonically w ith the increasing volume fraction of crowders?.This observation is due to the dom inating increased medium viscosity during the chain closure and is consistent w ith the work of Shinet al.[44].By comparison,for SC-type crowders,the dynam icsof chain closure getsmuch more com p licated as?increases and can be roughly divided into four dynam ic regim es.For?≤0.08,a slight decline inτcis observed.W ith a further increase in?,τcincreases sharp ly till?=0.36.A fterwards,a dramatic decrease inτcem erges and the chain closure at?=0.44 iseven faster than that at?=0.Finally,a sharp increase inτcappears again.

The com p licated dynam ic behaviors of chain closure proceeding in SC-type crowders reflect the comp lexity of the interp lay between the unfavorable increased medium viscosity and the favorable com pressed conformational space in this case.The strength of the dep letion attractions induced by the presence of S-type crowders is≈?kBT/σ2.However,this strength is≈?PkBT/σ2for SC-type crowders w ithPbeing the length of SC-type crowders[47].Therefore,a polymer chain in SC-type crowders is likely to bemore com pact than the one in S-type crowders at the same?.As shown in FIG.4,themean squared end-to-end distance of a chain in SC-type crowders?Ree?is always smaller than that of a chain in S-type crowders at the same?. W hen the increased m edium viscosity induced by the presence of crowders is not very striking at low?,thesmaller?Ree?of a chain in SC-type crowders accounts for the slight decline inτc.As?increases further,the increased m edium viscosity dom inates the dynam ics of chain closure so that a sharp increase inτcis observed. In contrast to the gradual decrease in?Ree?of a chain in S-type crowders at the whole range of?we have m easured,the chain in SC-type crowders undergoes a dramatic decrease in its size at?=0.36?0.44 as p lotted in FIG.4.This observation provides a phenomenological exp lanation about the significantly accelerating chain closure at the corresponding range of?.However, what is the underlying physical origin of the dramatic decrease in?Ree?at?=0.36?0.44?

FIG.4 Themean squared end-to-end distance of a polymer chain?Ree?as a function of the volum e fraction?of two kinds of shapes of crowders.

Unlike the isotropic S-type crowders,the SC-type crowders in this work have an aspect ratioδ=P/σ≈5 and are highly anisotropic.According to the Onsager theory[54],the rod-like SC-type crowdersmay undergo an isotropic to nem atic(I-N)transition w ith increasing?.To clarify whether theI-Ntransition would occur as?increases to a certain range,we have calculated the nematic order parameter of the SC-type crowders which is defined as

whereθiis the orientational angle of theith SC-type crowder.In the simulations,the nematic order parameterSis calculated by solving the largest eigenvalue of the orderingmatrix Q.Q is defined in the term s of the orientations of the crowder axes ui[55]

Here I is theunit tensor.The valueof thenematic order parameterSis close to zero in the isotropic phase,while it approaches to one in the nematic phase.

FIG.5 The nematic order parameterSof SC-type crowders as a function of?.

As shown in FIG.5,an initial decrease inSfollowed by a stab le value ofSis observed due to a sm all quantity of SC-type crowders.However,Sincreases dramatically as?≥0.32 which is indicative of theI?Ntransition.Concom itantly,the polym er chain is confined among SC-type crowders so that a sharp decrease in its?Ree?occurs.This sim ilar caging effect on a polymer chain has also been reported by Shinet al.[44].The diff erence lies in that the caging eff ect here is a result of theI-Ntransition of SC-type crowders,while in the work of Shinet al.[44],this effect is just caused by the size of crowders.In addition,both the strength of the dep letion attraction induced by SC-type crowders and their excluded volume interactions depend on the size of a SC-type crowder.If the size of themonomer of a SC-type crowder decreases,the strength of the dep letion attraction is expected to get enhanced,while the excluded volume interactions isweakened.As a consequence,therem ight be amore obvious decrease inτcat low?,and the accelerating eff ect of the chain closure due to theI-Ntransition of SC-type crowders at high?m ight be less pronounced.Nonetheless,the four dynam ic regimesof chain closure that proceeds in SC-type crowderswould be retained.

IV.CONCLUSIONS

We have investigated the eff ects of the shape of crowders on the dynam ics of a polymer chain closure by using 3D Langevin dynam ics simulations in the present work.We show that the chain closure in spherical crowders gets slower w ith the increasing volum e fraction of crowders?,which is sim p ly due to the dom inating increased medium viscosity.By contrast,the dynam ics of chain closure becomesvery comp licated w ith increasing?in spherocylindrical crowders.Notably,them ean closure timeτcis found to have a dram atic decrease at a range of?=0.36?0.44 in this case.The superficial reason is the much more rapid decrease in the mean squared end-to-end distance of a chain in spherocylindrical crowders at this range of?com pared w ith the case of spherical crowders.By calculating the nematic order parameterSof spherocylindrical crowders, we find that the crowders in the system undergo an isotropic to nematic transition w ith increasing?.It isthe occurrence of this transition that gives rise to the striking caging effects suffered by the polymer chain.In view of the com p lexity of the crowded cellu lar environm ents,our resultshere are ofgreat relevance to the loop formation of biopolymers in realistic cells.

V.ACKNOW LEDGM ENTS

This work is supported by the Fundam ental Research Funds for the Central Universities of China (No.WK 2060200020)and the China Postdoctoral Science Foundation(No.2015M 581998).

[1]J.A llem and,S.Cocco,N.Douarche,and G.Lia,Eur. Phys.J.E 19,293(2006).

[2]L.Lapidus,W.Eaton,and J.Hofrichter,Proc.Natl. Acad.Sci.USA 97,7220(2000).

[3]Y.Sheng,P.Hsu,J.Z.Y.Chen,and H.Tsao,Macrom olecu les 37,9257(2004).

[4]M.Sano,A.Kam ino,J.Okamura,and S.Shinkai,Science 293,1299(2001).

[5]S.Jun,J.Bechhoefer,and B.Ha,Europhys.Lett.64, 420(2003).

[6]A.Dua and B.Cherayil,J.Chem.Phys.116,399 (2002).

[7]R.A fra and B.Todd,J.Chem.Phys.138,174908 (2013).

[8]J.Stam p le and I.Sokolov,J.Chem.Phys.114,5043 (2001).

[9]P.Bhattacharyya,R.Sharma,and B.Cherayil,J. Chem.Phys.136,234903(2012).

[10]I.Sokolov,Phys.Rev.Lett.90,080601(2003).

[11]T.Gu′erin,O.B′enichou,and R.Voituriez,Nat.Chem. 4,568(2012).

[12]A.Rey and J.Freire,Macromolecu les 24,4673(1991).

[13](a)B.Friedm an and B.O’Shaughnessy,Phys.Rev.A 40,5950(1989); (b)B.Friedman and B.O’Shaughnessy,Europhys.Lett. 23,667(1993); (b)B.Friedman and B.O’Shaughnessy,Macrom olecu les 26,4888(1993); (c)B.Friedman and B.O’Shaughnessy,Macrom olecu les 26,5726(1993).

[14]A.Pod telezhnikov and A.Vologodskii,M acrom olecules 30,6668(1997).

[15]M.O rtiz-Repiso,J.Freire,and A.Rey,M acrom olecules 31,8356(1998).

[16]J.K im and S.Lee,J.Chem.Phys.121,12640(2004).

[17]J.K im,W.Lee,J.Sung,and S.Lee,J.Phys.Chem.B 112,6250(2008).

[18](a)G.W ilem ski and M.Fixm an,J.Chem.Phys.60, 866(1974); (b)G.W ilem ski and M.Fixm an,J.Chem.Phys.60, 878(1974).

[19]M.Doi,Chem.Phys.9,455(1975).

[20]A.Szabo,K.Schulten,and Z.Schulten,J.Chem.Phys. 72,4350(1980).

[21]A.Ansari,C.Jones,E.Henry,J.Hofrichter,and W. Eaton,Science 256,1796(1992).

[22]H.Neuweiler,M.L¨ollm ann,S.Doose,and M.Sauer,J. M ol.Biol.365,856(2007).

[23]A.M¨oglich,F.K rieger,and T.K iefhaber,J.Mol.Biol. 345,153(2005).

[24]M.Buscaglia,L.Lapidus,W.Eaton,and J.Hofrichter, Biophys.J.91,276(2006).

[25]J.Fern′andez,A.Rey,J.Freire,and I.de Pi′erola, Macromolecules 23,2057(1990).

[26]S.Chan and K.D ill,J.Chem.Phys.90,492(1989).

[27]P.Debnath and B.Cherayil,J.Chem.Phys.120,2482 (2004).

[28]N.Toan,G.M orrison,C.Hyeon,and D.Thirum alai, J.Phys.Chem.B 112,6094(2008).

[29]J.Z.Y.Chen,H.Tsao,and Y.Sheng,Phys.Rev.E 72,031804(2005).

[30]D.Doucet,A.Roitberg,and S.Hagen,Boiphys.J.92, 2281(2007).

[31]I.Yeh and G.Hummer,J.Am.Chem.Soc.124,6563 (2002).

[32]T.Uzawa,T.Isoshima,Y.Ito,K.Ishimori,and D. M akarov,and K.Plaxco,Boiphys.J.104,2485(2013).

[33]W.Yu and K.Luo,Sci.China Chem.58,689(2015).

[34]W.Yu and K.Luo,J.Chem.Phys.142,124901(2015).

[35]D.Sarkar,S.Thakur,Y.Tao,and R.Kapral,Soft M atter 10,9577(2014).

[36]R.Everaers and A.Rosa,J.Chem.Phys.136,014902 (2012).

[37]A.Am itai,I.Kupka,and D.Holcman,Phys.Rev.Lett. 109,108302(2012).

[38]A.Am itai and D.Holcman,Phys.Rev.Lett.110, 248105(2013).

[39]L.Lapidus,P.Steinbach,W.Eaton,A.Szabo,and J. Hofrichter,J.Phys.Chem.B 106,11628(2002).

[40]H.Neuweiler,A.Schulzn,M.B¨ohm er,J.Enderlein,and M.Sauer,J.Am.Chem.Soc.125,5324(2003).

[41]O.Stiehl,K.Weidner-Hertram p f,and M.Weiss,New J.Phys.15,113010(2013).

[42]J.Shin,A.G.Cherstvy,and R.M etzler,ACS M acro Lett.4,202(2015).

[43]N.Toan,D.M arenduzzo,P.Cook,and C.M icheletti, Phys.Rev.Lett.97,17830(2006).

[44]J.Shin,A.G.Cherstvy,and R.Metzler,Soft Matter 11,472(2015).

[45]J.Shin,A.G.Cherstvy,W.K.K im,and R.M etzler, New J.Phys.17,113008(2015).

[46]S.Kondrat,O.Zimm erm ann,W.W iechert,and E.von Lieres,Phys.Biol.12,046003(2015).

[47]H.Kang,N.M.Toan,C.Hyeon,and D.Thirum alai,J. Am.Chem.Soc.137,10970(2015).

[48]K.K rem er and G.G rest,J.Chem.Phys.92,5057 (1990).

[49]J.D.Weeks,D.Chand ler,and H.C.Andersen,J. Chem.Phys.54,5237(1971).

[50]M.P.A llen and D.J.Tildesley,Computer Simulation of LiquidsNew York:Oxford University Press,(1987).

[51]D.Chand ler,Introduction to M odern Statistical M echanics,New York:Oxford University Press,(1987).

[52]D.L.Erm ak and H.Buckholz,J.Com put.Phys.35, 169(1980).

[53]P.H¨anggi,P.Talkner,and M.Borkovec,Rev.M od. Phys.62,251(1990).

[54]L.Onsager,Ann.N.Y.Acad.Sci.51,627(1949).

[55]S.C.M cG rother,D.C.W illiam son,and G.Jackson,J. Chem.Phys.104,6755(1996).

ceived on March 1,2017;Accepted on April 17,2017)

?Author to whom correspondence shou ld be addressed.E-m ail: yw cheng@ustc.edu.cn