Method for Predicting the State of Association of Discretized Protein Models. Application to Leucine Zippers

Transcription

1 Biochemistry 1996, 35, Method for Predicting the Stte of Assocition of Discretized Protein Models. Appliction to Leucine Zippers Michl Vieth, Andrzej Kolinski,, nd Jeffrey Skolnick*, Deprtments of Chemistry nd Moleculr Biology, The Scripps Reserch Institute, North Torrey Pines Rod, L Joll, Cliforni 92037, nd Deprtment of Chemistry, UniVersity of Wrsw, Psteur 1, Wrsw, Polnd ReceiVed August 31, 1995; ReVised Mnuscript ReceiVed NoVember 13, 1995 X ABSTRACT: A method tht employs trnsfer mtrix tretment combined with Monte Crlo smpling hs been used to clculte the configurtionl free energies of folded nd unfolded sttes of lttice models of proteins. The method is successfully pplied to study the monomer-dimer equilibri in vrious coiled coils. For the short coiled coils, GCN4 leucine zipper, nd its frgments, Fos nd Jun, very good greement is found with experiment. Experimentlly, some subdomins of the GCN4 leucine zipper form stble dimeric structures, suggesting the regions of differentil stbility in the prent structure. Our clcultions suggest tht the stbilities of the subdomins re in generl different from the vlues expected simply from the stbility of the corresponding frgment in the wild type molecule. Furthermore, prts of the frgments structurlly rerrnge in some regions with respect to their corresponding wild type positions. Our results suggest for n Asn in the dimeriztion interfce t lest pir of hydrophobic intercting helicl turns t ech side is required to stbilize the stble coiled coil. Finlly, the specificity of heterodimer formtion in the Fos-Jun system comes from the reltive instbility of Fos homodimers, resulting from unfvorble intr- nd interhelicl interctions in the interfcil coiled coil region. Since the coiled coil motif ws first proposed by Crick (to explin the kertin diffrction pttern) (Crick, 1953, 1952), coiled coils hve been the subject of incresing ttention (Cohen & Prry, 1986; Cohen & Prry, 1990; Hrrison, 1991; O She et l., 1991). Coiled coils consist of two or more helices wound round one nother nd cn be found in the muscle protein tropomyosin (Johnson, 1975; Phillips, 1986), in blood clots s fibrin, nd in hir s kertin (Frser & McRe, 1971; Cohen & Prry, 1986; Cohen & Prry, 1990). Furthermore, coiled coils ply n importnt role in trnscriptionl ctivtors (Lndshultz et l., 1988; Hrrison, 1991) regulting cell growth, differentition, nd oncogenesis nd therefore re importnt mediclly (Perutz, 1992). In these proteins, coiled coils, nmed leucine zippers becuse they possess Leu in every seventh position, hve been proposed to form dimeriztion domins (Lndshultz et l., 1988; Hrrison, 1991). The biologicl importnce nd reltive simplicity of leucine zippers hve mde them the subject of extensive experimentl (Hodges et l., 1981; O She et l., 1989, 1991, 1993; Smel et l., 1989; Hrbury et l., 1993; Lovejoy et l., 1993; Lumb et l., 1994) nd theoreticl studies. They hve been used s models for studying vrious interctions responsible for driving protein folding (Hrbury et l., 1993; O She et l., 1993), for testing hypotheses bout the specificity nd stbility of protein structures (Hrbury et l., 1993; O She et l., 1993), nd for understnding fctors influencing oligomeric ssembly nd protein design principles (Hrbury et l., 1993; Lovejoy et l., 1993; O She et l., 1993). Becuse of the bundnce This work ws supported in prt by NIH Grnts GM nd 1R03-TW * Corresponding uthor. Phone: (619) ; Fx: (619) Deprtment of Chemistry, The Scripps Reserch Institute. Deprtment of Moleculr Biology, The Scripps Reserch Institute. University of Wrsw. X Abstrct published in AdVnce ACS Abstrcts, Jnury 1, of interesting experimentl dt nd their smll size nd reltive simplicity, leucine zippers re idel model systems for vrious theoreticl pproches to the protein folding problem (Krystek et l., 1991; Nilges & Brunger, 1993; Zhng & Hermns, 1993; DeLno & Brunger, 1994; Vieth et l., 1994). Tking dvntge once gin of the simplicity of leucine zippers, the present work focuses on developing theoreticl pproch to understnding the fctors responsible for the stbility of dimeric coiled coils reltive to the monomeric structures. The sequences of coiled coils hve chrcteristic heptd repet (bcdefg) n (McLchln & Stewrt, 1975). Residues t positions nd d re in generl occupied by hydrophobic residues nd form n interfce between two or more helices (in leucine zippers, position d is occupied by Leu residues). The e nd g positions re occupied mostly by chrged residues, nd the methyl groups of those residues form the edges of hydrophobic core. The b, c, nd f positions re mostly hydrophilic. In ddition to hving Leu in most d positions, leucine zipper frgments of DNA binding proteins re short (23-45 residues long) (Lndshultz et l., 1988), nd some of them, when excised from their prent proteins, hve tendency to dimerize (O She et l., 1989, 1991). A number of trnscription fctors form heterodimers, e.g., c-fos nd c-jun (Hi et l., 1989; Smel et l., 1989), with leucine zipper interfces. The isolted leucine zipper domins of c-fos nd c-jun lso hve tendency for heterodimeriztion (O She et l., 1989), similr to the intct proteins. The experimentl observtion tht ttrcted our ttention ws the phenomenon of subdomin folding in the GCN4 leucine zipper system (Lumb et l., 1994). The vrious subdomins (frgments) of the wild type GCN4 leucine zipper were synthesized, nd their therml stbility ws ssessed. The frgment corresponding to residues 8-30 of /96/ $12.00/ Americn Chemicl Society

2 956 Biochemistry, Vol. 35, No. 3, 1996 Vieth et l. the wild type GCN4 leucine zipper ws found to hve substntil stbility. In contrst, the closely relted frgment corresponding to residues in the wild type GCN4 ws predominntly unfolded. In order to investigte this problem, we need to develop methodology to extrct from simultions the equilibrium constnts for formtion of coiled coils from unfolded monomeric chins. Previously, the prediction of the folding pthwys nd structure of the GCN4 leucine zipper hs been reported by our group (Vieth et l., 1994). Subsequently, the clcultion of the oligomeriztion equilibri of GCN4 leucine zipper nd some of its mutnts ws then described (Vieth et l., 1995). However, unfolded monomeric chins cnnot be treted by this method. Therefore, in this pper, new more generl pproch is proposed to clculte free energies of the unfolded s well s folded chins. The outline of the pper is s follows. In the Method section, the generl expression for the equilibrium constnt of monomer-dimer equilibri is presented. The tretment of the equilibri between monomers nd dimers presented in this pper is in principle generl nd not limited to the lttice models of proteins. However, clcultion of different contributions to the free energy is presented in the context of lttice model of proteins (Kolinski & Skolnick, 1994; Vieth et l., 1995). Thus, the lttice model of proteins is briefly introduced nd the clcultion of different contributions to the dimeriztion free energy is described. In prticulr, trnsfer mtrix tretment is presented to evlute the internl contributions to the free energy of dimeriztion. In the ppendices, we present the detiled description of the tretment s well s our efforts to vlidte the methodology. The Results section focuses on the clcultion of the dimeriztion free energy for the GCN4 leucine zipper nd its frgments, the monomer-dimer equilibrium in Fos, nd the heterodimer-homodimer equilibri in the Fos-Jun system. In the Discussion section, the implictions of our results re presented, together with elbortion of the necessry condition for stbiliztion of single hydrophilic residues in the helicl interfce of coiled coils. METHOD In wht follows, we present n overview of the clcultion of the monomer-dimer equilibri for pir of chins. We first present the generl sttisticl mechnicl formlism. Then, the lttice model of proteins is introduced nd the clcultion of different components of free energy of dimeriztion re described. Formlism for Monomer-Dimer Equilibri Consider the equilibrium constnt for dimeriztion (Mc- Qurrie, 1976): 2M T 1D (1) K ) [D]/[M] 2 (2) with [M] nd [D] being the equilibrium concentrtions of monomers nd dimers, respectively. The totl concentrtion of individul chins C 0 (ssuming tht the only species in the system re monomers nd dimers) is expressed s: wheres the frction of individul chins in monomers nd dimers is given by: Obviously, x M + x D ) 1. Substituting x M nd x D into eq 2: nd: C 0 ) 2[D] + [M] (3) x M ) [M] C 0 K ) 2 2C 0 x M x D ) 2[D] C 0 (4) Thus, the clcultion of x M nd x D require vlues for both K nd C 0. In order to relte the equilibrium constnt K to the microscopic vribles, it is useful to rewrite eq 5 in the form of mole frction equilibrium constnt K x (McQurrie, 1976): Substituting the totl number of chins for the mole frctions: where N is the totl number of individul chins in box of volume V, nd N M nd N D re numbers of monomers nd dimers, respectively. N D /N M2 is simply the rtio of the totl prtition function for dimers Z D divided by the squre of the totl monomer prtition function Z M. Thus, K x cn be written in the following form (McQurrie, 1976; Skolnick, 1980): Z conf,m(d) re the configurtionl prtition functions for the monomers, M (dimers, D), tht include the rottionl nd internl contribution. V 0 is the volume ccessible to first tom in the second chin in the dimer, given tht the first tom in the first chin of the dimers is fixed, σ D is the symmetry number (d2! for homodimers nd 1 for heterodimers). Since we choose n internl coordinte system (fixed t the first bed of chin one) to clculte the prtition function, the fctor V comes from the integrtion over the degrees of freedom of the first bed in the first chin. As noted in series of ppers, the contributions from the moment degrees of freedom cncel in the numertor nd denomintor (Herschbch, 1959; Myer & Myer, 1963; Holtzer, 1995) (for this internl coordinte system), leving the rtio of configurtionl prtition functions to determine the equilibrium constnt. Rerrnging eq 7, we get Let us note tht V M ) N/V is the verge volume ccessible per chin nd is equl to 1/C 0. Equtions 6 nd 7b provide the bsic equtions for evlution of the frction of monomers nd dimers in the system t given concentrtion C 0. In order to dissect the x D x M ) KC 0 4KC 0 K x ) C 0 K ) x D 2x M 2 K x ) 2N D /N 2(N M /N) 2 ) N N D N M 2 K x ) N Z D Z M 2 ) N VV 0 Z conf,d σ D (VZ conf,m ) 2 K x ) V 0 V M Z conf,d σ D Z conf,m 2 (5) (5b) (6) (6b) (7) (7b)

3 Theoreticl Study of Leucine Zipper Dimeriztion Biochemistry, Vol. 35, No. 3, totl contribution to the free energy chnge for the rection in eqution 1, let us define the stndrd molr free energy chnge for the monomer-dimer equilibrium (McQurrie, 1976): Combining 8 with eq 7b: θ A MfD θ A MfD ) -kt ln K x (8) ) -kt ln (V 0 /V M ) - kt ln Z conf,d + 2kT ln Z conf,m + kt ln σ D (8b) The totl free energy of dimeriztion cn be written s sum of trnsltionl nd configurtionl prt: θ A MfD θ ) -T S tr (V M ) + A conf (8c) with S tr ) k ln (V 0 /V M ) nd A conf ) A int - T S rot. The totl free energy chnge from eq 8-c cn be viewed s three subprocesses corresponding to three different energy contributions (trnsltionl, rottionl, nd internl). Figure 1-d shows schemtic thought experiment tht cn be done to understnd the mening of different contributing terms to the free energy of dimeriztion. The first subprocess (Figure 1b) cn be viewed s the loss of trnsltionl degrees of freedom upon bringing the two first chins together. The methodology for clculting the trnsltionl entropy is shown in Figure 2. Next, n end is oriented, nd rottionl entropy is lost (Figure 1c). Finlly, the ntive interctions re formed (Figure 1d). In the following, method to clculte ech contribution to the free energy for the lttice model of proteins is described. Lttice Model of Proteins Since the detils of the lttice model hve been presented elsewhere (Kolinski & Skolnick, 1994; Vieth et l., 1995), we present here only brief description of the methodology. Additionl, slient detils re provided in Appendix A. The protein is modeled by n R-crbon representtion of the bckbone nd by single sphere, multiple rotmer side chins. The R-crbons re connected by set of vectors of the type 1.22*{(2, 2, 0), (2, 2, 1), (3, 0, 0), (3, 1, 0), (3, 1, 1)} (Kolinski & Skolnick, 1994; Vieth et l., 1995). The entire sttisticl interction scheme, derived from high resolution structures (see the smple derivtion in the recent pper by Godzik et l. (1995)) from the Protein Dt Bnk (Bernstein et l., 1977), is subdivided into short nd long rnge interctions. Among the short rnge interctions, there re locl conformtionl preferences of neighboring residues long the sequence together with rotmer energy. By the term long rnge interctions, we men ll interctions between residues tht re t lest 4 residues prt in sequence. All long rnge interctions re sequence dependent. They consist of side chin pir potentil, coopertive pir potentil (which fcilittes interctions between secondry structurl elements, i.e., helices nd β-sheets), nd contct bsed on body term (similr to solvtion energy). The model hydrogen bonds (derived in the spirit of Levitt nd Greer (1977)) cn be locl (helicl) or long rnge (in β-sheets) nd operte only between R-crbons. The conformtions re smpled using Monte Crlo procedure (Metropolis, 1953). Configurtionl Prtition Function for the Dentured Stte. In the unfolded stte, both chins re nonintercting FIGURE 1: A thought experiment tht dissects the different contributions to the free energy of dimeriztion. The shded circle represents the first bed of the reference chin (the first chin). The open circle represents the first bed of the second chin. The first chin hs freedom to trnslte nd rotte in both the unfolded nd folded stte. (A) The entire dimeriztion process under considertion. On the left-hnd side re the rectntsstwo independent chinssnd on the right-hnd side re the productss two chin, prllel coiled coil with specific interctions. (B) Reduction in trnsltionl entropy ssocited with bringing the first bed of the second chin (open circle) close to the first bed of the first chin (shded circle). On the left-hnd side, both beds hve n verge ccessible volume V M (V M is the verge volume per molecule, 1/C 0 ). After they re brought together, the first bed hs volume V M ccessible to it, wheres the second bed hs only smll vibrtionl volume V 0 in the dimer. (C) The chnge in rottionl prt of configurtionl free energy. The left-hnd side shows free rottion of both chins, wheres on the right-hnd side the two chins cn rotte only s unit. The reltive rottion of the second chin with respect to the first one is restricted. (D) The chnge in the internl free energy. On the left-hnd side, the two chins hve fixed reltive orienttion (determined by the first two vectors in both chins) but do not interct nd cn ssume ny conformtion. On the right-hnd side, two chin, coiled coil with specific interchin interctions is formed. nd, thus, cn be treted independently. For short chins (contining less thn 6 bonds), it is possible to do n exct enumertion of ll possible sttes to obtin the totl internl prtition function for the unfolded stte. Unfortuntely, for the longer chins, the exct enumertion is computtionlly intrctble. However, if one ssumes tht the totl energy of chin in prticulr conformtion cn be written s sum of energies of smll overlpping segments (i.e., 4 bond segments), then the prtition function cn be clculted using trnsfer mtrix tretment (Zimm & Brgg, 1959; Lifson & Roig, 1961; Flory, 1969; Polnd & Scherg, 1970) described in detil in Appendix B. This type of tretment ws extensively used in helix-coil trnsition theory (Zimm

4 958 Biochemistry, Vol. 35, No. 3, 1996 Vieth et l. seven residue chin (-Gly-Al 5 -Gly-), the exct enumertion gives slightly lower free energy (-17.1 kt) thn the trnsfer mtrix tretment (-16.3 kt) does, but both vlues re close enough to be considered similr. For longer chins, it is uncler whether the cncelltion in the free energy clcultion persists, but we use this method s first pproximtion to the free energy of the unfolded stte. The contribution of the long rnge interctions to the unfolded stte free energies for longer chins (up to 11 residues) is being evluted; however, ny specultion on this issue is beyond the scope of this pper. Configurtionl Prtition Function for Dimeric Molecules. The folded stte is considered to be subset of conformtions tht hve specific overll topology, hydrogen bond pttern, nd specific side chin contct mp (Ptitsyn, 1987). The entire configurtionl prtition function for the folded stte cn be written s product of the rottionl nd internl prts (McQurrie, 1976): n levels Z conf,d ) Z rot Z int,d ) N D1 N D2 n(e i )e -βe i (9) i)1 FIGURE 2: Schemtic representtion of clcultion of loss of trnsltionl entropy for the first (white) bed of the second chin. In the folded stte, the white bed hs n ccessible vibrtionl volume V 0. The bottom picture shows method of clculting this volume. The coordintes of the first bed of the second chin re first expressed in the coordinte system centered t the first bed of the first chin. Note tht the entire spce is discretized nd divided into smll cubic boxes of side of 2.6 Å. Then, the simultion of folded stte is run, nd the number of occurrences of the white bed is counted in ech box every 100 MC cycles. The mximlly occupied box is selected, nd the frequency of the white bed being in this box is clculted. For exmple, in this cse we hve 5 snpshots, nd the number of times the mximlly occupied box is visited is 3. The vibrtionl volume V 0 is computed s the rtio of the volume of the elementry box (2.6 Å 3 ) to the probbility of white bed being in the mximlly occupied box (P x ) 3 / 5 ). & Brgg, 1959; Lifson & Roig, 1961; Polnd & Scherg, 1970) s well s in polymer physics (Flory, 1969). The pproch we use in this pper is generliztion of the tretment proposed by Skolnick nd Kolinski (1992). The entire polypeptide chin is divided into N-4 four bond (five residue) overlpping segments. Ech segment is treted s being in one of severl discrete rottionl sttes (Flory, 1969). Only interctions within ech segment re considered. For ech of the overlpping four bond segments, we clculte the sttisticl weights of ll possible conformtions nd use these weights to build N-4 trnsfer mtrices. Multipliction of these mtrices nd summtion of the resulting elements give the prtition function for the system. This tretment neglects the long distnce excluded volume effects, s well s the sttisticl weight of intercting clusters of residues, which would be creted if the chin folded onto itself (Flory, 1969). Locl elements of secondry structure (turns nd helicl sttes) re ccounted for in this tretment. For short chins, long rnge, repulsive excluded volume effects nd the ttrctive contributions from hydrophobic clusters on verge cncel nd yield free energies which re close to those obtined by the exhustive enumertion. The exct enumertion ws compred to the trnsfer mtrix tretment for six (-Gly-Al 4 -Gly-) nd seven residue (-Gly- Al 5 -Gly-) chins. For the six residue chin, both tretments give identicl results with free energies kt. For the N D1 is the number of distinct conformtionl sttes for the first two vectors in the first chin (ssumed to be equl 4704, s determined by sttistics), wheres N D2 is the number of distinct sttes for the first two residues in the second chin. n(e i ) is the degenercy of energy level E i, n levels is the number of energy levels, nd β ) 1/kT. The probbility of being in ny energy level (for convenience, we choose the verge energy level E 0 ) is given by (McQurrie, 1976): P(E 0 ) ) n(e 0 ) exp(-βe 0 ) Z int (10) P(E 0 ) is the probbility of the system being in E 0. If we could get n estimte for the degenercy of one energy level (sy n(e 0 ) for convenience), then we would be ble to obtin the configurtionl prtition function for the system: n(e 0 ) Z conf,d ) N D1 N D2 P(E 0 ) exp(-e 0 /kt) (11) We cn now write the expression for the configurtionl free energy of the folded stte: A conf,d ) -kt ln Z conf,d ) -kt ln N D1 N D2 + E 0 - kt ln n(e 0 ) + kt ln P(E 0 ) (12) where k ln(n D1 N D2 ) is the rottionl entropy term, nd k ln(n(e 0 )) - k ln(p(e 0 )) is the internl entropy of the system rising from the degenercy of the verge energy level (k ln(n(e 0 )) nd the fluctution terms (-k ln(p(e 0 ))). The verge energy of the system together with the energy probbility distribution (Figure 8) cn be obtined from Monte Crlo simultions of the folded stte (see Appendix C). First, we generte strting structure for ech sequence in prllel coiled coil rrngement. Then, production runs re crried out to clculte the verge energy of the system together with the energy probbility distribution. The degenercy of the energy level E 0 n(e 0 ) is clculted from the ensemble of structures within 4 kt of the verge energy using similr trnsfer mtrix tretment s for the unfolded stte.

5 Theoreticl Study of Leucine Zipper Dimeriztion Biochemistry, Vol. 35, No. 3, FIGURE 3: Schemtic depiction of the frgments. The top lyer shows digrm of helicl wheel with the heptd residue repet indicted. The rectngles t the bottom show the sequences of the GCN4 wild type nd ll peptide frgments investigted in this study together with ssignment to the hydrophobic heptd positions nd d. Note tht ech peptide frgment strts with residue djcent to the hydrophobic core nd ends with residue tht would be in the core if the helices were longer. RESULTS Using the procedure described in the Method section, the free energies of the unfolded sttes nd the free energies of folded sequences in the structure of double strnded, prllel coiled coils hve been clculted. In ll cses studied here, we eliminted the possibility of formtion of ntiprllel species by energetic considertions (see fitness function in Figure 7). A number of sequences hve been investigted. First, there is the GCN4 leucine zipper wild type, shown experimentlly to be dimeric coiled coil (O She et l., 1991). Next, there re vrious 23 residue frgments (subdomins) of the GCN4 leucine zipper studied by Lumb et l. (1994) indicted schemticlly in Figure 3. Ech subdomin strts from the residue djcent to the hydrophobic core (t the g nd c positions) nd ends t the position tht ws in the hydrophobic core in the wild type. We expect tht, t both ends, the residues tht initite or terminte the hydrophobic core will be destbilized with respect to the identicl residues in the wild type. The reson for tht is loss of interctions with the preceding or subsequent residues (present in the wild type but bsent in the frgment) nd prtil exposure of those residues to solvent. The frgment corresponding to residues 8-30 of the wild type GCN4 ws found to be stble s coiled coil dimer, wheres the closely relted frgment corresponding to residues of GCN4 wild type ppered to be predominntly unfolded (Lumb et l., 1994). Lumb et l. (1994) concluded tht specific pcking interctions cn be energeticlly more importnt thn locl secondry structure propensities. Lumb et l. lso suggested tht the protein folding cn be understood in terms of formtion of coopertively folded domins. In ddition to these two peptides, we lso test nother 23 residue frgment corresponding to the residues 4-26 of the wild type GCN4, which hs not yet been studied experimentlly. However, bsed on the energetic profile of the wild type (Vieth et l., 1995), we would expect the 4-26 frgment to form the most stble 23 residue dimeric coiled coil. Monomer-Dimer Equilibri in the GCN4 Leucine Zipper. Over the experimentl concentrtion rnge, the predicted frction of chins in the dimeric structure is shown in Tble 1 for the GCN4 leucine zipper. Becuse of the limited ccurcy of our free energy estimtes, we restrict ourselves to the ssignment of the dominnt species. Our results re in greement with experiment nd indicte tht the wild type of GCN4 leucine zipper should populte dimeric chins. Tble 1 lso shows the clculted conformtionl (rottionl prts included) free energies of the GCN4 leucine zipper in the unfolded stte s well s in the coiled coil structure. The free energy chnges upon folding re lso presented. The trnsltionl entropy loss upon dimer formtion is shown in the second lst column for three different concentrtions (2 µm, 43 µm, 1 mm) (Lumb et l., 1994). In ddition, Tble 1 presents the verge RMS fluctutions from the verge structure in the coiled coil stte together with the volume occupied by the first bed of the second chin. In order to evlute the energetic nd entropic contributions to the free energy of the unfolded stte, unrestrined Monte Crlo simultions of GCN4 leucine zipper wild type monomer (in T ) 1.85 with no long rnge interctions) hve been performed. The difference between the free energy computed by the trnsfer mtrix pproch (see Appendix B) nd the verge energy computed from the Monte Crlo simultion ws considered to be the entropy of the unfolded stte. Tble 1 presents the dissection of the free energy of folded nd unfolded stte of the wild type GCN4 leucine zipper. As would be expected, the dominnt contribution to the unfolded stte configurtionl free energy comes from the entropy. The verge energetic contribution is smll, nd by our ssumption, there is no energetic contribution from the long rnge interctions. Upon folding, this sitution chnges drmticlly. The dominnt contribution to the free energy of the folded stte comes from the energy (-96.6 kt per monomer), nd the entropy drops down by roughly 50% in comprison to the unfolded stte. The mjor energetic contribution to the folded stte comes from hydrogen bonds, locl side chin orienttionl preferences, pirwise interctions, nd coopertive side chin pcking interctions. From the vlues of the internl entropy presented in Tble 1, we cn clculte the verge number of sttes per residue for the unfolded nd the folded stte. For the unfolded stte, the verge number of sttes per residue is close to 45. For the folded stte, this number decreses to 7. Thus, our model provides lmost 7-fold reduction in the number of sttes upon folding. This reduction is slightly lower thn the 8-fold reduction estimted by Privlov (1992). This reltively minor difference in entropy chnge cn be due to the fct tht coiled coils hve lrger exposed surfce re thn globulr proteins s well s to pproximtions of our pproch. It is noteworthy tht if the sequence of the GCN4 leucine zippers were in the tetrmeric stte, then the clculted reduction in the verge number of sttes per residue upon folding would be roughly 7.7; this is very close to the number estimted by Privlov. This my not be

6 960 Biochemistry, Vol. 35, No. 3, 1996 Vieth et l. Tble 1: Thermodynmicl nd Structurl Prmeters for the GCN4 Leucine Zipper dominnt species 43 µm 1 mm free energies monomer free energy dissection dimer free trnsltionl entropy loss h energy dissection 2 µm 43 µm 1 mm b (0) e (41) e c f 64.3 f (1.3) d 45 g 7 g The predicted dominnt species for concentrtions 43 µm nd 1 mm ( 2 indictes tht dimers only re present) re shown. Experimentlly, only dimers re present t these concentrtions. b The configurtionl free energy of the unfolded stte monomers in kt. c The configurtionl free energy of the folded stte (coiled coil dimer) in kt. d The configurtionl free energy chnge upon folding (the free energy of dimer minus twice the monomer free energy) in kt units. e The verge energy per chin in kt (percentge (%) of long rnge interctions is shown in prentheses). f The configurtionl entropy per chin multiplied by the reduced temperture (in kt). g The verge number of sttes per residue. h The trnsltionl entropy loss upon formtion of the dimer multiplied by the reduced temperture for different concentrtions (2 µm, 43 µm, 1 mm) in kt. The verge volume occupied by the first bed of the second chin V 0 is 67.6 Å 2. i The verge C R RMS devition of single structure from the verge structure (stndrd devition is shown in prentheses). RMS in Å i Tble 2: Comprison with Experiment of the Predicted Dominnt Species for GCN4 Leucine Zipper Frgments concn dependence of monomer/dimer rtio dominnt species protein 43 µm 1 mm theory expt GCN :100 0: GCN :0 99:1 1 1 GCN :91 2:98 2 NA 2 ( 1 ) indictes the presence of dimers (monomers) only, NA indictes tht experimentl dt re not vilble. Tble 3: Free Energy of the Unfolded Stte nd the Folded Stte (Coiled Coil Prllel Dimer) for Different Frgments of the GCN4 Leucine Zipper protein free energy of unfolded chin free energy of folded chin θ A conf trnsltionl entropy loss b 2 µm 43 µm 1 mm θ A conf is the configurtionl free energy chnge upon folding including loss of rottionl entropy kt ln(n D1/N D2). b Trnsltionl entropy loss, -k ln(v/v 0) multiplied by temperture. Tble 4: Structurl Prmeters for the Dimers of the GCN4 Leucine Zipper Frgments Computed from Monte Crlo Simultions protein v RMS vlues for structures for ll structures for structures within 4kT from the v energy b vol occupied by the first bed V 0 c (0.3) 1.0 (0.3) (1.1) 3.2 (0.9) (0.5) 1.1 (0.5) 57.1 Averge RMS (stndrd) devitions from the verge structures for ll structures occurring in the simultion. b Averge RMS devitions from the verge structures computed from those structures within 4kT of the verge energy. c Volume occupied by the first bed V 0 in Å 3 of the second chin (provided tht the first bed of the first chin is pinned) nd verged over ll simultions. surprising, since coiled coil tetrmers closely resemble the four helix bundle topology observed in globulr proteins. Wht is surprising, however, is tht the number of sttes per residue in the folded stte is so lrge. Most of the vrition tht we observe comes from the mobility of the side chins in the exposed prts of the molecule s well s in the exggerted bckbone mobility on the lttice. The ide of mny conformtionl substtes (distinguished by smll chnges in the protein structure) in the folded stte ws widely elborted on by Fruenfelder et l. (1988). Nevertheless, we feel tht our model overestimtes the number of these sttes; it lso overestimtes the rnge of conformtionl fluctutions in the folded stte. As reminder, let us note tht the inherent resolution of our lttice imposed by its geometry nd wide interction bsins is estimted to be round 2.6 Å (Kolinski & Skolnick, 1994). Also the verge RMS of C R toms with respect to the verge structure for the folded stte is round 2 Å, with fluctutions rnging from 0.8 to 3.5 Å (see Tbles 1, 4, nd 9). Thus, the lttice model hs folded stte defined s tube with 1.3 Å rdius; this is probbly why such lrge number of sttes per residue in the ntive conformtion is seen. Equilibri for GCN4 Leucine Zipper Subdomins. Tbles 2-6 summrize the results for the frgments of the GCN4 leucine zipper. The predicted monomer/dimer rtio for different sequences is shown in Tble 2. For ll of the experimentlly tested peptide frgments, the predicted dominnt species re in greement with experiment; i.e., frgment 8-30 is predicted to form stble dimer, nd is predicted to be predominntly unfolded. For the frgment corresponding to residues 4-26 of the wild type, we predict the preferentil formtion of dimers. Tble 3 shows the free energies of the unfolded nd folded sttes for the frgments. From these dt one cn infer which subdomin of the wild type GCN4 contributes mostly to the stbility of the prent structure. Focusing on the configurtionl free energy contribution, the lrgest stbility per residue is exhibited by the 8-30 frgment of the wild type ( -1kT per residue). Another feture of the dt from Tble 3 is tht the unfolded stte free energies for the frgments re very close to one nother nd tht the differences in stbility reside in the folded stte. Except for the frgment (whose first 10 residues re disordered), the vlues of the trnsltionl entropy loss on dimeriztion re lso similr for most tested sequences. Tble 4 shows the verge RMS fluctutions of single structure from the verge structure. The fluctutions of the frgment re noticebly lrger thn in either the 4-26 or 8-30 frgments. As noted bove, the different stbilities of the frgments rise mostly from the differences in folded stte free energies. Tble 5 shows the dissection of the folded stte free energies into the verge energy nd entropy. For the 4-26 nd 8-30 frgments, which by our prediction pper s stble dimers, the entropic contributions re similr within the error of our clcultions. For the frgment tht is predicted to be unfolded, the entropic contribution is lrger. This reltively lrge entropic contribution of the folded stte

7 Theoreticl Study of Leucine Zipper Dimeriztion Biochemistry, Vol. 35, No. 3, Tble 5: Dissection of Folded Stte Free Energy for the Frgments of GCN4 Leucine Zipper protein energy -TS int All vlues shown re in kt. Rottionl entropy TS rot is 14.4kT, 13.1kT, nd 17.1kT, for 4-26, 8-30, nd frgments, respectively. Tble 6: Comprison of the Estimted Configurtionl Free Energies of Dimer Formtion (from the Wild Type per Residue Plot) nd Clculted for Frgments protein θ estd A conf θ clcd A conf Indictes the configurtionl free energy of dimer minus twice the free energy of monomers obtined from the wild type energy per residue vlues. b Indictes the configurtionl free energy of dimer minus twice the free energy of monomers clculted explicitly from the simultion nd trnsfer mtrix tretment. All vlues in kt units. cnnot compenste for smll energetic stbiliztion; thus, unfolded monomers re preferred. In greement with the lrger configurtionl entropy for the frgment, the RMS fluctutions obtined from the Monte Crlo simultions, in Tble 4, lso show lrger vlues. This is in greement with the lrger configurtionl entropy vlues for this frgment. Tble 5 shows tht the 8-30 frgment is (consistent with overll free energy vlues) energeticlly the most stble. The unstble frgment shows substntilly lower energetic contribution thn others. These dt suggest tht nlysis of the differentil stbility of these frgments could be done bsed on the energetic considertions in the folded stte. Another interesting nlysis bers on questions of the difference between the wild type nd its constituent frgments. Could the different stbilities of frgments be rtionlized from the plots of free energy of dimeriztion (the free energy of dimer minus the free energy of two monomers) per residue using the wild type dt? Are there ny regions tht in the frgments (prt from the ends likely to destbilize the molecules in comprison to the corresponding residues in the wild type) re substntilly different in stbility in comprison to the wild type? Tble 6 shows the estimted vlues of the totl free energy of dimeriztion ssuming tht ll of the contributions for ll of the residues re s in the wild type. The estimted vlues predict tht frgment be unstble, wheres 4-26 nd 8-30 should exist s stble dimers, with the highest stbility ssigned to the The only frgment for which the difference between the estimted nd clculted vlues could be ssigned to the end effects is For this frgment, the clculted vlue is higher tht the estimte. The difference could be ssigned to the expected destbiliztion of the dimeric structure by prtilly unburied hydrophobic residues t both ends. The explicit clcultions show tht the 8-30 frgment is the most stble nd is even more stble thn would be estimted from the wild type dt. This suggests possible dditionl stbiliztion of some residues with respect to the wild type which my rise from slightly different side chin pcking. A similr but opposite effect is present in the lest stble frgment. FIGURE 4: Energies per residue per chin of prllel coiled coil dimers for the 8-30 frgment (A) nd frgment (B) in comprison to energy per residue for the wild type (solid line). Frgment plots re shown s thin lines. The energy per residue is clculted s the sum of ll of the energetic terms for given residue, verged over ll of the simultions. Ech energy term in which two (or more) residues prticipte is divided by two (or more), i.e., pir energy, hydrogen bond energy, nd templte energy. The sum of energies over ll residues gives the totl verge energy of the system. (A) Plot showing the energy per residue in the 8-30 frgment (bold line with filled squres) in comprison to the wild type (thin line with open circles). Note the reltive destbiliztion of the 8-30 frgment t the N- nd C-terminus with respect to the wild type. Also residues t old 12 nd 16 positions of frgment experience dditionl stbiliztion reltive to the wild type. (B) Plot of the energy per residue for the frgment (bold line with filled squres) in comprison to the wild type (thin line with open circles). For 12 lst residues, the two curves re identicl. However, destbiliztion of the N-terminus in the frgment extends for the first 11 residues. Figure 4A,B shows the energy per residue plots for two frgments 8-30 nd compred to the wild type. As seen in Figure 4A, the 8-30 frgment is, s expected, destbilized t both ends, probbly becuse residues t the edges of hydrophobic core lose interctions with prtners tht used to be there in the wild type. However, substntil dditionl stbiliztion is exhibited by two residues in the helicl interfce, i.e., 5 Leu, which occupies the d position (12 Leu in the wild type) nd 9 Asn t n position (16 Asn in the wild type). By exmining the plots of the vrious energy terms per residue, two types of interctions re found to be responsible for their dditionl stbiliztion (see Tble 7). The lrgest stbiliztion comes from the pir interction

8 962 Biochemistry, Vol. 35, No. 3, 1996 Vieth et l. Tble 7: Dissection of the Extr Energetic Stbiliztion of the 8-30 Frgment by Residues 5 Leu nd 9 Asn (d nd Positions) residue estd pir energy clcd pir energy estd locl E β clcd locl E β 5 Leu d Asn All vlues in kt units per monomer. energy, nd slightly smller contribution origintes from the locl side chin orienttionl preferences. The pir energy difference comes from the slight side chin rerrngement nd loss of interctions between 5 Leu nd 9 Asn (this interction is strongly repulsive in our sttisticl pir potentil (Vieth et l., 1995)). The RMS differences between the verge structures for the frgments nd the corresponding region in the wild type re 1.1, 1.5, nd 2.1 Å for the 4-26, 8-30, nd frgments, respectively. These vlues re on the order of the RMS fluctutions for Monte Crlo simultions nd do not preclude the possibility of side chin repcking. In Figure 4B, the first 10 residues of frgment show substntil energetic destbiliztion with respect to the wild type (Figure 4B) for the first 10 residues. Visul inspection of the Monte Crlo trjectories leds to the conclusion tht first 10 residues re highly disordered. This is in greement with quite lrge RMS fluctutions for the entire frgment shown in Tble 4. The residue tht is responsible for this series of observtions is Asn 16. A possible explntion is tht the hydrophilic Asn in the wild type is forced to be in the hydrophobic core of the coiled coil structure by the stbilizing interctions, nmely, coopertive hydrogen bonding nd fvorble hydrophobic interctions elsewhere in the molecule. The lck of n internl disordered region preceded nd followed by intercting helices rises from the loop entropy effect (Skolnick, 1983) (the configurtionl entropy loss ssocited with constrining two ends of rndom coil). In the cse of coiled coils, loop entropy elimintes rndom coiled residues in between intercting helicl stretches (Skolnick, 1983). Thus, coiled coils form single intercting helicl stretch, tht cn be preceded or followed by rndom coil residues. The lter condition occurs for the frgment where the short helicl stretch prior to hydrophilic Asn is not strong enough to hold Asn in helicl conformtion. On the contrry, the entire stretch of 10 residues prefers (due to the entropic resons) to be disordered rther thn to form n intercting helicl stretch, followed by n internl rndom coil bubble. This sitution is in contrst to tht in the wild type nd in frgment 8-30 where the stbiliztion of coiled coil structure strts (nd ends) t the second (second to lst) residue in the helicl interfce. This indictes tht, in order to stbilize unfvorble Asn residue in the helicl interfce, t lest two consecutive hydrophobic residues re required (s in the cse 8-30 frgment) to the left nd to the right side of this residue. Specificity of Coiled Coils. Equilibri in Fos-Jun system. In order to investigte coiled coil system hving the possibility of forming more thn one structure, we chose the Fos-Jun trnscriptionl ctivtor system. The experimentl system itself consists of unimolr mixture of Fos nd Jun peptides t different concentrtions (O She et l., 1989), where there is possibility of formtion of homodimers of Tble 8: Predicted Dominnt Species nd Thermodynmicl Prmeters for 41 Residue Fos Sequence without CGG Linker dominnt species concn dependence of monomer/dimer rtio b theory expt 43 µm 1 mm trnsltionl entropy loss f free energies 2 µm 43 µm 1 mm c 1 1, 2 99:1 89: d e The dominnt species ssigned for the entire concentrtion regime (43 µm-1 mm). 2 indictes the presence of dimers only, wheres 1 the presence of monomers. b The predicted rtio of the monomers to dimers clculted for different concentrtions. c The configurtionl free energy of the unfolded stte monomers in kt. d The configurtionl free energy of the folded stte (coiled coil dimer) in kt. e The configurtionl free energy chnge upon folding (the free energy of dimer minus twice the monomer free energy) in kt units. f The trnsltionl entropy loss upon formtion of the dimer multiplied by the reduced temperture for different concentrtions (2 µm, 43 µm, 1 mm) in kt. both Fos nd Jun s well s heterodimers of Fos with Jun. Jun homodimers re reltively stble, wheres Fos homodimers re of lower stbility (O She et l., 1989) (see below). Experimentlly, in the system contining unimolr mixture of Jun nd Fos, only heterodimers Jun-Fos re observed (O She et l., 1989). First, we performed the investigtion of the stbility of dimeric Fos-41 species reported to be minimlly stble (O She et l., 1989) nd used our method to estimte the stbility of Fos coiled coils. The Fos-41 sequence corresponds to 41 residues, , from c-fos oncoprotein with n dditionl H200Y muttion (O She et l., 1989; Schuermnn et l., 1991). Tble 8 presents the frction of monomeric nd dimeric chins t vrious concentrtions for Fos species without the CGG linker. Experimentlly, these shorter Fos leucine zippers form dimers only t high concentrtion (O She et l., 1989), nd their formtion is concentrtion dependent. In Tble 8, the free energies of unfolded monomers nd dimeric coiled coils re given. Our prediction is too corse to quntitte the concentrtion dependence of species; however, we do see noticeble increse in the dimer popultion upon incresing the concentrtion. We next present the results of our clcultions for n equimolr mixture of leucine zippers from Fos nd Jun oncoproteins. Becuse the experiment ws done under strong renturting conditions (O She et l., 1989) nd the dimers were cross-linked (we ssume tht the only effect of crosslinking is to constrin chins), the only species present in the system re dimers, nd the rection under considertion is: Fos-Fos + Jun-Jun T 2Fos-Jun (13) All of the peptides hve CGG linkers t their N-terminl ends, s described in the experimentl study (O She et l., 1989). In our sttisticl potentil, the Cys-Cys interction is the strongest. Even with no dditionl potentil for crosslink formtion, this interction is sufficient to keep the two Cys residues intercting t ll times (see the low vlues of ccessible volume V 0 in Tble 9). We use the pproch described in the Method section to obtin the relevnt configurtionl free energies of homodimeric Fos nd Jun nd heterodimeric Jun-Fos (note tht heterodimer is preferred by symmetryssee eq 7).

9 Theoreticl Study of Leucine Zipper Dimeriztion Biochemistry, Vol. 35, No. 3, Tble 9: Thermodynmicl nd Structurl Prmeters for the Fos-Fos, Jun-Jun nd Fos-Jun Dimers with CGG Linkers dominnt species protein P E b rel rtio of the dimers c free energies in coiled coil structure d free energy of unfolded chins vol occupied by the first bed V 0 g Fos-Fos e f (1.3) Jun-Jun e f (0.3) Fos-Jun X X e f (1.1) The predicted (clculted) dominnt species; X indictes which dimers re ssigned. b The dominnt species ssigned from the experiment. c The frctions of ech dimer. d The configurtionl free energy of the dimeric coiled coils in kt. e The configurtionl free energy of the unfolded, monomeric sttes in kt. f The configurtionl free energy chnge upon folding (the free energy of dimer minus twice the free energy of monomer) in kt units. g Volume occupied by the first bed V 0 in Å 3 of the second chin (provided tht the first bed of the first chin is pinned) nd verged over ll simultions. h The verge C R RMS devition of single structure from the verge structure (stndrd devition is shown in prentheses). RMS in Å h Tble 9 shows the predicted dominnt species for the unimolr mixture of Fos (corresponding to residues of c-fos (Schuermnn et l., 1991) with n dditionl H200Y muttion) nd Jun (corresponding to residues of c-jun (Schuermnn et l., 1991) with dditionl H318Y muttion) with CGG linkers t N-termini. The free energy of ech sequence in the coiled coil dimer nd verge RMS devitions round the verge structures obtined from MC simultions re lso presented in Tble 9. Only heterodimers of Jun-Fos re predicted to be present. This is consistent with the experimentl observtion (O She et l., 1989). Bsed on Tble 9, homodimers of Jun hve the highest reltive stbility, wheres Fos homodimers hve the lowest reltive stbility. Becuse the verge free energy of homodimers (hlf of the sum of the free energies of Jun nd Fos) is higher thn the free energy of heterodimers, heterodimers re preferred. The specificity of heterodimer formtion cn be ccounted for by the reltively low stbility of Fos-Fos homodimers. This observtion is in greement with the previous suggestion bsed on experiment tht heterodimers re formed becuse of the low stbility of Fos homodimers (O She et l., 1989). Thus, in this exmple the most stble species (Jun-Jun) is not dominnt, nd the preference for heterodimer formtion comes from the instbility of nother component of the system (Fos homodimers). The plots of the energy per residue presented in Figure 5 indicte tht the resons for the reltively low stbility of Fos homodimers with respect to the Fos-Jun heterodimer re quite complex. Bsed on our simultions, the residues tht contribute to the reltive destbiliztion of the homodimeric Fos structure through interhelicl interctions re locted in region 5-13 (in prticulr, residues in positions 5 nd 8d) nd residues t positions 22d, 30e, nd 33. Position 8d (Leu in both Fos nd Jun) is destbilized in Fos due to the interction with two Thr residues (5, 12), wheres in Fos- Jun, Leu 8d intercts fvorbly with Ile 5 nd Vl 12 from Jun monomer. Positions 22d (Leu in both Fos nd Jun) nd 30e (Leu in Fos nd Arg in Jun) re reltively destbilized in Fos-Fos homodimer due to unfvorble pcking interctions nd prtil exposure to solvent. Position 33 (Lys in Fos nd Vl in Jun) is destbilized in Fos-Fos due to the unfvorble interctions with the corresponding 33 from the second chin. In generl, the determinnts of lower reltive stbility of Fos-Fos re relted to both inter- nd intrhelicl interctions. It is interesting to compre the bsolute stbilities of dimers with GGC linkers. Tble 9 shows the configurtionl free FIGURE 5: Plot of the energy per residue difference between Jun- Fos nd Fos-Fos homodimers (computed s the energy per residue in Jun-Fos minus the energy per residue in Fos-Fos). The lbels for residue numbers indicte (from the top) the Fos residue t given position, the Jun residue, nd the heptd positions, respectively. The lrgest, energetic reltive destbiliztion of Fos-Fos homodimer ppers to come from residues 4-13, s well s residues 22, 30, nd 33. Most of those residues occupy positions t the helicl interfce (, d, e, g). energy differences (free energy of folded structures minus twice the free energy of unfolded monomers) for both homodimers nd heterodimer. The bsolute stbilities re consistent with the reltive stbilities. However, Jun-Fos heterodimer is now roughly of the sme stbility s Jun-Jun homodimer. DISCUSSION In this pper, we hve presented method bsed on trnsfer mtrix tretment to clculte the monomer-dimer equilibrium in leucine zipper systems. For ll cses exmined, the prediction of dominnt species is in good greement with experiment. A detiled nlysis of the simultions showed for some frgments of the GCN4 leucine zipper tht quite substntil rerrngement cn occur with respect to the wild type GCN4. Sometimes, s in the frgment, mny residues rerrnge becuse of lck of stbiliztion of the coiled coil structure which when combined with loop entropy fvors helix dissolution. In other cses (8-30), side chins slightly rerrnge to minimize the conformtionl free energy. Thus, becuse frgments cn loclly djust their conformtion, one cnnot simply ssess subdomin stbility bsed on the wild type dt lone. Our results confirm the observtion of Lumb et l. (1994) tht some subdomins (frgments) of the GCN4 cn form stble dimeric coiled

10 964 Biochemistry, Vol. 35, No. 3, 1996 Vieth et l. coils. On the bsis of our dt, we speculte tht, in order to stbilize hydrophilic Asn residue in helicl conformtion in the hydrophobic core, it is necessry to hve two sufficiently stble, intercting helicl turns t both sides of this residue. This observtion is n illustrtion of loop entropy, which in coiled coils prohibits rndomly coiled residues between intercting helicl stretches. If one of those helicl turns is not strong enough to force hydrophilic residue in the helicl interfce to be helicl, then the entire frgment will be disordered. In Fos-Jun system, heterodimers re formed becuse of the reltive instbility of Fos homodimers. This reltive instbility is predicted to come minly from the region comprising residues 5-13 (interfcil Thr 5, Leu 8d). Other interfcil residues (Leu 22d nd Leu 30e, Lys 33) lso destbilize the Fos-Fos homodimer. The source of the reltive instbility comes from interhelicl interctions, locl pcking, nd different buril preferences. Our findings point out tht the reltive instbility of Fos homodimers comes minly from those residues occupying positions nd d of the heptd repet. Residues t e nd g positions seem to be less importnt. In generl, our predictions re in greement with experiment (Schuermnn et l., 1989, 1991; O She et l., 1992); however, further investigtion is required to determine the specific role of ny given residue in the preferentil heterodimer formtion. Becuse preferentil heterodimer formtion results from the reltive instbility of Fos homodimers, ll of the residues responsible for the reltive instbility of Fos homodimers ultimtely drive the process of heterodimeriztion. Coiled coils re highly coopertive systems, nd mny of the phenomen observed in these systems cn be explined by either coopertive or nondditive effects. By coopertivity, we men the dditionl energy system gins if two or more events occur t the sme time. The stbiliztion of Asn 16 in the helicl interfce in the wild type ws explined by the necessity of mintining the coopertive network of hydrogen bonds nd by the stbilizing role of the coopertive pirwise interctions (Vieth et l., 1994, 1995). Similrly, the effect of single point N16V muttion ws rtionlized by the coopertive pirwise interctions nd nonlocl compenstion effects (Vieth et l., 1995). The difference between the energetic profiles of the wild type nd frgments cn lso be explined by smller or lrger structurl rerrngements. The lck of stbiliztion of the hydrophilic Asn in the interfce of the subdomin cn be explined by the insufficient coopertivity of the hydrogen bond network nd side chin interctions in the region djcent to this residue. Finlly, the stbiliztion of Lys residues in the helicl interfce of Fos-Fos nd Jun-Fos cn lso be explined by the stbiliztion resulting from the coopertive hydrogen bond network s well s the coopertive pcking interctions. The method we presented in this pper is, in principle, generl enough to clculte the free energy of folding of ny smll protein ssuming tht the finl structure is known or cn be deduced. Thus, the effect of the muttions could be studied for both folded nd unfolded sttes nd compred to experimentl dt. In principle, the method presented here is not limited to lttice models but could be pplied to ny protein model tht hs smll number of locl minim for the corresponding set of peptide frgments. Work in this direction is now in progress. The work presented in this pper sets the stge for the lttice free energy clcultions of proteins. It lso builds groundwork for fst coiled coil prediction lgorithms. In prticulr, this work provides the bsis for the utomted ssessment of the heterodimeriztion bility of leucine zippers tht is of gret importnce in studies of trnscriptionl regultion. While coiled coil systems hve been recognized s the simplest exmples of proteins nd the methodology presented here hs, even for these systems, limittions nd problems, this pper provides encourgement for further theoreticl studies long these lines of proteins nd biologicl systems in generl. ACKNOWLEDGMENT Helpful discussions with Prof. Chrles L. Brooks, III, re cknowledged. We thnk the referees for their useful suggestions. A.K. is n Interntionl Reserch Fellow of the Howrd Hughes Medicl Institute. APPENDIX A Description of the Lttice Model. (1) Geometric Representtion nd MoVe Set. There re in principle 90 possible wys of connecting two consecutive R-crbons on this lttice, but some geometric restrictions for consecutive two nd three sets of vector occurrences limit this number by roughly 60%. Crystl structures of proteins from the Brookhven Protein Dt Bnk, PDB, cn be represented by the lttice model with n verge, root men squre devition, RMS, of Å (Godzik et l., 1993). For the folded stte clcultions, the Monte Crlo move set consists of two bond moves, three bond rerrngements, smll shifts of lrger chin pieces, chin end modifictions, nd rotmer equilibrtion. One Monte Crlo cycle for this system is considered to hve (N - 2)M two bond moves, 2M two bond moves, M shifts of the chin pieces, nd M(N - 3) three bond moves, where M is the number of chins (in this clcultion M ) 2) nd N is the number of R-crbons. Ech simultion run consisted of (200000)M cycles (Vieth et l., 1995). (2) Interction Scheme. The entire potentil (with the exception of the hydrogen bond term) is bsed on sttisticl nlysis of set of high resolution crystl structures from the PDB dtbse. The use of sttisticl potentil to estimte protein stbility hs been ttempted previously. For exmple, Brynt nd Lwrence showed tht the frequency of occurrence of chrged residues in the proteins from PDB obeys Coulombs lw; however, the dielectric constnt is too high (Brynt & Lwrence, 1991). Thus, such sttisticl potentils my provide some insight towrd understnding protein stbility. Bsed on the folding of the Hodges sequences (Hodges et l., 1981) nd test of the dynmic stbility of ssembled dimers, the scling fctors for the different energy terms were chosen to keep the helix content of the nonintercting chins below 50%, s well s to mintin proper blnce of the short rnge nd long rnge interctions. The scling fctors for ll of the coiled coils systems studied below re the sme s in the previous study on the GCN4 leucine zipper folding from rndom chins (Vieth et l., 1994b) s well s in previous study of oligomeric equilibri (Vieth et l., 1995). A detiled description of ll energy terms is presented elsewhere (Vieth et l., 1995). The totl energy of the entire system is given by (Vieth et l., 1995):