Unexpected swelling of stiff DNA in a polydisperse crowded environment
UUnexpected swelling of stiff DNA in a polydisperse crowded environment
Hongsuk Kang, Ngo Minh Toan, Changbong Hyeon, ∗ and D. Thirumalai
1, 2, 3 Chemical Physics and Biophysics Programs, Institute of Physical Science and Technology,University of Maryland, College Park 20742, USA Korea Institute for Advanced Study, Seoul 130-722, Korea Chemistry and Biochemsitry, University of Maryland, College Park 20742, USA
We investigate the conformations of DNA-like stiff chains, characterized by contour length ( L )and persistence length ( l p ), in a variety of crowded environments containing monodisperse softspherical (SS) and spherocylindrical (SC) particles, mixture of SS and SC, and a milieu mimickingthe composition of proteins in E. Coli. cytoplasm. The stiff chain, whose size modestly increasesin SS crowders up to φ ≈ .
1, is considerably more compact at low volume fractions ( φ ≤ .
2) inmonodisperse SC particles than in a medium containing SS particles. A 1:1 mixture of SS and SCcrowders induces greater chain compaction than the pure SS or SC crowders at the same φ withthe effect being highly non-additive. We also discover a counter-intuitive result that polydispersecrowding environment, mimicking the composition of a cell lysate, swells the DNA-like polymer,which is in stark contrast to the size reduction of flexible polymer in the same milieu. Trappingof the stiff chain in a fluctuating tube-like environment created by large-sized crowders explainsthe dramatic increase in size and persistence length of the stiff chain. In the polydisperse medium,mimicking the cellular environment, the size of the DNA (or related RNA) is determined by L/l p .At low L/l p the size of the polymer is unaffected whereas there is a dramatic swelling at intermediatevalue of L/l p . We use these results to provide insights into recent experiments on crowding effectson RNA, and also make testable predictions. The recognition that the crowded cellular environ-ment can profoundly influence all biological processes,such as gene expression [1–3], protein [4–6] and RNAfolding [7–9] and protein-protein interactions [10], isgetting increasing attention recently although its im-portance was recognized decades ago [11]. A simplecalculation using the typical concentration of macro-molecules shows that the average spacing between pro-teins in the
E. coli is ∼ R g ≈ . N / nm of afolded protein [12] with N ∼
300 amino acid residues.Therefore, the cellular interior, replete with macro-molecules of different sizes and shapes (a polydispersesoup), is crowded, affecting the stability and shapesof the molecules of life. For example, compactionof DNA, relevant in a variety of biological processesranging from organization of nucleoid in bacteria toDNA packaging in phage heads, is greatly facilitatedin the presence of neutral osmotic agents. The effect ofneutral polymer (poly-ethylene oxide) (PEO) in com-pacting DNA (Ψ-condensation) was demonstrated ina pioneering study by Lerman [11], who showed thatDNA undergoes a dramatic reduction in size if theconcentration of PEO exceeds a critical value. Be-cause the interactions between DNA and PEO in theseexperiments was established to be repulsive [13], Ψ-condensation is determined solely by the volume ex-cluded to DNA by the crowding particles. Subse-quently, Post and Zimm [14] produced insightful the-oretical explanations based on further experiments oncrowding-induced compaction of DNA. In the inter- vening years a number of theoretical and experimentalstudies have explored various aspects of DNA com-paction [15–17].Despite these advances a molecular description ofhow macromolecular crowding, especially a milieu con-taining polydisperse crowders, affects the spatial orga-nization of DNA is poorly understood. Simulationsof DNA in the presence of explicit crowding particlesare computationally intensive, but their importancein describing the shapes of flexible polymers has beendemonstrated in a number of recent studies [18–22].Because of technical complications theories do not takeinto account the effects of polydispersity [15, 16, 23–30]. In addition, the extent of structural changes andhow they depend on mixtures of crowding particles ofdifferent shapes for realistic sizes of crowders found inthe cytoplasm are unknown. Most of the studies havefocused on intrinsically flexible polymers with very lit-tle work on relatively short stiff DNA-like polymers,and related RNA. Stiff short chains can display quan-titatively different behavior than the one associatedlong flexible polymers, as argued in the case of cy-clization of DNA [31–33]. These issues take on addedimportance because of potential relevance to genomeconfinement [4, 34, 35], where the structural organiza-tion is functionally related to gene expression.Inspired by these observations, we first performedsimulations of worm-like chain [36] (WLC - a reliablepolymer model for describing many of the propertiesof DNA at high salt concentration) in monodispersesolutions containing soft spherical (SS) and sphero- a r X i v : . [ c ond - m a t . s o f t ] A ug cylindrical (SC) crowding particles. Here, we focus onDNA-like chains in the limit where the contour length( L = N m l , N m is the number of monomers, and l is the bond length) is not significantly longer than thepersistence length, l p . The systematic study leads to anumber of unexpected predictions, which can be testedusing synthetic polymers in the presence of nanoparti-cles and DNA using crowding agents. (i) The SC crow-ders induce greater compaction than SS particles. Thecompaction is accompanied by substantial reduction of l p (by nearly a factor of two) similar in magnitude tothat observed by ion-induced compaction of DNA andRNA [37]. (ii) One of the most striking results of ourstudy is that stiff chains are more compact in a mix-ture of SS and SC, at physiologically relevant volumefractions, than in monodisperse crowders at the samevolume fraction. The substantial compaction of DNAin the mixture is due to ordering (dense packing) of theSS crowders around DNA due to depletion attractioninduced by the SCs. (iii) We also carried out sim-ulations of DNA in polydisperse crowders mimickingthe composition of macromolecules in E. coli . Sur-prisingly, we obtained a counter-intuitive and theoret-ically unanticipated result that polydisperse sphericalcrowding particles cause swelling of DNA at volumefraction ≈ E. Coli. cytoplasm. Sur-rounded predominantly by large-sized crowders, theswollen conformation of DNA (nearly three times in-crease in volume of the chain) is entropically driven.
RESULTS
Crowding-induced softening of DNA.
In orderto set the scale for bending energy of the polymer, wefirst calculated the end-to-end distance distribution, P ( R ee ), in the absence of crowders to obtain thebare persistence length, l p for φ = 0. By fitting thesimulated P ( r ) to an analytic expression for WLC(Eq.S3) [31], we obtained l p = (15 . ± . σ m ≈ σ m ≈ .
18 nm (1b), which coincides with l p for DNA in high monovalent salt concentration.Because l p /σ m >
1, the shape of the chain in the pres-ence of crowding particles should be determined by aninterplay of bending rigidity and attractive depletioninteraction due to crowders. P ( R ee ) of the WLC withvarying φ for both monodisperse SS and SC crowdersshow gradual shift to the smaller R ee with increasing φ from 0 to 0.4 (1b). At high φ , as φ increases, l p obtained from the fit of P ( r ) to Eq.S3 graduallydecreases for both SS and SC crowders, implyingthat the crowding particles induces compaction ofthe WLC polymer (1b, top panel). Interestingly, theWLC polymer exhibits a non-monotonic dependence P ( R ee / σ m ) R ee / σ m φ =0.0 0.2 (SS)0.4 (SS)0.2 (SC)0.4 (SC) m cyl c y l c (a) (b) l p / σ m φ SSSC
FIG. 1: (a) Coarse-grained models of spherical (SS) crow-der (red), spherocylinderical (SC) crowder (yellow) andWLC chain (blue) used to model DNA. The relevant di-mensions are labeled σ sph , σ cyl , and σ m . (b) End-to-enddistance distribution of WLC at two different volume frac-tions ( φ = 0 .
2, 0.4) of SS and SC crowders. The top panelshows the persistence length ( l p ) of the polymer as a func-tion of the volume fraction of SS and SC crowders. of size with increasing φ . The SS crowders induce aminor increase of l p (stiffening or expansion) of thechain (1b top panel and 2a) for 0 < φ ≤ .
1, followedby a decrease of l p (softening or compaction) at larger φ = 0 . − .
4. In contrast, the SC crowders reduces l p much more efficiently than the SS crowders at φ ≤ . φ ≈ − . φ is in the range from φ = 0 . φ = 0 .
4. We further substantiate this result below bycalculating the change in the polymer size ( R g ) andnematic order parameter of the crowders. The slightincrease in l p not withstanding, the overall trend isthat there is substantial softening ( l p decreases bynearly a factor of 2) as φ increases from 0 to 0.4. Dependence of R g on φ for monodispersecrowders. Snapshots of polymer conformations atdifferent values of φ show modest compaction as φ in-creases (2a). Similar to l p , the dependence of R g on φ for SS and SC crowders displays substantial difference.At φ = 0 . R g (0 .
2) is smaller than R g (0) by only 4% in SS crowders whereas R g (0 .
2) decreases by 17 %in SC crowders. Given that the volume of the chainis ∼ R g ( φ ) the extent of compaction induced by SCis substantial compared with SS crowders. The quan-titative difference between the effects of SS and SCcrowders on the chain compaction is explained usingthe depletion interaction (or Asakura-Oosawa (AO) in-teraction [38]) that produces an effective attraction be-tween monomers. The strength of the AO interaction (c)(b) -0.1 0 0.1 0.2 0 2 4 6 8 10 < P [ c o s θ (r) ] > r/ σ m φ =0.1 0.2 0.3 0.4 ✓ ij r ij P ( r ) r/ m Q ( . σ m ) φ (a)
30 32 34 36 38 40 0 0.1 0.2 0.3 0.4 R g ( φ ) / σ m φ SSSC
FIG. 2: (a) The change of R g of the polymer from R g (0) = 37 . φ . (b) Liquid crystal order parameterfor SC crowders, averaged over the ensemble of crowder particles, as a function of distance. (c) The extent of local nematicordering is quantified with Q ( r ) at r = 4 . σ m . for SS, roughly given by ≈ φk B T /σ sph , has to exceedthe energy ( ∼ ( L/l p ) k B T ) required to bend the poly-mer on scale l p for compaction to occur. For small φ , itis unlikely that the AO attraction can compensate forthe bending penalty. Thus, we expect little change in R g ( φ ) at small φ for SS crowders. On the other hand,the strength of the AO interaction on the WLC forSC crowders is ≈ φP ( σ m /σ cyl ) k B T where P (= 2 σ cyl )is the cylinder length. In both cases, the origin ofthe AO depletion interaction, which has to exceed thebending energy to compact the stiff chain, leading toan effective short range (on length scale ∼ σ m ) attrac-tion between monomers and polymer compaction, ispurely entropic. 2a shows that R g ( φ ) decreases mono-tonically till φ ≈ . ∼ l p .The strong effect of compaction of the WLC chaininduced by SC relative to SS crowders can also bequantified by comparing the volume excluded to thepolymer by the crowders. On the basis of scaled parti-cle theory, we can estimate the entropy cost of insert-ing a hard sphere of dimension σ HS in a box containinghard fluid particles. The entropy difference for insert-ing the hard sphere of diameter σ HS is related to C ( σ )[39], C ( σ HS ) = V cyl V sph = 3 (cid:2) ( σ HS + σ cyl ) P + ( σ HS + σ cyl ) (cid:3) σ HS + σ sph ) (1)where V cyl ( V sph ) is the volume excluded by rod-like(spherical) crowders. For the parameters listed in Ta-ble S1, we find that V cyl > V sph provided σ HS ≈ R g >σ cyl or σ sph . The entropic cost of inserting a sphericalparticle of size R g into a fluid of cylindrical crowdersexceeds that for inserting it into a system consisting ofspherical crowders. By achieving greater compaction of the WLC in SC crowders, the entropy difference isminimized, thus, explaining the results in 2a.The pattern of compaction in the two crowd-ing environment is qualitatively different. For SScrowders, compaction occurs only when φ exceeds ≈ . dR g /dφ ≈ φ → l p ( φ ) (1b, top panel), R g ( φ )(2a) shows a signature of minor swelling in therange of φ = 0 − .
1, which was also observed inexperiments on the effects of crowding on a ribozyme(see below) [8]. In contrast, for anisotropic crowdingagents (SC crowders), R g ( φ ) decreases monotonicallyfor φ ≤ . dR g /dφ < φ ) and increasesfrom φ = 0 . φ = 0 .
4. High volume fractionof SC results in the reswelling of WLC (Figure2a), showing that the shape of the crowding par-ticles can have a profound effect on size of a stiff chain.
Local nematic order and increase in R g . In-terestingly, as φ exceeds 0.3, R g ( φ ) of the chain in-creases in SC crowders (2a). The increase in R g ( φ )at higher φ is due to plausible development of lo-cal nematic ordering for SC crowders at φ > . φ ex-ceeds a critical value φ I → N . We calculated the liq-uid crystal order parameter, (cid:104) P (cos θ ) (cid:105) , where θ isthe angle between the long axes of any pairs of SCcrowders, P ( x ) = (cid:0) x − (cid:1) is the second Legen-dre polynomial, and (cid:104) . . . (cid:105) denotes the ensemble av-erage. (cid:104) P (cos θ ) (cid:105) is almost zero ( < .
05) for all φ ,which means that even the highest φ (= 0 .
4) is less than φ I → N . However, locally the crowders adopt nematic-like state. To ascertain if this is the case, we calcu-lated P ( r ) = (cid:104) (cid:80) i,j P (cos θ ij ) δ ( | (cid:126)r i − (cid:126)r j | − r ) (cid:105) as afunction of distance, r . 2b shows that the angularcorrelation of SCs becomes stronger as φ increases atshort distances.To quantify the extent of “local” nematic ordering,we calculated Q ( r ) = N − P (cid:82) rr min P ( r (cid:48) ) d (cid:126)r (cid:48) , where N P is the number of pairs separated by r and r min is the minimal distance between SC crowders. Atthe distance r = 2 . σ m where the pair correlationhas first peak for φ = 0 . P (2 . σ m ) increasessignificantly from 0.0 to 0.23 as φ increases from 0.1to 0.4 (2c). At r = 4 . σ m , Q ( r ) ≈ .
23 suggests thatthe local nematic ordering of cylindrical crowders isreinforced in the vicinity of the WLC. This resultimplies that relatively stiff chain induces ordering ofSC crowders along the polymer axis, and strengthensthe anisotropic interaction of rod-like particles. Athigh φ the chain may be thought of as being in a localnematic field, which elongates the polymer along thelocal direction of the nematic field, thus explainingthe increase in R g ( φ ) when φ exceeds 0.3. φ R g ( φ ) / σ m AdditiveMixture
FIG. 3: Compaction of stiff chain in a mixture of SS andSC crowders. R g of WLC as a function of φ of 1:1 SS andSC mixture (green). The orange line is the calculated R g at each φ value by assuming that the effects of SS and SCcrowders on WLC compaction are additive. The results ofmonodisperse SS and SC crowders in 2a are shown with thedashed lines to underscore the substantial enhancement ofthe chain compaction by the mixture. Non-additive effect in a mixture of spheresand spherocylinders.
The dependence of R g ( φ ) on φ of the WLC in the 1:1 mixture of SS and SC crowdersis shown in 3. The mixture has a profound effect on thesize of DNA. The value of R g (0 .
4) is reduced by over40% from R g (0), whereas the maximum compaction inmonodisperse SC at φ = 0 . φ = 0 . φ =0 .
4, the expected result for R Ag ( φ ) = R g (0) + δR Ag ( φ ),where δR Ag ( φ ) = δR SS g ( φ/
2) + δR SC g ( φ/ R g reduction is additive. Remarkably, R g ( φ )is significantlly lower than R Ag ( φ ) (3), (or | δR g ( φ ) | > | δR Ag ( φ ) | , | δR SC g ( φ ) | , | δR SS g ( φ ) | ), indicating that the (a)(b) (i)(ii) (iii) g (r) r/ σ m φ =0.2 g mixSS-SS (r)g mixSC-SC (r)g mixSS-SC (r) g (r) r/ σ m φ =0.4 g mixSS-SS (r)g mixSC-SC (r)g mixSS-SC (r) g (r) r/ σ m g m-SSmix ( φ =0.4)g m-SCmix ( φ =0.4) g m ono SS - SS (r) r/ σ m φ =0.10.20.30.4 g m ono S C - S C (r) r/ σ m φ =0.10.20.30.4 FIG. 4: (a) RDFs are calculated for all possible combi-nations of crowder-crowder pairs (at (i) φ = 0 . φ = 0 . σ sph = 4 σ m and σ cyl = 4 / σ m were used in actual simulations, we deliberately reducedthe sizes of SS and SC crowders for clear illustration ofthe crowders around the chain). (b) RDFs of SS-SS andSC-SC pairs at φ = 0 . − . mixture of SS and SC restricts the volume available tothe WLC to a much greater extent than the individualcomponents do.The surprising finding of significant compaction inthe mixture can be qualitatively explained using thenotion of depletion potentials for mixture of SS andSC. Consider the interaction between two sphericalparticles in the presence of rods (SC particles). If a SCparticle is spatially trapped then the rod loses transla-tional and rotational entropy because of orientationalrestrictions. The large unfavorable entropy loss resultsin the depletion of the SC from the space where the SCparticle is trapped. The result is that there would bean excess osmotic pressure due to the AO attractionthat pushes the SS and the monomers together. Twoconsequences of the entropy-driven depletion interac-tions are : (i) Due to the attractive interactions, the SSparticles are more closely packed than in the absence ofthe SC (compare g mixSS-SS ( r ) at φ from 4a and g monoSS-SS ( r )at φ/ σ cyl < P < σ sph , it has been experimen-tally shown that the addition of a small fraction (byvolume) of the SC can even lead to crystallization oflow density suspension of hard spheres [40, 41]. (ii)We also expect that the excess volume available tothe WLC should be greatly reduced compared to themonodisperse crowders. In such a confined space theWLC should be considerably more compact than inthe presence of monodisperse crowders at the same φ .The expected enhancement in the packing of the SSdue to the SC is evident in the pair distribution func-tion g ( r ). The results for φ = 0 . φ = 0 . g mixSC-SC ( r ), between SCs and g mixSS-SC ( r ),between SC and SS, do not exhibit significant struc-ture. In sharp contrast, g mixSS-SS ( r ) has the structurecorresponding to a high density liquid (especially at φ = 0 . φ/ φ SS = 0 . g monoSS-SS ( r ) (4b) is relatively fea-tureless.The much stronger depletion force due to theSC crowders results in a considerable reduction inthe volume accessible to the chain, which explainsthe dramatic reduction of R g ( φ ) compared to thepure component case. The WLC at φ = 0 . g mixm-X ( r ), the RDF between the monomerand the crowders ( X = SS or SC) (4a-(iii)). Thenumber of SS near the monomer calculated using N SS = 4 π (cid:0) N V (cid:1) (cid:82) r min r g m-SC ( r ) dr where r min is thefirst minimum in the g m-SS ( r ) at φ = 0 . .
4. Asimilar calculation for SC gives N SC = 4 . Polydisperse crowders mimicking the
E. coli cytoplasm composition.
To a first approximationthe
E. coli cytoplasm may be represented by a mixtureof spheres because majority of the crowders present inlarge numbers (ribosomes, polymerases and other largecomplexes, and smaller particles) are compact [42]. Inorder to assess the shape of DNA-like chain in sucha mixture, we investigated the effect of polydispersespherical particles on the conformational fluctuationsof the stiff polymer.Strikingly, the behavior of the stiff chain in a poly-disperse mixture of SS particles differs drastically fromthose in the mixture of SS and SC of the same volume(3). A few features in the non-monotonic dependenceof R g ( φ ) as a function of φ (5a) are worth pointingout. (i) There is a very modest reduction in R g ( φ ) at φ ≈ .
1, which has only large crowding particles. Sucha decrease is comparable to that found in 2a. (ii) Unex-pectedly, R g ( φ ) starts increasing in a mixture contain-ing large and medium sized crowders. Most strikingly,in the mixture roughly mimicking the E. coli cyto-plasm [42] there is a large increase ( ∼ R g ( φ )compared to φ = 0. The competition between bending free energy and depletion potential leading to a dra-matic swelling of the stiff chain is counter-intuitive.The ensemble of the chain conformations (5a) exhibit-ing the expansion of the chain captures these effectsvisually. (iii) The E. coli mixture dramatically stiffensthe polymer. The persistence length of the chain forthe swollen chain in the
E. coli mixture is about 2.6times larger than for the one at φ = 0. The stiffeningeffect of mixture of spherical crowders on WLC cap-tured the snapshots in 5. (iv) It is noteworthy thatin the polydisperse mixture with E. coli. compositionreduces the size of a flexible chain that lacks bendingpenalty (5a and Figure S3), underscoring the impor-tance of chain stiffness.The reswelling at high φ (5a, 5b) can be understoodqualitatively using the following arguments. At a spec-ified total volume fraction there are a lot more smallcrowding particles than large ones. Therefore, the en-tropy of the crowding particles is maximized if theWLC is surrounded by the larger sized particles withthe smaller ones being further away from the chain. Inthis picture, the WLC is localized in a region in whichthe larger sized particles are with higher probability inproximity to the monomers (Figure S4). Because theinteractions between the crowding particles and thestiff chain are repulsive the DNA chain would prefer tobe localized in a largely crowder-free environment. Ifwe assume that such a region is roughly spherical, cre-ated predominantly by the largest crowders, then itssize has to be on the order of R g to accommodate theWLC chain. In such a cavity there is an entropic costto confine the stiff chain. The probability of findingsuch a region decreases exponentially as R g gets large.In addition, in a spherical region the DNA would formspool-like structures requiring overcoming bending en-ergy. The combination of these effects makes it likelythat an optimal spherical regime can be found to min-imize the free energy of WLC. If the region is cylin-drical and large enough such that tight hairpins (cost-ing substantial bending penalty) are avoided then thechain free energy may be minimized by confining it ina roughly cylindrical cavity. Such a possibility is sup-ported by simulations, which show that on an averagethe shape of the depletion zone is aspherical resemblinga fluctuating tube (see images in 6a). As a result, wecan visualize the polymer to be essentially confined toan anisotropic (but fluctuating) tube in which trans-verse fluctuations of the chain are restricted but onein which tight hairpin turns cannot form because ofbending penalty. In such a cavity the chain stiffens,thus expanding in size in order to minimize both thebending penalty and entropy cost of confinement. Amore quantitative and accurate theory is difficult toconstruct because of the many body correlation amongthe polydisperse crowding particles.The plausibility of the physical picture given aboveis further substantiated by examining how the crowd-ing particles with different sizes are arranged in spaceand how they surround the extended WLC. Distribu-tions of the polydisperse crowding particles are notuniform, but exhibit local size ordering. This is evi-dent in the local size correlation function ( r -dependentsize variance), ξ ( r ) = (cid:104) d i d j (cid:105) r − (cid:104) d (cid:105) (6b) where (cid:104) d (cid:105) isthe mean diameter and (cid:104) d i d j (cid:105) r in the first term de-notes an average of the product of two diameters d i and d j taken over crowders ( j ) located at a distance r from a crowder i [43]. The local size correlation ξ ( r )(we set (cid:104) d i d j (cid:105) r = 0 when no pair exists), which isby definition zero for both monodisperse crowders andhomogeneously distributed polydisperse crowders, re-veals the presence of size ordering up to r ≤ − φ mix X ( r ) (cid:16) = π (cid:0) σ X (cid:1) × φ X g mix m − X ( r ) (cid:17) with X = 1 , ,
3) as a function of distance from the WLCmonomers in 6c further captures the non-uniform dis-tribution of crowders. The crowders with intermediatesize ( X = 2) occupy the largest volume near the WLC.In addition, the comparision of φ mix2 ( r ) with φ mono2 ( r )(the volume fraction of monodisperse 11 %-crowdersaround WLC monomer; the dashed line in 6c) showsthat the intermediate ( X = 2) and large ( X = 1) sizedcrowders are pushed closer to the monomers by thesmall crowders, which confines a segment of the poly-mer to a tube-like region (6a). The depletion forcesin a polydisperse solution give rise to a spatial inho-mogeneity of crowders, resulting in the chain beingconfined to a cylindrical region created by the large-sized ( X = 1 ,
2) crowders. The expansion of the chainin such a confined space [44] provides a plausible phys-ical explanation for the large increase in the size of theDNA.
DISCUSSION
Swelling and collapse of DNA.
The counter-intuitive finding that a stiff polymer, with γ = L/l p notlarge (see below), can swell relative in a polydispersemixture of spheres, is (to our knowledge) unprece-dented. There are simulation and theoretical studiespredicting the collapse of flexible polymers and pro-teins in mixed solvents due to volume exclusion effectsalone [45, 46]. However, the present study shows pre-cisely the opposite behavior for stiff chains whereas aflexible chain tends to become compact (not a glob- ule in the E. Coli. like milieu). We propose that thisunusual effect is related to an interplay of chain bend-ing and the complex depletion effect in a polydispersecrowding system. In order to substantiate our proposalwe carried out simulations for chains with γ varyingfrom 2 to 10 in the model E. Coli. -like system. Thesimulations show hardly any change in R g (see FigureS5). For all the values of γ ( ≤
10) the stiff polymer canbe accommodated in a large enough region in whichthe crowding particles do not suppress the conforma-tional fluctuations. Only when γ exceeds a minimumvalue, but is not too large, then chain swelling occursby formation of tight turns.We provide arguments that when γ exceeds a certainvalue the DNA-like polymer must undergo a transitionfrom the swollen state to a globule. In other words,there must be a sharp coil-globule transition inducedby the crowders. When γ (cid:29) γ has to exceed 40 to observe a genuinecoil-globule transition. The simulation results and thephysical arguments allows us to predict a rich depen-dence of R g in the E. Coli. environment (7a). (a) (b) P [ R g ( φ ) / R g ( ) ] R g ( φ )/R g (0) φ =0.00.3 P [ R ee ( φ ) / R ee ( ) ] R ee ( φ )/R ee (0) φ =0.00.3Fit R g ( φ ) / R g ( ) φ
11% 11%:11%11%:11%:8% φ = . F I G . : C r o w d i n g e ↵ ec t s o f p o l y d i s p e r s e SS o n t h e W L C . ( a ) R g f o r = . , . nd . . A t = . , O n l y . n m p a r t i c l e s a r e u s e d . A t = . b o t h . n m a nd . n m p r e s e n t i n t h e s y s t e m w i t h t h e s a m e v o l u m e f r a c t i o n o f % f o r e a c h . A t = . , t h e s y s t e m c o n s i s t s o f . n m , . n m a nd . n m s i ze d c r o w d e r s w i t h . : . : . v o l u m e f r a c t i o n . ( b ) E n s e m b l e o f t h e W L C a t = nd . . T h ee n s e m b l e v i s u a ll y c o nfi r m s t h ec h a i n i s e x t e nd e db y c r o w d i n g . ( c ) R a d i a l d i s t r i bu t i o n f un c t i o n o f l a r g ec r o w d e r s f r o m a m o n o m e r i n m o n o d i s p e r s e ( r e d ) a ndp o l y d i s p e r s ec r o w d i n g e n v r i o n m e n t s ( g r ee n ) . ( d ) P a i r c o rr e l a t i o n f un c t i o n s o f l a r g ec r o w d e r s i n m o n o d i s p e r s e ( r e d ) a ndp o l y d i s p e r s ec r o w d i n g e n v r i o n m e n t s ( g r ee n ) . φ = ( b )( d ) F I G . : C r o w d i n g e ↵ ec t s o f p o l y d i s p e r s e SS o n t h e W L C . ( a ) R g f o r = . , . nd . . A t = . , O n l y . n m p a r t i c l e s a r e u s e d . A t = . b o t h . n m a nd . n m p r e s e n t i n t h e s y s t e m w i t h t h e s a m e v o l u m e f r a c t i o n o f % f o r e a c h . A t = . , t h e s y s t e m c o n s i s t s o f . n m , . n m a nd . n m s i ze d c r o w d e r s w i t h . : . : . v o l u m e f r a c t i o n . ( b ) E n s e m b l e o f t h e W L C a t = nd . . T h ee n s e m b l e v i s u a ll y c o nfi r m s t h ec h a i n i s e x t e nd e db y c r o w d i n g . ( c ) R a d i a l d i s t r i bu t i o n f un c t i o n o f l a r g ec r o w d e r s f r o m a m o n o m e r i n m o n o d i s p e r s e ( r e d ) a ndp o l y d i s p e r s ec r o w d i n g e n v r i o n m e n t s ( g r ee n ) . ( d ) P a i r c o rr e l a t i o n f un c t i o n s o f l a r g ec r o w d e r s i n m o n o d i s p e r s e ( r e d ) a ndp o l y d i s p e r s ec r o w d i n g e n v r i o n m e n t s ( g r ee n ) . WLCSAW
FIG. 5: Effects of polydisperse soft sphere mixture on thesize of WLC (
L/l p = 20 at φ = 0). (a) R g is calculatedfor WLC (black solid line) and SAW (orange dashed line)polymers at (i) φ = 0.11 of crowders with r = 10 . φ = 0 .
22 with r = 10 . r = 5 . φ = 0 . r = 10 . r = 5 . r = 2 . φ = 0 and 0.3 areshown at the bottom, demonstrating the contrasting effectof polydisperse crowders on the conformations of WLC andSAW. (b) P ( R g ) (top) and P ( R ee ) (bottom) of WLC at φ = 0 and 0 .
3. The fit of P ( R ee ) at φ = 0 . l φ =0 . p = 41 σ m ≈
130 nm, which is ∼ . l φ =0 p ( ≈ σ m ≈ nm ). P ( R g ) of SAW(flexible self-avoiding chain) is shown in Figure S3. (a)(b) r ✓ m + X ◆ m i x X ( r ) v x g m - x (r) r/( σ m + σ x ) σ /2=10.4nm σ /2= 5.2nm σ /2= 2.6nm σ /2=10.4nm (mono) (c) -3.5-3-2.5-2-1.5-1-0.500.5 10 15 20 25 ξ (r) r(nm)Crowders onlyWLC and Crowders FIG. 6: (a) A snapshot from simulation, demonstrating(left) the polydisperse crowding environment, (middle) theWLC inside the crowders, and (right) the crowding par-ticles decorating the monomers. (b) Local size orderingcorrelation function, ξ ( r ), indicates a non-uniform, hetero-geneous size ordering of polydisperse crowders. (c) φ mix X ( r )( X = 1 , , Insights into crowding effects on RNA.
Re-cent experiments have examined the effects of poly-tehylene glycol (PEG) on the folding of a ribozyme[8]. It has been argued that the impact of PEG canbe understood based on the excluded volume effect.The SAXS experiments on
Azoarcus ribozyme with195 nucleotides shows that R g , near the midpoint ofthe Mg ion needed for the folding transition, ini-tially increases before becoming compact (7b). Fold-ing of this RNA is also accompanied by decrease in thepersistence length, which can be modulated by crow-ders. For Azoarcus ribozyme γ ≈
36 where we haveused L ≈ × .
55 nm and l p ≈ R g as PEG (assumed to be sphereical) concentra-tions increase. The experimental data is in qualitativeagreement with this expectation. It would be most in-teresting to examine the effects of polydisperse crowd-ing agents on the complex problem of RNA folding tofurther some of our predictions. CONCLUSIONS
In summary, using explicit simulations of crowdingparticles, we predict multiple and unexpected scenar-ios for the effects of polydisperse crowding environ-ment on the size and shape of a semiflexible polymer,which has served as a model for DNA and even RNA.Depending on the size, shape, and composition of themixtures of crowding particles we find evidence forboth compaction, and surprisingly dramatic increasein size as well. The results are of great relevance to therecent explosion of interest in the behavior of RNA [8],DNA [26, 48], and proteins [49, 50] in macromolecularcrowding conditions both in vitro and in vivo . Theprediction that shape of chains, such as DNA, RNA,and F-actin [51], can be dramatically altered in a poly-disperse milieu can be tested in experiments.
METHODS
Model.
To study the effects of crowding particleson a stiff chain, we used a coarse-grained model ofWLC polymer ( N m = 300), SS and SC crowding par-ticles (Figure 1a). The length scales, σ sph , σ cyl , and σ m ( ≈ .
18 nm for DNA) denote the size (diameter)of the SS, SC crowding particles and the monomer ofWLC polymer, respectively. We set the aspect ratioof the SC to be 2 and σ sph = 2 / σ cyl = 4 σ m , so thatthe volumes of the individual SS and SC crowdingparticles are identical. In the WLC model, chainconnectivity, with a fixed bond length, is maintainedusing a large spring constant connecting two consec-utive beads. The bending rigidity of the chain wasimplemented by quadratic bond angle potential. Wechose Weeks-Chandler-Andersen (WCA) potential forinteractions between monomers, and r − soft-spherepotential is employed for excluded volume interac-tions for crowder-crowder and crowder-polymer beads. Soft Spheres (SS)
The energy function for the sys-tem consisting of the WLC and soft spherical crowdersis, E = E S + E B + E WCA + E R , (2) E S = N m − (cid:88) i =1 K ( | (cid:126)r i +1 − (cid:126)r i | − l ) l , (3) E B = N m − (cid:88) i =1 G ( θ i − θ ) , (4) φ R g ( φ ) / R g ( ) A z oa r c u s , R g ( φ ) / R g ( ) (b) ( L/l p ⇡ R g ( ) R g (0) L/l p ⇡⇡ (a) φ = . φ = φ = . φ = ( ⇡ ⇡ ⇡ . . g (r) r / σ m φ = . g m i x SS - SS (r) g m i x S C - S C (r) g m i x SS - S C (r) . . . . R g ( φ ) / σ m φ A dd i t i v e M i x t u r e ( a ) ( b ) ( c ) . . . . . g S C S C - S C (r) r / σ m φ = . . . . g m o n o S C - S C ( r ) r / m . . . g SSSS - SS (r) r / σ m φ = . . . . g m o n o SS - SS ( r ) r / m . .
52 12345678910 g (r) r / σ m g m - SS m i x ( φ = . ) g m - S C m i x ( φ = . ) . . . . g (r) r / σ m φ = . g m i x SS - SS (r) g m i x S C - S C (r) g m i x SS - S C (r) ( i ) ( ii )( iii ) FIG. 7: (a) A schematic of the expected changes in the size of biopolymers in the milieu of
E. coli -like polydispersecrowding environment with φ = 0 . L/l p is varied. Depending on the parameter value γ = L/l p , which characterizesthe chain length and stiffness, the polymer undergoes swelling ( γ ≥
10) or coil-to-globule transition ( γ (cid:29) Azoarcus ribozyme in 0.56 mM-Mg ion solution (blue circles). For comparison we show insimulation (black circles) results for R g ( φ ) changes in monodisperse SS crowding from 2a. E WCA = (cid:88) i,j
We modelthe spherocylindrical crowder by connecting fivespherical crowders allowing for overlap (Figure 1a).Five beads in the anisotropic crowders are connectedusing E S and E B in Eqs. (3) and (4) with verylarge values for the spring and the bending rigidityconstants ( K sp and G sp , the analogues of K and G in3 and 4) in order to maintain the cylindrical shape.We ignore excluded volume interaction between the beads within a particular cylindrical crowder,because the parameter G sp is sufficiently large. Bychoosing the diameter of the cylinder σ cyl = 2 . σ m ,the volumes of SS and SC crowders are identical( πσ sph = πσ cyl + πσ cyl × σ cyl ) (Figure 1a). Theparameter values are given in Table S1. Mixture of SS and SC.
To examine if the crowdersof different shape have additive effects on the sizeof the WLC, we also considered a system containingspheres and spherocylinders. We chose equimolarmixture containing N = N = N/ φ = N ( v SS + v SC ) = N v SS with v SS = v SC . Thus, in a mixture φ SS = φ SC = φ/ φ SS and φ SC are the fractions of volumeoccupied by SS and SC, respectively. Modeling the E. coli environment.
Using theapproximate composition of the crowding particles in
E. coli in terms of sizes of crowders [42] we mimicthe cytoplasm as a mixture of spheres containingthe three largest particles. They are the ribosomewith radius r = 10 . r = 5 . r ≈ . Simulation details.
In order to obtain adequatesampling of the conformational space of the system,we used low friction Langevin dynamics (LFLD). Itcan be shown rigorously, and has been confirmed insimulations, that the thermodynamic properties of thesystem do not depend on the choice of the friction coef-ficient [7]. In the LFLD, the diameter of the monomer σ m , τ = ( mσ m /(cid:15) ) / , and (cid:15) were chosen as the unitsof length, time, and energy, respectively. The valueof σ m suitable for DNA is ≈ .
18 nm. Friction coef-ficients, ζ m , for monomers and ζ c for crowders were ζ m = 0 . mτ − and ζ c = ζ m σ sph /σ m [52] (Table S1).The duration of each trajectory ranges from 2 × to5 × ∆ t where ∆ t = 0 . τ .Initially, a semi-flexible chain was placed in a sim-ulation box without the crowders. We performed theLFLD for 10 time steps to equilibrate the system.The crowding particles were added to generate a sam-ple with φ = 0 .
05. Higher volume fractions werereached by inserting additional crowding particles tothe simulation box. Subsequently, Lennard-Jones in-teraction annealing (adiabatic increase of (cid:15) ) was car-ried out for φ ≥ . (cid:15) = 0 . k B T and0 . t for the integration time step of LFLD. First, theLennard-Jones interaction parameter (cid:15) was increasedby 0 . k B T for every 10 time steps until it reaches (cid:15) = 1 . k B T . Next, we increased the simulation timestep by 0 . t for every 10 time steps until the timestep reaches ∆ t .For purposes of efficient computation we devised amethod (see also SI in Ref. [19]) in which crowdingparticles are added on the fly and the volume of thesimulation box is adjusted to keep the volume frac-tion constant. During the simulations we adjusted thesize of the simulation box according to the chain con-formation at a given time to minimize the number ofcrowders. At every time step, we checked if the chainis enclosed in the simulation box. If any monomer andthe boundary of the box is closer less than three timesthe average distance between the crowders, we resizedthe box and added crowding particles to the newly ex-tended empty spaces. As a result of constantly resizingthe box (a cuboid with changing dimensions) the num-ber of crowders varies. The volume of the cuboid andthe number of crowders are varied in such a way that φ is a constant. The average number of crowders inour simulations varies from 4000 to 8000 depending on φ . The particular ensemble used in these simulation isnot used frequently although it is discussed by Callen(see page 148 in [53]). If the system consists of mono-disperse particles (generalization to multi-componentsystem follows readily as explicitly shown in Ref[53])in which the number of particles ( N ) and volume( V ) fluctuate, the independent variables conjugates tothese two variables are chemical potential ( µ ) and pres-sure ( p ). The thermodynamic potential in this ensem-ble is Ω( µ, p, T ) = U − T S + pV − µN = G − µN = 0(follows from Euler relation), which means that vari-ations in N and V do not change the thermodynamicpotential Ω( µ, p, T ). This is precisely what is desiredin these simulations.To ensure that the results do not depend on thechoice of ensemble we also repeated the simulationsin the canonical ensemble for the polydisperse case.As expected on theoretical grounds the results for R g , P ( R g ), and the energy per particle (thermodynamicquantity) in the two ensembles are the same. Thecomparison is given in the Fig.S6.In total, we generated 25 trajectories at eachvolume fraction to obtain statistical properties. Wecollected data for analysis after a minimum of 10 simulation time steps. Acknowledgements.
This work was supported inpart by the National Science Foundation (CHE 13-61946). ∗ [email protected][1] Morelli, M. J, Allen, R. J, & Rein ten Wolde, P. Bio-phys. J. (2011) , 2882–2891.[2] Tabaka, M, Kalwarczyk, T, & Ho(cid:32)lyst, R.
NucleicAcids Resesarch (2014) , 727–738.[3] Ge, X, Luo, D, & Xu, J. PLoS One (2011) , e28707.[4] Zhou, H. X, Rivas, G, & Minton, A. P. Annu. Rev.Biophys. (2008) , 375.[5] Elcock, A. Curr. Opin. Struct. Biol. (2010) , 196–206.[6] Cheung, M. S. Curr. Opin. Struct. Biol. (2013) ,212–217.[7] Denesyuk, N & Thirumalai, D. J. Am. Chem. Soc. (2011) , 11858.[8] Kilburn, D, Roh, J. H, Guo, L, Briber, R. M, & Wood-son, S. A.
J. Am. Chem. Soc. (2010) , 8690–8696.[9] Pincus, D. L, Hyeon, C, & Thirumalai, D.
J. Am.Chem. Soc. (2008) , 7364–7372.[10] Schreiber, G, Haran, G, & Zhou, H.-X.
Chem. Rev. (2009) , 839–860.[11] Lerman, L.
Proc. Natl. Acad. Sci. U.S.A. (1971) ,1886–1890.[12] Dima, R. I & Thirumalai, D. J. Phys. Chem. B (2004) , 6564–6570. [13] Hyeon, C, Dima, R. I, & Thirumalai, D. J. Chem.Phys. (2006) , 194905.[14] Post, C. B & Zimm, B. H.
Biopolymers (1979) ,1487–1501.[15] Grosberg, A. Y, Erukhimovitch, I. Y, & Shakhnovitch,E. Biopolymers (1982) , 2413–2432.[16] Castelnovo, M & Gelbart, W. Macromolecules (2004) , 3510–3517.[17] Ramos, J. ´E. B, de Vries, R, & Ruggiero Neto, J. J.Phys. Chem. B (2005) , 23661–23665.[18] Kudlay, A, Cheung, M. S, & Thirumalai, D.
J. Phys.Chem. B (2012) , 8513–8522.[19] Kang, H, Pincus, P. A, Hyeon, C, & Thirumalai, D.
Phys. Rev. Lett. (2015) , 068303.[20] Kim, J, Jeon, C, Jeong, H, Jung, Y, & Ha, B.-Y.
SoftMatter (2015) , 1877.[21] Shin, J, Cherstvy, A. G, & Metzler, R. Soft matter (2015) , 472–488.[22] Shendruk, T. N, Bertrand, M, de Haan, H. W, Harden,J. L, & Slater, G. W. Biophys. J. (2015) , 810–820.[23] Frisch, H & Fesciyan, S.
J. Polym. Sci.: Polym. Lett.Ed. (1979) , 309–315.[24] Naghizadeh, J & Massih, A. R. Phys. Rev. Lett. (1978) , 1299.[25] D.Thirumalai. Phys. Rev. A. (1988) , 269–276.[26] Vasilevskaya, V. V, Khokhlov, A. R, Matsuzawa, Y,& Yoshikawa, K. J. Chem. Phys. (1995) , 6595.[27] van der Schoot, P.
Macromolecules (1998) , 4635–4638.[28] Krotova, M. K, Vasilevskaya, V. V, Makita, N,Yoshikawa, K, & Khokhlov, A. R. Phys. Rev. Lett. (2010) , 128302.[29] Diamant, H & Andelman, D.
Physical Review E (2000) , 6740.[30] Shaw, M. R & Thirumalai, D. Phys. Rev. A. (1991) , R4797.[31] Hyeon, C & Thirumalai, D. J. Chem. Phys. (2006) , 104905.[32] Vafabakhsh, R & Ha, T.
Science (2012) , 1097–1101.[33] Le, T. T & Kim, H. D.
Nucleic Acids Research (2014) , 10786–10794.[34] Lieberman-Aiden, E, van Berkum, N, Williams, L,Imakaev, M, Ragoczy, T, Telling, A, Amit, I, Lajoie,B, Sabo, P, Dorschner, M, Sandstrom, R, Bernstein,B, Bender, M, Groudine, M, Gnirke, A, Stamatoy-annopoulos, J, Mirny, L, Lander, E, & Dekker, J. Sci-ence (2009) , 289.[35] Parry, B. R, Surovtsev, I. V, Cabeen, M. T, O’Hern,C. S, Dufresne, E. R, & Jacobs-Wagner, C.
Cell (2014) , 183–194.[36] Marko, J. F & Siggia, E. D.
Macromolecules (1995) , 8759–8770.[37] Moghaddam, S, Caliskan, G, Chauhan, S, Hyeon, C,Briber, R, Thirumalai, D, & Woodson, S. J. Mol. Biol. (2009) , 753–764.[38] Asakura, S & Oosawa, F.
J. Polym. Sci. (1958) ,183–192.[39] Ogston, A. G. J. Phys. Chem. (1970) , 668–669.[40] Vliegenthart, G & Lekkerkerker, H. J. Chem. Phys. (1999) , 4153–4157. [41] Oversteegen, S, Wijnhoven, J, Vonk, C, & Lekkerk-erker, H.
J. Phys. Chem. B (2004) , 18158–18163.[42] Roberts, E, Magis, A, Ortiz, J. O, Baumeister, W,& Luthey-Schulten, Z.
PLoS Comp. Biol. (2011) ,e1002010.[43] Williamson, J. J & Evans, R. M. L. Soft Matter (2013) , 3600–3612.[44] de Gennes, P. G. (1979) Scaling Concepts in PolymerPhysics . (Cornell University Press, Ithaca and Lon-don).[45] Brochard, F & De Gennes, P.
Ferroelectrics (1980) , 33–47.[46] Xia, Z, Das, P, Shakhnovich, E. I, & Zhou, R. J. Am.Chem. Soc. (2012) , 18266–18274.[47] Caliskan, G, Hyeon, C, Perez-Salas, U, Briber, R. M,Woodson, S. A, & Thirumalai, D.
Phys. Rev. Lett. (2005) , 268303.[48] Chen, Y, Abrams, E. S, Boles, T. C, Pedersen, J. N,Flyvbjerg, H, Austin, R. H, & Sturm, J. C. Phys. Rev.Lett. (2015) , 198303.[49] Guzman, I & Gruebele, M.
J. Phys. Chem. B (2014) , 8459–8470.[50] Politou, A & Temussi, P. A.
Curr. Opin. Struct. Biol. (2015) , 1–6.[51] Frederick, K. B, Sept, D, & Enrique, M. J. Mol. Biol. (2008) , 540–550.[52] Veitshans, T, Klimov, D, & Thirumalai, D.
FoldingDes. (1997) , 1–22.[53] Callen, H. B. (1985) Thermodynamics and an Intro-duction to Thermostatistics . (Wiley).[54] Thirumalai, D & Mountain, R.
Phys. Rev. A. (1990) , 4574–4578.[55] Hyeon, C, Lee, J, Yoon, J, Hohng, S, & Thirumalai,D. Nat. Chem. (2012) , 907–914.[56] Kang, H, Kirkpatrick, T. R, & Thirumalai, D. Phys.Rev. E (2013) , 042308. SUPPORTING INFORMATION
Multiple layered neighbor list.
The large sys-tem size needed to reliably simulate the WLC chainin the presence of explicit crowders is computationallydemanding. To To circumvent the system size prob-lem, we develop a computer code that minimizes thenumber of operations to compute interaction poten-tial. In order to acheive this goal, we devised and im-plemented the multiple layered neighbor list (MLNL)technique. To our knowledge, this methodology hasnot been used in simulations before. In conventionalVerlet algorithm, the list of neighbors, which are theparticles located within a cut-off distance, R c , is cre-ated for each particle at the beginning of the simu-lation. When a pair-interaction is needed, we searchonly the neighbors instead of computing interactionsbetween all pairs of particles. Thus, the computa-tional cost decreases as R c gets smaller. However,since the positions of the particles are constantly evolv-ing, we have to update the neighbor list with a cer-tain frequency. In conventional Verlet list, all neigh-bor lists are updated whenever the maximum displace-ment of any particle exceeds R c . The frequency of up-dates increases as R c becomes smaller, thus increas-ing the computational costs for updating the neighborlist. The two competing demands (frequent update forsmall R c and infrequent update for computations in-volving larger number of interaction pairs) requires anoptimal value of R c .MLNL is designed to reduce the computational costsfor calculating interaction potentials and updatingneighbor lists by using multiple numbers of neighborlists. It consists of several neighbor lists each witha different cut-off distances, R (1) c < R (2) c · · · < R ( n ) c .Interaction potentials are only calculated using theupper-most layer, which has the smallest R c = R (1) c .This strategy minimizes the cost of calculating inter-action potentials. When the maximum displacementof a particle exceeds R (1) c , instead of calculatingthe distance between all pairs of particles as in theconventional algorithms, we update the upper-mostneighbor list using the neighbor list with R (2) c . Thus,the requirement of computing O (cid:0) N (cid:1) interactionsis avoided to a large extent by using this technique.The disadvantage of the MLNL is that memoryrequirement can be quite large especially when thesystem size is large. For N = 300, three layers suffice.With this choice we were able to perform convergedsimulations. Shape anisotropy of polymer.
Asphericity (∆)and shape parameters ( S ) are used to characterize the ∆ S φ ∆ (SS) ∆ (SC)S (SS)S (SC) FIG. S1: Asphericity (∆) and shape parameter ( S ) ofWLC as a function of SS and SC volume fraction, φ . shape anisotropy of polymer. The asphericity param-eter (∆) and a shape parameter ( S ). Both ∆ and S ,which are rotationally invariant, are defined using theinertia tensor, T αβ = 12 N N (cid:88) i,j ( r iα − r jα ) ( r iβ − r jβ ) (S1)where r iα is the α (= x, y, z ) component of bead i ofthe WLC chain. The eigenvalues of T αβ are related to R g via R g = Tr( T ) = (cid:80) j λ j . The anisophicity (∆)and shape ( S ) parameters are∆ = 32 (cid:88) i =1 (cid:0) λ i − ¯ λ (cid:1) (Tr( T )) , S = 27 (cid:81) i =1 (cid:0) λ i − ¯ λ (cid:1) (Tr( T )) (S2)where ¯ λ = Tr( T )3 . For a globule, ∆ = S = 0. Thus,we expect that if crowders induce compaction then ∆and S should decrease monotonically as φ increases.Compaction in R g for φ ≤ . S ) as a function of crowdercharacteristics (Fig.S1). For spherical crowders, both∆( φ ) and S ( φ ) decrease with increasing φ , implyingthat the WLC becomes more spherical with increasingcompaction. In contrast, there is a sharp increasein ∆( φ ) and S ( φ ) as the volume fraction of the SCcrowders increases beyond φ ≈ . . ≈ ∆(0) and S (0 . ≈ S (0), whichshows that at both φ = 0 and high φ the WLC is ananisotropic ellipsoid ( S > R g ( φ ) when φ > . Analytical expression for the end-to-end dis-tance distribution of WLC.
The end-to-end dis-tance distance distribution of WLC, obtained in Ref.2 σ m σ c K G K sp G sp l θ k B T ∆ t ζ m ζ c σ m (cid:15) . (cid:15) (cid:15) (cid:15) σ m (cid:15) τ . mτ − ζ m (cid:16) σ c σ m (cid:17) TABLE S1: Parameters characterizing the model. Lennard Jones energy constant (cid:15) , the diameter of monomer σ m and τ = (cid:113) mσ m (cid:15) are used as the fundamental units for energy, length and time scales. K and G define the strength of bondand angle potentials (Eqs. 2 and 3); K sp and G sp are the corresponding parameters for SC crowders. l is a bond lengthbetween monomers of a chain, k B T is a temperature, ∆ t is a simulation time step, ζ m and ζ c are the friction coefficientsfor monomer and crowders, respectively. d ( t ) / d ( ) t (10 ∆ t) d ( t ) / d ( ) t (10 ∆ t) F q ( t ) t (10 ∆ t) a b d ( ) / d ( t ) FIG. S2: a. The time-dependence of the intermediatedynamic scattering function (Eq.S4). b. Energy metric(Eq.S5) as a function of t . We calculated these quantities toensure that F q ( t ) → d ( t ) /d (0) → d (0) /d ( t ) ∼ t in the inset). Because the averages are computed using t > ∆ t , whereas measure of ergodicity decay on t ≈ ∆ t ,the chain is fully equilibrated at φ = 0 . [31], P ( r ) = 4 π N r (1 − r ) / exp (cid:20) − t (1 − r ) (cid:21) (S3)with r = R ee /L , t = L/l p , N = π − / c / e c c − +15 c − and c = t where L = ( N m − σ m is the contour lengthwas used to obtain numerical values of the persistencelength, l p . Pair correlation function of polydispersedcrowders.
Homogeneously distributed, the averageseparations ( D ) between crowders in cell lysate wouldbe D X /σ X = (4 π/ / (1 / φ − / =1.68, 1.68, 1.87for X = 1, 2, and 3, respectively [19]. In contrastto this expectation, the pair correlation between thepolydisperse crowders of each size indicates local sizeordering, a major population at r/σ X ≈
1, whichis more pronounced than in the monodisperse case(Fig.S4).
Evidences of equilibration.
Because the crowd- ing particle sizes are large care must be taken to ensurethat the system is well equilibrated. To provide ex-plicit evidence that our computational results ( φ = 0 . q = | (cid:126)q | = 2 π/r s where r s is the position ofthe peak in the total pair distribution function, tells ushow our system relaxes from an initial configuration: F (cid:126)q ( t ) = 1 N N (cid:88) k =1 e i(cid:126)q · ( (cid:126)r k ( t ) − (cid:126)r k (0)) , (S4)where (cid:126)r k ( t ) is the position of k -th monomer of oursemiflexible chain at time t . As shown in Fig.S2a, F (cid:126)q ( t ) → d ( t ) = 1 N N (cid:88) i =1 ( E α,i ( t ) − E β,i ( t )) (S5)where E α,i ( t ) = t (cid:82) t E α,i ( τ ) dτ is the energy of particle i averaged over time t from the trajectory generatedfrom two different initial condition α and β , showsthat d ( t ) /d (0) → d (0) /d ( t ) ∼ t , a hallmark of ergodicity, is explicitlyshown (Fig.S2b inset). The results show the stiff chainin the polydisperse crowding environment ergodicallyexplore the conformational space, thus ensuring thatthe results in Figs.5 and 6 represent converged results.3 P [ R g ( φ ) / R g ( ) ] R g ( φ )/R g (0) φ =0.00.3 FIG. S3: Distributions of gyration radius of SAW chainin the absence of crowders ( φ = 0 .
0) and polydispersedcrowder environment ( φ = 0 .
3) mimicking the cytoplasmiccondition of
E. Coli . Note that unlike WLC in the maintext (Figure 5), SAW chain is compacted in the polydis-persed crowding environment. g x - x (r) r/2 σ x σ /2=10.4nm σ /2= 5.2nm σ /2= 2.6nm σ /2=10.4nm (mono) r/ X g m i x X X ( r ) FIG. S4: Crowder-crowder pair correlation functions be-tween crowders with the same kind in the polydispersecrowding solution. R g ( φ ) / R g ( ) φ N= 3364158300 = 2 = 4 = 10 = 20 FIG. S5: Effect of polydisperse crowding environment onthe size of polymers with different γ = L/l p P [ R g ( φ ) / R g ( ) ] R g ( φ )/R g (0) a b largest mid-sized smallest E ne r g y pe r pa r t i c l e ( k T ) µ pTNVT FIG. S6: Comparison of simulation results of WLC underpolydisperse mixture environment in µ PT and NVT en-sembles. a. R g -distributions of WLC in NVT (circles) and µ pT ensembles (red line). b.b.