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Critical aspects of hierarchical protein folding Alex Hansen, ∗† Mogens H. Jensen,‡ Kim Sneppen§ and Giovanni Zocchi∗∗ Niels Bohr Institute and NORDITA, Blegdamsvej 17, DK-2100 Ø, Denmark (February 1, 2008) We argue that the first order folding transitions of proteins observed at physiological chemical conditionsendinacriticalpointforagiventemperatureandchemicalpotentialofthesurrounding 8 water. Weinvestigate thiscritical pointusing ahierarchical Hamiltonian anddetermineits univer- 9 sality class. This class differs qualitatively from those of other known models. 9 PACS: 05.70.Jk,82.20.Db,87.15.By,87.10.+e 1 n Proteins are complex macromolecular objects [1] whose structure is determined by the evolutionary process that a J formed them [2,3]. In particular this means that the folding of proteins into its three-dimensional structure must be efficient. There have been several attempts to grasp aspects of the protein folding process [4], in enumeration of 3 1 configurations [5], in description of folding pathways [6] and in discussing the influence of water on protein structure [7]. ] Presumable proteins evolved by subsequently adding properties to simpler structures which also fold. This has t f led us to propose a hierarchical description of the folding process, where each added subunit folds conditionally on o s the folding of structures above it in the hierarchy [8]. Formulating this in the framework of statistical mechanics we t. suggestthat the folding canbe parametrizedin a number of variablesϕ1,...,ϕN each taking the value 0 or 1 where 1 a correspond to a correctly folded substructure [8,9]. The hierarchy is implemented through the Hamiltonian m d- H = − E0 (ϕ1+ϕ1ϕ2+ϕ1ϕ2ϕ3+···+ϕ1ϕ2···ϕN) . (1) n The interactions between the protein and the surrounding water may be takeninto account by adding to this Hamil- o c toniana coupling parametrizedthroughthe water variablesw1,w2,...,wN [9]. These variablescouple to the unfolded [ protein degrees of freedom because they expose hydrophobic amino acids to the water. The resulting Hamiltonian is 1 v H = −E0 (ϕ1+ϕ1ϕ2+ϕ1ϕ2ϕ3+···+ϕ1ϕ2···ϕN)−[(1−ϕ1)w1+(1−ϕ1ϕ2)w2+...+(1−ϕ1ϕ2···ϕN)wN] 4 (2) 2 1 where the hydrophobic effect is taken into account by letting each of the w’s take a value from the set, Emin+s∆, 1 s=0,1,...,g−1. ∆ is the spacing of the energy levels of the water-protein interactions. 0 8 The partition function is 9 / N 1 r−N −1 t Z = eE0/T + 1 (3) a (cid:16) (cid:17) (cid:18)2 1−r (cid:19) m - The variable r is the ratio of statistical weights of unfolded to folded state, per variable: d n g 1−e−∆/T o r = e−µ/T (4) 2 1−e−g∆/T c : v with µ=−E0−Emin being the chemical potential of the surrounding water. Xi The physical meaning of this model is that the water molecules in contact with an unfolded portion of the protein has lower entropy than when not in contact (thus in the our model, hydrophobicity is caused by ordering of water r a and not by repulsive potentials, as is usually believed [10]). In the model one finds that a first order transition takes place when the parameter r switches between r < 1 and r > 1. Plotting r against T one obtains a non-monotonic ∗Permanent address: Department of Physics, Norwegian University of Science and Technology, NTNU, N–7034 Trondheim, Norway †Electronic Address: [email protected] ‡Electronic Address: [email protected] §Electronic Address: [email protected] ∗∗Electronic Address: [email protected] 1 function which for small µ values passes r = 1 twice, corresponding to unfolding at both low and high temperature, as indeed seen in experiments [11,12]. The mechanism for the transitions is the following. At high temperature the entropy gain of the protein chain causes the unfolding. As temperature is lowered the system gains more entropy by shielding the hydrophobic residues from the water. This leads to folding. As the temperature is loweredeven further the cold unfolding transition occurs. Below this transition entropy is insignificant and the dominating effect is the attractive coupling between the water and the unfolded protein. For an intermediate value of the chemical potential, r just touches the line r = 1, that is dr/dT = 0 when r = 1, corresponding to a merging of two first order transitions. This defines a critical point. Around this point, r varies quadratically in T −T and linearly in µ−µ , as seen from expanding Eq.(4). In experiments of protein folding this c c point is accessibleby changingthe pH value ofthe solution. Infact, Privalov’sdataon lowpH values indeed indicate that such a critical point exists. The scaling properties around this point thus opens for a possibility to gain insight into the nature of the folding process, in particular whether the hierarchicalscheme we suggest can be falsified. InFig.1aweshowheatcapacityasafunctionoftemperatureforchemicalpotentialbelow,atandabovethecritical value µ = µc. For the chosen values of E0 = 1 and level density ∆ = 0.02 and g = 350 the critical point is situated at T = 1.33303..., µ = 1.2838.... That is, it is situated at a minimum of the heat capacity curve. This is at c c first sight surprising, usually heat capacity has a pronounced increase at the critical point. The minimum reflects a partialorderingofthe hierarchy,asenvisionedinFig.1bwhereweshowthe degreeoffolding,countedbythe average number offoldedvariablesϕi =1,i=1,...,n fromi=1until the firstvariablei=n+1 whichtakesvalue ϕn+1 =0. The averagevalue of this hni is N/2 at the critical point, reflecting that the system is on averagehalf orderedat this point. Correspondinglythe heatcapacitydips to a value inbetweenthe value ofanunfoldedanda completely folded state. Tocharacterizethefunctionalformofthedipintheheatcapacity,weinvestigateanalyticallyC (T)=C(T,µ)− sing C(T,µ ) with µ>>µ for different values of hierarchy size N. For finite N we may express the singular part of the c c heat capacity in the form: C = |T −T|−α g (T −T)N1/ν (5) sing c c (cid:16) (cid:17) where g(x) → const when x → ∞ and g(x) ∝ xα when x → 0. We find analytically α = ν = 2 from differentiating the partition function (3). Fig. 2a demonstrate this finite size scaling. Similarly we in Fig. 2b show the behavior of the order parameter hni as function of T −T and N: c hni = |T −T |β f (T −T )N1/ν (6) c c (cid:16) (cid:17) with f(x) → const when x → ∞ and f(x) ∝ x−β when x → 0 where exponents β = −2, also found analytically. It may be surprising that β is negative, but this reflect in part the unusual use of an extensive (in N) order parameter, in part that for µ=µ then the order parameter only obtains a non-zero value at T =T when N →∞. c c Likewise,wefindthatthesusceptibilityχ=dhni/dµscalesas|T−T |−γ whereγ =4andthathni∝(µ−µ )1/δ for c c µ>µ whereδ =−1. Thustheusualexponentrelations,α+2β+γ =2,α+β(δ+1)=2,andγ(δ+1)=(2−α)(δ−1) c are fulfilled [13]. However the hyperscaling reation dν = 2−α, where d is the dimensionality of the system, is not fulfilled. However,this relation has no meaning, as there are no spatial degrees of freedom. Intermsofexperimentsonproteins,therelevantscalingbehaviouristhehowthedegreeoffolding(orderparameter) andthe heatcapacitybehavesasfunction oftemperature, whenonechangeschemicalpotentialawayfromits critical value. The qualitative prediction is that the width of the singular part of the heat capacity has a minimum at the critical value µ=µ . The broadening of the heat capacity is c T −T C (T −T )2 = h c for µ>µ (7) sing c (cid:18)∆µ1/2(cid:19) c where h(x) ∝ x−2 for x →∞ and h(x) = const for x→ 0 and where ∆µ= max(µ−µc,∆µmin) with ∆µmin ∝ 1/N takes into account the finite size sensitivity of the scaling. We show in Fig. 3a, an example of such a data collapse. Thesepredictionsareexperimentallyaccessiblethroughtheuseofstandardcalorimetrictechniques,whereoneshould seek to obtain a data collapse above the critical point, i.e. the point of minimal width. The heat capacity below the critical µ is complicated by the merging of two first order transitions. However, the distance between these moves away from each other in T as ∆µ1/2. Likewise, we expect the degree of folding hni to show data collapse of the form 2 T −T hni(T −T )2 = k c for µ>µ (8) c (cid:18)∆µ1/2(cid:19) c where k(x) behaves asymptotically as h. We show this in Fig. 3b. This quantity can be observed experimentally through fluoresence measurements. In summary, we have proposed that the hierarchical ordering of folding pathways implies a critical point with a diminishing heat capacity at criticality. We have determined all critical exponents, and proposed two experiments that could confirm or falsify the concept of hierarchicalfolding. [1] B. Alberts, D. Bray, J. Lewis, M. Raff, K. Roberts and J. D. Watson, Molecular biology of the cell (Garland Publ., New York,1994). [2] H.Neurath,Science, 224, 350 (1984). [3] W. Gilbert, Science, 228, 823 (1985). [4] K.A. Dill, S. Bromberg, K. Yue,K.M. Fiebig, D.P. Yee,P. D. Thomas and H.S. Chan, Protein Science 4, 561 (1995). [5] P.A. Lindg˚ard and H. Bohr, Phys. Rev.E 56, 4497 (1997). [6] K.A. Dill, K. M. Fiebig and H. S.Chan, Proc. Natl. Acad.Sci. 90 1942 (1993). [7] H.Li, C. Tang and N. S.Wingreen, NEC preprint(1997). [8] A.Hansen, M. H. Jensen, K. Sneppenand G. Zocchi, Physica A, in press (1998). [9] A.Hansen, M. H. Jensen, K. Sneppenand G. Zocchi, submitted to Proc. Natl. Acad. Sci. [10] J. T. Edsall, J. Am. Soc. 57 1506 (1935). [11] P.L. Privalov, Biochem. and Molecul. Biol. 25, 281 (1990). [12] G. I. Makhatadze and P. L. Privalov, Adv.Prot. Chem. 47, 307 (1995). [13] H.E. Stanley, Phase Transitions and Critical Phenomena (Cambridge Univ.Press, Cambridge, 1971). FIG.1. a) Heat capacity, C, as a function of T. b) Degree of folding, hni, as a function of T. Here g =350, ∆=0.02 and N =100. Thevalue N =100 has been chosen as to beclose to realistic valuesfor thisparameter. FIG.2. a) Finite size scaling of the heat capacity for µ=µ , g =350 and ∆=0.02. Here α=2 and ν =2. b) Finite size c scaling of degree of folding, hni. Hereβ =−2. FIG.3. a) C (T −T )2 vs. (T −T )/∆µ1/2. b) hni(T −T )2 vs. (T −T )/∆µ1/2. Wehavechosen N =100, g=350 and sing c c c c ∆=0.02. Note thegood quality of thedata collapse in spite of smallness of thesystem. 3 800 µ=1.0 600 µ=µ c µ=1.5 C 400 200 0 0 1 2 3 4 T Figure 1a A. Hansen, M.H. Jensen, K. Sneppen and G. Zocchi Critical aspects of hierarchical protein folding 100 µ=1.0 µ=µ 80 c µ=1.5 60 <n> 40 20 0 0 1 2 3 4 T Figure 1b A. Hansen, M.H. Jensen, K. Sneppen and G. Zocchi Critical aspects of hierarchical protein folding −0.1 −0.3 α T −0.5 ∆ g n si C N = 100 −0.7 N = 200 N = 400 N = 800 −0.9 N = 1600 −1.1 −3 −2 −1 0 1 2 3 1/ν T N Figure 2a A. Hansen, M.H. Jensen, K. Sneppen and G. Zocchi Critical aspects of hierarchical protein folding 3 N = 100 N = 200 N = 400 N = 800 2 N = 1600 β − T ∆ > n < 1 0 −3 −2 −1 0 1 2 3 1/ν T N Figure 2b A. Hansen, M.H. Jensen, K. Sneppen and G. Zocchi Critical aspects of hierarchical protein folding 0.0 2 ) c T − T −0.1 ( g µ = 1.34 n si µ = 1.36 C µ = 1.40 µ = 1.44 µ = 1.48 −0.2 −0.5 0.0 0.5 (T−T )/∆µ1/2 c Figure 3a A. Hansen, M.H. Jensen, K. Sneppen and G. Zocchi Critical aspects of hierarchical protein folding µ = 1.34 0.9 µ = 1.36 µ = 1.40 µ = 1.44 µ = 1.48 2 ) c T − T ( 0.4 > n < −0.1 −1 0 1 (T−T )/∆µ1/2 c Figure 3b A. Hansen, M.H. Jensen, K. Sneppen and G. Zocchi Critical aspects of hierarchical protein folding