Engineering of stable and fast-folding sequences of model proteins

Transcription

1 Proc. Natl. Acad. Sci. USA Vol. 90, pp , August 1993 Biophysics Engineering of stable and fast-folding sequences of model proteins E. I. SHAKHNOVICH AND.A. M. GUTIN Department of Chemistry, Harvard University, 12 Oxford Street, Cambridge, MA Communicated by Peter G. Wolynes, April 29, 1993 (received for review December 29, 1992) ABSTRACT The statistical mechanics of protein folding implies that the best-folding proteins are those that have the native conformation as a pronounced energy minimum. We show that this can be obtained by proper selection of protein sequences and suggest a simple practical way to find these sequences. The statistical mechanics of these proteins with opimized native structure is discussed. These concepts are tested with a simple lattice model of a protein with full enumeration ofcompact conformations. Selected sequences are shown to have a native state that is very stable and kinetically accessible. How and why proteins fold to their native structure is an intriguing unsolved problem in molecular biophysics. From the theoretical side there are two different approaches to this problem (1-10). The first approach-initiated by Taketomi and Go (1) and then, with several significant modifications, continued by several groups (2-5)-was to define some special model with certain biases to the known native state and to investigate its folding properties from both thermodynamic (1, 2, 4) and kinetic (3, 5, 11) perspectives. The basic feature of these models is "ultraspecificity," which means the introduction of some special force fields biasing the polypeptide chain to the native state. Investigation of such models provided many interesting insights and allowed analysis of the sufficient conditions for folding. Another approach is the investigation of the necessary conditions for folding using simple, completely unbiased protein models. Such an unbiased model is a random heteropolymer. Statistical mechanics of random heteropolymers has been considered (refs. 7, 8, and 12; see the review in ref. 13). Molecular dynamics simulations of folding in a simplified heteropolymer model were studied in ref. 14. It was shown in refs. 7, 8, and 12 that many thermodynamic properties of real proteins could be understood on the level of simplest random heteropolymers without introducing ultraspecific force fields. It was also shown (8) that a large enough fraction of random sequences form unique structure at physiological temperature. Studies of heteropolymers have revealed that their energy spectrum (i.e., set of conformations and their energies) consists of two parts: the "continuous" part, to which the majority of random conformations belong, and the discrete part, representing a few conformations with best-fit contacts. In the continuous part, energy levels are highly populated (so that an exponentially large number of conformations belongs to each such level). More important, this part is selfaveraging; i.e., its features do not depend on specific realization of a sequence but rather on gross properties of the sequence ensemble (such as composition). However, the bottom part of the spectrum is very sequence-sensitive, so that different sequences deliver significantly different energies to their native conformations. Then, as temperature The publication costs of this article were defrayed in part by page charge payment. This article must therefore be hereby marked "advertisement" in accordance with 18 U.S.C solely to indicate this fact decreases the system undergoes a freezing folding transition passing from the continuous to the discrete part of the spectrum (a detailed discussion of the analytical theory of heteropolymers is in refs. 7 and 8; a less mathematical review is in ref. 13). Monte Carlo simulations of the kinetics of folding of heteropolymers in a model with fully enumerated compact conformations (9, 10) have shown that there exist further kinetic requirements for protein sequences to fold. A detailed analysis (10) has suggested that a necessary condition for rapid folding is that the native state represents a pronounced energy minimum with energy which is significantly lower than energies of random conformations from the continuous part of the spectrum: the spectrum of fast-folding sequences is "wide." This also provides thermodynamic stability to the native state. In this work we go beyond analysis of random heteropolymers and discuss how to optimize sequences to obtain better stability and folding. The above discussion shows that such optimization is a choice of sequences with native conformation significantly "pulled" down in energy scale relative to the energies of random conformations from the continuous part of the spectrum. Then the question arises how this optimization can be done with "biologically legitimate" rules of the game, i.e., on the level of a sequence search. Thermodynamics of heteropolymers suggests a simple way to approach this: since the continuous part of the spectrum is self-averaging-i.e., only composition-dependent-whereas the energy of the native conformation is sequence-dependent, optimization of this energy with respect to sequences by keeping the amino acid composition unchanged will lead to the desired goal, making the energy spectrum "wider." The direct and universal way of sequence optimization is to use the Metropolis Monte Carlo algorithm (15) in sequence space (16). This means that instead of setting the energy and searching for sequences with this energy, we should set a selective temperature T.., and run a Monte Carlo optimization process in sequence space with T,,, and with the additional condition that the amino acid composition is unchanged. (Optimization without this condition would rapidly converge to a homopolymer, which obviously does not have a unique structure.) Optimization at a given selective temperature has the advantage that it allows fluctuations of energy with respect to sequences and therefore facilitates overcoming local minima of energy in sequence space. Optimization at low T,,l provides sequences with low energy of the target structure. In the subsequent discussion we will consider the simplest model of sequences consisting of monomers of two types. Of course this simplified representation misses several features of proteins, which are 20-letter sequences, with side chains, several types of interactions, etc. However, the simple models which we discuss below are aimed not at the description and prediction of all complicated features of proteins but rather at clarifying the essential physics of protein folding and design. This is the competition between the tendency to minimize the number of unfavorable contacts and polymeric bonds. For this reason a potential should be chosen which

2 7196 Biophysics: Shakhnovich and Gutin provides this feature and simultaneously is most logical for the theory. Two-letter sequences can be encoded simply as sequences of integer numbers {ao}, where vi = 1 or -1 depending on monomer type. The simplest interaction potential in this system is a contact potential (17). The native structure is then defined via its atomic coordinates, {ri}. The energy of a sequence {a,}, fitted into this structure is 1 N ElQul) = - > U(ori, oj)a(r? - r.), [1] 2 i,ji where N is the total number of monomers and A defines the contact potential between them: A(r) = 1 if rlo, < r < rhigh and O otherwise. We take the strength of contact potential in a simple form: U(oi, oj) = BO + Boia-. [2] Bo < 0 provides average attraction between monomers; this sequence-nonspecific term shifts equilibrium toward globular states. B < 0 is a strength of sequence-specific interactions, it favors interaction between like monomers and shifts equilibrium toward separation. For the present study the choice of potential in the form of Eq. 2 has an advantage over the more commonly used potential when only monomers of one type attract each other (18). With the potential chosen in the form of Eq. 2, one can guarantee that the native states are compact, thus justifying the computationally feasible search over only compact conformations. Since the Monte Carlo procedure in sequence space converges to a canonical ensemble the probability of a sequence {IO} is: where P{ai'} = 38[ vi- (N. - Np)]exp(- ) [3] N~~~~~~ z=>l exp(e)181 cri -(NA -NB) [4] is the partition function in sequence space. NU and NA (= N - NB) are the total numbers of monomers of A and B types, respectively. 8 functions in Eqs. 3 and 4 allow only those sequences with a given amino acid composition. The rules of sequence optimization given by Eqs. 1, 3, and 4 are isomorphic to the Ising model of ferromagnetism, with the additional condition that the total magnetic moment is conserved. This magnetic model is on an inhomogeneous "lattice" {rn} and is unfrustrated (all bonds are ferromagnetic). The analogy between sequence optimization described by Eq. 3 and ferromagnetism is very helpful for understanding the key features of the process of sequence selection. The average energy of sequences that fit to the native structure at a given selective temperature Tsei is directly related to the energy of a magnetic system at the same temperature and can be found by using the usual thermodynamic relation: a E(Tsel) =-2sel a Tsel T 's [5] where I'=-Tsei In 2 is a free energy of the magnetic system and its sequence analog. Since "spin flips" in the magnetic system are related directly to mutations in the protein system, the number of sequences having energy E in the target Proc. Natl. Acad. Sci. USA 90 (1993) structure is directly related to the number of spin configurations at a given energy. The latter is exp(s), where S = -af/atsel is the entropy of a spin system. As TSe decreases, the magnetic system exhibits a secondorder phase transition at some Tfei; a fact that is well known from the theory of ferromagnetism (19). In ferromagnets, with the total magnetic moment conserved, the ferromagnetic ordered phase corresponds to macrodomains with spins having a preferred direction ("up" or "down" in each domain). In protein terms this means that sequences selected below TeLI are such that in the native structure, monomers of different types are strongly segregated from each other. The "left-right" rather than "in-out" is a feature of the symmetry of the potential but it does not change the basic physics. The statistical mechanics of transitions to the native state in the model with optimized sequence is illustrated in Fig. 1. Analysis given in Fig. 1 suggests that the folding transition occurs between the native state and a multitude of states belonging to the continuous part of the spectrum having energy ED. However, the continuous part is self-averaging; i.e., energy of the denatured state is not sequence-specific (only composition-specific). It follows from this scheme that ordinarily mutations will affect the native state more strongly than the denatured state. The reason for this conclusion is that in the optimized native state, the majority of contacts are best fit and therefore each contact "costs" more than in the denatured state, where fluctuations between folds lead to self-averageness (i.e., sequence independence of thermodynamic quantities). Mutations affect energy of the denatured state (provided that it does not have unique structure) only via change of composition and therefore average properties of a sequence. This does not contradict the assertion (20) that occasionally changes in the denatured state may be dominant. The simple graphical analysis of Fig. 1 is based on analytical theory whose details will be published elsewhere. The main goal of more rigorous theoretical calculations is to prove that Fig. 1 is realistic; i.e., that optimization leads indeed to this type of energy spectrum with the bottom part concave. / FIG. 1. Mechanism of the transition to the native conformation. The curve is entropy S(E) = In w(e), where w(e) is the density of states with energy E from the spectrum. In its continuous part this quantity is self-averaging (i.e., nonspecific to sequences; see above and Fig. 3). Ec denotes the boundary of the continuous part, and A is the energy gap between the native state and the continuous spectrum (8). The slope of the dashed line is 1/TC, where Tc is the temperature of the glass transition in a random heteropolymer (7). Since the thermodynamic relation ds/de = 1/Tholds for our system, construction of the tangent corresponds to the condition that the free energies of the native state and the denatured state are equal. Therefore, the slope of the tangent is 1/Tf, where Tf is the temperature of the folding transition to the native state. The energy jump is EN - ED, which gives the latent heat of transition to the native state; the entropy jump is SD. /

3 Biophysics: Shakhnovich and Gutin This appears to be the case in three dimensions and may be wrong in two dimensions. [The reason why 2 is the marginal dimension for linear compact polymers has been discussed (12, 13).] The theory is based on calculation of the average free energy of a protein: (F) = -kt X In Z{cr,}P{oa}, [6] {ad} where ( ) denotes sequence averaging with probability P{cr,} for a given sequence {or,} (as in Eq. 3). Factor P{a,} in Eq. 3 biases summation toward selected sequences-i.e., takes nonrandomness into account. Z is the configurational partition function of a protein with a given sequence: Z r,} exp - kbt Hg(ri-r)) [7] This is a usual expression for the partition function of a heteropolymer (e.g., see refs. 7, 8, 21, and 22). Summation is over all conformations of a protein which are expressed through Cartesian coordinates of the atoms. The g functions describe the covalent structure of the chain and impose restrictions on the mutual positions of monomers which are nearest neighbors along the chain. E({r,}) is the nonbonded energy of a protein; it depends on the conformation {r,} and, of course, on the protein sequence {fi}: i N E({r,}) = - X U(oi, ao)a (ri - rj) * [8] 2 i,j This expression for energy is the same as Eq. 1. However, Eq. 1 describes the energy of the native conformation, {r?}, whereas Eq. 8 relates to an arbitrary conformation of the protein with the same sequence {fl}. The thermodynamics of such proteins with evolutionarily selected sequences is defined by Eqs. 6-8, and allows analytical solution with the aid of the replica approach (7, 12). Results of this theory pertinent to the present discussion are as follows. (i) At high "selective" temperature, the energy of the native structure is higher than Ec (the boundary of the continuous part of the energy spectrum of a protein; see Fig. 1). As the real temperature decreases, the protein folds into some conformation from the bottom, discrete part of the spectrum, which is not related to the target structure. (ii) Decrease of the selective temperature provides sequences that deliver lower energy to the target structure. Below critical Tc 1, Eo(Tc 1) = Ec provides sequences with a large gap between the native structure and conformations from the continuous spectrum. When the selective temperature falls below Tfel (the temperature at which the ferromagnetic transition takes place in the magnetic counterpart), sequences fit to the target structure have strong separation between unlike monomers. The folding transition for such sequences which takes place when real temperature decreases involves a direct jump from random conformations belonging to the continuous energy spectrum to the target structure; this transition occurs as a first-order one (see Fig. 1 for more details). The condition for this scenario to be valid is that energy of one (native) conformation be significantly smaller than the energy corresponding to the boundary of the continuous spectrum, EC (an estimate for EC is given in the caption of figure 1 in ref. 8): ENAT << Ec = NzBo + NB[z(ln y)]1/2, [9] Proc. Natl. Acad. Sci. USA 90 (1993) 7197 where Bo and B are from Eq. 2, 'y is a measure of chain flexibility which has the meaning of effective number of conformations per monomer, and z is the average number of contacts per monomer. (Results of ref. 24 suggest that this estimate is valid also for the "two-letter" proteins which we consider here.) The minimal possible energy of the native state which is obtained at "ideal" separation (or at Tse, = 0) is obviously Emin = N(Bo + B)z - BO(N213). [10] The second term corresponds to the surface of the boundary between hydrophobic and polar groups. In some cases (smaller proteins or low coordination number of a lattice) this boundary may not exist. The condition of Eq. 9 with Eq. 10 implies that ln y << z. This means that a large energy gap between the selected ground state and other conformations is possible only for chains that are stiff enough. However, it is not very restrictive and may be satisfied for compact polypeptide chains for which y 1.4. (Overall compactness of polypeptides can be reached due to average attraction between monomers, reflected by the term Bo in our potential.) These concepts are tested by using a model with exhaustively enumerated conformations. This is a 27-mer on a 3 x 3 x 3 fragment of a simple cubic lattice (25). The total number of compact self-avoiding conformations unrelated by symmetry is 103,346 in this model. We consider for comparison two ensembles of sequences: random and selected. The first ensemble consists of random sequences of 14 A and 13 B monomers each having a nondegenerate ground state. To create it we generated 500 random sequences and chose 20 that had a nondegenerate lowest energy compact state. For these sequences three to four A-B contacts were usually observed in the ground state. For "selected sequences" we used the following procedure. A structure was picked randomly from the list of 103,346 compact structures. The sequence search was done by using the Monte Carlo sequence algorithm to provide a minimal number (not more than one) of AB contacts in a chosen structure; therefore T,ei was taken to be sufficiently low: TS,, = 0.1. Composition was fixed to be 14 A monomers and 13 B monomers. Of 20 structures, 8 allowed us to find sequences with complete separation between A and B monomers (like the one shown in Fig. 2). The remaining 12 FIG. 2. Ground-state conformation of a 27-mer on a 3 x 3 x 3 cubic lattice for a selected sequence (ABABBBBBABBABA- BAAABBAAAAAAB). Note that this conformation has no contacts between unlike monomers.

4 7198 Biophysics: Shakhnovich and Gutin Proc. Natl. Acad. Sci. USA 90 (1993) 0.3 co z w a U- LL 0.1 U'' E E FIG. 3. Density of states (entropy per monomer) for different energies for a random (open bars) and a selected (filled bars) sequence. A total of 103,346 compact conformations were taken for this plot. The scale for nondegenerate native structure for random and selected sequences (where S = 0) is slightly distorted to enable us to show these states. provided sequences with one A-B contact. We used the interaction potential given by Eqs. 1 and 2 with Bo = -2 and B = -1. This choice of parameters may seem not very realistic for proteins, as it provides separation of "left side" from "right side" rather than interior from exterior. However, as we mentioned before, the main process in protein folding is competition between separation and polymeric bonds, and this potential provides this feature. On the other hand this potential is much better than the "hydrophobic" potential (-1,0,0) used in ref. 18, because it guarantees that the ground-state conformation is maximally compact; therefore it is included in the enumerated set and is known. This is very important for kinetic studies (see below). The density of states for a selected and a random sequence is shown in Fig. 3. The spectrum for the selected sequence is wider, but differences between the two sequences are pronounced only in the bottom part, whereas the central part is indeed selfaveraging. The bottom part of the S(E) dependence for selected sequence is concave, which validates the idealized representation given in Fig. 1. Lower native energy makes the selected sequence more stable (Fig. 4). The summation in the partition function Z(T) is taken over all 103,346 compact configurations. This means that we S~ FOLDICITY FIG. 5. Distribution of "foldicities" for 20 selected sequences (filled bars) and 20 random sequences (open bars) with nondegenerate ground state. For each sequence 10 runs of 20,000,000 Monte Carlo steps were made. The foldicity of a sequence is defined as the fraction of runs at which ground state was found (10). consider stability against transition from the state with unique structure to the homopolymer-like compact state where multiple conformations are allowed. The possibility of these two states was pointed out in theoretical studies (7) and in experiments (26, 27). It can be seen that the native state for the selected sequence is twice as stable as for a random sequence: it has a transition temperature twice that of the random sequence. The ability of selected sequences to fold rapidly to the native state was analyzed by a Monte Carlo folding algorithm in conformational space (9). Selected sequences without or with only one AB contact deliver minimal theoretically possible energies to their target structures. In this case the conformation of the global minimum of energy is known for each selected sequence. With regard to random sequences, it is less clear whether their global minimum is fully compact (and therefore known) or not. However, in all our studies we never observed for these sequences conformations with energy lower than the one which we identified as the global minimum based on enumeration of compact conformations. "Foldicities" of two sets of sequences are compared in Fig. 5. Selected sequences have overall higher foldicity than random ones. An important question concerns the statistical properties of selected sequences. They are definitely nonrandom, but can this nonrandomness be revealed from statistical analysis of the primary structures of selected sequences? Fig. 6 gives a negative answer to this question. The distribution of lengths of poly(a) and poly(b) blocks in selected sequences is indistinguishable from those of completely random sequences. This type of analysis is often used to analyze the I I 1% 0.6 z w 5U024 Cc T FIG. 4. Plot ofthe stability ofthe ground state versus temperature for a "selected" sequence (solid line) and for a random sequence with nondegenerate ground state (dashed line). Stability is defined as the Boltzmann probability to be in the native state, PNAT(T) = exp(-e /kt)/z(t) BLOCK LENGTH FIG. 6. Length distribution for poly(a) and poly(b) blocks found in selected sequences (bars) and in random sequences (dots and line). The distribution of block lengths is exponential.

5 Biophysics: Shakhnovich and Gutin randomness of protein sequences (23); we see, however, that lack of such sequence correlations may not exclude nonrandomness, or uniqueness, of a primary structure. (This resembles the "magic glass" used to read scrambled texts. Meaningful text is recovered only when the glass is applied, or, in our terms, when a sequence is folded into its native conformation.) We formulated and investigated an ultraspecific model obtained by sequence selection. The main result of this work is that it is possible to design a sequence with stability and folding kinetics comparable with ones observed in ultraspecific models where the native state was singled out by special parameters of the force field. Many important properties of proteins with selected sequences were predicted by Bryngelson and Wolynes (3), who discussed folding of ultraspecific models. Of special importance is the introduction of two temperatures, Tf and Tc, which characterize the folding transition and the glass transition, respectively. This is directly applicable to our sequence model as well (see Fig. 1). Proper choice of a sequence with the low-energy native state ensures that Tf > T,. The glass transition for a heteropolymer means that at low temperatures few conformations from the bottom, discrete part of the energy spectrum dominate thermodynamically. When the native energy is significantly less than the energies of other conformations, the folding transition takes place at temperature above the glass transition and the system jumps to the native state directly from the continuous part of the spectrum. It is important to note that at any temperature "parasitic" conformations, which have low energies but are structurally very different from the native state, are unstable. This is because the same energy parameter (B) characterizes the native and the low-energy nonnative states. As was pointed out in ref. 3, such low-energy parasitic states may serve as kinetic traps with high energetic barriers for escape from them. That such traps are thermodynamically unstable at any temperature may be the main reason for rapid folding of selected sequences. Our "kinetic" statements are based on lattice studies of relatively short chains (27 monomers). Conclusions from lattice models not supported by analytical theory may be misleading in several cases. Much more work is necessary to reach a real understanding of the kinetics of protein folding. We are very grateful to Alexey Finkelstein and Oleg Ptitsyn for numerous fruitful discussions. E.I.S. has benefited from frequent illuminating discussions with Peter Wolynes and Martin Karplus about subjects raised in this work. We thank Andrej Sali for discussion of the protein folding problem and providing us with his graphics program ASGL. E.I.S. is a Packard Fellow, and financial Proc. Natl. Acad. Sci. USA 90 (1993) 7199 support from the David and Lucille Packard Foundation is gratefully acknowledged. Acknowledgement for partial support of this research is made also to the Donors of Petroleum Research Fund, administered by the American Chemical Society. 1. Taketomi, Y. H. & Go, N. (1975) Int. J. Pept. Protein Res. 7, Bryngelson, J. D. & Wolynes, P. G. (1987) Proc. Natl. Acad. Sci. USA 84, Bryngelson, J. D. & Wolynes, P. G. (1989) J. Phys. Chem. 93, Shakhnovich, E. I. & Gutin, A. M. (1989) Studia Biophys. 132, Skolnick, J. & Kolinski, A. (1990) Science 250, Garel, T. & Orland, H. (1988) Europhys. Lett. 6, Shakhnovich, E. I. & Gutin, A. M. (1989) Biophys. Chem. 34, Shakhnovich, E. I. & Gutin, A. M. (1990) Nature (London) 346, Shakhnovich, E. I., Farztdinov, G. M., Gutin, A. M. & Karplus, M. (1991) Phys. 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