Helix Coil Theory

Biophysical Chemistry 101–102(2002) 281–293

0301-4622/02/$ - see front matter� 2002 Elsevier Science B.V. All rights reserved.PII: S0301-4622Ž02.00170-9

Recent advances in helix–coil theory

Andrew J. Doig

Department of Biomolecular Sciences, UMIST, P.O. Box 88, Manchester M60 1QD, UK

Received 18 December 2001; received in revised form 1 March 2002; accepted 1 March 2002

Abstract

Peptide helices in solution form a complex mixture of all helix, all coil or, most frequently, central helices withfrayed coil ends. In order to interpret experiments on helical peptides and make theoretical predictions on helices, itis therefore essential to use a helix–coil theory that takes account of this equilibrium. The original Zimm–Bragg andLifson–Roig helix–coil theories have been greatly extended in the last 10 years to include additional interactions.These include preferences for the N-cap, N1, N2, N3 and C-cap positions, capping motifs, helix dipoles, side chaininteractions and 3 -helix formation. These have been applied to determine energies for these preferences from10

experimental data and to predict the helix contents of peptides. This review discusses these newly recognised structuralfeatures of helices and how they have been included in helix–coil models.� 2002 Elsevier Science B.V. All rights reserved.

Keywords: a-Helix; Random coil; Protein stability; Protein structure; Zimm–Bragg; Lifson–Roig

1. Introduction

It is a pleasure to be a part of this tribute toJohn Schellman. He has made numerous funda-mental contributions to biophysical chemistry. Inparticular, he was the first person to analyse thethermodynamics of the helix–coil transitionw1x,thus founding the field that is the subject of thisreview. Peptides that form helices in solution donot show a simple two-state equilibrium betweena fully folded and fully unfolded structure. Insteadthey form a complex mixture of all helix, all coil

E-mail address: [email protected](A.J. Doig).

or, most frequently, central helices with frayed coilends. In order to interpret experiments on helicalpeptides and make theoretical predictions on heli-ces, it is therefore essential to use a helix–coiltheory that considers every possible location of ahelix within a sequence. The purpose of this reviewis to cover how helix–coil theories have beendeveloped in the last 10 years, principally byincluding additional structural features of the helix,such as the distinct preferences that amino acidshave for particular locations in the helix, or helixdipole interactions. I do not consider in detail theresults obtained from applying the models toempirical data, such as the values of preferences

282 A.J. Doig / Biophysical Chemistry 101 –102 (2002) 281–293

for helix interiors, or the origins of helix energet-ics, such as the influence of solvation on helixpreferences. These areas are well covered inreviews w2–8x, as well as numerous papers citedlater in the text.The first wave of work on helix–coil theory

was in the late 1950s and early 1960s, and thishas been reviewed in detail by Poland and Scher-aga w9x. Scheraga and co-workers, in particular,continued to use and develop these models todetermine helix preferences with hostyguest poly-peptides. In their work, long polypeptides ofhydroxybutyl-L-glutamine or hydroxypropyl-L-glu-tamine containing 100s of amino acids were ran-domly substituted with low amounts of a guestamino acid. Application of helix–coil theory gavehelix preferences of the guestw10x.In 1992, Qian and Schellmanw11x reviewed

current understanding of helix–coil theories. Sincethis time there has been a great deal more interestin the field, primarily driven by two kinds ofexperimental result. Firstly, following the pioneer-ing work of Marqusee and Baldwinw12x, peptidemodels have become available that form stablea-helices, despite typically having only 15–20 aminoacids. Their sequences are mostly alanine, thusgiving a relatively context-free environment toinsert interactions of interest. Secondly, examina-tion of protein crystal structures has shown thatamino acids have distinct preferences for differentenvironments within the helix. For example, Argosand Palauw13x, Richardson and Richardsonw14xand Presta and Rosew15x showed in the 1980sthat amino acid preferences for the first positionsin the helix are entirely different from interiorpreferences. Helix–coil theory was thereforedeveloped to(i) provide a quantitative interpreta-tion of the new experimental results from shortpeptide models and(ii) include the new interac-tions missing from the older models. It is thepurpose of this review to examine this recent work.First, I briefly cover the problem of the helix–coiltransition and the older Zimm–Bragg(ZB) andLifson–Roig(LR) models. A more detailed expla-nation can be found in the Poland and Scheragaw9x or Qian and Schellmanw11x reviews. Then Icover new structural features of the helix and howthey are included within helix–coil models.

2. Structure of the a-helix

Proteins are built of regular local folds of thepolypeptide chain called secondary structure. Thea-helix was first described by Pauling, Corey andBranson in 1950w16x, and their model was quicklysupported by X-ray analysis of haemoglobinw17x.Irrefutable proof of the existence of thea-helixcame with the first protein crystal structure ofmyoglobin, in which most secondary structure ishelical w18x. a-Helices were subsequently foundin nearly all globular proteins. It is the mostabundant secondary structure, with;30% of res-idues found ina-helicesw19x.A helix combines a linear translation with an

orthogonal circular rotation. In thea-helix thelinear translation is a rise of 5.4A per turn of the˚helix and a circular rotation is 3.6 residues perturn. Side chains spacedi,iq3, i,iq4 and i,iq7are therefore close in space, and interactionsbetween them can affect helix stability. Spacingsof i,iq2, i,iq5 and i,iq6 place the side chainpairs on opposite faces of the helix, avoiding anyinteraction. The helix is primarily stabilised byi,iq4 hydrogen bonds between backbone amidegroups.The conformation of a polypeptide can be

described by the backbone dihedral anglesf andc. Mostf,c combinations are sterically excluded,leaving only the broadb region and narroweraregion. One reason why thea-helix is so stable isthat a succession of the sterically alloweda f,cangles naturally position the backbone NH andCO groups towards each other for hydrogen bondformation. It is possible that a succession of themost stable conformation of an isolated residue ina polymer with alternative functional groups couldpoint two hydrogen-bond donors or acceptorstowards each other, making secondary structureformation unfavourable. One reason why polypep-tides may have been selected as the polymer ofchoice for building functional molecules is thatthe sterically most stable conformations also givestrong hydrogen bonds.The residues at the N-terminus of thea-helix

are called N9–N-cap–N1–N2–N3–N4, etc., wherethe N-cap is the residue with non-helicalf,cangles immediately preceding the N-terminus of

283A.J. Doig / Biophysical Chemistry 101 –102 (2002) 281–293

Fig. 1. Zimm–Bragg and Lifson–Roig codes and weights for thea-helix.

ana-helix and N1 is the first residue with helicalf,c angles w14x. The C-terminal residues aresimilarly called C4–C3–C2–C1–C-cap–C9, etc.The N1, N2, N3, C1, C2 and C3 residues areunique because their amide groups participate ini,iq4 backbone–backbone hydrogen bonds usingeither only their CO(at the N-terminus) or NH(at the C-terminus) groups. The need for thesegroups to form hydrogen bonds has powerfuleffects on helix structure and stabilityw15x.

3. Helix–coil theory

The simplest way to analyse the helix–coilequilibrium, still occasionally observed, is the two-state model, in which the equilibrium is assumedto be between 100% helix conformation and 100%coil. This is incorrect and its use gives seriouserrors. This is because helical peptides are gener-ally most often found in partly helical conforma-tions, often with a central helix and frayed,disordered ends, rather than in the fully folded orfully unfolded states.

3.1. Zimm–Bragg model

The two major types of helix–coil model are(i) those which count hydrogen bonds, principallyZB w20x, and (ii) those that consider residueconformations, principally Lifson–Roigw21x. Inthe ZB theory the units being considered arepeptide groups and they are classified on the basisof whether their NH groups participate in hydrogenbonds within the helix. The ZB coding is shownin Fig. 1. A unit is given a code of 1(e.g. peptideunit 5 in Fig. 1) if its NH group forms a hydrogen

bond, and 0 otherwise. The first hydrogen-bondedunit proceeding from the N-terminus has a statis-tical weight of ss, successive hydrogen bondedunits have weights ofs, and non-hydrogen-bondedunits have weights of 1. Thes-value is a propa-gation parameter ands is an initiation parameter.The most fundamental feature of the thermody-namics of the helix–coil transition is that theinitiation of a new helix is much more difficultthan the propagation of an existing helix. This isbecause three residues need to be fixed in a helicalgeometry to form the first hydrogen bond, whileadding an additional hydrogen bond to an existinghelix requires that only one residue is fixed. Theseproperties are thus captured in the ZB model byhavings smaller thans. The statistical weight ofa homopolymeric helix of aN hydrogen bonds isss . The cost of initiation,s, is thus paid onlyNy1

once for each helix, while extending the helixsimply multiplies its weight by one additionals-value for each extra hydrogen bond. The completehelix–coil equilibrium is handled by determiningthe statistical weight for every possible conforma-tion that contains a helix, plus a reference weightof 1 for the coil conformation. The population ofeach conformation is given by the statistical weightof that conformation divided by the sum of thestatistical weights for every conformation(i.e. thepartition function). Thus, the greater the statisticalweight, the more stable the conformation. Partitionfunctions are extremely powerful concepts in sta-tistical thermodynamics, since they include allproperties of an equilibrium. Any property of theequilibrium can be extracted from the partitionfunction by applying the appropriate mathematical


function. This is analogous to quantum mechanics,where any property can be determined from asystem by applying an operator function to awavefunction. In this case the properties could bethe mean number of hydrogen bonds, the meanhelix length, the probability that each residue iswithin a helix, etc. Statistical weights can beregarded as equilibrium constants for the equilib-rium between coil and the structure(as the refer-ence coil weight is defined as 1). They cantherefore be converted to free energies asyRTln(weight).

3.2. Lifson–Roig model

In the LR model, each residue is assigned aconformation of helix(h) or coil (c), dependingon whether it has helicalf,c angles. Every con-formation of a peptide ofN residues can thereforebe written as a string ofN ‘c’s or ‘h’s, giving 2N

conformations in total. Residues are assigned sta-tistical weights depending on their conformationsand the conformations of surrounding residues. Aresidue in an h conformation with an h on eitherside has a weight ofw. This can be thought of asan equilibrium constant between the helix interiorand the coil. Coil residues are used as a referenceand have a weight of 1. In order to form ani,iq4 hydrogen bond in a helix, three successiveresidues need to be fixed in a helical conformation.M consecutive helical residues will therefore haveMy2 hydrogen bonds. The two residues at thehelix termini (i.e. those in the centre of chh orhhc conformations) are therefore assigned weightsof v. The ratio ofwyv gives the approximate effectof hydrogen bondingw1.7:0.036 for Alaw22x oryRTln(1.7y0.036)sy2.1 kcal mol x. A helicaly1

homopolymer segment ofM residues has a weightof v w and a population in the equilibrium of2 My2

v w divided by the sum of the weights of2 My2

every conformation(i.e. the partition function). Inthis way the population of every conformation iscalculated and all properties of the helix–coilequilibrium are evaluated. The LR model is easierto handle conceptually for heteropolymers, sincethew and v parameters are assigned to individualresidues. The substitution of one amino acid at acertain position thus changes thew- and v-values

at that position. In the ZB model, the initiationparameters is associated with several residuesand s with a peptide group, rather than a residue.It is therefore easier to use the LR model whenmaking substitutions. Indeed, most recent workhas been based on this model. A further differenceis that the ZB model assigns weights of zero to allconformations that contain a chc or chhc sequence.This excludes a very large number of conforma-tions that contain a residue with helicalf,c anglesbut with no hydrogen bond. In LR theory, theseare all considered. The ZB and LR weights arerelated by the following formulaw11x: sswy(1qv); ssv y(1qv) .2 4

The treatment of peptide conformations is basedon Flory’s isolated-pair hypothesisw23x. This statesthat while f and c for a residue are stronglyinterdependent, giving preferred areas in a Rama-chandran plot, eachf,c pair is independent of thef,c angles of its neighbours. Pappu et al. exam-ined the isolated-pair hypothesis in detail byexhaustively enumerating the conformations ofpoly(Ala) chains w24x. Each residue was consid-ered to populate 14 mesostates, defined by rangesof f,c values. By considering all 14 mesostateN

strings, all conformations were considered for upto seven alanines. The number of allowed confor-mations was found to be considerably fewer thanthe maximum, thus showing that the isolated-pairhypothesis is invalid. The chains mostly populatedextended or helical conformations, as many partlyhelical conformations are sterically disallowed.Such effects are not included in helix–coil theo-ries, thus presenting a considerable challenge forthe future. Helix–coil theories assign the sameweight (1) to every coil residue; steric exclusionmeans that this should vary and be lower than 1in many cases.

3.3. Single sequence approximation

Since helix nucleation is difficult, conformationswith multiple helical segments are expected to berare in short peptides. In the one-, or single-helical-sequence approximation, peptide conformationscontaining more than one helical segment areassumed not to be populated and are excludedfrom the partition function(i.e. assigned statistical


weights of zero). As peptide length increases, theapproximation is no longer valid, since multiplehelical segments can be long enough to overcomethe initiation penalty. The single-sequence approx-imation also breaks down when a sequence with ahigh preference for a helix terminus is within themiddle of the chain. The error from using thesingle-sequence approximation therefore shows awide variation with sequence and could be poten-tially serious if a sequence has a high preferenceto populate more than one helix simultaneously.Conformations with two or more helices may alsooften include helix–helix tertiary interactions thatare ignored in all helix–coil models.

4. Extension of the helix–coil models

The majority of recent work has been based onthe LR modelw21x. In the original model, weightsare assigned to residues in the centre of hhh triplets(a weight ofw for propagation) or in the centreof chh or hhc triplets(a weight ofv for initiation).Residues in all other triplets have weights of 1.LR-based models have been extended by assigningweights to additional conformations. In general,this work has been motivated by the discovery ofadditional features that affect helix stability inprotein crystal structures. Their inclusion withinhelix–coil theory has allowed the measurement ofthe effect of these features on helix stability.

4.1. N- and C-caps

The first residue in a helical(h) conformationat the N-terminus is called N1 and the precedingresidue, the last in a non-helical(c) conformationbefore the beginning of the helix, is called the N-cap. Argos and Palauw13x first showed that N-cappreferences differ from other helical positions,although the terminology was originated by Rich-ardson and Richardsonw14x. N-capping can there-fore be included in LR theory by assigning aweight of n to the central residue in a cch tripletw25x. Application of the model to experimentaldata for which the N-terminal amino acid of ahelical peptide was variedw26x allowed the deter-mination of then-values, and hence free energy of

N-capping(asyRTln n) of all the amino acidsw27x.Similarly, the C-cap is the first residue in a non-

helical conformation(c) at the C-terminus of ahelix. C-cap weights(c-values) are assigned tocentral residues in hcc triplets. Application of themodel to experimental data for which the C-terminal amino acid of a helical peptide was variedallowed the determination of thec-values, andhence free energy of C-capping(asyRTln c) ofall the amino acidsw27x.A problem with the original definitions of the

capping weights above is that they apply to isolat-ed h or hh conformations that are best regarded aspart of the random coil. A helical hydrogen bondcan only form when a minimum of three consec-utive h residues are present. The most stabilisingclass of N-caps(Asp, Asn, Ser, Cys and Thr)accept hydrogen bonds to the NH groups of theN3 residue. This is clearly impossible if an N3residue does not exist, which is the case if thereare not three or more consecutive h residues. TheDoig et al. modelw25x has the flaw that sequencessuch as Asp–Ala–Gly have a high population forthe chc conformation, as Asp has a high N-cappreference. This is incorrect, however, as the AspN-cap preference results from hydrogen bondingto the N3 position, so should not be apparent inchc and chhc conformations. Andersen and Tongw28x and Rohl and Baldwinw6x therefore changedthe definition of the N-cap to apply only to the cresidue in a chhh quartet. The N-cap residues inchc or chhc conformations have weights of zeroin the Andersen–Tong model and 1 in the Rohl–Baldwin model. These modifications give differentN-cap and helix interior energies after fitting.

4.2. Capping boxes

The N-terminal capping boxw29x includes a sidechain–backbone hydrogen bond from N3 to theN-cap (i,iy3). For example, a Ser–X–X–Glusequence has a high preference for a helix N-terminus, as the Ser side chain accepts a hydrogenbond from the Glu backbone NH, while the Serbackbone NH donates a hydrogen bond to the Gluside chain. This is included in the LR model byassigning a weight ofw=r to the chhh confor-


mation, wherer is the weight for the Ser backboneto the Glu side-chain bondw6x.

4.3. Side chain interactions

As helices have 3.6 residues per turn, side chainsspacedi,iq3 or i,iq4 are close in space. Sidechain interactions are thus possible when four orfive consecutive residues are in a helix. They areincluded in the LR-based model by giving a weightof w=q to hhhh quartets andw=p to hhhhhquintets. The side chain interaction is between thefirst and last side chains in these groups; thewweight is maintained to maintain the equivalencebetween the number of residues withw weightingand the number of backbone helix hydrogen bonds.Scholtz et al.w30x used a model based on the

one-helical-sequence approximation of the LRmodel to quantitatively analyse salt-bridge inter-actions in alanine-based peptides. Only a singleinteraction between residues of any spacing wasconsidered, although this was appropriate for thesequences they studied.Shalongo and Stellwagenw31x also proposed

incorporating side-chain interaction energies intothe LR model, using a clever recursive algorithm.In our model we have considered onlyi,iq3 andi,iq4 side chain interactionsw32x; Shalongo andStellwagen more generally consider side-chaininteractions of any spacing. In their implementa-tion of the Lifson–Roig formalism, however, theychange the definition of the propagating and initi-ating (capping) weights, such that the physicalmeaning of these parameters is lost and the numberof propagating residues(w) no longer correlateswith the number of hydrogen bonds formed. Theircapping parameters are associated with residues inhelical conformations, instead of coil. In proteins,however, N- and C-cap residues have non-helicaldihedral anglesw14x. This loss of physical meaningmeans that their energies for substituting residuesat capping and interior positions in peptide helicesare not directly transferable to helices in proteins.

4.4. N1, N2 and N3 preferences

The N1, N2 and N3 helix positions are uniquebecause their amide NH groups do not participate

in i,iq4 backbone–backbone hydrogen bonds. Thehelix N-terminus shows significantly different res-idue frequencies for the N-cap, N1, N2, N3 andhelix interior positionsw13,14,33x. Penel et al.w34xmade a detailed survey of the structures adoptedby the amino acids at N1, N2 and N3 and identifiedmany new structural features. They found that themost significant structural preferences can beexplained by short-range hydrogen bonding to thefree N-terminal NH groups, as foreseen by Prestaand Rosew15x, with the strongest trends being theN2 amino acids Gln, Glu, Asp, Asn, Ser, Thr andHis preferentially formingi,i or i,iq1 hydrogenbonds to the backbone. A complete theory for thehelix should therefore include distinct preferencesfor the N1, N2 and N3 positions.In the original LR model, the N1 and C1

residues are both assigned the same weight,v.Shalongo and Stellwagenw31x separated these asv and v . Andersen and Tongw28x did the sameN C

and derived complete scales for these parametersfrom fitting experimental data, although some val-ues were tentative. The helix initiation penalty isv =v , and sov - and v -values are all smallN C N C

(;0.04).We added weights for the N1, N2 and N3(n1,

n2 andn3) positions as follows. Then1-value isassigned to a helical residue immediately followinga coil residue. The penalty for helix initiation isnow n1Øv, instead of v , as v remains the C12

weight. An N2 helical residue is assigned a weightof n2Øw, instead ofw. The weightw is maintainedin order to keep the useful definition of the numberof residues, with aw weighting being equal to thenumber of residues with ani,iq4 main-chain–main-chain hydrogen bond. Then2-value is anadjustment to the weight of an N2 residue thattakes into account the structures that can be adopt-ed by side chains uniquely at this position. Simi-larly, an N3 residue is now assigned the weightn3Øw, instead ofw. Residues in the centre of therare hch conformation have a weight of 1. Appli-cation of the model to peptides designed to probethe N1 position(CH CO–XAAAAQAAAAQAA-3

GY–NH w35x) and the N2 position(CH CO–2 3

AXAAAAKAAAAKAAGY–NH w36x) has given2

N1 and N2 preferences for most amino acids forthese positions, and these agree well with prefer-


ences observed in protein structures. Petukhov etal. similarly obtained N1, N2 and N3 preferencesfor non-polar and uncharged polar residues byapplying AGADIR (see below) to experimentalhelical peptide data, and found almost identicalresultsw37,38x.

4.5. Helix dipole

The three non-hydrogen-bonded NH groups atthe N-terminus of the helix give a net positivecharge; similarly, the three CO groups of C1, C2and C3 give a negative charge at the C-terminusw39–41x. There is therefore a general trend fornegative side chains to be favoured at the N-terminus of the helix and positive groups at the C-terminus. Helix dipole effects were added to theLR model by Scholtz et al.w30x, although theyused the one-sequence approximation, so that onlyone or no dipoles in total are present. In LRmodels, helix dipole effects are subsumed withinother energies. For example, N-cap, N1, N2 andN3 energies include a contribution from the helixdipole interaction, so the energy of interaction ofcharged groups at this position with the dipoleshould not be counted in addition.

4.6. chc and chhc conformations in the randomcoil

The assignment of weights to very short helicalsegments chc and chhc requires careful consider-ation. While each such conformation is disfavou-red, as they are residues that pay the entropic costof restriction to helicalf,c space without formingstabilising bonds, the large number of places inwhich they can form in the peptide means thatthey make a significant contribution to the randomcoil population. The original LR model assignsthese conformations weights ofv andv , implicitly2

assuming that the initiation penalty for helix for-mation also gives the correct weights for thesecoil conformations. If, however, residues havemore conformational freedom within a chc confor-mation than within the N1 or C1 positions of thehelix, which seems reasonable, then the weightsof chc and chhc should be higher. The root of theproblem is that the c and h conformations are not

well defined. The helix–coil parameters are deter-mined by fitting a model to experimental data,rather than by rigorously defining an area of theRamachandran map as helical and assessing thepopulation within that area.In our N1N2N3 model, an isolated helical resi-

due, in the centre of a chc conformation, isassigned a weightn1, rather thanv. This is becausethe most important structures adopted by residuesat N1 are hydrogen bonds between the N1 sidechain and the N1 backbone NH groupw34x. Theseinteractions can form if only a single residue is inan h conformation. All N1 residues commencingany helical segment have the weightn1. The N2residue in a chhc sequence is assigned the weightn2Øv for the same reason: N2 structures are typi-cally hydrogen bonds between the N2 side chainand the N2 backbone NH groupw34x. These cantherefore form even if only two consecutive helicalresidues are present.Andersen and Tongw28x gave weights of zero

to chc and chhc conformations. While it seemsstrange to state that such conformations are neverpopulated, they did find a better fit to experimentaldata than when using the Doig et al. modelw25x.This may be because they also assigned N-capweights to a minimum of three helical segments,as discussed above. Weights for these coil confor-mations thus vary greatly. As all the models fitwell to experimental data, it is unclear at presentwhich choice is best.

4.7. 3 - and p-helices10

The ideala-helix has a periodicity of 3.6 resi-dues per turn, encloses 13 atoms in a ring byformation of an i,iq4 C_O H–N hydrogen∆

bond, and is thus a 3.6 -helix. The 3 -helix is a13 10

more tightly wound helix, stabilised byi,iq3C_O H–N hydrogen bonds, while thep-(4.4 )-∆ 16

helix is wound less tightly, with i,iq5C_O H–N hydrogen bonds. The Lifson–Roig∆

formalism can easily be adapted to describe helicesof other co-operative lengthsw42x. To treat the3 -helix–coil transition, the helical conformation,10

h , is replaced with h , reflecting the differentf,ca t

angles of 3 -helical residues. The fundamental10

difference between a 3 -helix and ana-helix is10


that the 3 -helix has ani,iq3 hydrogen bonding10

pattern, rather than thei,iq4 pattern characteristicof the a-helix. For a given number of units inhelical conformations, a 3 -helix consequently has10

one more hydrogen bond than ana-helix. Anequivalent description of this difference is thatinitiation is easier for a 3 -helix than for ana-10

helix: one fewer unit needs to be fixed in a helicalconformation before the first hydrogen bond isformed. To include this difference in the 3 -helix10

theory, one of thea-helical initiating residues(i.e.the central unit of either the h h c or the ch ha a a a

triplet) must become a 3 -helix-propagating resi-10

due. We arbitrarily chose to assign the propagatingstatistical weight,w , to the central unit of thet

h h c triplet such that the helix-propagating unitia a

is associated with the hydrogen bond formedbetween the CO of peptideiy2 and the NH ofpeptide iq1. The remainder of the statisticalweights applicable to thea-helix–coil theory aremaintained.The models described above for thea-helix–

coil and 3 -helix–coil transitions can be combined10

to describe an equilibrium including purea-heli-ces, pure 3 -helices and mixeda-y3 -helices10 10

w42x. In this model, three conformational states arepossible, 3 -helical(h ), a-helical (h ) and coil10 t a

(c). Stretches of residues in h conformation area

treated as in the purea-helix model, and stretchesof residues in h conformation are treated as in thet

pure 3 -helix model. Mixed helices consist10

regions ofa- and regions of 3 -helical structure,10

and transitions between the two types of helices.We defined two additional parameters,t and tN C,

to describe the junction from 3 - toa-helix and10

from a- to 3 -helix, respectively. The pure 3 -10 10

helix and the mixeda-y3 -helix models were10

subsequently extended to include side chain inter-actions w43x. 3 -Helices have onlyi,iq3 side10

chain interactions, while bothi,iq3 andi,iq4 arepossible in mixed helices.Sheinerman and Brooksw44x independently pro-

duced a model for theay3 ycoil equilibrium,10

based on the ZB formalism rather than the LRmodel. They similarly extended the classificationof conformations fromaycoil to ay3 ycoil. Their10

model differs from ours primarily in that it doesnot include additional parameters for junctions

betweena- and 3 -helical segments, and that it10

allows a 3 -helix to extend only from the C-10

terminus of an a-helix. N-terminal 3 -helical10

extensions toa-helices are often observed in crys-tal structures, howeverw19,45x.In a p-helix, formation of ani,iq5 hydrogen

bond requires that four units be constrained to thep-helical conformation, h . Thep subscript des-p

ignates the conformation and weights describingthe p-helix, the dihedral angles of which aredistinct froma- and 3 -helices. Assigning statis-10

tical weights to individual units requires consider-ation of the conformations of the unit itself and itsthree nearest neighboursw42x. The initiating statis-tical weight,v , is assigned to a helical unit whenp

one or more of its two N-terminal and nearest C-terminal neighbours are in the coil conformation.The definition of helix-initiating units as the twoN-terminal and one C-terminal units of each helicalstretch is again arbitrary. Units in ap-helicalconformation with three helical neighbours areassigned the propagating statistical weight,w . Ap

p-helix-propagating residue,i, is thus associatedwith the hydrogen bond between the NH of residueiq2 and the CO of residueiy3. Some recentwork suggests that thep-helix may be of morethan theoretical interestw46–49x.

4.8. AGADIR

AGADIR is an LR–based helix–coil model devel-oped by Serrano, Munoz and co-workers. The˜original modelw50x included parameters for helixpropensities excluding backbone hydrogen bonds(attributed to conformational entropy), backbonehydrogen bond enthalpy, side chain interactionsand a term for coil weights at the end of helicalsequences(i.e. caps). The single-sequence approx-imation was used. The original partition functionassumed that many helical conformations did notexist, as all conformations in which the residue ofinterest is not part of a helix were excludedw32,50x. These were corrected in a later version,AGADIRms, which considers all possible confor-mationsw51x. If AGADIR and LR models are bothapplied to the same data, to determine a side chaininteraction energy, for example, the results aresimilar, showing that the models are now not


significantly different w51,52x. The treatment ofthe helix–coil equilibrium differs in a number ofrespects from the ZB and LR models, and thesehave been discussed in detail in by Munoz and˜Serranow51x. The minimal helix length inAGADIRis four residues in an h conformation, rather thanthree. The effect of this assumption is to excludeall helices that contain a single hydrogen bond;only helices with two or more hydrogen bonds areallowed. In practice, this probably makes littledifference, as chhhc conformations are usuallyunfavourable and hence have low populations.Early versions ofAGADIR considered that residuesfollowing an acetyl at the N-terminus or precedingan amide at the C-terminus were always in a cconformation; this was changed to allow these tobe helicalw53x.The latest version ofAGADIR, AGADIR1s-2w53x,

includes terms for electrostaticsw53x, the helixdipole w53,54x, pH dependencew54x, temperaturew54x, ionic strengthw53x, N1, N2 and N3 prefer-encesw37x and capping motifs, such as the cappingbox, hydrophobic staple, Schellman motif andPro-capping motif w53x. The free energy of ahelical segment,DG , is given byhelical-segment

DG sDG qDG qDG qDGhelical-segment Int Hbond SD dipole

qDG qDG , which are terms for thenonH electrost

energy required to fix a residue in helical angles(with separate terms for N1, N2, N3 and N4),backbone hydrogen bonding, side-chain interac-tions, excluding those between charged groups,capping and helix dipole interactions, respectively.Electrostatic interactions are calculated with Cou-lomb’s equation. Helix dipole interactions were allelectrostatic interactions between the helix dipoleor free N- and C-termini and groups in the helix.Interactions of the helix dipole with chargedgroups located outside the helical segment werealso included. pH dependence calculations consid-ered a different parameter set for charged anduncharged side chains and their pK values. Thea

single-sequence approximation(see above) is usedagain, unlike inAGADIRms.AGADIR is at present the only model that can

give a prediction of helix content for any peptidesequence, thus making it very useful. It can alsopredict NMR chemical shifts and coupling con-stants. In order to do this, it must include estimates

of all the terms that contribute to helix stability,notably the 400 possiblei,iq4 side-chain interac-tions. Since only a few of these interactions havebeen accurately measured, the terms used cannotbe precise. Further determination of energetic con-tributions to helix stability is therefore still needed.

4.9. Lomize–Mosberg model

Lomize and Mosberg also developed a thermo-dynamic model for calculating the stability ofhelices in solutionw55x. Interestingly, they extend-ed it to consider helices in micelles or a uniformnon-polar droplet to model a protein core environ-ment. Helix stability in water is calculated as thesum of main-chain interactions, which is the freeenergy change for transferring Ala from coil tohelix, the difference in energy when replacing anAla with another residue, hydrogen bonding andelectrostatic interactions between polar side chainsand hydrophobic side-chain interactions. Anentropic nucleation penalty of two residues perhelix is included. Different energies are includedfor N-cap, N1–N3, C1–C3, C-cap, hydrophobicstaples, Schellman motifs and polar side-chaininteractions, based on known empirical data at thetime (1996). Hydrophobic interactions were cal-culated from decreases in non-polar surface areawhen they are brought in contact. Helix stabilityin micelles or non-polar droplets is found bycalculating the stability in water, then adding atransfer energy to the non-polar environment.

4.10. Extension of the Zimm–Bragg model

Following the discovery of short peptides thatform isolated helices in aqueous solution, Vasquez´and Scheraga extended the ZB model to includehelix dipole and side chain interactionsw56x. Themodel is very general, as it can include interactionsof any spacing within a single helix. It was appliedto determinei,iq4 and i,iq8 interactions. Long-range interactions, beyond the scope of LR models,can thus be included. Robertsw57x and Gans et al.w58x also refined the ZB model to include sidechain interactions.


5. Applications

The use of a helix–coil model is essential toquantitatively interpret results on helical peptides.They have therefore been widely used to determinethe forces that affect helix stability, includinginteriors w5,22,59–62x, N-caps w25–27x, C-capsw27,63,64x, N1 w35x, N2 w36x, N3 w37,38x, cappingboxes w65–67x and side chain interactionsw30–32,52,68–75x. The results compare well to thosemade in proteins by site-directed mutagenesis.In general, helix–coil parameters are extracted

from experimental data by measuring the helix–coil content of a peptide that contains the interac-tion of interest, and in which the parameters forall other terms in the sequence are known. Forexample, the sequence Ac–AKAVAAAKAVAAA-KAKAGY–NH was designed to measure the2

side-chain interaction energy between Val and Lysw74x. Two Val–Lys i,iq4 side chain interactionsare present. This peptide had a helix content of34%. The control sequence, Ac–AKVAAAAK-VAAAAKAKAGY–NH , is identical, except the2

two Val residues are moved one place towards theN-terminus, thus removing the Val–Lys interac-tions. It had a helix content of 25%, thus showingthat the Val–Lys interactions are stabilising, asthey increase the helix content by 9%. All theparameters required to predict the helix content ofthe control peptide using an LR-based modelw22xwere already measured, namely thew-, n-, c- andv-values of Ala, Lys, Val, Gly and Tyr. Thepredicted helix content of 23% was in excellentagreement with experiment, providing confidencethat the model and parameters are correct. Thehelix content of sequence with the Val–Lys inter-actions could not be predicted, since the weightfor this interaction(p-value) was unknown. It wasfound by determining the Val–Lysp-value thatgave a prediction in agreement with the experi-mental helix content. In this case it was 1.6, givinga Val–Lys energy ofyRTln 1.6sy0.25 kcalmol .y1

The models can predict helix contents for indi-vidual residues, as well as the mean helix content(typically measured by circular dichroism), sohave been used to interpret hydrogen exchangedata w76x and C13 chemical shifts at individual

residuesw77x. Knowledge of all the energies pres-ent in a sequence allows the prediction of the helixcontent of a peptide. In our experience, LR modelsare accurate to within a few percent.AGADIR isthe only current method that can give a predictionfor any sequence.

6. The future

There are several ways in which helix–coilmodels can continue to be developed. Firstly,additional interactions can be included, such asi,iq7 side chain interactions between side chainsseparated by two turns of the helix, or morestructurally complex C-capping motifs. Secondly,conformations in addition to helix and coil can beincluded. This approach has begun by consideringa-helix, 3 -helix and coil, although it would10

clearly be preferable to consider theb conforma-tion, as this is the next most frequent. Sheet–coilmodels are inherently more difficult to develop, asessentially any residue can form an interactionwith any other in a sheet, unlike helices thatcontain only short-range interactions. Even thesimplest sheet–coil modelsw78–81x are far morecomplex than helix–coil models. Thirdly, the sim-ple division into helix or coil conformations couldbe made more sophisticated by considering rota-meric states for each residue. Fourthly, tertiarystructure could be included. Qian made a start tothis problem by developing a model for coiled-coils that included a parameter for the interactionbetween two helicesw82x. Finally, the treatment ofthe random coil needs to be improved, as theisolated-pair hypothesis appears to be invalidatedby local steric effectsw24x. This may be possiblethrough the use of simulations of helical peptidesor denatured proteins(for examplew83–93x).Ultimately, we would like to be able to calculate

the stability of every possible conformation of apeptide or protein, thus solving the protein foldingproblem. The problem that all of these develop-ments run into is that the calculation of thepartition function rapidly becomes very complexand unwieldy. Time will tell whether advances intheory and computer power will allow us toapproach the partition function of a small protein


with tertiary structure, or whether the method isalready near to its practical limit.

Acknowledgments

I thank all the people who I have worked within this field, namely Avi Chakrabartty, Carol Rohl,Buzz Baldwin, Tod Klingler, Ben Stapley, JimAndrew, Eleri Hughes, Simon Penel, DuncanCochran and Jia Ke Sun. Ben Stapley is thankedfor reading the manuscript.

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