Statistical Mechanics of the Combinatorial Synthesis and Analysis of Folding Macromolecules

Transcription

1 J. Phys. Che. B 1997, 101, Statistical Mechanics of the Cobinatorial Synthesis and Analysis of Folding Macroolecules Jeffery G. Saven*, and Peter G. Wolynes School of Cheical Sciences, UniVersity of Illinois at UrbanasChapaign, Urbana, Illinois ReceiVed: May 22, 1997; In Final For: July 25, 1997 X Cobinatorial cheistry techniques provide a proising route to the design of acroolecules that acquire predeterined folded conforations. A library of sequences based on a pool of different onoer types can be synthesized, where the sequences are partially designed so as to be consistent with a particular target conforation. The library is screened for folding olecules. The nuber of sequences grows rapidly with the length of the polyer, however, and both the experiental and coputational tabulation of sequences becoe infeasible. For polyers and libraries of arbitrary size, we present a self-consistent, ean-field theory that can be used to estiate the nuber of sequences as a function of the energy in a target structure. The theory also yields the probabilities that each position in the sequence is occupied by a particular onoer type. The theory is tested using a siple lattice odel of proteins, and excellent agreeent between the theory and the results of exact enuerations are observed. The theory ay be used to quantify particular design strategies and the facility of finding low-energy sequences for particular structures. The theory is discussed with an eye toward protein design and the utability of particular residues in known proteins. 1. Introduction Cheists have long been able to design and synthesize sall copounds, even when the olecular echaniss of such syntheses are not well-understood. Using such synthetic ethods, researchers can craft new olecules and probe their cheical properties, thus shedding light on the echaniss involved. For exaple, the physical organic cheist ight ake a series of copounds all based on a coon thee in her search for quantifiable trends. 1 The cheist is guided by cheical theory both in her choice of olecules and in her interpretation of the experients, experients which in turn infor the theory. This sybiotic pairing of synthetic cheistry and cheical theory has proven fruitful in such areas as cheical bonding, conforational analysis, and reactivity. Clearly, our cheist is ore likely to arrive at soe sort of generalized understanding if she can access and analyze a large nuber of copounds. Facilitated by recent advances in anaging cheical diversity, researchers can now synthesize, catalog, and assay huge libraries of olecules, where each eber of the library is a different cobination of a particular set of cheical building blocks. These cobinatorial cheistry experients are typically used to search for olecules having a desired property, i.e., the binding of a drug candidate to a receptor. By noting the nubers and coon features of olecules with the desired property, researchers can also use these sae experients to reveal uch about the underlying cheistry. Soe researchers in protein cheistry have begun to take a siilar cobinatorial approach to the design of particular folded architectures. For ost proteins, the polyer s three-diensional structure is deterined by its aino acid sequence. The protein cheist should be able then to design sequences that fold to a particular structure, once the physical cheistry of the folding is understood. While it is straightforward to ake linear polyeric sequences, the protein cheist is troubled by a nuber of coplications. Due to the size and coplexity of proteins, any Present address: Departent of Cheistry, University of Pennsylvania, Philadelphia, PA X Abstract published in AdVance ACS Abstracts, Septeber 15, S (97) CCC: $14.00 of the theoretical ethods used to study sall olecules, e.g., ab initio ethods and coplete conforational search, becoe infeasible. Siply synthesizing a representative nuber of sequences is a daunting task, since the nuber of possible peptide sequences is exponentially dependent on, the nuber of residues. Choosing an appropriate target structure is also not without abiguity. One recent study has found that although nearly half of all nonhoologous, globular proteins adopt one of nine failies of folds, any protein folds have little siilarity with any of the reaining protein structures. 2 In addition, in ost proteins no single interaction doinates the folding energetics; hydrogen bonds, hydrophobic effects, van der Waals interactions, and steric packing effects all act in concert to stabilize the folded structure, but the relative agnitudes of their respective contributions are still controversial. Much has been done in the design of particular sequences, but as the size of the peptides in the library grows, the explicit tabulation of all sequences quickly becoes intractable, experientally as well as coputationally. Yet surveying the whole library can reveal trends as to what interactions are likely to destabilize as well as stabilize particular structures. We discuss in this report a workable theory that subtends all ebers of a given library. The theory can aid in the analysis and partial (cobinatorial) design of proteins and other intraolecularly self-organizing olecules, and the theory can be used as a tool to winnow such libraries to a anageable size. In the design of foldable protein sequences, we ust first address why a particular aino acid sequence should fold to a unique structure. We consider this question fro the viewpoint of the energy landscape theory of protein folding. 3-6 Aino acid sequences are heteropolyeric and typically coprise a large nuber of the available aino acids in an apparently rando order. Upon nonspecific collapse of the rando heteropolyer, incoensurate residues are likely to be brought into contact with one another. In addition, charged residues ay be transiently buried in the interior of the globule, or hydrophobic residues ay be exposed to solvent. For nearly all such rando, collapsed states, not all of the interactions within the globule will be satisfied. The energy surface is said 1997 Aerican Cheical Society

2 8376 J. Phys. Che. B, Vol. 101, o. 41, 1997 Saven and Wolynes to be frustrated, and the connectivity of the polyer chain is largely responsible for this frustration. 7 In addition, the nuber of collapsed conforational inia is large; it is exponential in the length of the chain. As in spin glasses, the lowest energy states for such rando, frustrated systes need not be structurally siilar. 8 At low teperatures the rando heteropolyer ay becoe trapped in any one of these low-energy inia. On the other hand, proteins are able to negotiate this rugged terrain and do fold reversibly to unique structures. At physiological teperatures, a protein ust therefore have an additional feature to its energy landscape. The energy of the native (folded) state is sufficiently low such that it copensates for the free energy of the nonnative conforational states of the polyer. The free energy of the unfolded states is a function of the average energy of these conforations, the distribution of their energies, and their nuber (entropy). Due to the energetic ruggedness and the entropy of the nonfolded states, the energy of the folded structure ust be uch lower in energy than is typical for a collapsed, rando heteropolyer of the sae coposition. The structure of the folded state is low in energy because it has a large nuber of favorable interactions, both between residues and between a particular residue type and its local environent. This energetic consistency between a protein sequence and its folded conforation has been tered the principle of inial frustration. 7,9 A protein s aino acid sequence deterines its structure, and at present any different protein topologies are known. Given this inherent plasticity of peptides, researchers have attepted to sculpt particular protein architectures by careful choice of the aino acid sequence. Soe researchers interested in de novo protein design have focused on iniizing frustration, through hierarchical design schees. 10 For exaple, in designing four helix bundles, workers have used peptide sequences that are known to for stable helixes. These helixes are then engineered to have copleentary hydrophobic faces that interact upon aggregation. Appropriate linker sequences are added to connect the these helixes into one contiguous sequence. In one case, a disulfide bond connecting two helixes has been introduced to yield a conforationally defined, though sall, two-helix peptide. 11 Using these ethods, researchers have attepted to engineer helical bundles, R/β proteins, and β sheet proteins. In each case the global tertiary fold being designed is siilar to a naturally occurring one, but recently, Dolgikh et al. have aied for a truly novel protein structure. 12 The proteins so developed have any of the properties of natural proteins in that they are copact and have substantial aounts of secondary structure, but typically what fors is a olten globule state that has no well-defined tertiary fold. 13 otwithstanding, these efforts appear very proising, and this de novo design of particular sequences is likely to lead to artificial proteins soon. Alongside these experiental efforts, researchers have been tackling the issue of protein design fro a coputational viewpoint. Methods for scoring trial sequences are required, and researchers have developed a nuber of ways of quantifying which aino acid sequences ight be copatible with a chosen structure. The chosen structure specifies a three-diensional skeleton for the trial sequences. Richards and co-workers have deterined what utations are copatible with a given structure. 14,15 These authors used an atoistic approach in which they utated side chain residues and then iniized the energy. These ethods reveal uch about the interactions between side chains and the effect of utation on the packing of the individual residues. The coputational deands of such ethods, however, liit the nuber of sequences that ay be considered. Other researchers have developed eans of scoring trial sequences based upon the coparison of sequences that are known to fold to a particular structure in the Protein Data Bank (PDB) Focusing priarily on backbone degrees of freedo, researchers have also considered siplified odels of proteins, where reduced energy functions that involve effective residue-residue interaction energies have been used. The ost coonly used energy functions are the inforationbased potentials, those that are derived statistically fro the probability of occurrence of a particular residue contact pair in the PDB While uch concern reains about the reliability, eaning, and accuracy of such potentials, these potentials are at present the ost useful reduced description for the free energies involved in folding. General issues regarding design using reduced descriptions have been addressed More concretely, soe workers have used such potentials to verify design algoriths using both off-lattice, 32 and ore coonly, lattice odels of proteins. 33,34 Generally, such algoriths search for low-energy structures using a siple energy function, where utations are randoly introduced along the chain. The search ay be done by screening against a pool of known sequences 17 by using genetic algoriths 35,36 or by using Monte Carlo ethods. 33,37-41 Other researchers have considered the polyerization of sequences on a lattice in the presence of an object to which the polyer binds as a route to designed ( iprinted ) structures. 42 In one design schee, the energy of the sequence is iniized at constant coposition, under the assuption that the free energy of the unfolded enseble of states is solely a function of the total nubers of each type of residue. 33 (If the coposition were not constrained, a hoopolyer would be the lowest energy sequence, which we know does not fold to a unique structure.) We ust ention that a sequence with a low-energy in the target structure does not necessarily fold uniquely to that structure; that sequence ay acquire a different conforation that is still lower in energy. 43 Researchers have addressed these issues by using alternate design criteria that take into account structures other than the target ,40 All these ethods are liited, however, in that they rely upon the explicit tabulation of sequences and the subsequent evaluation of each sequence s energy in a particular structure (and in soe studies nonnative structures). While uch can be learned about the sequence-structure apping, 44,45 with the exception of very siple systes, only a sparse sapling of sequence space of real proteins is possible. Our knowledge of the principles of protein design reains incoplete, a fact which otivates the use of a cobinatorial approach to protein design. By synthesizing large nubers of peptide sequences, researchers not only enhance their chance of discovering sequences that fold to a particular structure but they also stand to learn what properties foldable sequences in a library share. Cobinatorial ethods can reveal generalizable principles about the forces that stabilize protein structures. The design of specific sequences typically only probes portions of the sequence landscape for a chosen target structure. Given that any sequences can share a coon structure, we seek to understand as large a tract of the sequence energy landscape as possible. In addition, in designing novel proteins, dictating the specific contacts between residues or the precise packing aong residues ay be unnecessary. This fact is underscored by the presence of peptides with a nuber of protein-like properties that have been isolated fro rando sequence libraries. 46,47 Cobinatorial surveys have also been used to search for sequences that are copatible with a given structure. Katekar et al. have synthesized a library of peptide sequences, where a

3 Folding Macroolecules J. Phys. Che. B, Vol. 101, o. 41, binary patterning of hydrophobic and hydrophilic residues was chosen to be consistent with a four-helix bundle. 48 The sites on the interior (exterior) of the structure were chosen to be hydrophobic (hydrophilic). The precise aino acids at the these positions, however, was allowed to fluctuate. At each hydrophobic and hydrophilic position, five and six possible residue types, respectively, were peritted. Of a sapling of 48 sequences that were correctly expressed, 29 of these were protein-like in that they were resistant to proteases. Soe of these sequences also folded to copact structures and had significant secondary structure. This partial design, based on a siple binary pattern, not only yielded olecules with proteinlike properties but the experient confired that nonspecific hydrophobic interactions, with concoitant secondary structure foration, are iportant deterinants in specifying the folded structure. The issue of which aino acids in a sequence are necessary to for a stable structure and which are not ay be addressed by studying utations in naturally occurring proteins. The Matthews group has synthesized any stable utants of T4 lysozye 49 and even a few stable double utants. 50 The core residues of λ repressor can be very robust with respect to utation; 70% of 125 cobinations of the hydrophobic residues yield biologically active proteins. Axe et al. have found that 12 of the 13 core residues of barnase ay be substituted and it still retains its biological activity, 51 thus indicating that siply aintaining a hydrophobic core is a doinant factor in the stability of the enzye. Itzhaki et al. have found that chyotrypsin inhibitor 2 can support 100 different utations. 52 In addition to laboratory utagenesis, researchers have also considered the utability of the residues in structures contained in the PDB. 53 Often, though not always, these conserved sites are crucial to the function or stability of the protein. Fro the viewpoint of designing new proteins and understanding utational variability using cobinatorial libraries, the synthetic protein cheist would prefer to know a priori the nuber of sequences that are likely to fold to a given structure and the identities of those sequencessat least in soe average sense. Given the nuber of possible sequences of even a oderately sized protein of 100 residues, , obtaining an understanding of even this library sees at first glance to be an insuperable task. onetheless, a serviceable theory for the distribution of the energies of different sequences in the target structure can guide the cheist in his design and interpretation of his experients. Katekar et al. used qualitative cheical concepts to guide the design of their four-helix bundle library, but a quantitative theory has the potential to have even a stronger and ore detailed predictive power. In this paper, we present such a theory. Counting the nuber of sequences as a function of the energy is a task well-suited to statistical echanics. We are further aided by the observation that the protein design proble has uch in coon with the Ising agnet of condensed atter theory. 33 A coonly used approach in condensed atter physics is ean-field theory, wherein we consider average local energies associated with each site that are deterined by that site s local environent. Meanfield theory has seen extensive application to protein folding and the exploration of conforations, but here we use it to quantify the characteristics of sequence space. Here the internal variables are not the conforational states of the onoers but, rather, the type of aino acid that is present at each sequence position. The theory can also accoodate restrictions iposed by the researcher, such as constraints on the nuber of each type of onoer and on the residue identities allowed at particular positions in the sequence. The theory provides us with a way to estiate not only the nuber of sequences for a given overall energy but also the probability that each sequence position is occupied by a given onoer type. The theory provides a convenient eans to evaluate different cobinatorial design strategies, where we can copare the effects of changing residue pattern and target structure. In this report, we present the odel and copare it with results of an exactly solvable lattice polyer. 2. Theory for the uber and Coposition of Sequences Copatible with a Chosen Structure In this section we present a ean-field theory for counting the nuber of sequences copatible with a particular structure. The theory also yields the probability that each site is occupied by a particular type of onoer. Here we consider only the distribution of energies for different sequences in a chosen structure. Although siply finding the iniu energy sequence can be a isleading design strategy for soe energy functions, 58 sequences that fold to the chosen structure ust surely reside aong those that are sufficiently low in energy. Thus the theory is useful in that, for a particular design strategy, it provides estiates of the nuber and average identities of the low energy sequences. In our developent of the theory, we neglect sapling issues. All possible sequences of a typical protein cannot be synthesized, e.g., a polypeptide of length of 100 residues has ore than possible sequences. Thus any laboratory experient ust reflect a sapling of all possible realizations. In what follows, we assue that the space of all possible sequences is sapled uniforly, so that the probabilities presented below should iic those seen experientally. Since we are interested in characterizing the full sequence space, we neglect variation in the nubers of sequences due to incoplete sapling of the sequence space and non-unifor distribution of the onoers available at each sequence position that ay be present in actual experients. 48 We outline the presentation of the theory. In section 2.1 we show how the probability that sequences are less than a certain energy ay be calculated using the nuber of sequences as a function of the energy. Thus we need to deterine a icrocanonical entropy S(E), which is generally an involved calculation. For one-body energy functions, however, we can write siple expressions for the entropy (section 2.2) that are valid for arbitrary (section 2.2.1) and fixed (section 2.2.2) total copositions. We review how effective one-body energies ay be obtained using a siple ean-field theory in section 2.3. Last, we discuss the quantities of cheical interest that the theory yields, the ost iportant of which are S(E), the logarith of the nuber of sequences, and the individual probability w i (R) that a site i is occupied by a particular onoer type R Distribution of Energies Over Sequences. Here we let E be the energy of a sequence when it assues the folded or target conforation. Ω s (E) is the nuber of sequences having energy E in the chosen structure, and Ω s is the total nuber of sequences. Both Ω s (E) and Ω s satisfy any constraints concerning the total nuber of each type of onoer or the allowed onoer identities at particular sites. The sequence entropy S(E) is the logarith of the nuber of states having energy E in the folded structure. S(E) ) k B ln Ω s (E) (1) The sequence entropy S(E) is defined in a way that is exactly analogous to Boltzann s epitaphic equation for the entropy.

4 8378 J. Phys. Che. B, Vol. 101, o. 41, 1997 Saven and Wolynes In calculating the nuber of sequences of a particular energy, we focus on estiating S(E), exploiting the tools of statistical therodynaics. For low energies, S(E) an increasing function of E. AsE increases, the frustration present in the folded structure increases. As the nuber of unfavorable interactions increases, there are ore ways to distribute the, and the sequence entropy increases. To find the probability of finding a sequence with energy less than E in the chosen target structure, we integrate the Ω s (E) up to the chosen energy E. E f(e) ) - de Ωs (E )/Ω s (2) Alternately, if the allowed energies are discrete, as they are in a lattice odel, we su over allowed energies E < E. E f(e) ) Ω s (E )/Ω s (3) E ) The Sequence Entropy for S(E) for One-Body Energy Functions. We now turn to the estiation of S(E). Generally, for systes coprising any interacting sites, calculating the icrocanonical entropy exactly is nontrivial. If the energy of the syste is the su of individual energies of each coponent, however, then equations for the entropy ay be straightforwardly obtained. The ideal gas is one exaple. In our case, the interacting eleents are the individual residues, as they occur the target structure. Effective energy functions, i.e., the profile scoring functions, have been developed by Bowie et al. that depend only on the identity and location of a residue in a particular structure. 16 Hence these types of energy functions are of a purely one-body for, for which the theory presented here is exact. For ore coplicated energy functions that involve any-body interactions, we can use the ean-field theory presented in section 2.3 to obtain a self-consistent effective one-body energy function. For such an energy function, the energy of a particular sequence in the chosen structure is given by E ) ɛ i (R i ) (4) Here is the length of the polyer in onoer units, and ɛ i is the effective one-body energy at site i in the structure. The sequence is denoted by an ordered list of onoer identities {R 1... R }, where R i is the onoer type present at sequence position i. Since the conforation of the polyer is fixed, the index i labels both a particular onoer s (one-diensional) position in the sequence and its (three-diensional) position in the target structure. ɛ i is a function of the onoer identity R i present at site i. The sequence entropy is obtained by axiizing the entropy S(E) with respect to any unconstrained internal paraeters. Since the energy ay be written as a su of individual one-body ters, the S(E) ay be expressed as S(E)/k B )- R w i (R i )lnw i (R i ) (5) where k B is Boltzann s constant, and is the nuber of onoer types. In the case of peptides, is just the nuber of different aino acids used in synthesizing the sequences. Here w i (R) is the probability that residue type R is at position i in the structure. The sequence entropy S(E) is axiized subject to the constraint that the total energy is conserved. We incorporate this constraint by restricting the value of the internal energy U such that U ) E, where U ) R ɛ i (R i )w i (R i ) (6) An additional constraint is that the su of the identity probabilities on each site is unity, i.e., each site ust be occupied by at least one onoer 1 ) w i (R) (7) R)1 If by design, onoer R is precluded fro occupying site i, then we have additional constraints of the for w i (R) ) 0, if R is not allowed at site i (8) There are - i such constraints for each sequence position i, where i is the nuber of allowed residue identities at position i. As is done in conventional statistical therodynaics, 59 the values of the individual identity probabilities are those that axiize the entropy, eq 5, subject to the constraints in eqs 6-8. We perfor this axiization using the calculus of variations and introduce Lagrange ultipliers for each of the constraints. The ultipliers that arise fro the constraints given in eq 7 are easily evaluated using the constraint conditions eq Arbitrary Total Coposition. Here we consider the case where the only restrictions on the total nubers of each type of aino acid in a sequence are those dictated by the allowed aino acids at each site (see eq 8). For such a syste, after axiization of eq 5 subject to the constraints eqs 6-8, the sequence entropy ay be written as S(E) k B ) β U + ln z i (9) where β is an unevaluated Lagrange ultiplier, and z i ) ξ i (R i ) exp(-β ɛ i (R i )) (10) R i )1 ξ i (R i ) arises due to possible constraints on the allowed residue types at each sequence location (see eq 8), and it obeys ξ i (R i ) ) { 1 if residue R i is allowed at site i 0 if not (11) The site identity probabilities are given by a for that looks very uch like a Boltzann weight, w i (R) ) z ī 1 ξ i (R) exp(-β ɛ i (R)) (12) The Lagrange ultiplier β satisfies the constant energy constraint (see eq 6). ɛ i (R i ) exp(-β ɛ i (R i )) U ) R ξ i (R i ) (13) z i The constant energy constraint eq 6 introduces the Lagrange

5 Folding Macroolecules J. Phys. Che. B, Vol. 101, o. 41, ultiplier β. Using the analogy to statistical therodynaics, we see that β -1 has all the properties of an effective teperature, the conjugate variable of the energy in therodynaics, e.g., k B β ) S/ E. As the energy increases, so does β -1. Other researchers have referred to β -1 as the selection teperature. 29,31 In the language of therodynaics, β -1 is the teperature at which the average internal energy of the syste is U. For large Ω s (E), sequences sharing the sae effective teperature β -1 will have the sae internal energy. For large β -1 (high E near the axiu of S(E)), any different rearrangeents of the onoer types are consistent with these high energies, and S(E) is relatively insensitive to the precise location of each onoer type. At low effective teperatures β -1 (low E), the requireents on the sequences having these low energies are severe, and the energy of a sequence in the chosen structure is acutely sensitive to the particular ordering of the onoer types. We noralize the icrocanonical entropy so that it yields the correct total nuber of possible sequences. For an arbitrary total coposition, when no restrictions are placed on the nuber of each type of onoer, Recall that i is the nuber of onoer identities allowed at position i. The sequence entropy is noralized so that ) Ω s (15) Here E in and E ax are the iniu and axiu allowed values for a given energy function Fixed Coposition. Here we treat the case where the nuber of each type of onoer n(r) is the sae for each sequence. This we refer to as the constraint of constant coposition. The locations of the individual identities, however, can fluctuate and are constrained only by eq 8. Let n(r) bethe total nuber of residues of onoer type R. Recall that w i (R) is the probability that site i has residue type R. In applying this constraint, we see that for each sequence the su of the individual identity probabilities over sequence positions ust equal the nuber of onoers of that type. Recall that if the residue identity R is not allowed at site i, i.e., ξ i (R) ) 0, then w i (R) ) 0, as in eq 12. As in section 2.2.1, we axiize the entropy S(E) with respect to the constraints of constant energy (eq 6) and probability conservation (eq 7). We also include the constant coposition constraints eq 16, each of which has a corresponding Lagrange ultiplier we denote as the product β µ R. The resulting icrocanonical entropy is S(E) is where Ω s ) i (14) E axde Ein exp ( S(E) k B ) n(r) ) w i (R) (16) S(E)/k B ) β U + ln z i + β µ Rn(R) (17) R)1 z i ) ξ i (R) exp(-β (ɛ i (R) + µ R)) (18) R)1 The sequence identity probabilities becoe w i (R) ) z ī 1 ξ i (R) exp(-β (ɛ i (R) + µ R)) (19) where z i is given in eq 18. Using the constant coposition constraint in eq 16 we can write down a self-consistent equation that µ R ust satisfy. β µ R ) ln[ z 1 ī exp(-β ɛ i - ln n(r) (20) (R))] We can again draw the analogy to statistical therodynaics. As we discussed previously, β -1 has the properties of an effective teperature. The constant coposition constraints eq 16 require that the nuber of each onoer type is constant. The therodynaic conjugate variables of the nubers of each coponent are their effective cheical potentials µ R. The effective cheical potential µ R is an effective free energy per particle for onoer type R. For a specified overall coposition, soe onoer types ay be ore favorable than others on average. The effective cheical potentials readjust to aintain constant coposition in such cases. When the nuber of each onoer type is predeterined, the total nuber of sequences is given siply by the corresponding ultinoial coefficient 59,60 Ω s )! n(r)! R)1 (21) where the reader will recall that is the chain length in onoers, is the total nuber of available residues, and n(r) is the nuber of residues of type R. Equation 21 holds when all onoer types are allowed at each of the sequence positions, i.e., i ) for all i. The sequence entropy S(E) in eq 17 is noralized according to eq 15, where eq 21 is used to define Ω s Mean Field Treatent of the Molecular Energy Function for a Given Target Structure. In section 2.2, we presented the sequence entropy S(E) and the sequence identity probabilities w i (R) for arbitrary and constant copositions. The foralis applies when the energy ay be written as a su of one body ters. Though soe authors have chosen to describe sequence-structure copatibility in ters of one-body functions 16 for which the theory is exact, the ajority of energy functions involve two-body interactions in the for of a contact potential. 17,19,22 These are paraetrized such that the probability that two residues are within a prescribed distance of one another yields an effective contact energy. Typically, such potentials are derived fro the frequencies with which each type of contact occurs in a representative set of known protein structures. Soe authors have also even included three-body interactions between residues. 17,61 The siple foralis presented below ay be easily extended to energy functions involving arbitrary anybody interactions. For the sake of siplicity, we consider here only energy functions that involve a su of one- and two-body ters. The siple route to an effective one-body energy is based upon a well-known tool in condensed atter physics, that of a ean-field theory. The interactions aong the coponents of a particular syste ake the syste s statistical behavior nontrivial; the energy present at a particular site depends on the states, identities, and positions of the other coponents with which it interacts. In a siple for of ean-field theory, 62-65

6 8380 J. Phys. Che. B, Vol. 101, o. 41, 1997 Saven and Wolynes the fluctuating local energy at a particular position is replaced by the average energy of the interactions of the particle with its neighbors. The averaging is done over different realizations that are consistent with a particular set of therodynaic conditions. We take a siilar approach here. We note that any other researchers have applied the technique to biocheical probles, but usually with the intent of understanding conforational statistics Shakhnovich and Gutin have noted that protein design has any features in coon with the rando field Ising agnet, 33 and this approach otivated their design algorith. Other physicists have developed design algoriths with this analogy in ind. 37,40 Though ean-field theory does poorly in the critical region for three-diensional systes, 64,65 the critical region is not of interest in design probles where we are concerned priarily with configurations (sequences) of low energy or low effective teperature. Furtherore, ean-field theory is known to iprove as the diensionality of the syste increases, that is as each particle interacts with a larger nuber of the reaining particles. The average coordination nuber c i in proteins can fluctuate and depends on the prescribed interaction radius r c for two-body interactions, but in general it can be quite large: when r c ) 12 Å, as it does for soe energy functions, 22,61 then c i 20. (In this crude exaple, r c is the distance between alpha carbons, and we have used the PDB structures of yoheerythrin, yoglobin, and ribonuclease A). The energy function ost coonly applied to proteins includes one- and two-body ters. For a sequence whose site identities are specified by the configuration {R 1... R }, where R i is the identity of site i, the energy of the sequence when it takes on the native conforation is E ) γ i (R i ) + γ ij (R i,r j ;r ij ) (22) i*j where is the total nuber of residues, γ i (R i ) is an energy contribution due to the presence of residue type R i at site i, and γ ij (R i,r j ;r ij ) is the interaction between sites i and j when their onoer types are R i and R j, respectively. Typically, γ ij is a function of r ij the distance of separation between sites i and j. The one-body ter γ i quantifies the propensity of onoer types to reside in a particular structural context. 16,22 For exaple, different aino acids will have different propensities to reside in an alpha helix or other secondary structure. Also included in γ i are any propensities for onoers to be buried within the globule or exposed to solvent at the target structure s surface, the surface accessibility. The two-body ter γ i,j can be used to quantify interresidue contact propensities, 19,20,22,23 as well as excluded volue interactions. For the sequence design proble discussed here, all the r ij are deterined by the chosen target conforation and do not fluctuate. Therefore we will suppress the dependence of γ ij on r ij. We define ɛ i (R i ) as the local field (energy contribution) at site i when it is occupied by residue type R i due to any local energy contribution plus that is due to its interaction with its neighbors. ɛ i (R) ) γ i (R) + γ ij (R,R j ) (23) j)1 In our ean field treatent, we assue that ɛ i (R i ) can be replaced by its average value. The best choice for an effective one-body site energy is well-known and ay be obtained using the Gibbs-Bogobiulov inequality 66 ɛ i (R) ) γ i (R) + j)1 The average denoted by... is over all the possible identities of the neighboring residues at a given energy. The joint probability for a particular arrangeent of onoer identities aong site i s neighbors is siply the product of the individual site identity probabilities w j (R j ). Since w j (R j ) is noralized to unity (see eq 7), the average contribution due to the two-body ter is γ ij (R,R j ) j)1 ) j)1 Recall that the γ ij are nonzero only if residues i and j interact with one another. Therefore, the su only has nonzero contributions fro the neighbors of residue i. The resulting effective energy function is a su of one-body ters ɛ i (R), ɛ i (R) ) γ i (R) + j)1 Equation 26 provides us with an expression for the effective local field in ters of the putative one- and two-body interactions γ i and γ ij. The reader will notice, however, that the w j (R j ) also are functions of the ɛ i (R) (see eqs 12 and 19). Thus for a given energy function, we solve self-consistently for the weights w i (R) and the effective site energies ɛ i (R). ote that, with this for of the effective site energy eq 26, the contribution due to two-body interactions in eq 6 is double counted. 54 Therefore, we ust subtract one-half the value of this two-body energy contribution. E ) U - 1 / 2 R j)1 where U is defined as in eq 6. Alternately, we could have also subtracted a constant ter fro each of the energies ɛ i (R). Regardless, no correction is necessary for the individual site identity probabilities w i (R). ote that this developent of an effective one-body energy function is siilar in spirit to the quasi-cheical approach in the theory of the Ising agnet and binary alloys. 63,64,66 In the protein case considered here, however, all the sites are not equivalent. The equivalence aong sites in the lattice Ising agnet siplifies the solution of the that proble, but it is the nonequivalence of the sites in the protein proble, the heterogeneity of the sequence and structure, that is paraount for the ability of the protein to fold to a unique structure Suary of Theory. We now suarize the ipleentation of the theory. In section 2.3 we presented how an effective one-body energy (eq 26) ay be obtained fro an energy function that involves a one-body (profile) and twobody (interonoer contact) interactions. The effective site energies ɛ i (R) and site identity probabilities w i (R) can be solved for self-consistently using eqs 26 and 12, or, if the coposition is fixed, eqs 26 and 19. The icrocanonical sequence entropy γ ij (R,R j (24) ) Rj)1 γ ij (R,R j )w j (R j ) (25) Rj)1 γ ij (R,R j )w j (R j ) (26) Rj)1 γ ij (R i,r j )w i (R i )w j (R j ) (27) U ) R w i (R i ) ɛ i (R i ) (28)

7 Folding Macroolecules J. Phys. Che. B, Vol. 101, o. 41, S(E) is then obtained using eq 9 or eq 17. In each case, the sequence entropy is noralized according to the total nuber of possible sequences (see eq 21). We ay then use eq 2 to obtain the cuulative probability that a sequence having energy less than E in the target structure exists. In practice, we solve for the ɛ i (R k ) and µ k for a given value of β, and then use eq 27 to deterine the corresponding energy for these paraeters. In therodynaic ters, this is equivalent to choosing a teperature for the syste β -1 such that the internal energy is equal to the energy of interest. For a given target structure, the theory yields the probability w i (R) that a site i in the sequence is occupied by the onoer type R at a given energy. The theory also provides the logarith of the nuber of sequences S(E) that have a particular energy in the target structure. The theory ay be used to calculate a quantity that reflects the utational variability of each site. We will refer to this quantity s i as the local sequence entropy. 45,53 s i )- w i (R)lnw i (R) (29) R)1 where the su is over the residue identities at site i. Depending upon the structure, soe sites are ore likely to have a particular identity than others. For a given value of the overall energy E, the local sequence entropy s i is a easure of how likely utations are at a particular residue position. ote that if only one residue type is peritted at site i, then s i ) 0. If all possible residue types are equally likely at site i (w i (R) ) 1/ i ), then s i ) ln i. Recall that i is the nuber of onoer identities allowed at site i. At a given total energy E, a site having a sall value of s i has an identity that is likely to be conserved across different sequences. 3. Results and Applications of the Model to Lattice Models of Proteins The focus of this study is to illustrate the validity of the selfconsistent ean-field theory presented in section 2. For natural protein structures, there is currently soe controversy about the proper choice of energy functions, 27 and the nuber of available onoers (the aino acids) is large (20), iplying that the nuber of possible sequences is gigantic (20 ). For these reasons, we initially apply the theory to a syste that ay be uch ore easily understood. We choose the well-studied 27- er cubic lattice polyer. 34,67-70 In such lattice odels, siple energy functions ay be used for which soe sequences exhibit protein-like behavior: any conforations are possible but one conforation is therodynaically preferred. The siplicity of the odel akes it easy to discuss quantities that appear in the theory in ters of particular structural features. Furtherore, if the nuber of allowed onoer types is sufficiently sall, all possible sequences ay be enuerated for any given target structure, and hence the lattice polyer provides an exactly solvable odel that we can copare with the theory presented in section 2. We choose energy functions that are of a pure two-body (contact) for. In the theory presented in section 2, no siplifying approxiations are ade with regard to the onebody ter in a given energy function (see eq 22). In addition, ost of the energy functions that have been used for the lattice odels involve only interresidue contacts. Therefore, in illustrating the theory, we consider energy functions that involve only two-body interactions, i.e., all the γ i (R) ) 0. In the inialist odel we consider here, there are just two types of residues, H and P, which crudely iic hydrophobic and Figure 1. Copact structures of the 27-er cubic lattice odel. (A) The structure of Li et al. that is the conforational ground state of 3794 sequences. 70 (B) The ground state conforation of the 002 sequence of Socci and Onuchic. 71 hydrophilic residues in proteins. 5 we consider is of the for E ) i j>i Each energy function that γ ij (R i,r j ) (30) where R i, the aino acid identity, is either H or P. The γ ij are nonzero only if the sequence positions i and j are nearest neighbors in the target structure and if i and j are not adjacent in the sequence ( i - j > 1). In addition, the contact energies between dissiilar residues are syetric γ ij (H,P) ) γ ij (P,H). Thus the energy of a particular sequence is solely a function of the nuber of H-H, H-P, and P-P contacts that are ade within the target structure. The two energy functions that we consider in sections 3.1 and 3.2 differ only in the relative strengths of these three types of contacts. In each case, the energy function is known to possess therodynaically foldable sequences. That is, for soe sequences, at sufficiently low teperatures one conforation has the doinant Boltzann weight. Due to the discrete nature of both the energy functions and the lattice conforations, the exact total energy of each sequence ay only take on specific discrete values. We consider two copact lattice conforations of recent interest. For a specific choice of the contact energy function, a conforation recently highlighted by Li et al. (structure A in Figure 1) is the lowest energy conforation for 3794 sequences. 70 In contrast, soe conforations are the lowest energy state of only a few sequences, or none at all. For a different choice of energy function, structure B in Figure 1 is the lowest energy conforation of just two sequences. One of these, the 002 sequence, has been studied extensively with respect to its folding kinetics and therodynaics. 71,72 In the reainder of this section, we copare the exact results with those of the theory, which takes as input only the target structure and a given energy function. The exact results are obtained by enuerating all sequences for each target conforation. With only two types of onoer, the total nuber of possible sequences for the 27-er is large, 2 27 ) , but is still easy to enuerate coputationally. 67,70,71 For a given target structure, we group sequences according to their energy. In so doing, we obtain the exact S(E) that ay be copared with the self-consistent theory. Fro the explicit tabulation, we ay also obtain the exact individual site identity probabilities w i (R) and, using the w i (R), the exact local sequence entropies s i. We copare these three quantities with the values that are estiated by the theory Energy Function that Favors Buried Hydrophobic Residues. In this section we consider an energy function of the pure contact for in eq 30. Here we choose a for of the

8 8382 J. Phys. Che. B, Vol. 101, o. 41, 1997 Saven and Wolynes Figure 2. The sequence entropy S(E)vsEfor structure A. The energy function is that specified by eq 31. The heterogeneity of the sequences are varied by changing the ratio of polar (P) to hydrophobic (H) onoers. The following total P:H copositions were used: 14:13 (circles), 17:10 (squares), 20:7 (diaonds), 24:3 (triangles), and 26:1 (inverted triangles). The curves are the theoretical results for each coposition. energy function that is very siilar to one used recently by Li et al. 70 γ ij (H,H) )-3ɛ, γ ij (P,P) ) 0ɛ, and γ ij (H,P) )-ɛ (31) For this choice of the energy function, copact conforations have lower energies than extended ones. In addition, it satisfies the inequality γ ij (H,H) < γ ij (H,P) < γ ij (P,P), and the energy decreases for those sequences that have larger nuber of contacts involving H onoers. Hence, the energy function favors placing H onoers in the interior of a particular structure, since positions in the interior have the largest coordination nubers. Sequences with any H onoers in the interior are ore likely to have large nubers of the lowest energy H-H contacts. Thus the energy function favors placing H onoers on the interior of a conforation, which confors with the burial of hydrophobic residues seen in folded protein structures. This energy function also satisfies γ ij (H,H) + γ ij (P,P) < 2γ ij (H,P), which iplies that dissiilar onoers favor segregation within the collapsed globule. In the study of Li et al., an effective H-H contact strength of γ ij (H,H) )-2.3ɛ was used, but these authors ention that their results were not sensitive to the precise value of γ ij (H,H), as long as the above inequalities are satisfied. For structure A, we consider the nuber of sequences having a given energy in this target conforation. In Figure 2, we plot the sequence entropy S(E) for five different overall onoer copositions. The heterogeneity of the sequences decreases as the ratio of polar to hydrophobic residues (P:H) increases. The copositions considered have the following values (P:H): 14:13, 17:10, 20:7, 24:3, and 26:1. In each case there is excellent agreeent between the theory and the exact results. The predicted entropy S(E) is slightly lower than the exact result for low and high energies, but it covers the sae range of energies as the exact result with rearkable fidelity. The axiu value of S(E) decreases with increasing hoogeneity, since there are fewer sequences for the ore hoogeneous copositions (see eq 21). ote that, as the sequences gain increasing P content and becoe ore like a hoopolyer, the energies at the axia of these distributions increases. With Figure 3. The fraction of sequences f(e) having energy less than E for structure A. The energy function in eq 31 was used. The following total P:H copositions were used: 14:13 (circles), 17:10 (squares), 20:7 (diaonds), 24:3 (triangles), and 26:1 (inverted triangles). The curves are the theoretical results for each coposition. increasing P content, the nuber of stabilizing H-P and H-H contacts decreases. Furtherore, as the heterogeneity of the sequences decreases, the width of each distribution also decreases. The ost heteropolyeric coposition (14:13) spans the largest range of energy values. The corresponding P- hoopolyer distribution (27:0) has zero width at E ) 0 (not shown). In Figure 3 we present the cuulative probability f(e) for each of the five chosen copositions (see eq 3). Since f(e) is siply the integral of S(E), close agreeent with the exact result is again obtained. ote that at low energies, S(E) drops rapidly as the energy decreases. It is this low-energy region that is ost likely to contain sequences that fold to the target structure. The identity probabilities w i (R) reveal which residue types at each position are responsible for stabilizing these low energy sequences. In Figure 4, we plot the probability w i (P) that each site of conforation A is occupied by a polar residue for different energies. The P:H coposition in this case is 14:13. By coparing with Figure 2, we see that the nubers of sequences at each energy are Ω s (E )-50ɛ) ) 118, Ω s (E )-45ɛ) ) , and Ω s (E )-45ɛ) ) ote that at each of the energies, there is excellent agreeent between the theory and the exact tabulation. At the lowest energy considered E ) -50ɛ, any of the sites are predoinantly occupied by one type of residue; w i (P) 1 or 0. For the given coposition, there are severe restrictions on the allowed residues at each position if a sequence is to have this low energy in the target structure. As the energy is increased, these restrictions are relaxed, and a few high-energy contacts are present. The site identity probabilities take on values interediate between 0 and 1. At the energy that axiizes S(E) nearly all residue types are allowed at each position, subject only to the constraint of constant coposition: w i (P) 14/ (not shown). On the low-energy side of S(E), the residues that are ost likely to reain hydrophobic with increasing energy are those in the center of this conforation, residues 1, 26, and 27 (see Figure 1). Each of these residues is coordinated with four other nonbonded neighbors, a coordination nuber that is larger than that of any of the other residues. For low-energy sequences, their high coordination nubers iply that these residues are likely to be hydrophobic (H). Hence w i (P) should be sall. The energy function favors hydrophobes at sites of high coordination, which are likely to be buried within the structure.