Protein Engineering, Vol. 14, No. 8, 521-523,
August 2001
© 2001 Oxford University Press
COMMUNICATION |
Cunning simplicity of protein folding landscapes
1 Institute of Protein Research, Russian Academy of Sciences, 142290, Pushchino, Moscow Region, Russia 2 Samara State University, Mathematical Department, 1 Acad. Pavlov str., 443086, Samara, Russia
 |
Abstract |
|---|
 Top Abstract IntroductionDiscussion of the protein...ConclusionReferences |
|---|
Funnel-like landscapes are widely used to visualize protein folding. It might seem that any funnel-like energy landscape helps to avoid the `Levinthal paradox', i.e. to avoid sampling the impossibly large number of conformations for a folding protein. This cunning suggestion, reinforced by beautiful drawings of the energy funnels, stimulated some simple models of protein folding; one of them [D.J. Bicout and A. Szabo (2000)
Protein Sci., 9, 452–465] is especially straightforward and instructive. A thorough analysis of this strict funnel model (which does not consider a nucleation of phase separation in the course of folding) shows that it cannot provide a simultaneous explanation for both major features observed for protein folding: (i) folding within non-astronomical time, and (ii) co-existence of the native and the unfolded states during the folding process. On the contrary, the nucleation mechanism of protein folding can account for both these major features simultaneously.
Keywords: co-existence of the native and the unfolded phases/folding nucleus/protein folding/rate of folding/thermodynamic mid-transition/transition state/two-state kinetics
|
Introduction |
|---|
|
TopAbstractIntroductionDiscussion of the protein...ConclusionReferences |
|---|
The attractive idea of funnel-like protein folding landscapes came from the understanding that a large energy difference between the unfolded and the native states provides a sufficient bias of the conformational space, where both the energy and the entropy decrease as the protein folds to its lowest-energy (native) state (Bryngelson et al., 1995
; Chan and Dill, 1998; Dobson and Karplus, 1999).
It might seem that any smooth funnel-like energy landscape always (or at least under some temperature and solvent conditions) helps to avoid sampling the impossibly large number of conformations by the folding protein [this sampling problem is known as the Levinthal (1968) paradox]. This cunning suggestion, reinforced by beautiful drawings of the energy funnels, stimulated some simple models of protein folding; one of them, especially straightforward and therefore especially instructive, has been published recently (Bicout and Szabo, 2000; see also Zwanzig et al., 1992; Zwanzig, 1995). The advantage of this model is that it is strictly described, and therefore more open to analysis and criticism than general speculations of many other people. Because of this it is possible to show that a straightforward funnel model (which does not consider a nucleation of phase separation in the course of folding) is not adequate for protein folding, since it cannot provide a simultaneous explanation for both major features observed (Fersht, 1997) for protein folding: (i) folding within non-astronomical time, and (ii) co-existence of the native and the unfolded protein molecules during the folding process (and specifically, near the point of thermodynamic mid-transition between these two states).
|
Discussion of the protein folding models |
|---|
|
TopAbstractIntroductionDiscussion of the protein...ConclusionReferences |
|---|
The model of Bicout and Szabo describes protein folding as a diffusion within a d-dimensional sphere of radius R = 1. The chain configuration is specified by a point r within this sphere. The region 0
r a represents the native state; the region a < r 1 represents the denatured state of the protein. Thus, d x kBln(a) (where kB is the Boltzmann constant) is the entropy difference between the folded and the completely unfolded states of the chain. Since Bicout and Szabo assume a = 0.1, and since the entropy difference between the folded and unfolded states is 
2.3kBxnumber_of_residues (Privalov, 1979), d must be close to the number of chain residues, if the Bicout–Szabo model is applied to proteins. The energy landscape is specified by a spherically symmetrical potential U(r), that depends only on the distance r from the origin and has a form of the funnel U(r>a) = A(r–1), where A 
0, and U(ra) is sufficiently low (Figure 1A). The folding corresponds to the energy-driven diffusion of point r from the outer sphere towards the origin. This energy-driven diffusion is hampered (at r>a) by the entropy constituent of the free energy, –TS(r) = –NkBTln(r), where N = d–1 and T is the temperature.
|
is the free energy barrier and N the number of degrees of freedom. (B) The nucleus of protein structure, and the energy and entropy terms for the nucleation mechanism of folding. n is the number of unfolded residues, shown by dashed lines, and N–n residues are in the folded structure, shown by solid lines. The deviations of U and TS from the straight lines result from the surface tension; the coefficients 
and µ are small (<1); the entropic surface tension is produced by the Flory's entropy loss of the closed loops, protruding from the compact nucleus (Finkelstein and Badretdinov, 1997aSo far, so good. Now, however, we face a dilemma. (i) Either A is small (or temperature is high), so that A does not exceed
NkBT. In this case the diffusion encounters the free energy (actually, the entropy) barrier as high as NkBT in the order of magnitude (where N is close to the number of chain residues, if the Bicout–Szabo model is applied to proteins), and we meet the Levinthal paradox in all its strength: the time of transition [which is known (Moore and Pearson, 1981) to be proportional to exp(barrier_free_energy/kBT)] reaches exp(N) and, since N is 100 (the number of protein's residues), the folding is impossibly slow. (ii) Alternatively (in the low-temperature case), A is large relative to NkBT. If A > NkBT/a, the diffusion is fast, downhill in free energy all the way, but the co-existence of the native and the unfolded phases during the folding process (which is so well seen in the bottom of kinetic chevrons; see e.g. Fersht, 1997) is impossible. In addition, in the case of the moderately high A, A 
NkBT/a, the barrier can be small and the folding can be sufficiently fast, but the denatured state is very dense, rdenat a, which means that its entropy is much lower than the entropy of the unfolded state. Thus, this case also excludes co-existence of the native and the unfolded states. The above analysis shows that a straightforward (and hence very instructive) funnel model of Bicout and Szabo cannot simultaneously, under the same conditions, account for both major features observed for protein folding.
What is the problem with this strict funnel model? Looking carefully, one can see that the entropy S(r) = NkBln(r), following from this model, is given by the same, in essence, equation as the conventional entropy of N molecules of uniform ideal gas, confined in the volume V: S(V) = NkBln(V). In addition, the form of the potential U, assumed by Bicout and Szabo, corresponds to a constant (up to a certain volume threshold a, when the piston sticks to the cylinder) pressure, compressing this gas. But the ideal gas does not undergo a phase transition (while the protein chain does; see Privalov, 1979). Such a transition is a feature of a real gas, the molecules of which attract each another.
A real gas can indeed be converted (at low temperature) to a dense phase (solid or liquid) by compression, and both its energy and entropy decrease during this process. But does the real gas remain uniform during this transition (as an ideal gas under compression does)? No, it divides into the gas and the dense phases, and the phase transition goes via nucleation (Landau and Lifshitz, 1959). The nucleation consists in formation of a droplet (‘critical nucleus’) of a ‘new’ phase within the ‘old’ one. When the droplet grows, both the energy and the entropy decrease nearly linearly with its size; the deviations from the linearity are connected only with the free energy excess at the surface of the droplet (Figure 1B
).
The nucleation events are also observed in folding of proteins (Fersht, 1997) and of their simplified models (Abkevich et al., 1994): the semi-folded protein molecule contains a piece of the ‘native phase’ and a piece of the ‘unfolded phase’.
Although both the in vitro (Fersht, 1997) and in silico (Gutin et al., 1996) experiments show that the further destabilization of the unfolded state leads to even faster folding, the nucleation of protein folding is especially easily treated (Finkelstein and Badretdinov, 1997a; see comments on this paper in Wolynes, 1997) in a vicinity of the point of thermodynamic equilibrium between the native and the unfolded phases, where it also occurs rather rapidly (Fersht, 1997). Since the free energies of both phases are equal in the point of equilibrium between them, the free energy of the molecule with the nucleus increases only at the cost of free energy of the boundary between these phases (Figure 1B). Since the maximal size of this boundary is N 2/3 for a compact nucleus (where N is the number of residues in the chain), the maximal nucleation free energy is also kBT0N 2/3 (where T0 is the transition temperature). The subsequent estimate of the time of folding to the global energy minimum is exp(N2/3) ns [actually, from exp(0.5N 2/3) to exp(1.5N 2/3) ns, depending on protein topology; see Finkelstein and Badretdinov, 1997a,b for details]. This estimate is consistent with observed times of protein folding (Galzitskaya et al., 2001); it is zillion times smaller than the estimate exp(N) (i.e. billions of years for a 100-residue protein), which follows from the models of Levinthal and Bicout and Szabo (if the latter is applied to proteins).
Thus, the energy landscape, compatible with the major features of protein folding, must contain a very specific funnel: the funnel, where, in a result of phase separation, the entropy and the energy decrease nearly in parallel (i.e. where the maximal difference U – TS << NkBT0). Not every simple model of protein folding [in particular, not the Bicout–Szabo model, where the maximum of U(r) – TS(r) reaches NkBT0 at mid-transition] can embody such a funnel.
|
Conclusion |
|---|
|
TopAbstractIntroductionDiscussion of the protein...ConclusionReferences |
|---|
The above analysis is focused on only one, but the crucial, range of conditions for protein folding: on the vicinity of thermodynamic mid-transition between the unfolded and native states, where protein folding exhibits its universal features most clearly (Fersht, 1997
; Privalov, 1979; Galzitskaya et al., 2001). An overview of the folding scenario taking part far from this point can be found in many papers, both experimental and theoretical (see Bryngelson et al., 1995; Mirny et al., 1996; Fersht, 1997; Wolynes, 1997; Chan and Dill, 1998; Dobson and Karplus, 1999). Some of these papers stress a conceptual role of folding funnels, but do not specify which kinds of funnels are consistent with the main observed features of protein folding. The presented considerations give an important criterion of correctness of any protein-folding model (and specifically of a funnel model): whether or not this model can explain fast protein folding near the point of thermodynamic mid-transition between the folded and unfolded phases, and stresses that the nucleation phenomenon is of a key importance for this explanation.
|
Notes |
|---|
3 To whom correspondence should be addressed. E-mail: afinkel{at}vega.protres.ru
|
Acknowledgments |
|---|
This work was supported by an International Research Scholar's Award to A.V.F. from the Howard Hughes Medical Institute.
|
References |
|---|
|
TopAbstractIntroductionDiscussion of the protein...ConclusionReferences |
|---|
Abkevich,V.I., Gutin,A.M. and Shakhnovich,E.I. (1994) Biochemistry, 33, 10026–10036.[Medline]
Bicout,D.J. and Szabo,A. (2000) Protein Sci., 9, 452–465.[Web of Science][Medline]
Bryngelson,J.D., Onuchic,J.N., Socci,N.D. and Wolynes,P.G. (1995) Proteins, 21, 167–195.[Web of Science][Medline]
Chan,H.S. and Dill,K.A. (1998) Proteins, 30, 2–33.[Web of Science][Medline]
Dobson,C.M. and Karplus,M. (1999) Curr. Opin. Struct. Biol., 9, 92–101.[Medline]
Fersht,A.R. (1997) Curr. Opin. Struct. Biol., 7, 3–9.[Web of Science][Medline]
Finkelstein,A.V. and Badretdinov,A.Y. (1997a) Fold. Des., 2, 115–121.[Web of Science][Medline]
Finkelstein,A.V. and Badretdinov,A.Y. (1997b) Mol. Biol. (Russia, Engl. Ed.), 31, 391–398.
Galzitskaya,O.V., Ivankov,D.N. and Finkelstein,A.V. (2001) FEBS Lett., 489, 113–118.[Web of Science][Medline]
Gutin,A.M., Abkevich,V.I. and Shakhnovich,E.I. (1996) Phys. Rev. Lett., 77, 5433–5436.[Web of Science][Medline]
Landau,L.D. and Lifshitz,E.M. (1959) Statistical Physics. Pergamon Press, Oxford, UK.
Levinthal,C. (1968) J. Chim. Phys. Chim. Biol., 65, 44–45.
Mirny,L.A., Abkevich,V. and Shakhnovich,E.I. (1996) Fold. Des., 1, 103–116.[Web of Science][Medline]
Moore,J.W. and Pearson,R.G. (1981) Kinetics and Mechanism. John Wiley, New York.
Privalov,P.L. (1979) Adv. Protein Chem., 33, 167–241.[Medline]
Wolynes,P.G. (1997) Proc. Natl Acad. Sci. USA, 94, 6170–6175.
Zwanzig,R. (1995) Proc. Natl Acad. Sci. USA, 92, 9801–9804.
Zwanzig,R., Szabo,A. and Bagchi,B. (1992) Proc. Natl Acad. Sci. USA, 89, 20–22.
Received January 12, 2001; revised April 17, 2001; accepted May 18, 2001.
CiteULike 
Connotea 
Del.icio.us What's this?
| ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||