The conformation and dynamics of a protein chain with hydrophobic and polar nodes are examined by the bond-fluctuation model using Monte Carlo simulations on a cubic lattice. The minimal (nearest neighbor) interaction leads to standard (self-avoiding walk) conformation, i.e., the scaling of the radius of gyration Rg with the molecular weight N Rg ∝ Nγ with γ ≃ 3/5/ Specific interactions with longer range and higher strength are needed to approach the native globular conformations with γ \u3c 3/5. Relaxation into the globular ground state shows a weak power-law decay, i.e., Rg ∝ t-α, α ~ 0.06-0.12

Bjursell, Johan

Pandey, Ras B.

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The University of Southern MississippiThe Aquila Digital CommunityFaculty Publications11-1-2004Relaxation to Native Conformation of a Bond-Fluctuating Protein Chain With Hydrophobic andPolar NodesJohan BjursellGeorge Mason UniversityRas B. PandeyUniversity of Southern Mississippi, ras.pandey@usm.eduFollow this and additional works at: http://aquila.usm.edu/fac_pubsPart of the Physics CommonsThis Article is brought to you for free and open access by The Aquila Digital Community. It has been accepted for inclusion in Faculty Publications byan authorized administrator of The Aquila Digital Community. For more information, please contact Joshua.Cromwell@usm.edu.Recommended CitationBjursell, J., Pandey, R. B. (2004). Relaxation to Native Conformation of a Bond-Fluctuating Protein Chain With Hydrophobic andPolar Nodes. Physical Review E, 70(5).Available at: http://aquila.usm.edu/fac_pubs/2988Relaxation to native conformation of a bond-fluctuating protein chainwith hydrophobic and polar nodesJohan BjursellGeorge Mason University, MS 5C3, Fairfax, Virginia 22030-4444, USAR. B. PandeyDepartment of Physics and Astronomy, University of Southern Mississippi, Hattiesburg, Mississippi 39406-5046, USA(Received 3 October 2003; revised manuscript received 10 September 2004; published 30 November 2004)The conformation and dynamics of a protein chain with hydrophobic and polar nodes are examined by thebond-fluctuation model using Monte Carlo simulations on a cubic lattice. The minimal (nearest neighbor)interaction leads to standard (self-avoiding walk) conformation, i.e., the scaling of the radius of gyration Rgwith the molecular weight N Rg~Ng with g.3/5. Specific interactions with longer range and higher strengthare needed to approach the native globular conformations with g,3/5. Relaxation into the globular groundstate shows a weak power-law decay, i.e., Rg~ t−a, a,0.06–0.12.DOI: 10.1103/PhysRevE.70.052904 PACS number(s): 87.14.Ee, 36.20.Ey, 87.15.Aa, 87.15.HeThe structural stability of protein [1–4] chains has beenstudied extensively in recent years [5–19], primarily by com-putational methods. How the protein chain relaxes to its na-tive conformations is one of the main questions addressed bymany researchers [5–19]. A native structure evolves into astable configuration (in steady state or equilibrium) as thechain explores its conformational phase space and is ex-pected to be globular in appropriate solvent conditions. Aprotein is a large polymer consisting of 20 amino acids in aspecific sequence. These amino acid groups are similar ex-cept for their side chains that distinguish their characteristics.They are roughly divided into three categories: hydrophobicsHd, polar sPd, and charged sCd groups. Hydrophobic andpolar groups are considered to be the main constituents inmost coarse-grained models [20] and as the main constitu-ents to describe the general characteristics of the protein.In a coarse-grained model, a polymer chain is describedby nodes consecutively connected by bonds in a linear fash-ion. Primary chain models [21,22] are (i) constant bond (CB)chains with consecutive nodes connected by a constant bondlength on lattice, (ii) bond-fluctuating (BF) chains with fluc-tuating bond lengths on lattice, and (iii) bead-spring (BS)chains (and variants) off lattice. While the CB methods (i)are efficient in probing the equilibrium properties such asconformation of polymer chains, some microscopic detailsare usually missed in many simulations due to limited de-grees of freedom with fast but somewhat artificial segmentaldynamics. The off-lattice approaches (iii), on the other hand,are excellent in probing the microscopic details but generallytoo slow to reach equilibrium due to the long relaxation timein many complex systems, primarily because of large (prac-tically infinite) degrees of freedom. In order to examine theapproach to asymptotic global properties, resorting to simpli-fications [20] is almost unavoidable with either method un-less one develops hybrid simulation approaches that incorpo-rate the efficiency, effectiveness, and accuracy of bothmethods [23–25]. The BF model (ii) lies in between (i) and(iii) as it captures more microscopic details with a consider-ably larger number of degrees of freedom than the CB model(i) without significantly compromising the efficiency of adiscrete lattice. A chain node in the BF model occupies anelementary cube, i.e., a node is represented by eight latticenodes in contrast to a single node in the CB model on a cubiclattice [21]. Due to excluded volume constraints of the node(cube) the bond length fluctuates between 2 and ˛10 with theexception of ˛8 and involves as many as 108 vectors [21] toaccess it. Banavar and co-workers [8,9] have recently arguedthat the thickness of the bonds that tether the nodes are veryimportant to correctly take into account structural features ofthe protein. The bonds are usually very thin (negligiblysmall) in both CB (i) and BS (iii) models but they possessfluctuating length and thickness in the BF model (ii) which isour choice here to study the conformation and dynamics of aprotein chain model.Despite the limited degrees of freedom, the constant bondlattice model has the advantages of simplicity and computa-tional efficiency, which are useful for exploring issues suchas the energy landscape for stable structures of proteins. Us-ing the CB description of the HP protein chain, Dill andco-workers [10,11] have successfully described the core as-sembly and protein folding into native structure via funnelpathways [12]. Lattice models provide useful insight intosome of the basic characteristics of proteins [13,14]. Thedynamics of the HP chain and its relaxation to the nativestructure is severely limited due to relatively few choices(degrees of freedom) for the node to move. In addition, it isnot clear which combination of local segmental moves (i.e.,kink-jump, crankshaft, reptation, etc.) [21] should be used tocapture appropriate dynamical modes [23–25]. Off-latticemethods have contributed considerably in understanding theevolution of a helices and b sheets [15–18] where micro-scopic details are crucial. Incorporation of local interactionswith large numbers of degrees of freedom is very importantin such analysis around stable structures even with crudeapproximations [20]. The large-scale dynamics involving re-laxation from one structure to another is, however, severelyrestricted due to the long time needed with small movementof nodes in off-lattice simulations [23–25].PHYSICAL REVIEW E 70, 052904 (2004)1539-3755/2004/70(5)/052904(4)/$22.50 ©2004 The American Physical Society70 052904-1Very recently, Chen and Chen [19] have used the bond-fluctuation model to study the folding and native structuresof a specific protein, sensory rhodopsin I. They have usedinteractions between nodes, empty sites (to represent the hostmedium, i.e., a membrane between the water planes at theopposite sides of the lattice), the bending energy for thechain bonds, and the interaction energy with the waterplanes. The simulations seem to be performed in three stageswith appropriate interactions to avoid the long relaxationtime and to capture desired segmental packing. This mayappear somewhat ad hoc or arbitrary, but necessary to over-come the energy barriers and technical difficulties. We wouldlike to investigate the conformational relaxation and segmen-tal mobility of proteins in a single domain (keeping the sameinteraction throughout) in detail with a somewhat simpler setof interactions without imposing the constraints of a specificprotein. We focus on the general features of a simplifiedbond-fluctuating protein chain model by a large-scale com-puter simulation study: we encounter major technical prob-lem in reaching the native globular structure even with large-scale simulations but we are able to show how the radius ofgyration relaxes sRg~ t−ad with a revealing conversion ofconformational populations into their native state.We consider a cubic lattice of size L3 with L=30–200.The protein chains of length N=50–400 are considered withhydrophobic sHd and polar sPd nodes connected by fluctuat-ing bonds. A node occupies a cube (eight lattice sites) andthe bond length l in units of the lattice constant can vary,l2=4, 5, 6, 9, 10 with 108 vectors connecting consecutivemonomers [21]. Initially, a chain of length N is randomlyplaced in the lattice. Apart from the excluded volume effect,we consider a short-range interaction sUd among nodes andbetween nodes and empty sites which represent the effectivesolvent sSd components,U = oiokJsi,kd , s1dwhere the index i runs over all constituents sH , P ,Sd and kover all their neighboring sites within a range ri of site i. Theinteraction energy between the constituents sA ,Bd at thesesitesJsA,Bd = eAB. s2dThe range of interaction is varied, i.e., ri2= l2, where ri2=4, 5,10 represent nearest neighbor, next nearest neighbor sites,etc. The set of interaction matrix elementseHS = − ePS = e1, eHP = e2, eHH = ePP = e3, s3dwith a range of interaction strengths ei, i.e., e1=e2=0 ,1 ,2 ,3 , . . .; e3=0. The energy is measured in units of kBT.The chain nodes are moved randomly to their neighboringsites (i.e., cubes) with the METROPOLIS algorithm and an at-tempt to move each node once defines one Monte Carlo step(MCS) as the time unit [21]. The simulation is performed fora long time with a number of independent runs for averagingthe radius of gyration Rg and mean square displacements ofeach node skRn2ld and of their center of mass skRc2ld.A variety of random and ordered sequences includingdiblock copolymers of H and P are considered for compari-son. In general, the segmental and global motion of the pro-tein chain depend on the fraction of H and P groups and theirsequences. For example, a chain with primarily H groupstends to be less mobile than a chain dominated by P groups.Hydrophobic interactions seem to pin down the configura-tions leading to reduced mobility. The polar groups, on theother hand, enhance the segmental mobility and thereforeaccelerate the equilibration. The ordered sequences in blocksof H’s and P’s have lower energy than random sequences.However, the data presented below are generated from simu-lations performed with equal numbers of hydrophobic andpolar groups in random sequences.The protein chain equilibrates well with the short-rangeinteraction sri2=4d; therefore it is easier to study the scalingof the radius of gyration with the molecular weight sNd. Fig-ure 1 shows Rg2 versus N plots on a log-log scale. Accord-ingly,Rg2 ~ Ng, s4dwith the exponent g.0.6. Note that the radius of gyration ofeach chain has relaxed well which is confirmed by the varia-tion of energy with the time steps (see inset figures). Thedata for higher interaction strength seid are more fluctuating,but equilibration is achieved by increasing the simulationtime. Thus, with the minimal interactions, the equilibriumconformation of the HP protein chain is more like a self-avoiding walk than a compact globular form generally ex-pected in a native state.Figure 2 shows the variation of the gyration radius withthe time steps for higher range sri2=5 ,10d with different in-teraction strengths. For a relatively weak interaction se1=1d,we see that the protein chain has relaxed well for ri2=5. Theconformation is relatively spread as in Fig. 1. Increasing theinteraction strength to e1=2 ,3 ,5 leads to the onset of relax-FIG. 1. Rg2 versus chain length N on 2003 samples for random Hand P sequences. Nearest neighbor interaction sri2=4d with interac-tion strengths e1=1 si1d, 5 si5d are used. Simulations are performedfor up to 107 time steps, each with 100 independent runs. Insetfigures show the variation of Rg2 (for N=50–300) and energy (E forN=300 for clarity) with the time steps.BRIEF REPORTS PHYSICAL REVIEW E 70, 052904 (2004)052904-2ation into conformations with smaller sizes (presumably theglobular). These relaxations show a power-law decay of theradius of gyration,Rg2= R02s1 + Ct−ad , s5dwhere R0 is the radius of gyration of the chain in globularconformation and C is a constant. The decay exponent a=0.06–0.12 depends on the range and magnitude of the in-teraction. The decay of energy with the time steps is alsoconsistent with the relaxation of protein chains into theirglobular structures.It would be interesting to examine the variation of meansquare displacements with the time steps in order to probethe segmental mobility as the protein chains relax into theirglobular conformation. Figure 3 shows such a variation. Wesee that for a relatively low interaction strength se1=1d, bothnodes and their center of mass reach a diffusive power-lawbehavior,Rn,c ~ tn, s6dwith n.1/2 in the asymptotic regime. At higher interactionstrengths se1=2 ,3 ,5d, on the other hand, a critical slowingdown seems to occur. It becomes difficult to describe thedependence of Rn,c2 on time by a single power law in ourobservation time. While stronger interactions beyond thenearest neighbor range are needed for the protein conforma-tions to reach their globular states, the relaxation time is toolarge to reach such states within our computational resourcesat present. The necessity to resort to more simplifications (orincreased computational resources) is unavoidable.With such a large amount of data, it is desirable to digfurther into the conformational relaxation. In Fig. 4, wepresent the histogram of the radius of gyration. At a rela-tively short time st=104d from the beginning of simulation,we see a rather large spread in the magnitude of Rg from onesample to another sri2=5 ,e1=2d. Toward the end of the simu-lation st=107d, on the other hand, values of Rg in mostsamples have fallen to a very low value sRg2.30–40d. Sucha trend in population inversion from extended conformationinto a globular form is also seen with higher interactionstrengths. Thus, in order to analyze the native configuration,one has to selectively use those configurations which havereached their ground state. Such a crude sampling of theground state conformations (after 107 time steps) for differ-ent chain lengths shows signs of globular conformations (seethe inset of Fig. 4). In summary, the conformation of theprotein chain depends on the interaction (i.e., nature of thesolvent) and the sequence. We are not able to distinguishdifferences in data with different random sequences due tolarge fluctuations. However, we have verified the changes byexamining blocked sequences. While the appropriate interac-FIG. 2. Variation of Rg2 with time steps on a log-log scale forN=100 with next neighbor interaction sri2=5d with different inter-action strengths fe1=1 si1d ,2 si2d ,3 si3d ,5 si5dg and longer-rangeinteraction fri2=10 si6dg with strength e1=1 si1d (down triangles).The inset figure shows corresponding energy variation on a semilogscale. Sample size 2003 is used with 100 independent runs.FIG. 3. Mean square displacement of each node kRn2l and itscenter of mass kRc2l (inset) versus t for N=100 chain with randomsequences of H and P. Statistics is the same as in Fig. 2.FIG. 4. Conformational histogram, i.e., variation of Rg2 with in-dependent runs for N=100 with next neighbor interaction sri2=5d attime steps t=104, strength e1=2 si2d, and t=107, e1=2 si2d ,3 si3d.Inset figure is Rg2 versus N plot on a log-log scale for e1=2 si2d witheach data point generated for t=107 MCSs with a selective sam-pling of 100 independent runs; y-axis range is 35–80. Sample size2003.BRIEF REPORTS PHYSICAL REVIEW E 70, 052904 (2004)052904-3tions are necessary to reach the native structures, the relax-ation depends on the quality of the solvent. Despite a majorproblem with a very long relaxation time, our simulationreveals a clear population conversion (“funneling”) of HPprotein chains into their globular ground state. Relaxation ofthe radius of gyration shows a slow power-law decay [Eq.(5)] with a nonuniversal power-law exponent sad into a na-tive structure with Rg~Ng, g,3/5.This work was supported in part by NSF-EPSCoR GrantNo. EPS-0312618 and a basic research grant from the Uni-versity of Southern Mississippi. Participation in the NationalScience Foundation MRSEC program under the Grant No.DMR-0213883 was helpful. We thank Dietrich Stauffer forvaluable comments and corrections. The visualization is per-formed by an animation program ANIMP developed by Dr.Ray Seyfarth; we thank him for his help.[1] C.K. Mathews, K.E. van Holde, and K.G. Ahern, Biochemis-try, 3rd ed. (Addison-Wesley/Longman, Reading, MA, 2000).[2] P.C. Turner, A.G. McLennan, A.D. Bates, and M.R. H. White,Molecular Biology, 2nd ed. (BIOS/Springer-Verlag, New York,2000).[3] C. Branden and J. Tooze, Introduction to Protein Structure,2nd ed. (Garland, New York, 1999).[4] D. Baker, Nature (London) 405, 39 (2000).[5] A.I. Jewett, V.S. Pande, and K.W. Plaxco, J. Mol. Biol. 326,247 (2003).[6] B. Gillespie and K.W. Plaxco, Annu. Rev. Biochem. 73, 837(2004).[7] J.E. Kohn et al. (unpublished).[8] J.R. Banavar, O. Gonzalez, J.H. Maddocks, and A. Maritan, J.Stat. Phys. 110, 35 (2003).[9] J.R. Banavar and A. Maritan, Rev. Mod. Phys. 75, 23 (2003),and references therein.[10] K.M. Fiebig and K.A. Dill, J. Chem. Phys. 98, 3475 (1993).[11] K. Yue et al., Proc. Natl. Acad. Sci. U.S.A. 92, 312 (1995).[12] J.D. Bryngelson, J.N. Onuchic, N.D. Socci, and P.G. Wolynes,Proteins 21, 167 (1995).[13] C. Tang, Physica A 288, 31 (2000).[14] E.I. Shakhnovich, Phys. Rev. Lett. 72, 3907 (1994).[15] U.H.E. Hansmann and L.T. Wille, Phys. Rev. Lett. 88, 068105(2002).[16] U.H.E. Hansmann and Y. Okamoto, Curr. Opin. Struct. Biol.9, 177 (1999).[17] U.H.E. Hansmann, Physica A 321, 152 (2003).[18] S. Takada, Proteins 42, 85 (2001).[19] C.-M. Chen and C.-C. Chen, Biophys. J. 84, 1902 (2003).[20] U.H.E. Hansmann, Comput. Sci. Eng. 5, 64 (2003).[21] Monte Carlo and Molecular Dynamics Simulations in PolymerScience, edited by K. Binder (Oxford University Press, NewYork, 1995).[22] A. Baumgaertner, in The Monte Carlo Method in CondensedMatter Physics, 2nd ed., edited by K. Binder, Topics in Ap-plied Physics Vol. 71 (Springer, Berlin, 1995).[23] G.M. Foo and R.B. Pandey, Phys. Rev. E 57, 5802 (1998).[24] G.M. Foo and R.B. Pandey, J. Chem. Phys. 109, 1162 (1998).[25] G.M. Foo and R.B. Pandey, Macromol. Theory Simul. 8, 571(1998).BRIEF REPORTS PHYSICAL REVIEW E 70, 052904 (2004)052904-4

Relaxation to Native Conformation of a Bond-Fluctuating Protein Chain With Hydrophobic and Polar Nodes

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Relaxation to Native Conformation of a Bond-Fluctuating Protein Chain With Hydrophobic and Polar Nodes

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