raindrop coalescence of polymer chains during coil globule...

“Raindrop” Coalescence of Polymer Chains during Coil−GlobuleTransitionAnna Lappala and Eugene M. Terentjev*

Cavendish Laboratory, University of Cambridge, JJ Thomson Avenue, Cambridge CB3 0HE, U.K.

ABSTRACT: We approach the problem of coil−globuletransition dynamics numerically by Brownian dynamicssimulations. This method allows us to study the behavior ofpolymer chains of varying stiffness and the effects of bendingstiffness on chain morphology during the process of coil−globule collapse, imitating globule formation in poor solventconditions. We record and analyze a three-stage process ofglobule formation for flexible chains: (1) nucleation, (2)coalescence of nuclei, and (3) collapsed globule formation.Stiffer chains undergo similar formation stages; however, the“raindrops” formed by these chains are elongated (unlikespherical structures formed by flexible chains) and exhibitregular packing of chains into antiparallel hairpin structures. Inorder to assess the transition dynamics quantitatively, polymer chain configurations were analyzed by generating contact mapsand contact frequency histograms for all given configurations. These clusters are initial-configuration-dependent, and theirgrowth and intercluster contacts have direct analogy with the process of raindrop coalescence.

■ INTRODUCTIONThe problem of polymer chain collapse in poor solvents hasreceived much attention over the past decades, from bothanalytic1−6 and numerical approaches.7,8 This collapse, oftencalled the “coil−globule transition”, is well-understood as alocal version of the polymer demixing Flory has originallyformulated for polymer solutions.9 As the second virialcoefficient, representing nonlocal pair interaction betweenchain segments, becomes sufficiently negative (reflecting aneffective attractive pair potential), the chain suddenly shrinks insize and collapses into a condensed globular form with a localstructure more like that of a melt than a freely solvatedpolymer. A generally positive (repulsive) third virial coefficientbecomes important in the condensed globule and prevents thechain from nominally collapsing to a point. The coil−globuletransition is closely associated with the initial stages of proteinfolding,10 particularly secondary structure formation,11 andDNA condensation.12,13 Collapse can occur as a result ofeffective intrachain interactions, e.g., due to changing solventquality, but also as a consequence of interactions with a randomexternal potential which localizes the chain within a givenregion.14−17 In this work, we analyze the first classical scenariowhere collapse takes place due to long-range attractiveinteraction between monomers.Langevin and phenomenological models are the main two

theoretical directions that have been used to study the kineticsaspect of the polymer collapse. Mostly stemming from the earlywork of de Gennes,18 numerous phenomenological kineticmodels produce scaling laws for several regime characteristicsof collapse. An alternative theoretical model to the oneproposed by de Gennes was developed by Halperin and

Goldbart;19 it differs in their interpretation of collapse: the deGennes model is known as the “expanding sausage model” inwhich the chain forms an elongated structure that becomesthicker upon collapse, finally condensing into a globular state.The Halperin−Goldbart model is known as the “pearl-necklacemodel”, reflecting the morphological features of the tran-sitionsmall nuclei forming globules (pearls) that arerandomly formed along the chain (necklace)and thesecollapse by collision analogous to droplet coalescence in fluiddynamics. In contrast, Klushin20 did not view the problem asbased on fluid dynamics and nucleation as Halperin andGoldbart describe it later, but from the viewpoint of clustergrowth and aggregation where clusters are connected bystrands and the aggregation is driven by the entropic tension inthe strands. One of the main assumptions that differ in the twointerpretations is that a globule subject to end stretching isnonuniformly deformed but partially unfolds through differentpossible mechanisms segregating into two strands and aspherical core. These models propose varying scaling exponentsfor collapse time τc depending on the degree of polymerizationN: The de Gennes model predicts a single scaling exponent τc∼ N2 (ref 18); the Halperin−Goldbart model describes a three-stage nucleation and coalescence process with τ(1)c ∼ N0, τ(2)c∼ N1/5, and τ(3)c ∼ N6/5 (ref 19), while according to Klushin,20

τc ∼ N1.6. Various inconsistencies also exist between themodels, such as whether collapse is a three- or a four-stageprocess. Our simulations appear to confirm the collapse process

Received: November 17, 2012Revised: December 28, 2012Published: January 18, 2013

Article

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© 2013 American Chemical Society 1239 dx.doi.org/10.1021/ma302364f | Macromolecules 2013, 46, 1239−1247

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as described by Halperin and Goldbart:19 (1) pearling, (2)bridge stretching, (3) collapse of the pearl necklace, and (4)equilibrium collapsed configuration in a poor solvent (Figure1). It is important to mention that there is a possibleterminological confusion between theoretical models andnumericsin most numerical simulations, the “pearl necklace”term is used to describe a structure with globules formed byelectrostatic contacts,21 whereas a pearl necklace in Halperin−Goldbart terminology19 is one of the stages of polymer collapsethat we confirm in simulations; we therefore use theterminology of fluid dynamics, “raindrop formation”, that atleast qualitatively appears to describe the physics of theprocesses that we observe in our work. Despite the fact thatequilibrium polymer configurations in good and poor solventsare well understood, important factors that play a critical role inthe transition process such as chain length, flexibility of thechain, and viscosity of the solvent have not been wellcharacterized, and we try to address these points here.According to Halperin and Goldbart terminology, the

configurations of a collapsed chain are characterized by acorrelation length ξc, which is formulated as gc

1/2a ∼ ξc, wheregc is the number of monomers in a blob or a “pearl” and a is thesize of a monomer. The collapsed blobs (pearls), of size ξc,attract each other and aggregate densely, so the radius of thefinal collapsed state is R ≈ (N/gc)

1/3ξc (see Figure 8 below). Inthe early “sausage” model of de Gennes18 the initial state is alsorepresented as a string of blobs which upon coil−globuletransition experience attractive forces, hence minimizing thesurface energy between an inner domain and solvent. However,in that model, the shape of the intermediate state is believed tobe approximated better by a cylinder of length L and a radius ofr. Upon compaction, the “sausage” becomes thicker, finallycollapsing into a spherical lowest energy state.18 Our simulationfor a flexible chain confirms the Halperin−Goldbart morphol-ogy.Earlier numerical work on polymer collapse can be broadly

divided into three approaches: Metropolis Monte Carlosimulations, molecular/Brownian dynamics, and mesoscalesimulation techniques. Monte Carlo simulations22 produce

lowest energy state conformations without the time-dependentdynamics that can be observed in other algorithms, which is as adrawback if one needs to study the evolution of the processwith time. Molecular and Brownian dynamics techniques arefollowing the motion of particles: molecular dynamicsalgorithms integrate Newton’s equations of motion, whereasin Brownian dynamics one uses the Langevin equation todescribe the effects of thermal noise and Brownian motion.Mesoscale simulation techniques are useful to study hydro-dynamic flow phenomena, and there are variations insimulations of this type: (1) dissipative particle dynamics, (2)multiparticle collision dynamics, and (3) the lattice Boltzmannapproach.23 These are very efficient considering the time andlength scales greatly surpass those that can be studied inmolecular dynamics. However, a drawback of this approach isthat the concept of a “particle” in simulations of this typerepresents a whole molecule or fluid regions, and hence theatomistic details are omitted. Mesoscale techniques are flexibleand can therefore be coupled to atomistic algorithms such asmolecular dynamics, which explains the increasing popularity ofthe approach in the field of numerical analysis. One of theeminent works in the hybrid simulation approach wasperformed by Kikuchi et al.24 where it was used to investigatepolymer collapse and the role of hydrodynamics on collapsekinetics. In that simulation, solvent was modeled by a mesoscalealgorithm known as stochastic rotation dynamics and thepolymer by molecular dynamics: this separation of algorithmsallows one to overcome the limitation of computationallyintensive hydrodynamic time scales. Kikuchi et al. found thathydrodynamic interactions accelerate polymer collapse andappear to change the dynamics of collapse. However, thesimulation results are most often presented in terms of timesteps, whereas the conversion of these into real time may bequite different in cases of Brownian vs hybrid simulations. Infact, whether the collapse pathway is altered by hydrodynamicsis still debated.25 According to Cieplak et al.,26 hydrodynamicinteractions are found to affect time characteristics in thevicinity of the native state but have no impact on thesecharacteristics of structural fluctuations around the native state.

Figure 1. The summary of stages of chain coalescence into a globular state, via a sequence of steps resembling the nucleation and growth of liquidout of a gas. Parts A−D are sketches from ref 19, while the series E−G are the snapshots from our simulation at different times of collapse.

Macromolecules Article

dx.doi.org/10.1021/ma302364f | Macromolecules 2013, 46, 1239−12471240

Early models gave polymer research a new perspective;however, they also led to numerous diverging interpretations.One of the reasons for misleading results was the fact that thelength of the studied chains, N, was often not sufficient enoughto produce a distinct dynamics behavior. For example, theconclusion of ref 24 that the collapse starts from chain endsmay simply be a result of N < gc (in terminology of Halperin−Goldbart). Another debated argument in the polymersimulations literature is the representation of the solvent.Some argue that implicit and explicit solvent representationsproduce different results;24,27 this might however be an artifactthat again arises from an insufficient length of a simulatedchain, when solvent molecules can indeed be expected to play asignificant role in chain morphology.In this work, we closely follow the setup of Rosa and

Everaers28 and perform numerical simulations using the LargeScale Atomic/Molecular Massively Parallel Simulator(LAMMPS)29 to reproduce the dynamics of a polymer chainover time (Figure 1). For simplicity, our coarse-grainedsimulations do not include hydrodynamics. In order to obtaina comprehensive interpretation of polymer collapse dynamics,we designed further methods to analyze the collapse-associatedprocesses at a more detailed quantitative level. These include anextensive use of contact and contact-frequency maps,representing graphically the chain segments (monomers, orresidues) that are judged to be in direct contact in a foldingchain, and the analysis of time evolution of the dense“raindrops” (collapsed blobs separated by regions of a free-fluctuating chain). We performed simulations for a number ofdifferent initial configurations of expanded-coil chain in orderto avoid confusion of initial-configuration-dependent processeswith the real universal dynamics of the polymer. Further, wecompare the effects of bending stiffness on the morphology of acollapsing chain and find that above a certain stiffness thesemiflexible chain folds into “nematic droplets” with straightsegments and sharp hairpins.30 We also demonstrate that thedynamics depends on the viscosity of the solventfor lowerviscosities, the distinct morphological behavior associated withchain stiffness appear to remain intact; however, the process ofglobule formation takes place on a shorter time scale.

■ SIMULATION OF POLYMER CHAINSThe model used for numerical simulations in this work is basedon the bead−spring model described in molecular-dynamiccontext by Kremer and Grest.31 The system was simulated in astandard way, with a fixed number of particles at a fixed volumein a cubic cell with periodic boundary conditions, which allowsparticles to exit and re-enter the simulation box. A polymerchain is composed of connected monomeric units consisting ofN = 2000 monomersa chain length sufficient to demonstratethe stiffness-dependent dynamics of individual polymer chainsduring coil−globule transition. Each monomer has a diameterof σ, and interactions between monomers can be described bythe following potentials:(1) Finitely extensible nonlinear elastic (FENE) potential for

connected monomers (residues):

=− − ≤

>

⎧⎨⎪⎩⎪

UkR r R r R

r R

12

ln[1 ( / ) ],

0,FENE

02

02

0

0 (1)

The FENE potential is harmonic near its minimum, so that theeffective spring constant between monomers is k. However, a

FENE bond cannot be stretched beyond a maximum lengthdetermined by R0 (see below for the discussion of numericalvalues choice).(2) The Lennard-Jones potential for nonconsecutive pairs of

monomers: ULJ = 4ε[(σ/r)12 − (σ/r)6], where σ is the distanceat which the interparticle potential is zero, thus separating therepulsive and attractive regions, and ε the potential well depth.This potential reaches its minimum ULJ = −ε at r = 21/6σ. Fornumerical simulations, it is common to use a shifted andtruncated form of this potential which is set to zero past acertain separation cutoff. For a self-avoiding random-walk chain(i.e., swollen in a good solvent), the potential is set to zero atrcutoff = 21/6σ, representing only the soft repulsion betweenmonomers close-up:

ε σ σ σ

σ

=− + ≤

>

⎧⎨⎪

⎩⎪

⎡⎣⎢

⎤⎦⎥U

r r r

r

4 ( / ) ( / )14

, 2

0, 2LJ

12 6 1/6

1/6(2)

In order to simulate poor solvent conditions when there is aneffective attraction between monomers, the cutoff value was setto for rcutoff = 3σ (Figure 2). In each case the potential is alsoshifted up or down to ensure the continuity at the cutoff radius.

(3) Bending elasticity of semiflexible chain is described by apotential Ustiff = Kθ(1 − cos θ), where θ is the angle formedbetween two consecutive bonds and Kθ is the correspondingbending modulus.For simulated chains for N = 2000 monomers, parameters

used for simulations are as described in ref 32 with themonomer diameter σ chosen to be 0.3 nm (which is a typicalsize of an aminoacid residue in proteins). Parameters for k andR0 were set so as to avoid any bond crossing. For oursimulations, the values were used as in ref 31; the maximumbond length R0 = 1.5σ, and the spring constant k = 30ε/σ2. Thespring constant was studied by Kremer and Grest31 and wasfound to be strong enough such that the maximum extension ofthe bond was always less than 1.2σ for kBT = 1.0ε, making thebond breaking energetically unfeasible. The important ratio Kθ/kBT, which is the measure of the chain persistence length, lp =σ(Kθ/kBT), was set to vary from lp = σ (or Kuhn’s length of 0.6nm) to lp = 10σ (or Kuhn’s length of 6 nm) depending on theflexibility of the simulated chain. Integration time step waschosen to be tint = 0.012τ where τ is the reduced (Lennard-Jones) time, defined as τ = σ(m/ε)0.5, as discussed in refs 28and 31. Taking the values of σ and ε above, and the typical

Figure 2. Graphical representation of the shifted and truncatedLennard-Jones potential for good solvent conditions (cutoff r = 1.12σ,indicated by an arrow), so the effective potential between monomers ispurely repulsive, and the poor solvent conditions (cutoff r = 3σ, alsoindicated by an arrow), so that there is an attractive potential well ofapproximately kBT deep.



mass of an aminoacid residue (molecular weight ≈100), weobtain the estimate of τ ≈ 0.03 s. Hence, with 106 time steps ofthis coarse-grained simulation we are able to follow thedynamics of a polymer chain for 360 s.In order to study the coil−globule transition, the pair-

interaction (Lennard-Jones) potential is altered as shown inFigure 2: at t = 0 the “good-solvent” repulsive potential withrcutoff = 21/6σ is switched to the attractive potential with rcutoff =3σ, which represents an instantaneous quenching of the system.We then simulate the process of collapse over time, asdiscussed, and observe the characteristic “pearling” for bothflexible and semiflexible chains (Figure 3).

■ ANALYSIS OF CONFORMATIONS BY CONTACTMAPS

In order to analyze the conformational dynamics of a chain, oneneeds a method that would let us follow the monomer contactevolution over time. This can be achieved by generating so-called contact maps, which are essentially a representation of the3D connectivity in a two-dimensional map. This method hasbeen used to represent proteins; however, this type of analysishas not been performed on idealized chains, the conformationsof which exist as a result of Brownian dynamics. The algorithmfor generating a contact map for a given set of coordinates of allchain units is simple: for each consecutive monomer we list all

other monomers that are within the chosen contact radius.Obviously, the choice of this radius is somewhat arbitrary, andwe were guided simply by the visual clarity of features on themaps: we regarded a monomer mi and a monomer mj in contactwhen they are found within the radius of 2σ. In this way there isalways the trivial contact on, or near, the main diagonal, whichrepresents mi = mi and the chain connectivity mi ÷ mi±1. Theof f-diagonal contacts, by symmetry always of the square shapesymmetric about the main diagonal, represent the essentialfeatures of the collapsing structure.

Flexible Chain (lp = σ). As discussed earlier, oursimulations indeed confirm the phenomenological modelpresented by Klushin,20 as well as Halperin and Goldbart,19

and are in good agreement with earlier simulation work.24

Using our contact map approach, we can further analyze thestructures formed within the globule, which, to our knowledge,has not been done before. We demonstrate that for flexiblechains first small clusters are formed (Figure 5), and there is noregular packing within a cluster (or a “pearl” in Halperin−Goldbart terminology); these clusters evolve and merge eitherby coalescence and by various other mechanisms described indetail by Klushin,20 such as one cluster pulling another clusterto unwind and therefore grow. Interestingly, the early processesof coil−globule transition indeed are more likely todemonstrate the process of coalescence, whereas other

Figure 3. Flexible chain (A) showing the characteristic “pearling” stage of the collapse process and a semiflexible chain (B) showing characteristicelongated structures formed along the chain.

Figure 4. Contact map representation of a flexible chain (A) and a semiflexible chain (B): both exhibit cluster formation; however, the semiflexiblechain appears to have a more ordered internal structure of each cluster.



mechanisms (pulling/unwinding) are more likely to take placein the final stages of collapse.Semiflexible Chain (lp = 10σ). Perhaps unsurprisingly, the

dynamic behavior of collapsing semiflexible chains is differentfrom that of flexible chains (Figures 4, 6, and 7). The chaininitially becomes elongated upon compaction, and the pearl-necklace structure is not present as we observe it in flexiblechains, even though some characteristics of that behavior arestill present, such as cluster formation and merging of clusters.However, simply observing a cylindrically shaped cluster doesnot reflect how intrachain contacts within the cluster are

formed. Using the contact map analysis, we identify a veryregular pattern that resembles a diamond shape. Furtheranalysis revealed that each “diamond” represents an antiparallelhairpin structure,30 and these are regularly packed into higher-order structures where multiple antiparallel hairpins mergetogether in both parallel and antiparallel fashion, depending onthe geometric constraints of the complex structure. It has to benoted that there exists a variation in hairpin size as well as in thehairpin density. In order to obtain a clearer representation ofthe coil−globule transition in this case, we performed analysisof each individual bundle of hairpins (which can be identified as

Figure 5. Series of snapshots for a flexible chain and corresponding representations of the structures in the contact map space. The time scale t hereis represented in units of 1000 simulation time steps.



squares that consist of multiple diamond-like shapes) withrespect to time. It is apparent from analysis of this type that acritical bundle size has to be reached before (1) nucleationtakes place and (2) two hairpin clusters merge into one largerstructure, which will be discussed in the next section.

■ BOX-PLOT ANALYSIS

In order to quantify the evolution of contact maps and detectthe emergence of clusters, as well as present the process ofcluster coalescence, one can perform the following analysis. In a

contact map, as described in earlier sections, we can identify asquare-shape region that is present around the main diagonal ofa given map. Every square that emerges from a small group ofcontacts known as nucleation points gives us the informationabout how many monomers are present in a given cluster. Theevolution of size s of these clusters (“raindrops”) is shown inthe box-plot diagrams in Figure 8, and the size of each raindropis related as R(s) ≈ s1/3σ. These raindrops, once formed, willgrow and merge with other contact-map squares, producinglarger clusters of monomers, until the single collapsed globularstate is reached, and the square occupies the whole map (s =

Figure 6. Series of snapshots for a semiflexible chain and corresponding representations of the structures in the contact map space. The time scale there is represented in units of 1000 simulation time steps.



N). These squares emerge at different times, and each remainsrelatively constant in size until coalescence with another. Weobserve that in order to form a bigger cluster and for it tomerge with another group of monomers, one requires the initialcluster size of at least ∼100 monomers for flexible chains(Figure 8A) and closer to 200 for stiffer chains (Figure 8B). Inthis representation, we observe that the critical nucleationoccurs when Rc ≈ 3−4σ for flexible chains and Rc ≈ 6σ for stiffchains. We also find that cluster formation is initial-configuration-dependent. This was confirmed by studying theevolution of five different initial configurations for each chainquenching episode; in all of these, we observed that theaccidental initial off-diagonal contacts gave rise to a square inthe region where these contacts existed initially (as one expectsin nucleation). From our results, we also find that for flexiblechains (lp = σ) the number of clusters in the initial stage of theevolution process is higher, and these are also smaller in sizecompared to the semiflexible chain (lp = 10σ), while the latestage is longer and more pronounced for a flexible chain, withjust two big droplets persisting for a long time (see Figure 1Gor 5 for t = 300).

■ DYNAMICS OF COIL−GLOBULE TRANSITIONIn order to understand the impact of solvent viscosity on theprocess of coil−globule transition, we performed simulations ofpolymer chains under different conditions ranging from low tohigh viscosities. This issue was originally raised in the work ofAnsari and Kuznetsov,33 who experimentally studied the

process of hairpin formation in single-stranded nucleotidechains and demonstrated the viscosity dependence of theprocess of hairpin formation; they conclude that diffusion of thechain through the solvent is involved in the rate-limiting step ofhairpin formation. For different viscosities, we studied the timeof collapse that is defined by the time at which the finalelongated bridge between two globules is no longer detected oncorresponding contact maps. On the box plot of Figure 8 thistime is clearly visible as the moment when the cluster sizejumps to s = N. We confirm the observations of Ansari andKuznetsov (Figure 9) and therefore extend this idea to thefundamental concept of coil−globule transition for all chains.

Figure 7. A closer look at the ordered structures present in the contact maps for semiflexible chains: (A) a series of diamond-like structures each ofwhich represents an antiparallel hairpin as shown in the inset of (B).

Figure 8. An illustration of box-plot analysis for a flexible (A) and a semiflexible (B) chain. The size of the square side s on the contact map is animportant parameter equivalent to gc, the number of monomers in a blob, in Halperin−Goldbart terminology. Box-plots demonstrate the evolutionof cluster domains with time. Each line represents a separate cluster, which continues until it merges with another.

Figure 9. Power-law dependence of the collapse time on the viscosityof the solvent for a flexible chain. The solid line represents a power lawof −1/2 (i.e., the time is proportional to the square root of viscosity).



Interestingly, our simulations demonstrate that this dependencecan be described by a power law: the time of chain collapse isproportional to the square root of viscosity.

■ BIOLOGICAL PERSPECTIVE

In this work, we specifically examined hairpin formation insemiflexible polymers; however, in order to demonstrate thebiological importance of this phenomenon, we also introducethe concept of β-hairpin formation in proteins (Figure 10).Hairpin formation is a topic that has received attention fromthe field of biophysics due to its direct connection to theproblem of protein folding. Let us consider the secondarystructure features of a generic protein. These can generally begrouped into three categories: (1) β-sheet, (2) α-helices, and(3) unstructured amorphous regions. Each these has their ownrole in the dynamics and the functionality of a protein, by eitherstabilizing it or making the structure more flexible (unstruc-tured regions can bring residues that are far apart along thelinear chain closer, which will allow the protein to perform aspecific function).For most proteins, the structures of which have been solved

experimentally, atomic coordinate data have been madepublicly available. In order to show the biological relevance ofthe contact maps created from simulations in this work, weanalyze a widely used, mainly β-sheet protein: the greenfluorescent protein (PDB ID 1RXX). In its globular state it isfound to form a β-barrel structure that consists of numerous β-hairpins.34 As with antiparallel hairpin structures described forthe simulated chain (Figure 7), these exhibit similar character-istic diamond-shaped form, where the line that crosses the maindiagonal represents one of the hairpins.It is likely that the β-sheet structure, due to stable hydrogen

bonding, is less flexible than the α-helical secondary structure,and it could be speculated that β-hairpin formation is aproperty that follows from this increased stiffness of the chain.From the static representation of the 3-dimensional proteinstructure, it is not possible to study hairpin formation overtime. However, it has been observed experimentally that β-hairpin formation depends on the viscosity of the solventtheless viscous the solvent, the faster the formation takes place.Our simulations indeed demonstrate this dynamical property.Viscosity gradients vary within the cell, and hence this property

might be of biological importance to polymer-associatedprocesses such as DNA folding in the nucleus and proteinfolding dynamics.

■ DISCUSSION

We have demonstrated that our Brownian dynamicssimulations of polymer chains are in agreement withphenomenological models, other existing simulation algorithmssuch as Monte Carlo and hybrid mesoscale/moleculardynamics methods, and the experimental results available tous. We perform quantitative data analysis of chain config-urations by generating dynamic contact maps and study theevolution of cluster sizes with the box-plot analysis to identifythe time- and globule-size-dependent properties of the coil−globule transition. This analysis allows us to study the effect ofchain bending stiffness on its morphological propertiessemiflexible polymer chains undergo the transition processthrough a number of antiparallel hairpin structures that mergeinto elongated cylinder-like structures with parallel (nematic-like) folding, which later collapse into a highly anisotropicglobular ground state. We also demonstrate a biologicalconnection by highlighting similar properties of contact mapsfor model semiflexible polymers and for a real β-hairpin protein,which suggests that hairpin formation in proteins is mainly dueto the higher stiffness that arises from β-sheet hydrogenbonding properties in polypeptide chains and not necessarilythe chemical details of individual amino acids that form thehairpin.It is important to note that the qualitative nature of our

results is quite universal. This is due to the underlying scaling ofparameters in the simulation, based on the “monomer” size σ.For instance, Rosa and Everaers28 used a value for σ = 30 nm,which corresponds to the diameter of experimentally observedchromatin fiber (DNA−protein complex) that would pack intoa chromosome in the nucleus of a cell. To simulate asemiflexible nature of such a fiber, they used the persistencelength lp = 10σ, which is exactly our semiflexible case. Since allother units are scaled proportionally, for the chromatinsimulation, one also has the Lennard-Jones time τ ≈ 0.02 s,and therefore, our predictions hold quantitatively.

Figure 10. A cartoon representation of a folded 1RRX protein (yellow: β-sheet; green: unstructured regions; magenta: α-helix; blue: π-helix) and acorresponding contact map representation of this protein for residue contacts within 15 Å.



■ AUTHOR INFORMATIONCorresponding Author*E-mail [email protected] authors declare no competing financial interest.

■ ACKNOWLEDGMENTSThis work was funded by the Osk. Huttunen Foundation(Finland) and performed using the Darwin Supercomputer ofthe University of Cambridge High Performance ComputingService, provided by Dell Inc. using Strategic ResearchInfrastructure Funding from the Higher Education FundingCouncil for England.

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