We explore the coupling between the helix-coil and coil-globule transitions of a helical polymer using Monte Carlo simulations. A very rich state diagram is found. Each state is characterized by a specific configuration of the chain which could be a helix, a random coil, an amorphous globule, or one of various other globular states which carry residual helical strands. We study the boundaries between states and provide further insight into the physics of the system with a detailed analysis of the order parameter and other properties

Carri, Gustavo A.

Varshney, Vikas

English

The University of Akron

The University of AkronIdeaExchange@UAkronCollege of Polymer Science and Polymer Engineering10-14-2005Coupling Between Helix-Coil and Coil-GlobuleTransitions in Helical PolymersVikas VarshneyGustavo A. CarriUniversity of Akron Main Campus, gac@uakron.eduPlease take a moment to share how this work helps you through this survey. Your feedback will beimportant as we plan further development of our repository.Follow this and additional works at: http://ideaexchange.uakron.edu/polymer_ideasPart of the Polymer Science CommonsThis Article is brought to you for free and open access by IdeaExchange@UAkron, the institutional repository ofThe University of Akron in Akron, Ohio, USA. It has been accepted for inclusion in College of Polymer Science andPolymer Engineering by an authorized administrator of IdeaExchange@UAkron. For more information, pleasecontact mjon@uakron.edu, uapress@uakron.edu.Recommended CitationVarshney, Vikas and Carri, Gustavo A., "Coupling Between Helix-Coil and Coil-Globule Transitions in HelicalPolymers" (2005). College of Polymer Science and Polymer Engineering. 9.http://ideaexchange.uakron.edu/polymer_ideas/9Coupling between Helix-Coil and Coil-Globule Transitions in Helical PolymersVikas Varshney and Gustavo A. Carri*The Maurice Morton Institute of Polymer Science, The University of Akron, Akron, Ohio 44325-3909, USA(Received 2 November 2004; published 12 October 2005)We explore the coupling between the helix-coil and coil-globule transitions of a helical polymer usingMonte Carlo simulations. A very rich state diagram is found. Each state is characterized by a specificconfiguration of the chain which could be a helix, a random coil, an amorphous globule, or one of variousother globular states which carry residual helical strands. We study the boundaries between states andprovide further insight into the physics of the system with a detailed analysis of the order parameter andother properties.DOI: 10.1103/PhysRevLett.95.168304 PACS numbers: 82.35.Lr, 82.35.Jk, 82.35.PqProteins exist in one of three possible states: as a col-lapsed globule, as a random coil, or in a well-defined nativestate [1]. Transitions among these configurations can beinduced by the application of an external stimulus like adecrease in temperature or solvent quality. Among all thepossible transitions, one stands out due to its commonoccurrence in biological and synthetic macromolecules:the coil-globule transition [2]. During this transition, themacromolecule changes its configuration from a randomcoil to a compact globule upon a decrease in temperature.However, while this transition is continuous for syntheticpolymers, the case is more complex for biopolymers likeproteins [3] where residual secondary structures persisteven in the globular state [4]. This residual ordering is aconsequence of a competition between the native foldedstate and the collapsed amorphous globular structure, bothbeing possible equilibrium states at low temperatures.Hence, understanding this competition between variouspossible equilibrium, low-temperature states in proteinsis a first class challenge in polymer physics. However,due to the structural complexity of proteins, we restrictourselves to the study of simpler molecular structures. Wefocus on the case of homopolypeptides which are known toform  helices and undergo the helix-coil transition [5,6].Consequently, the main purpose of this Letter is to studythe coupling of the helix-coil and coil-globule transitionsin homopolypeptides, a problem that belongs to the moregeneral classification of possible couplings of transitions inmacromolecular systems proposed by Di Marzio [7]. Thisproblem is also similar to the collapse transition of poly-mers with variable stiffness studied by Stukan et al. re-cently [8]. However, it should be noted that the latter paperadopts a purely bending potential for introducing chainstiffness, which is not identical to the torsional potentialadopted in the present Letter which we describe below.The polypeptide is described with a coarse-grainedmodel which maps the polymer onto a freely rotating chainwhere each residue is represented by a bead [6]. The co-operative nature of the helix-coil transition is captured us-ing a geometric criterion based on the concept of ‘‘torsion’’of a curve. This cooperativity is known to emerge from theformation of hydrogen bonds between pairs of residues iand i4 which, in turn, constrain the spatial positions andorientations of residues i 1, i 2, and i 3. Explicitly,the torsion determines the conformation (helix or coil) ofeach bead. If the torsion of a bead differs from the one ofthe helix by less than a cutoff value then the bead is in thehelical state; otherwise, it is part of a random coil which isthe reference state. Helical beads carry a negative enthalpy,C 1300 K [6] that models the formation of a hydro-gen bond and determines the transition temperature of thehelix-coil transition, Thc; 312 K for our case. These con-cepts were implemented in a Monte Carlo simulation basedon the Wang-Landau (WL) algorithm and their consequen-ces were explored [6,9].In this Letter, we generalize the model to account for thecoil-globule transition in a similar manner to previousworks [8,10] and explore the physical consequences ofthe simultaneous presence of both transitions. This is ac-complished through an implicit-solvent Monte Carlo simu-lation based on the WL algorithm which computes thedensity of states (DOS) of the system by sampling all thephase space. Solvent effects are modeled as an effectiveinteraction between beads described by a modifiedLennard-Jones (LJ) potential of the formVij  4"rij12 rij6rij  ; (1a)Vij  1 rij < ; (1b)where rij is the distance between beads i and j, " is thestrength of the interaction, and  the bead diameter. Theeffect of solvent is taken into account via a compactnessparameter  defined as   PiPj>i Vij=". Hence, amore compact configuration corresponds to a larger valueof . For each configuration of the chain we record thenumber of beads in the helical state, Ns, and the value of .Therefore, the energy of the configuration isE  C:Ns  ":: (2)Since E is a function of Ns and , we define the DOS of thesystem as a function of same parameters, i.e., gNs;  andPRL 95, 168304 (2005) P H Y S I C A L R E V I E W L E T T E R S week ending14 OCTOBER 20050031-9007=05=95(16)=168304(4)$23.00 168304-1 © 2005 The American Physical Societyuse the WL algorithm to compute it. Once gNs;  isknown, the equilibrium properties are analyzed using for-mulas of statistical mechanics.Figure 1 shows the state diagram for a polymer with 60beads together with typical configurations of the chain. Thedark and light segments represent helical and coil beads,respectively. The boundaries between states were obtainedfrom the peak in the heat capacity. Figure 1 shows someboundaries (small peaks) ending before merging with theother boundaries (large peaks). This occurs when thesmaller peaks approach the large ones; they becomeshoulders or are absorbed by the large peaks; hence, theycould not be resolved. At low temperatures and for small "the chain is in the helical state which, upon an increase intemperature, undergoes the helix-coil transition [5,6].Moreover, at low temperatures an increase in " results ina cascade of continuous conformational transitions startingin the helical state and ending in a collapsed globular statewith residual local order as depicted by the dark helicalstrands. In addition, for any ", the residual order is meltedby an increase in temperature. Indeed, at high tempera-tures, the chain adopts either a random coil or an amor-phous globular state, which are separated by the coil-globule transition boundary. Figure 1 also shows how thetransition temperature changes with ".Figures 2– 4 show the mean square radius of gyration,hR2gi, order parameter of the helix-coil transition, i.e., thefraction of helical beads, hi, and the number of helicalstrands, hNhi, for values of temperature, T, and " in theintervals (50 K, 500 K) and 0; 1000, respectively. Wehave explored this large region of parameter space toidentify as many states as possible.When "  0, the helix-coil transition is recovered [5,6].Figure 2 shows the decrease in hR2gi with increasing tem-perature suggesting a cooperative transition. This is con-firmed by Fig. 3 which shows the typical sigmoid-typeshape of hi as a function of temperature, implying itscooperative nature. The transition occurs at Thc which ismostly determined by the parameter C [6]. At low tem-peratures, the helical content is one implying an all-helicalconfiguration but, as the temperature increases the helicalcontent decreases to zero indicating the melting of thehelical strands. Figure 4 shows hNhi as a function oftemperature. The peak around Thc indicates that the helicalstructure breaks into more than one helical strand duringthe transition, in agreement with helix-coil transition theo-ries [5].Figure 2 also shows that upon a small increase in ",hR2gi decreases. This decrease is clearly observed at hightemperature and is a consequence of the stronger attrac-tion among beads which tries to collapse the chain irre-spective of temperature. At low temperatures the decreasein hR2gi is not observed because the chain prefers the helicalconformation.At high temperatures the chain does not form helices andthe model predicts a transition from the random coil to theglobular state as " increases (Fig. 2). This transition shouldalso occur with decreasing temperature at fixed ". Figure 2shows an initial decrease of hR2gi with decreasing tempera-ture for temperatures higher than Thc and " 200.However, the formation of the helical structure interfereswith the globular state at temperatures close to Thc.At low temperatures the model predicts a cascade ofsteplike, continuous conformational transitions with in-creasing ". The sharpness of these transitions is a resultof the competition between various stable and metastablestates present at low temperatures. Moreover, it indicatesthat each transition is cooperative as the helix-coil transi-tion. This observation agrees with the experiments ofMoore and collaborators who studied the helix-coil tran-sition of phenylene ethynylene oligomers at constant tem-perature as a function of solvent quality and observed theFIG. 1. State diagram for a chain with 60 beads. Helical beads(dark segments), coil beads (light segments).FIG. 2. hR2gi as a function of " and temperature for a chain with60 beads.PRL 95, 168304 (2005) P H Y S I C A L R E V I E W L E T T E R S week ending14 OCTOBER 2005168304-2typical sigmoid-type shape of hi which indicates cooper-ativity [11]. However, they observed only one transitionwhich is due to the low degree of polymerization (DP) usedin their studies. We also studied short chains (10 beads) andobserved only one transition. However, longer chains, e.g.,DP  20, 30, 45, and 60, showed a cascade of stepliketransitions as discussed below.For small values of " and temperature, the chain is in thehelical state (Fig. 3). However, as " increases, hi de-creases in a steplike manner at very specific values of ".This decrease is due to the breaking of the helical structurecaused by the LJ attractions and was observed for all thechain lengths studied (20, 30, 45, and 60 beads). Thismechanism is similar to the one present in the standardhelix-coil transition of long enough chains where the poly-mer undergoes the transition by breaking the all-helicalconformation into shorter helical strands as the tempera-ture is increased. However, the helix-coil transition doesnot consider solvent effects and the helical strands do notaggregate like in our case. Since the underlying mecha-nisms for breaking the helical structure are the same,cooperativity is expected to occur in the various, low-temperature transitions. Along the same lines, Fig. 2 showsa steplike decrease in hR2gi for the same values of " im-plying that the breaking of the helix also changes thedimensions of the chain, making it more compact. Inaddition, Fig. 4 shows that hNhi also changes at eachtransition. First it increases from one to six in steps and,afterward, decreases to five at higher values of ". Thesechanges in hNhi along with the decrease in hi and hR2gisupport the proposed mechanism for breaking the helicalstructure illustrated in Fig. 1 with several snapshots of thepolymer chain.During the first steplike transition, the all-helical con-formation breaks into two smaller, but relatively long,helical strands which may have any relative orientationin space. However, we argue that the most probable con-figuration is the one where they are aligned parallel to eachother and are located side by side, as shown in Fig. 1. Thisrelative alignment is favored by the increase in the numberof contacts between the beads of both helical strands whichdecreases the LJ energy and the helical conformation ofboth strands which decreases the torsional energy. This firsttransition was observed for all the DP studied making it acharacteristic feature of the model. An increase of " gen-erates a transition to a conformational state where threehelical strands are parallel to each other which was alsoobserved for all the chain lengths studied. As " increasesfurther, the model predicts states with 4 and 6 helicalstrands parallel to each other. However, the parallel align-ment of the helices is disturbed by the large number of coilbeads at higher values of ". For example, for a 60-beadchain large values of " generate a transition between twostates both with six helical strands. In this case, the latterconformation with lower hi has a lower degree of align-ment of the strands, supporting the previous argument. Afurther increase of " leads to a decrease to 5 helical strandsin a nonparallel configuration. Similar behavior was ob-served for 45 beads. Interestingly, for the 30-bead case theresidual helical strands were found to be parallel to eachother. The same energetic argument used to rationalize theformation of the first state is also applicable to the otherFIG. 4. hNhi as a function of " and temperature for a chainwith 60 beads.FIG. 3. hi as a function of " and temperature for a chain with60 beads.PRL 95, 168304 (2005) P H Y S I C A L R E V I E W L E T T E R S week ending14 OCTOBER 2005168304-3states until the LJ attractive interactions overwhelm theformation of helices and control the behavior of the chain.We also studied the effect of chain length on the statediagram and found significant changes only in the low-temperature regime. An increase in chain length has twomain consequences. First, new states with more helicalstrands appear between the all-helical conformation andthe collapsed globular state with residual order. Therefore,more and sharper transitions are observed. However, due tofinite size effects, they are not transitions in the thermody-namic sense. This dependence on chain length is oppositeto the one reported by Stukan and collaborators for semi-flexible chains where an increase in chain length reducesthe number of stable states of the chain [see Fig. 1 inRef. [8] ] because no disklike globular structures arepresent; only toroidal globules are predicted. These toroi-dal states are akin to the various globular states reported inthis Letter. Indeed, several toroidal states with differentwinding numbers can be found with increasing chainlength and strength of the attractive interactions [12].Second, the LJ attractive interactions become stronger[they increase as DP2 [2] ] and overwhelm the enthalpygain due to the formation of longer or more helices (itincreases as DP). Thus, smaller values of " are required tobreak the various structures. For example, for a 30-beadchain at 200 K the all-helical conformation breaks when"  200 but, for a 60-bead chain the same transition occurswhen "  100.At low temperatures hNhi increases with increasing "due to the breaking of the helices into shorter helicalstrands and then decreases due to the monotonic decreasein hi (Figs. 3 and 4). This leads to a maximum in hNhiwhich is determined by the chain length. In our case, thishappens when hNhi is equal to 3, 5, and 6 for chain lengthsof 30, 45, and 60. However, hNhi does not approach zero onfurther increase of " implying that some residual secondarystructure remains even in the collapsed globular state. Thisresidual structure melts with increasing temperature(Fig. 1). However, transitions from one globular state toanother one with different number of helical strands arepossible (Fig. 1) because the transition temperatures de-pend on ". The lower the transition temperature, the morestable the helical structure. Hence, the stronger the inter-action needed to break the helical strands which results in atransition at larger values of ".In summary, we have investigated the coupling betweenthe helix-coil and coil-globule transitions. We found that,in addition to the helix-coil and coil-globule transitions, themodel predicts a series of helix-globule and globule-globule transitions at low temperatures where an all-helicalstate collapses into a globular state with residual secondarystructures through a cascade of continuous conformationaltransitions. Furthermore, the residual order disappearsgradually with an increase in temperature leading to tran-sitions between collapsed globular states and the amor-phous globular or random coil states.This material is based upon work supported by theNational Science Foundation, Grant No. CHE-0132278.We also acknowledge The Ohio Board of Regents,Action Fund, Grant No. R566.*To whom any correspondence should be addressed.Electronic mail: gac@uakron.edu[1] M. Karplus and E. I. Shakhnovich, in Protein Folding,edited by T. Creighton (Freeman, San Francisco, 1992).[2] P. J. Flory, J. Chem. Phys. 17, 303 (1949); E. A. Di Marzio,Macromolecules 17, 969 (1984); Y. A. Grosberg and D. V.Kuznetsov, ibid. 25, 1996 (1992).[3] C. B. Anfinsen, Science 181, 223 (1973).[4] P. Romiszowski, Biopolymers 54, 262 (2000).[5] D. Poland and H. A. Scheraga, Theory of Helix-CoilTransition in Biopolymers (Academic, New York, 1970).[6] V. Varshney, T. E. Dirama, T. Z. Sen, and G. A. Carri,Macromolecules 37, 8794 (2004); V. Varshney and G. A.Carri, ibid. 38, 780 (2005).[7] E. A. Di Marzio, Prog. Polym. Sci. 24, 329 (1999);J. Chem. Phys. 119, 6378 (2003).[8] M. R. Stukan, V. A. Ivanov, A. Y. Grosberg, W. Paul, andK. Binder, J. Chem. Phys. 118, 3392 (2003).[9] F. Wang and D. P. Landau, Phys. Rev. Lett. 86, 2050(2001); B. J. Schulz, K. Binder, M. Mu¨ller, and D. P.Landau, Phys. Rev. E 67, 067102 (2003).[10] A. Kolinski, J. Skolnick, and R. Yaris, Proc. Natl. Acad.Sci. U.S.A. 83, 7267 (1986).[11] R. B. Prince, J. G. Saven, P. G. Wolynes, and J. S. Moore,J. Am. Chem. Soc. 121, 3114 (1999); J. C. Nelson, J. G.Saven, J. S. Moore, and P. G. Wolynes, Science 277, 1793(1997).[12] Y. A. Kuznetsov and E. G. Timoshenko, J. Chem. Phys.111, 3744 (1999).PRL 95, 168304 (2005) P H Y S I C A L R E V I E W L E T T E R S week ending14 OCTOBER 2005168304-4

Coupling Between Helix-Coil and Coil-Globule Transitions in Helical Polymers

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Coupling Between Helix-Coil and Coil-Globule Transitions in Helical Polymers

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