Favorable domain size in proteins Dong Xu 1 * and Ruth Nussinov 1,2

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1 Research Paper 11 Favorable domai size i proteis Dog Xu 1 * ad Ruth Nussiov 1,2 Backgroud: It has bee observed that sigle-domai proteis ad domais i multidomai proteis favor a chai legth i the rage 1 15 amio acids. To uderstad the origi of the favored size, we costruct a empirical fuctio for the free eergy of ufoldig versus the chai legth. The parameters i the fuctio are derived by fittig to the eergy of hydratio, etropy ad ethalpy of ufoldig of ie proteis. Our eergy fuctio caot be used to calculate the eergetics accurately for idividual proteis because the eergetics also deped o other factors, such as the compositio ad the coformatio of the protei. Nevertheless, the eergy fuctio statistically characterizes the geeral relatioship betwee the free eergy of ufoldig ad the size of the protei. Results: The predicted optimal umber of residues, which correspods to the maximum free eergy of ufoldig, is 1. This is i agreemet with a statistical aalysis of protei domais derived from their experimetal structures. Whe a chai is too short, our eergy fuctio idicates that the chage i ethalpy of iteral iteractios is ot favorable eough for foldig because of the limited umber of iter-residue cotacts. A log chai is also ufavorable for a sigle domai because the cost of cofiguratioal etropy icreases quadratically as a fuctio of the chai legth, whereas the favorable chage i ethalpy of iteral iteractios icreases liearly. Coclusios: Our study shows that the eergetic balace is the domiat factor goverig protei sizes ad it forces a large protei to break ito several domais durig foldig. Addresses: 1 Laboratory of Experimetal ad Computatioal Biology, IRSP, SAIC Frederick, NCI-FCRDC, Frederick, MD , USA. 2 Sackler Istitute of Molecular Medicie, Tel Aviv Uiversity, Tel Aviv 69978, Israel. *Preset address: Computatioal Bioscieces, Life Scieces Divisio, Oak Ridge Natioal Laboratory, Oak Ridge, TN , USA. Correspodece: Ruth Nussiov ruth@cifcrf.gov Key words: domai, etropy, free eergy, protei foldig, statistical aalysis Received: 15 September 1997 Revisios requested: 8 October 1997 Revisios received: 15 October 1997 Accepted: 22 October 1997 Published: 27 November Foldig & Desig 27 November 1997, 3:11 17 Curret Biology Ltd ISSN Itroductio It has bee widely recogized that proteis are ofte divided ito domais havig 1 15 amio acids [1 4]. Give the regular size of these protei buildig blocks, a periodicity i protei sizes is expected. Ideed, clusterig of molecular sizes aroud multiples of a uit size, ~14 kda, has bee observed i Escherichia coli proteis by gel electrophoresis [5]. A statistical aalysis of a large collectio of oredudat protei sequeces idicates that proteis cosist of sequece uits with characteristic legths of 125 residues for eukaryotes ad 15 residues for prokaryotes [6]. It has bee proposed that multidomai proteis may have evolved from proteis havig sigle domais via domai isertio [7]. For example, mammalia aspartic proteases, which cotai two structurally homologous lobes, are suggested to have arise durig evolutio from a homodimer ezyme via gee duplicatio ad fusio [8,9]. Similarly, the reverse trascriptase may have evolved from domai fusio of acestors of the type I ribouclease H ad the polymerase domai [1]. Durig the foldig process, large proteis are thought to form several stable collapsed hydrophobic foldig uits or domais, which the assemble [11,12]. The foldig of each stable domai i a multidomai protei is similar to the foldig of a sigle-domai protei [13]. The rather uiform size of this structural uit suggests that a geeral priciple such as geometrical or physical optimizatio at the DNA or protei level is resposible [5]. Berma et al. [6] proposed a possible recombiatioal origi of the domai structure. Because the optimal umber of residues i protei domais correspods to the optimal size for circularizatio of DNA circles [14], the authors suggest that a DNA circle could be a elemetary recombiatioal uit i the early evolutio of proteicodig sequeces. Although this hypothesis may be valid, the optimizatio at the gee level durig evolutio has bee uder the pressure of the gees products (i.e. proteis). Hece, it is reasoable to assume that there is a fudametal base i eergetics for protei domais to favor a particular size. Dill developed a lattice model to accout for the eergetics of protei foldig [15]. The model predicts a optimal chai legth for maximal protei stability. If a chai is too short, the protei would have too little iterior volume to form eough favorable cotacts betwee hydrophobic residues. If a chai is too log, the protei is forced to bury may ufavorable hydrophilic residues i the iterior of the protei, give a particular ratio betwee the umber of hydrophobic ad hydrophilic residues. But because the theory idicates that the optimal domai size ca icrease rapidly with icreasig the ratio betwee hydrophobic ad hydrophilic

2 12 Foldig & Desig Vol 3 No 1 residues, it failed to explai why durig evolutio favorable large protei domais have ot bee geerated by icreasig the fractio of hydrophobic residues i a chai. I order to study the foldig thermodyamics as a fuctio of protei chai legth it is importat ad feasible to employ a more realistic model of protei eergetics tha the highly simplified lattice model. Such a study will provide a quatitative ratioale for protei domais to have a preferable size. It will also shed some light o domai formatio i large proteis. Here, we provide a statistical descriptio of the eergy terms as a fuctio of chai legth, matched through thermodyamic data characterized by Makhatadze ad Privalov [16]. It is ot our itetio to derive a uiversal free eergy fuctio for the chai legth for idividual proteis because foldig thermodyamics ot oly depeds o the chai legth, but also o other factors such as the shape of the protei ad its disulfide bods. Nevertheless, the statistical eergy fuctio of chai legth reflects the average tred i protei eergetics. I particular, as we show below, the optimal chai legth i protei domais, predicted from the statistical eergy fuctio, is i agreemet with the observatio o experimetal protei structures. I this paper, we first itroduce the theory ad the methods. The we preset the results of the derived eergy fuctio ad its predictio of optimal domai size, compared with the statistical aalysis of protei structures. Fially we discuss the validity ad implicatios of our results. Results Theory ad methodology We first describe our eergy terms as a fuctio of chai legth ad the assumptios that they etail. Next we provide the data used i fittig the eergy fuctios. Fially we itroduce the method employed i the statistical aalysis of the protei structures. Thermodyamics of protei ufoldig Let us cosider a protei ufoldig at temperature T. The overall chage i free eergy for the protei ufoldig, G, ca be described by [16]: G = + H it T S cof (1) where is the chage i the free eergy of hydratio o ufoldig, H it is the chage i the ethalpy of iteral iteractios o ufoldig i the gaseous phase, ad S cof is the cofiguratioal cotributio to the etropy chage durig the ufoldig. The sig of G govers the foldig process. If G >, the peptide chai ca fold ito a folded protei; if G <, the chai caot fold. is the free eergy chage of ufoldig due to the trasfer of the protei from the gaseous phase to water. If the chage i free eergy is G i the gaseous phase for the hypothetical ufoldig process that forms the same ative protei structure the: = G G (2) where G is the chage i free eergy for the ative protei ufoldig, as used i Equatio 1. The value of is proportioal to ASA, the chage i the solvet accessible surface area o ufoldig [16]. For globular proteis, ASA, i uits of Å 2, ca be writte as [17]: ASA = 1.48 M w 6.3 M.73 w (3) where M w is the molecular weight of the protei i uits of Dalto. The average molecular weight of the residues i globular proteis is Da [18]. Hece, M w ca be approximated by: M w = (4) where is the umber of residues. Take together: = a ( ) (5) where a is a costat to be determied. The chage i the ethalpy of itramolecular iteractios o ufoldig i the gaseous phase, H it, ca be writte as: H it = b+ c (6) where b ad c are costats. As show i [16], the specific cofiguratioal etropy of protei ufoldig, S cof / icreases with icreasig. Thus: S cof / = f+ g (7) where f ad g are costats. We shall come back to justify the fuctioal forms of H it ad S cof /. Thermodyamic data To determie the parameters a, b, c, f ad g i Equatios 5 7, we have employed a data set of ie sigle-domai proteis ad we use, H it ad S cof, all of which were derived from the experimetal free eergy of ufoldig G [16]. The physiological temperature (37 C) is used: T = 31K (8) The thermodyamic quatities i [16] were measured at 25 C ad 5 C. Because the iterval betwee 25 C ad 5 C is small, we liearly extrapolated a quatity at 37 C [Q(37)] from the correspodig values at 25 C [Q(25)] ad 5 C [Q(5)]: Q(37) = Q(25) + Q(5) (9) The values of, H it ad S cof / at 37 C i ie proteis are listed i Table 1. Domai size Islam et al. [19] assiged the domais i the 284 oredudat protei chais based o iter-residue cotacts

3 Research Paper Favorable domai size i proteis Xu ad Nussiov 13 Table 1 Thermodyamics characteristics of the studied proteis at 37 C. H it S cof / Protei (kj mol 1 ) (kj mol 1 ) (JK 1 mol 1 ) BPTI ( 1467) 2349 (2345) (49.9) Ubiquiti ( 29) 296 (3246) (5.49) RNase T ( 2877) 4338 (4648) (52.68) Cytochrome c ( 2877) 5282 (4648) (52.68) Barase ( 365) 4939 (4949) (53.15) RNase A ( 359) 5658 (565) (54.24) Lysozyme ( 3669) 6151 (59) (54.63) Iterleuki 1β ( 4442) 6296 (712) (56.51) Myoglobi ( 4442) 7618 (712) 57.6 (56.51), Number of amio acids;, chage i free eergy of hydratio o ufoldig; H it, chage i the ethalpy for iteral iteractios o ufoldig; S cof, cofiguratioal cotributio to the etropy for protei ufoldig. The umbers that are ot i brackets derive from the data i [16] at 37 C; the umbers i brackets were calculated usig Equatios 5 7. i experimetal protei structures. There are 197 sigledomai proteis ad 22 domais i multidomai proteis. We have employed the sizes of the domais i their domai assigmet. Assume there are M(i) domais, where i is the umber of residues ad i = 1, 2, 3,..., N. The distributio of protei domais aroud a domai of m residues i legth, P(m), ca be calculated by: 1 i= m+ v P ( m) = M ( i) (1) (2v + 1) K i= m v where K is the total umber of protei domais, ad 2v +1 is the widow size, i umber of amio acids. We ow preset the empirical free eergy of ufoldig as a fuctio of the umber of residues. From this fuctio, we will derive the optimal legth of amio acids i a protei domai ad compare it with the statistical aalysis of experimetal protei structures. Matchig the parameters The data i Table 1 are matched by Equatios 5 7 through the least squares fit. The parameters obtaied are as follows: a =.2353 kj mol 1 (11) b = 5.7 kj mol 1 c = kj mol 1 (12) f =.789 J K 1 mol 1 g = J K 1 mol 1 (13) Figure 1 ad Table 1 show the compariso betwee the experimetal values ad the fittig results derived from Equatios 5 7 ad the parameters give i Equatios The correlatio coefficiets betwee the values i [16] ad those calculated by our eergy fuctio are.934 for ad.964 for H it, both values idicatig a very good fit. The correlatio coefficiet of S cof / is.655, which is ot as good as the values for ad H it but is still sigificat. The relatively weak correlatio of S cof / is due to the less sesitive depedece of S cof / o. Overall, the quality of the fittigs shows the validity of the fuctioal forms i Equatios 5 7. I particular, there is a costat term c i H it as show i Equatio 6. The fuctioal form of H it ca be uderstood from a heuristic illustratio. H it describes the chage i ethalpy of iteral iteractios o ufoldig i the gaseous phase. If a peptide chai is too short, it caot form eough iter-residue cotacts for itramolecular iteractios to cotribute a positive H it. This is reflected i the egative costat c i H it, which esures the miimum umber of residues to establish stable iteractios (i.e. H it > oly if > 11; H it < i the rage of 11 is a artifact of the fittig, but it is irrelevat to our subject). Oce the stable iteractios are established, the cotributio of a particular residue to H it is mostly from its eighborig residues. Because the desity is homogeous i proteis, the umber of eighborig residues is basically costat. Hece, it is ot surprisig that H it liearly correlates with the umber of residues. Optimal domai size derived from the fittig fuctio By usig Equatio 1 ad Equatios 5 7 together with the fittig parameters, the free eergy of ufoldig ca be writte as: G() = (kj mol 1 ) (14) Figure 2a shows G() ad its three eergy compoets,, H it ad T S cof, derived from Equatios 5 7. ad T S cof destabilize the folded protei. H it cotributes to the protei stability whe > 11. The three compoets are very large compared to G() itself.

4 14 Foldig & Desig Vol 3 No 1 Figure 1 (a) S cof / (kj K 1 mol 1 ) H it (kj mol 1 ) (kj mol 1 ) (b) (c) Foldig & Desig The data derived from [16] (filled circles) ad theoretical fittig curves of Equatios 5 7 (lies) as a fuctio of the umber of residues (), at 37 C. (a) The chage i the free eergy of hydratio o ufoldig, ; (b) the chage i the ethalpy of iteral iteractios o ufoldig, H it ; ad (c) the cofiguratioal compoet of the chage i etropy of protei ufoldig per residue ( S cof /). Figure 2b idicates that there is oly a arrow regio of (i.e. mi < < max ) i which G() >. G() reaches its maximum value at = opt. The values obtaied for mi, max ad opt are: mi = 6.4, max = 143.1, opt = 1.4 (15) Hece, the optimal legth of a sigle-domai protei is 1 residues, ad folded sigle-domai proteis are stable i the rage residues. Statistical aalysis of protei domais The distributios of protei domai sizes are show i Figure 3. Figure 3a compares the distributios of sigledomai protei chais with those of domais i multidomai chais, over itervals of 5 amio acids. The figure illustrates that the two distributios are similar. Whe the two distributios are combied, we obtai the average domai size: ave = 148 (16) To determie the optimal value ( dom ) for the maximum P(), we have calculated P() for cotiguous with a widow size of 21, as show i Figure 3b. The optimal value is: dom = 16 (17) Figure 3b idicates that P() is asymmetrical. Discussio We ow discuss the validity of our eergy fuctio, the optimal chai legth of proteis based o the free eergy fuctio, ad the etropic cotributio, which is crucial for determiig the rage of favorable chai legths i a domai. We also aalyze the thermodyamic origi of the divisio of a large protei ito several domais. Fially, we address some limitatios of our study ad possible further exploratios alog our approach. Eergy fuctios We have derived the eergy fuctios for the chage i free eergy of hydratio for ufoldig ( ), the chage i ethalpy of iteral iteractios for ufoldig ( H it ), ad the cofiguratioal cotributio to the etropy of protei ufoldig ( S cof ) at 37 C for ie sigle-domai proteis. There are alterative fittig approaches establishig the relatioship betwee the eergetics of the foldig of a protei sequece ad its correspodig protei structure. For example, oe may fit the eergy terms by the chage of the hydrophobic ad hydrophilic surface areas durig ufoldig [2] ad the covert the surface areas to the umber of residues. Such idirect coversios may, however, lead to a ureliable depedece of due to the strog correlatio betwee the hydrophobic surface area ad the hydrophilic oe. For studyig -depedet eergetics, it is importat to fit the experimetal thermodyamic data directly to, as we have doe here. Our eergy fuctios do ot presume a dielectric costat of the protei, a particular shape of its domais, or a particular ratio betwee the umber of hydrophobic ad hydrophilic residues. Our assumed eergy fuctios fit well with the data of, H it ad S cof / i ie proteis. O the other had, the absolute value of the free eergy of ufoldig is much less tha its compoets, H it ad T S cof. Typical values for the free eergy of ufoldig are i the order of tes of kj mol 1, whereas the three compoets

5 Research Paper Favorable domai size i proteis Xu ad Nussiov 15 Figure 2 (a) 75 (b) 8 Eergy (kj mol 1 ) H it G() T S cof G (kj mol 1 ) max mi opt Foldig & Desig Eergy terms as a fuctio of the umber of residues, at 37 C. (a) G() based o Equatio 14 (dashed lie) ad its compoets (, H it ad T S cof ; solid lies) based o Equatios 5 7. (b) The solid lie is a magificatio of (a) for G(). The dashed lie marks G =. mi ad max are the cross poits betwee the two lies (i.e. G() > for mi < < max ) ad opt is the optimal value at which G() reaches its maximum. are usually two orders higher. Hece, oe caot use our free eergy fuctio to calculate the free eergy of ufoldig for a idividual protei. Nevertheless, the free eergy fuctio, derived from the three compoets, reflects well the statistical profile of domai stability versus chai legth, as well as the ivolvemet of each free eergy compoet. Optimal chai legth i proteis Our free eergy fuctio predicts that the optimal umber of residues i a sigle-domai protei is 1. This is i agreemet with a statistical aalysis of protei domais, which idicates a optimal legth of 16. The quatitative agreemet substatiates the otio that the foldability of a chai govers the preferred legth of protei domais. The optimal free eergy of ufoldig at = 1 origiates from the balace of three compoets,, H it ad T S cof. Both ad T S cof destabilize folded proteis. As becomes larger, H it gradually overcomes the two ufavorable free eergy compoets ad drives the total free eergy to be favorable for foldig. If a peptide chai is too short, there are isufficiet iter-residue cotacts to have a large eough H it for a stable folded protei, similar to the sceario depicted by Dill [15]. Whe Figure 3 (a) 7.5 (b) P() (1 3 ) P() (1 3 ) dom Foldig & Desig Domai legth distributio P() as a fuctio of the umber of residues. (a) P() for residues 1 5, 51 1, 11 15,... with a widow of 5 amio acids. The solid lie represets the sigle-domai chais (197 chais) ad the dashed lie represets the domais i the multidomai chais (22 domais). (b) P() versus the domai size of residues, for the domais i both the sigle-domai chais ad the multidomai chais (399 domais). P() is calculated for a widow size of 21 (i.e. i the rage of ± 1). dom idicates the value at which P() reaches its maximum.

6 16 Foldig & Desig Vol 3 No 1 > 1, because T S cof has a quadratic form as a fuctio of, T S cof decreases the free eergy of ufoldig faster with respect to tha H it icreases the free eergy of ufoldig; H it icreases oly liearly versus. Hece, a sigle-domai protei caot be too large. This explaatio differs from Dill s suggestio that a log chai is ufavorable for a sigle-domai protei because it has to bury may ufavorable hydrophilic residues i the iterior [15]. Compared with Dill s theory, our model is based o a more realistic eergy fuctio rather tha a simplified lattice model, ad our quatitative predictio does ot require ay presumed compositio betwee the hydrophobic residues ad the hydrophilic residues. The domiat factor i goverig the size of sigle-domai proteis is the quadratic form of S cof (). If S cof () were liear, the larger (or the smaller) the protei size the better. The origi of the 2 depedece of S cof () has bee suggested to accout for the surface residues, which are less compact tha the iteral residues [16]. The umber of surface residues is proportioal to 2/3 rather tha 2, however. We therefore propose aother mechaism for the 2 depedece i S cof (). As a protei folds, the first etropy loss is the restrait i the movemet of each residue, for both its backboe ad its sidechai. This term is proportioal to. I additio, a sequece has to fold ito a uique cofiguratio. There is oly a limited umber of eergetically allowed coformatioal categories or protei folds for a sigle domai [21,22]. The protei folds impose restraits o the iter-residue distaces. The loger a sequece, the more restraits it has. This additioal cost i cofiguratioal etropy durig foldig icreases dramatically as icreases, ad may result i the 2 depedece i S cof (). Our free eergy fuctio predicts that folded sigledomai proteis oly stabilize i the rage residues. Although most protei domais i the statistical aalysis of protei structures fall ito this rage, there are some sigle-domai proteis beyod it. We suggest the followig reasos for the discrepacy. First, there may be errors origiatig from the data i [16] ad the fittig of the eergy terms, so that the free eergy fuctio is ot accurate eough. Secod, some of the proteis i our statistical aalysis are trasmembrae or membrae-boud proteis. For example, the M segmet of the trasmembrae protei photosythetic reactio ceter has 333 residues; ad cytochrome P45, which is a membraeboud protei, has 457 residues. Because our eergy fuctio was derived from globular proteis, it caot describe trasmembrae or membrae-boud proteis. Third, disulfide bods ca reduce the etropy cost so that the rage of favorable chai legth ca be wideed. Small proteis, i particular, usually have disulfide bods ([18]; e.g. there are three disulfide bods i the protei BPTI with 58 residues), but very large proteis also have disulfide bods (e.g. there are ie disulfide bods i the protei euramiidase with 388 residues). Fourth, some large sigle-domai proteis cosist of loops which do ot pack compactly aroud the protei core. For example, i the large protei purie ucleoside phosphorylase (289 residues; PDB code 1ula), the loop regios 59 65, ad hag aroud the protei core ad oly loosely coect to other parts of the protei. Such ucompact packig ca also reduce the etropic cost of protei foldig so as to icrease the upper boud of sigle-domai protei size. Fially, the method of Islam et al. [19] may ot be sesitive eough to fid all the possible domais i proteis. Some of the assiged large sigle-domai proteis may have two or more domais i the actual foldig. Multiple domais i large proteis Aalysis based o kow protei structures shows that a peptide chai larger tha 25 amio acids barely folds cooperatively ito just oe domai [23]. Multidomai proteis form through the dockig of domais, which are the compact, hydrophobic, idepedetly foldig uclei [12] durig the protei foldig process. The distributio of the chai legths i sigle-domai proteis is similar to the distributio of the umber of amio acids i domais of multidomai proteis, (Figure 3a). The periodicity observed i protei sizes [5,6] reflects the distributio of domai sizes. The period of chai legth, accordig to the distributio i Figure 3b, should be betwee the optimal value for maximum P() ( dom = 16) ad the average domai size ( ave = 148). This is i agreemet with the studies i [5,6]. The thermodyamic origi for large proteis to divide ito several domais i foldig is revealed i our eergy fuctios. The etropic cost for protei foldig icreases quadratically with icreasig chai legth i a sigle-domai protei. By dividig the protei ito several domais, the sum of the etropic cost of each domai is substatially less tha the cost i formig a sigle-domai protei. Some limitatios Our model provides a ew perspective for further studies o the structure eergetics relatioship of proteis, i particular o the thermodyamic origi of protei sizes. The coceptual framework of our study does ot deped o the details of the eergy fuctios. Nevertheless, there are some limitatios i our studies due to the limited size ad quality of the thermodyamic data. Furthermore, differet methods vary i partitioig the free eergy of ufoldig, especially for the assessmet of the cofiguratioal etropy [24 28]. Although our eergy fuctios fit the data well, the model is based o statistics rather tha o a ab iitio method. Hece, the fuctioal forms of eergetics ca be improved from further studies. As more data become available, the ew eergy fuctios may result i more accurate estimates of the domai size ad eable studies of additioal aspects of protei sizes, such as the relatioship betwee the size ad the compositio of a protei, its shape, or its disulfide bods.

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