protein structure prediction by threading. why it works and why...

Protein Structure Prediction by Threading.Why it Works and Why it Does Not

Leonid A. Mirny and Eugene I. Shakhnovich*

Harvard UniversityDepartment of Chemistry andChemical Biology, 12 OxfordStreet, Cambridge, MA 02138USA

We developed a novel Monte Carlo threading algorithm which allowsgaps and insertions both in the template structure and threadedsequence. The algorithm is able to ®nd the optimal sequence-structurealignment and sample suboptimal alignments. Using our algorithm weperformed sequence-structure alignments for a number of examples forthree protein folds (ubiquitin, immunoglobulin and globin) using both`ìdeal'' set of potentials (optimized to provide the best Z-score for agiven protein) and more realistic knowledge-based potentials. Two physi-cally different scenarios emerged. If a template structure is similar to thenative one (within 2 AÊ RMS), then (i) the optimal threading alignment iscorrect and robust with respect to deviations of the potential from the`ìdeal'' one; (ii) suboptimal alignments are very similar to the optimalone; (iii) as Monte Carlo temperature decreases a sharp cooperative tran-sition to the optimal alignment is observed. In contrast, if the templatestructure is only moderately close to the native structure (RMS greaterthan 3.5 AÊ ), then (i) the optimal alignment changes dramatically when an`ìdeal'' potential is substituted by the real one; (ii) the structures of sub-optimal alignments are very different from the optimal one, reducing thereliability of the alignment; (iii) the transition to the apparently optimalalignment is non-cooperative. In the intermediate cases when the RMSbetween the template and the native conformations is in the rangebetween 2 AÊ and 3.5 AÊ , the success of threading alignment may dependon the quality of potentials used.

These results are rationalized in terms of a threading free energylandscape. Possible ways to overcome the fundamental limitations ofthreading are discussed brie¯y.

# 1998 Academic Press

Keywords: threading; Monte Carlo procedure; protein structure prediction*Corresponding author

Introduction

The problem of predicting protein conformationfrom sequences is of great importance and hasdrawn a lot of attention recently (see e.g. Moultet al., 1997; Shakhnovich, 1997a; Finkelstein, 1997;Jones, 1997; Levitt, 1997) with hundreds of papersfrom dozens of groups.

A most desirable solution to the problem is to®nd a model and an algorithm that stimulate fold-ing of a protein pretty much in a way that mimicsnatural protein folding and converges to the nativeconformation. While some success along these lineshas been documented (Kolinski & Skolnick, 1994),

this approach encounters a number of serious tech-nical dif®culties, making ab initio structure predic-tion hardly feasible now and perhaps in theforeseeable future (Finkelstein, 1997; Ortiz et al.,1998; Mirny & Shakhnovich, 1996). The mainreason for such a reclusion was discussed byShakhnovich (1997a), Finkelstein (1997) and Mirny& Shakhnovich (1996): a ``good'' folding modelmust be detailed enough to reproduce energeticsfaithfully and yet simple enough to be computa-tionally feasible, a blend that has not been reachedyet.

The energetics requirement is a very importantone as far as folding is concerned: the energy func-tion must be precise enough to single out theunique native structure as the global energy mini-mum among an astronomically large number ofdecoys, some of which have very low energy(Shakhnovich, 1994). The precision of potentials

E-mail address of the corresponding author:[email protected]

Abbreviation used: MC, Monte Carlo.

Article No. mb982092 J. Mol. Biol. (1998) 283, 507±526

0022±2836/98/420507±20 $30.00/0 # 1998 Academic Press

required to achieve this goal was analyzed byBryngelson (1994) and Pande et al. (1995) for latticemodel chains and by Mirny & Shakhnovich (1996)for real proteins. The analysis by Mirny &Shakhnovich (1996) suggests that at the level of asimple two-body approximation of energetics andstructure-less amino acids there may not exist anypotential which is able to fold real proteins intotheir native conformations. Adding more detailsinto the model is probably the way to go. How-ever, this complicates the search in conformationalspace, making computations far more demanding.Thus, ab initio folding success is contingent on ®nd-ing a safe pathway between the Scilla of incorrectenergetics and the Kharibda of a too complicatedand thus computationally infeasible model.

The complications inherent in ab initio foldingwere realized early by a number of workers in the®eld, and alternative approaches were suggested;the most notable of them is threading (Finkelstein& Reva, 1991; Finkelstein, 1997). The key idea ofthe threading method is to decrease dramaticallythe number of decoys. This is achieved by con-straining all protein conformations to a smallersubset of conformations obtained by threadingthrough known protein structures that serve as ascaffold for the protein sequence in question and®nding the energetically optimal alignment of thesequence to the scaffold structure. From the physi-cal point of view the threading problem is some-what equivalent to folding, because it also requiressearching over a large set of possible alignmentsfor the one that delivers minimum `ènergy''. Itwas shown (Lathrop, 1994) that such a search is anNP complete problem (i.e. that there is an apparent``Levinthal'' paradox in threading). As in folding,the search in threading is biased by the energyfunction, so that the related key issue is the pre-cision of the energy function. The rationale forusing threading rather than folding is the hopethat a less precise energy function will suf®ce forthe search over a more constrained conformationalset: in this case the native state should be distin-guished as having the lowest energy among thesmaller number of alternatives. However, suchsimpli®cation of the conformational space comes ata serious price. The reason is that the native struc-ture itself may not belong to the constrained con-formational set! In this case, threading seeks anapproximate solution, i.e. the one that is closest tothe native state in the conformational set of align-ments. However, if this best solution is relativelydistant structurally from the native state, its energymay be considerably higher than the energy of thenative state (even with the `ìdeal'' potential thatstrongly favors the native state). This factor clearlymay decrease the energy gap, balancing on thenegative side the gain achieved due to the restric-tion of conformational space. Obviously, the con-formational space restriction, which is the basis ofthe threading approach, is not an `ìnnocent''approximation that is guaranteed to work almostby de®nition. Clearly this issue requires a detailed

study that aims to address a question of gains andlosses made by threading approximations and addto our intuition about which factors are moreimportant for particular models and when shouldwe expect success in threading simulations andwhen we cannot.

As pointed out above, threading is very muchlike folding in terms of key questions and dif®cul-ties. In folding one asks basically two questions:are energy functions correct? and is a confor-mational search ef®cient enough to ®nd the globalminimum? An important advance in protein fold-ing theory, which started from the seminal work ofGo (Taketomi et al., 1975), is understanding thatthose two questions can be studied separately. The®rst approach proposed by Go (Taketomi et al.,1975) was to design an energy function that favorsthe native contacts and disfavors non-native ones.Such an energy function gives rise to fast folding(Gutin et al., 1996). However, the arti®cial penaltiesimposed on non-native interactions make themodel somewhat unphysical for studying thephysical principles of folding, because in real lifestrong non-native contacts cannot be excluded apriori. In fact they occur in some proteins (Lacroixet al., 1997). A more physical model for folding isbased on sequence design, which generates, forany given potential function, special sequences forwhich the native structure is guaranteed to be theglobal minimum. Then the same potential is usedfor folding as the one used to design sequences(Shakhnovich, 1994; Shakhnovich et al., 1996a). Inthis case folding simulations quickly converge tothe native state (Gutin et al., 1996; Shakhnovich,1994). The properties of energy landscape anddynamics that lead to the native state can be stu-died in detail with implications for folding andevolution of real proteins (Shakhnovich et al.,1996a). An approach that is very similar in spirit isto design a potential which provides low energy toa natural sequence in its native structure(Goldstein et al., 1992; Mirny & Shakhnovich, 1996;Koretke et al., 1996; Hao & Scheraga, 1996; Ortizet al., 1998). While this energy function may not betransferable to other proteins (Mirny &Shakhnovich, 1996) it serves its purpose by provid-ing a free energy landscape with a large gap andhence reasonably fast folding. One conclusion fromthe analysis carried out for a number of foldingmodels suggests that Monte Carlo (MC) simulationrepresents a powerful search strategy that is veryef®cient in ®nding the native state on a physicallyreasonable landscape.

Here we take a similar systematic approach tostudy threading. First we develop and present aMonte Carlo threading algorithm, which allowsgaps and insertions both in structure and insequence. The advantage of the Monte Carloapproach is that it converges to the Boltzmann dis-tribution. This feature makes this method a valu-able tool to map and characterize a free energylandscape and outline the physical requirements ofconvergence to the global minimum solution.

508 Monte Carlo Threading

To test and rationalize the MC threading methodwe apply it to a number of problems of increasingcomplexity. The reason behind this ``gradual''approach is the need to differentiate between limi-tations intrinsic to the threading approach, assuggested above, and the ones that originate fromuncertainties in potential functions and the form ofthe Hamiltonian (scoring function) used.

Following this program, ®rst we use our algo-rithm for the structure-structure alignment. It turnsout that in the framework of our approach thestructure-structure alignment is a counterpart ofthe Go model studied in folding: it provides the``ideal'' potential function in which only the nativeinteractions are favorable. This simple implemen-tation of the method makes it straightforward tocompare it with existing heuristic approaches suchas the structure alignment algorithm Dali (Holm &Sander, 1993) used to build the FSSP database. Thecomparison shows that the MC procedure pro-posed in this work gives a more optimal alignmentthan Dali (with the same scoring function as usedin Dali).

Next, we turn to a more realistic two-bodyenergy function where interaction energy dependson amino acid types, and distances between themrather than on their location in the native structure.We design an ``ideal'' parameter set for this Hamil-tonian scoring function and study how theapproximate character of the potential functionaffects the results of threading at various degreesof similarity between the template and the nativeconformations.

Moving closer to the realm of structure predic-tion, we explore the accuracy of threading, usingone of the knowledge based potentials that are cur-rently available in the literature (Miyazawa &Jernigan, 1996).

In order to make our conclusions signi®cant andgeneral we carried out the analysis for threefold classes: a/b (ubiquitin and its structuralhomologues), all-b (class I immunoglobulin fold)and all-a (globin fold). The results are consistentbetween the fold classes studied. This allows us toarrive at quantitative conclusions concerning thedegree of similarity between the native structureand the template that is required for successfulsequence-structure alignment.

In what follows we provide a detailed discussionof the ubiquitin superfamily followed by the data(Tables 4 and 5) with comments for the immuno-globulin and globin fold.

Model

MC threading

To sample possible sequence-structure align-ments and search for the alignment with the mini-mal energy we use the Monte Carlo (MC)procedure. The power of the MC procedure is thatit allows us to ®nd a global minimum on a varietyof rough landscapes (Allen & Tildesley, 1987). In

the search for a minimum, it samples possiblealignments and allows us to study statistical prop-erties of the energy landscape. This made the MCprocedure extremely useful in the study of variousdisordered physical systems such as spin-glasses(Binder, 1995), liquids (Allen & Tildesley, 1987)and proteins (Shakhnovich, 1997b; Abkevich et al.,1995; Pande et al., 1997; Sali et al., 1994).

We develop the MC procedure to samplesequence-structure alignments. In contrast to pre-vious work (Bryant, 1996; Lathrop & Smith, 1996),the whole space of possible alignments is sampled,allowing arbitrary gaps in both sequence and struc-ture and arbitrary fragments of matching sequenceand structure. Gap penalties are not used.

General scheme

To construct the MC procedure one needs to®nd a suitable representation of sequence-structurealignment and design a move set which couldmake sampling effective, i.e. provide a high accep-tance ratio for the trial moves. In Methods wedescribe the alignment representation and themove set we designed.

Simulation starts from a random alignment. Ateach step of the procedure we make a trial movewhich slightly changes the alignment. Then wecompute how energy changes, �E, upon thismove. The move is accepted or rejected accordingto the Metropolis scheme (Metropolis et al., 1953).A move is always accepted if �E 4 0, and isaccepted with probability P � exp(ÿ�E/T) if�E > 0, where temperature T is a parameter of theprocedure. Irrespective of whether the move isaccepted or rejected another step is done and soon.

This procedure was proven to converge to theBoltzmann ensemble (Allen & Tildesley, 1987). Inthe simulation, the equilibrium ensemble isobtained by recording an alignment every 1000steps. The interval is important to avoid correlationbetween sampled alignments (Binder, 1986). Thetypical length of the run is 106-107 steps which is10-100 times longer than the typical time requiredto reach the alignment with the minimal energy (atthe optimal temperature). An ensemble obtained inthis way is used to compute average values ofdifferent quantities, such as energy and variousdistance measures (see below).

Energy function

The energy of an alignment is given by the sumof pairwise residue-residue interactions. If a struc-ture of protein I is represented by a set of pairwisedistances between residues rIii0 ; i; i0 � 1; . . . ; I and asequence of protein J aligned to this structure isai, j � 1, . . . ,J then:

E �XI

i<i0�1

V�pi; pi0 ; rIii0 � �1�

Monte Carlo Threading 509

where V ( j, j0, R) is the energy of interactionbetween residues j and j0 located at distance R inspace, and pi is the number of the residue in thesequence aligned with position i in the structure(see Methods: Alignment representations). If aj anda0j are identities of residues in positions j and j0,then:

V� j; j0;R� � U�aj; a0j;R�; �2�where U(x, Z, R) is the potential which gives theenergy of interaction between residue types x andZ located at distance R in space. Particular formsof V( j, j0, R) and U(x, Z, R) used for threading aredescribed below.

To write the energy function in this form wemade the following assumptions.

(1) Energy is a sum of pairwise residue-residueinteractions.

(2) Interaction between residues is a function oftheir identities only (not of the distance betweenthem along the chain (Sippl, 1995) or their localenvironments).

(3) Energy of interaction between residuesdepends on the distance between some ``represen-tative'' points of the residues (Ca, Cb, center ofmass, etc.) and not on their relative orientation(Berriz et al., 1997; Bahar & Jernigan, 1996).

In this study all the distances are measuredbetween Cb atoms. We do not use penalties forgaps. Instead, similar to Holm & Sander (1993), weconstrain the length of a fragment to be greaterthan or equal to Lmin � 6.

Note that threading with a pairwise potential isequivalent to alignment of two matrices: rii0 andV( j0, j, � ) under the constraint that diagonals of thematrices coincide.

Results

Testing MC search strategy with an`ìdeal'' potential

Our ®rst goal is to evaluate the proposed MCthreading as a search strategy for threading as wellas to probe a possible alignment energy landscape.The results of threading, however, always dependon both the potential and the search strategy. Inorder to test the search strategy and eliminate theproblem of inaccurate potential we use an `ìdeal''potential.

`Ìdeal'' potential is the one which guarantees thelowest energy to the native conformation. UsingdRMS (distance RMS, see Methods) as the energyfunction is an example of an `ìdeal'' potential(Elofsson et al., 1996). Clearly, the native confor-mation has dRMS � 0 and all other conformationshave dRMS > 0. In the case of contact potential, theso-called Go-matrix (Go & Abe, 1981), which wedescribed in the Introduction, can serve as an`ìdeal'' potential. Namely, if i and i0 are in contact(i.e. rii0 < Rcut) in the native structure, then

V( j, j0, r < Rcut) � ÿ1 and 0 otherwise, where Rcut isa contact cutoff. Then all conformations which donot have all the native contacts have higher energythan the native conformation. Below we use differ-ent `ìdeal'' potentials to test the MC threadingalgorithm.

Since an `ìdeal'' potential guarantees the lowestenergy to the native conformation, success ofthreading and quality of alignment depend onlyon the search strategy (threading algorithm).

Threading is equivalent tostructure comparison

When the energy V( j, j0, R) of interactionbetween residues j and j0 is a function of rJjj0 (dis-tance between j and j0 in the native structure J ),threading becomes equivalent to structure-struc-ture comparison. For example, when:

V� j; j0;R� ��rJjj0 ÿ R�2

�Nalg ÿ 1��Nalg ÿ 2� �3�

the energy function becomes equal to dRMS2:

E � 1

�Nalg ÿ 1��Nalg ÿ 2�XI

i�2<i0�1

�rJjj0 ÿ rIii0 �2

� dRMS2�I ;J � �4�where Nalg is the length of alignment, i.e. numberof matched residues. Hence, we can use ourthreading algorithm for structure-structure align-ment as well as for threading. In general anythreading algorithm which allows pairwise dis-tance-dependent potential can be used for structurecomparison.

Then the ®rst test for a threading algorithm is toapply it to protein structure-structure alignmentsand compare the results with those obtained by awidely used structure comparison algorithm(Holm & Sander, 1993).

Testing alignment procedure: structure-structure alignment

Here we compare our results with the structurealignments made by Holm & Sander available inthe FSSP database (Holm & Sander, 1993). FSSPwas built using the Dali algorithm (Holm &Sander, 1993), which utilizes the following scoringfunction for structure comparison:

V� j; j0;R� � 0:2ÿ 2�rJjj0 ÿ R

rJjj0 � R

!exp�ÿ�rJjj0 � R�=40�

�5�We use the negative of this Dali scoring function asa potential for MC threading. In fact, this potentialrepresents an `ìdeal'' Lennard-Jones-like potentialfor threading. It has a deep minimum at distanceR � rJjj0 , rapidly increases at small R < rJjj0 and


slowly increases at R > rJjj0 , approaching a constantvalue. (The exponent leads to a very slow decreaseof the potential at large R � 40 AÊ .) This shapemakes the Dali potential equivalent to an `ìdeal''distance-dependent potential, which has a mini-mum when two residues are at the separation thatthey have in the native structure.

Finding the optimal alignment

To assess our Monte Carlo threading as a searchstrategy we apply it to structure-structure compari-son. Optimal alignments obtained by our pro-cedure are then compared with alignmentsprovided in the FSSP database. Importantly, weuse exactly the same measure of fold similarity asthe one used by Holm & Sander to build the FSSPdatabase.

Here we consider a few examples of structureswhich have similar folds and no signi®cantsequence similarity:

ubiquitin-ubiquitin self-alignment �1ubi� : Ubi-Ubi

ubiquitin-cRaf1 �1gua� : Ubi-Gua

ubiquitin-G-protein �1igd� : Ubi-Igd

ubiquitin-ferrodoxin; chain A�1frr� : Ubi-Frr

PDB identi®ers are taken for abbreviation of eachpair. As a control we also use a pair of proteinswith no structural similarity:

ubiquitin-plastocyanin �1plc� : Ubi-Plc

This control mimics the case when the wrongstructure has been chosen as a template for thread-ing.

For every pair, we compare the optimal align-ment found by our MC threading and the align-ment reported in the FSSP. Table 1 summarizes theresults of this test.

For all three cases and for the self-alignment we®nd alignments that are very close to the FSSPalignments. Moreover, our procedure clearly `òut-performs'' the Dali algorithm used to build theFSSP. Using the same scoring (energy) as Dali, ourprocedure ®nds alignments which have betterscores (lower energy) than alignments reported byFSSP. FSSP was built using a heuristic algorithm,which found alignments close to, but different

from the optimal ones (Holm & Sander, 1993). Incontrast, our procedure uses the Monte Carlosearch strategy and is known to ®nd the globaloptimum when simulated annealing is properly setup (Kirkpatrick, 1984).

From these results we conclude that our pro-cedure successfully passed the ®rst test. Theseresults also demonstrate the superiority of theMonte Carlo procedure in comparison to a heuris-tic algorithm in the search for the global optimum.Clearly, Monte Carlo threading can easily ®nd theright alignment when an exact distance-dependentpotential of residue-residue interactions is pro-vided. Can similar success be achieved when acontact potential (also an `ìdeal'' one) is usedinstead?

Contact potential versus distance-dependentpotential

A vast majority of protein models currently usedfor threading rely on contact approximation, i.e.two residues are said to interact with each other ifthe distance between them is less than a cutoff dis-tance. The energy in contact approximation isgiven by:

E �XI

i<i0�1

Bpi;p0i �Iii0 �6�

where Bjj0 is the energy of a contact between resi-dues in positions j and j0 of protein J , and �Iii0 � 1if positions i and i0 are in contact in protein Iand �Iii0 � 0 otherwise. Two residues i and i0 aresaid to be in contact if the distance between theirCb atoms rii0 < Rcut � 8.0 AÊ . If aj, j � 1, . . . ,J is thesequence of the protein J , then the energy of acontact Bjj0 � U(aj, aj0), where U(z, Z), x, Z � 1, . . . ,20is a potential of interactions between residues oftypes x and Z. The long-standing question is howthis approximation affects the accuracy of thread-ing alignments.

To address this question we make threadingusing both contact and distance-dependent poten-tials. Comparing optimal alignments obtained withthese models demonstrates the impact of contactapproximation on the accuracy of alignments.

Table 1. Comparison of the optimal structure-structure alignments obtained by minimizationof the Dali function by MC threading and by the Dali algorithm (Holm & Sander, 1993) (asreported in FSSP (Holm & Sander, 1997))

FSSP MC threadingProteins Dali dRMS (AÊ ) Lalg Dali dRMS (AÊ ) Lalg Qalg

Ubi-Ubi ± 0.00 76 ÿ2832 0.00 76 1.00Ubi-Gua ÿ729 2.05 68 ÿ1047 2.03 58 0.82Ubi-Igd ÿ344 2.34 46 ÿ362 2.58 50 0.87Ubi-Frr ÿ579 2.79 61 ÿ648 2.40 60 0.62

The distance between alignments is given by Qalg (see Methods).


An `ìdeal'' potential for the contact approxi-mation is given by:

Bjj0 � ÿ�Jjj0 �7�Threading with `ìdeal'' contact potential is equiv-alent to structure-structure alignment, which maxi-mizes the number of common contacts betweenthe two structures. In other words, we align con-tact maps of the two proteins to maximizeQ � �Ii<i0�1�Jpi;p0i

�Iii , which is the overlap betweencontacts in the matrices. Table 2 compares optimalalignments with contact and distance-dependentapproximations for the same ®ve cases.

Clearly, using contacts instead of a distance-dependent potential introduces very small changesin the alignments. These changes are predomi-nantly shrinkage/expansion of fragments by oneor two residues. We also observed that the closerthe two proteins are, the smaller the effect on con-tact approximation on the accuracy of the align-ment.

The major advantage of the contact approxi-mation, compared to the distance-dependent one,is the small number of parameters required tode®ne a potential. Since contact approximationdoes not change the optimal alignment very muchit can be used ef®ciently for threading. Below weuse the contact approximation for all threadingexperiments.

Thermodynamics of alignment

Here we study the generic properties of thethreading energy landscape. Following theapproach of `ìdeal'' potential we make structuralalignments of ubiquitin with cRaf1, G-protein, fer-rodoxin, plastocyanin and ubiquitin itself. Notethat these alignments are equivalent to threadingof the ubiquitin sequence through correspondingstructures when an `ìdeal'' potential is used.

For every case we make several MC runs, eachat different constant temperature T. Each runyields an equilibrium ensemble of alignments.Averaging over this ensemble we obtain hEi,Cu � (hE2i ÿ hEi2)/T2, and hQalgi as a function of T(see Figures 1 and 2). The temperature at which Cuhas the main peak is the transition temperature Tf.

In order to study suboptimal alignments we per-form a long MC run at transition temperature Tf .Alignments sampled every 1000 MC steps consti-tute an equilibrium ensemble of suboptimal align-ments. For this ensemble of alignments wecompute frequency wij of a match between everypair of residues i and j as:

wij � 1

M

XMm�1

dj;pmi

�8�

where pm is the mth alignment out of M in theensemble. Figures 3A, 4A and 5A present wij forthree pairs of proteins considered above.

Figure 1. The average energy inequilibrium ensemble as a functionof temperature for different pairs ofaligned proteins.

Table 2. Comparison of the optimal structure-structure alignments which minimize distance-dependent function (columns 2 to 4) and overlap between contacts (columns 5 to 6)

Contact Distance-dependentProteins Dali dRMS (AÊ ) Lalg Dali dRMS (AÊ ) Lalg Qalg

Ubi-Ubi ÿ2832 0.00 76 ÿ2832 0.00 76 1.00Ubi-Gua ÿ1029 2.16 69 ÿ1047 2.03 58 0.96Ubi-Igd ÿ239 3.15 52 ÿ362 2.58 50 0.90Ubi-Frr ÿ591 3.06 66 ÿ648 2.40 60 0.84Ubi-Plc(control) 592 5.54 65 ÿ272 2.67 42 0.30

The distance between alignments is given by Qalg.


After normalization:

~wij �wij

�Jj�1wij

we compute the positional entropy of alignments

Si as:

Si � ÿXJ

j�1

~wij log ~wij: �9�

Positional entropy Si introduced in this way clearlymeasures the degree of uncertainty in matching theresidue i. If i is always matched with j, then pos-

Figure 2. Heat capacity Cu as afunction of temperature. The mainpeak on Cu corresponds to the tran-sition temperature (Tf).

Figure 3. Self-alignment of ubiquitin. Equilibrium ensemble of alignments for structure-structure (A and C) andsequence-structure alignments (B and D) obtained at Tf . A and B, Distribution of matches wij in the alignments. Thecontinuous line shows the optimal alignment. C and D, Positional entropy Si, which is the measure of reliability ofalignment.


itional entropy Si � 0. In contrast, Si has its maxi-mum if i is matched equally often with all j terms.Figures 3C, 4C and 5C present Si for the threecases studied.

Self-threading

First we consider the case of self-alignment, i.e.the structural alignment of ubiquitin with itself(Ubi-Ubi). Figures 1 and 2 present average energyE and heat capacity Cu � dE/dT as a function of T.

As we raise the temperature, T, a sharp tran-sition in E is observed (see Figure 1). The jump inE shows that at high T the ensemble consists pre-dominantly of incorrect high-energy alignments,whereas at low T the optimal (``native'') alignmentdominates in the ensemble. What is more import-ant is that this transition is very cooperative (it hasa clear ®rst order-like type). The cooperative natureof the transition can easily be seen from the sharp-ness of transition in E, the peak on Cu (Figure 2)and the bimodal histogram (Figure 6A) obtained attransition temperature (Tf). Drawing a parallelwith protein folding, we can say that the twopeaks on the energy histogram correspond to twodistinct coexisting states: ``folded'' (correct optimalalignment) and `ùnfolded'' (random alignments).Both ``folded'' and `ùnfolded'' states are freeenergy minima. As T decreases the `ùnfolded''

state is destabilized. What is more important, as inthe ®rst-order transition scenario, the incorrectalignments with relatively low energy have highfree energy and are not populated in the equili-brium ensemble obtained by the MC procedure,because equilibrium ensemble should be domi-nated by the correct alignment and a few subopti-mal alignments with very similar structure.

Figure 3A presents alignments constituting theequilibrium ensemble. Clearly, the optimal align-ment (main diagonal, for this case) dominates overall alternative alignments. Suboptimal alignments,which contain incorrect i,j matches, constitute atiny fraction of the ensemble, since wii � 0.6 andwij 6�i < 0.01. This domination of the optimal align-ment is also manifested in a uniformly low pos-itional entropy Si (see Figure 3C).

Analysis of the self-alignment test brings us tothe following conclusions: (i) the optimal align-ment can be easily found by the MC procedure; (ii)the transition for random alignments to the opti-mal alignment is cooperative; (iii) the optimalalignments dominates in the ensemble. Althoughself-alignment is an essential test for any threadingprocedure, it de®nitely represents the simplestpossible case. Now we consider more realisticcases when the structure of ubiquitin is alignedagainst various similar folds.

Figure 4. Ubiquitin-cRaf1 protein. Equilibrium ensemble of alignments for structure-structure (A and C) andsequence-structure alignments (B and D) obtained at Tf . A and B, Distribution of matches wij in the alignments. Thecontinuous line shows the optimal alignment. C and D, Positional entropy Si.


High structural similarity: ubiquitin-cRaf1 protein

Now we make an alignment of ubiquitin andcRaf1 structures (Ubi-Gua). For this pair the FSSPdatabase (Holm & Sander, 1997) reportsdRMS � 2.2 AÊ for 58 atoms, providing Z � 5.5.

As in the case of self-alignment the transition isvery sharp (see Figure 1) yielding a peak of Cualmost as high as for the self-alignment (seeFigure 2). Another observation is that self-align-ment has a transition temperature Tf higher thanall other cases. The transition temperature for ubi-quitin-cRaf1 Tubi-Gua

f is the next highest. To ensurethat transition in this case retains its ®rst-ordercharacter we make a histogram of the energy ofthe equilibrium ensemble at Tf (see Figure 6B). Theobserved bimodal distribution is a clear indicationof the ®rst-order transition.

As in the case of self-alignment, the optimalalignment dominates over the alternative align-ments in the ensemble (see Figure 4A). Positionalentropy, however, reveals a region of alignment(residues 50 to 60), which has a high degree ofuncertainty in placing a gap and a fragment (seeFigure 4C). The rest of the alignment has very lowentropy and, hence, high certainty.

As with self-alignment, the ubiquitin-cRaf1 pairexhibits a ®rst-order-like transition, which is associ-ated with high accuracy and is manifested by ahigh peak in Cu and a bimodal energy distribution.

Figure 5. Ubiquitin-G-protein. Equilibrium ensemble of alignments for structure-structure (A and C) and sequence-structure alignments (B and D) obtained at Tf . A and B, Distribution of matches wij in the alignments. The continuousline shows the optimal alignment. C and D, Positional entropy Si.

Figure 6. Distribution of alignment energies at Tf .A, Self-alignment of ubiquitin; B, ubiquitin-cRaf1;C, ubiquitin-cRaf1 protein. Bimodal distribution is aclear indicator of cooperative ®rst-order-like transition.


Does every pair of aligned proteins exhibit the®rst-order transition or is it an indicator of a good``match'' between the two structures (or a sequenceand a structure in the case of threading)? Toaddress this question we consider two other pairsof proteins with a moderate structural similarityand a control case with no structural similarity.

Low structural similarity: ubiquitin-G-proteinand ubiquitin-ferrodoxin

Both pairs of proteins share a common fold, butdo not exhibit a great deal of similarity. Accordingto the FSSP database (Holm & Sander, 1997) theubiquitin-G-protein pair has dRMS � 2.8 AÊ for 46atoms providing very moderate Z � 2.8, and theubiquitin-ferrodoxin pair has dRMS � 3.6 AÊ for 63atoms, providing a better Z � 3.6.

Both of these proteins exhibit no ®rst-order tran-sition in structural alignments with ubiquitin ascan be seen for small Cu peaks (Figure 2) and amonomodal distribution of energies in the equili-brium ensemble at Tf (see Figure 6C). As expected,moderate similarity between structures leads to theabsence of cooperative transition.

Importantly, accuracy of the alignment also suf-fers a lot, as ¯uctuations in several positions ofalignment increase. There is a substantial degree ofuncertainty in placing every residue. A fuzzy pat-tern of wij shows that in most suboptimal align-ments every residue is shifted two or threepositions from its optimal match (see Figure 5A).These alternative alignments are very frequent inthe equilibrium ensemble and have an energy onlyslightly above the optimal energy. A higher levelof Si is another indicator of this uncertainty (seeFigure 5C).

No structural similarity: ubiquitin-plastocyanin

To consider the limiting case when proteins havealmost no structural similarity we make an align-ment of ubiquitin and plastocyanin structures. Thetransition in this case is very smooth and the peakin Cu is smaller than for all other cases (seeFigures 1 and 2). Alignment exhibits a great deal ofuncertainty in several places (data not shown).This case represents a ``reference point'' oppositeto that of self-alignment. For any pair of alignedproteins we can measure the peak of Cu and com-pare it with two other peaks: the peak of Cu forself-alignment and the peak for this reference, pooralignment.

Summary

Pairs of proteins, which are similar to each other,exhibit cooperative transition and provide the opti-mal alignment with a high reliability. In contrast,proteins which share a smaller degree of similarityhave substantial uncertainties in alignments andexhibit no ®rst-order transitions.

Absence of the ®rst-order-like transition can be asensitive indicator of a moderate similaritybetween the native structure of a protein and thestructure used for threading. What is more import-ant, it indicates that there is a huge number of sub-optimal alignments which are different from theoptimal alignments, but have the energy very closeto the optimal one. This represents a very dif®cultcase for threading. In fact, small (inevitable!) errorsin the potential can decrease the energy of subopti-mal alignments and increase the energy of the opti-mal alignment making it a non-optimal one. Weconsider this effect in more detail below.

Threading with a real potential

No residue-residue potential is able to providean `ìdeal'' pattern of interactions, where all nativecontacts are attractive and all non-native areambivalent (see equation (7).) In fact, a potentialassigns interaction energies to residue types, notpositions! Here we study how threading alignmentchanges when we use a full residue-residue poten-tial instead of an `ìdeal'' one. In other words,instead of making structure-structure alignmentwe make a sequence-structure alignment with areal potential. We, however, chose a residue-resi-due potential U(xZ), x, Z � 1, . . . ,20 to provide apattern of interactions Bjj0 � U(aj, aj0), whichresembles the `ìdeal'' pattern as closely as possible.Essentially, we use the ``best'' potential in place ofan `ìdeal'' one and study changes in alignmentsassociated with this minimal change in potential.In other words, we introduce a minimal inevitable``noise'' in the `ìdeal'' potential and study the stab-ility of the optimal alignment to this noise.

The ``best'' potential is obtained by an optimiz-ation procedure described by Mirny &Shakhnovich (1996) and, brie¯y, in Methods. Weoptimize the residue-residue potential (20 � 20matrix) for a single protein (ubiquitin in this case)in order to make as many native contacts attractiveand non-native contacts repulsive as possible. Thisoptimal potential provides the lowest energy to theubiquitin sequence in the native structure of ubi-quitin and maximizes the energy gap between thisstructure and the bulk of alternative ones (Mirny &Shakhnovich, 1996). Using this potential we makeMC threading of the sequence of ubiquitin throughthe four alternative structures (ubiquitin itself,cRaf1, G-protein and ferrodoxin). Optimal align-ments obtained in this way are then comparedwith the optimal alignments obtained by structure-structure comparison (see above).

Self-alignment

Self-alignment of ubiquitin using residue-residuepotential shows that the native structure remainsat the global energy minimum and the optimalalignment does not change. Although sevenmatches are lost from the optimal alignment (seeTable 3) there are no displacements in the align-


ment. A low level of positional entropy Si (seeFigure 3B and D) demonstrates domination of theoptimal alignment over the alternative ones.

Ubi-Gua

The optimal structure-structure alignment is nolonger the lowest energy. Another alignmentbecomes the optimal one when we make threadingwith the ``best'' potential. This optimal threadingalignment is slightly different from the optimalstructure-structure alignment (see Table 3). Particu-larly, the alignment gets shorter as some matchesare lost. More important, the last fragment ofthreading alignment (eight residues) shifts by oneresidue (compare continuous lines in Figure 4Aand B). This shift leads to a slight increase ofdRMS, but still yields absolutely accurate mount-ing of the ubiquitin sequence on the cRaf1 for therest 87% � (62 ÿ 8)/62 of the residues.

Comparing the density of matches wij for struc-ture-structure (Figure 4A) and structure-sequencealignments (Figure 4B), we observe an increasedcontribution from wrong alignments in thesequence-structure case. The major part of the opti-mal alignment, however, sustains this competitionbrought about by deviation from `ìdeality'' in theresidue-residue potential.

Ubi-Igd

For this case, the best structure-structure align-ment is by far not the lowest one in energy.Threading with the ``best'' residue-residue poten-tial changes the optimal alignment drastically.Three out of six fragments are misplaced. Allboundaries of the fragments are changed. Only53% of the matches in the optimal threading align-ment are present in the structural one. Alignmentgets shorter and provides RMS � 4.17 AÊ for 46 Ca

atoms, compared to RMS � 3.01 AÊ for 46 bestmatching Ca atoms for the structure-structurealignment.

Importantly, threading suboptimal alignmentsbecomes very different from the optimal one andare present with substantial frequency in the equili-brium ensemble (Figure 5B). These lead to lowreliability of the optimal alignment (Figure 5D).

Note that this result is obtained using the ``best''residue-residue potential, optimized for thesequence of ubiquitin.

Threading with a knowledge-based potential

We showed that when a minimal possible``noise'' is introduced into potential, the optimalalignment with a distant template changes substan-tially, yielding RMS > 5 AÊ . In contrast, alignmentswith close templates sustain minimal possible``noise'' in the ``best'' potential. How does an opti-mal alignment with different templates changewhen a realistic knowledge-based potential isused?

Table 3 presents the results of threading with thepotential derived by Miyazawa & Jernigan (Table 4,upper half of Miyazawa & Jernigan, 1996) (MJ96).As expected, the accuracy of alignment decreaseddramatically yielding RMS > 5 AÊ for all pairs. Evenself-alignment of ubiquitin becomes inaccuratewith RMS(Ca) � 2.36 AÊ over 74 residues. Impor-tantly, threading with the close template (Ubi-Gua)was rather accurate with the ``best'' potential butbroke down with MJ96.

In other words, while threading through a dis-tant template becomes inaccurate, even under aminimal possible ``noise'', threading through aclose template can sustain minimal ``noise'', break-ing down under a higher level of ``noise'' in poten-tial.

However, there is a possibility that an accurateenough potential can provide rather good align-ment on a very close template.

We conclude that further deviation of potentialfrom the `ìdeal'' one leads to the situation thatlow-energy decoys become energetically optimalrather than the structurally optimal alignment. Tounderstand this phenomenon more deeply westudy the energy landscape of threading.

Energy landscape of threading

The aim now is to study the general propertiesof the alignment energy landscape, i.e. how theenergy of an alignment changes as it approachesthe optimal one. The degree of similarity betweentwo alignments is measured by Qalg (see Methods).

Table 3. Comparison of the optimal structure-structure and optimal sequence-structure alignments for ubiquitin

Structural alignment(`ìdeal'' potential) Threading the ``best'' potential Threading MJ96 potential

Proteins Lalg RMS dRMS Lalg RMSa dRMSa Qalg Lalg RMSa dRMSa Qalg

Ubi-Ubi 76 0.00 0.00 69 0.00 0.00 0.92 74 1.83 1.79 0.82Ubi-Gua 69 2.00 1.80 62 2.76 2.65 0.75 74 5.16 4.21 0.33Ubi-Igd 52 3.01 2.49 46 4.17 3.40 0.53 55 6.92 4.98 0.36Ubi-Frr 66 3.03 2.68 64 4.49 3.96 0.61 71 5.42 4.87 0.26Ubi-Plc 65 5.62 6.07 54 10.23 10.62 0.09 ± ± ± ±

RMS is computed over Ca atoms after optimal superposition of the matched residues. dRMS is computed over Cb atoms. RMSCa

is a better measure of the distance between the backbones, whereas dRMSCb is better in characterizing similarity between folds.a For fair comparison of three alignments (structural, ``best'' and MJ96) dRMS and RMS were computed over the same length (the

length of the shortest alignment). Qalg is the distance between the threading and structural alignments. It measures the degree of suc-cess in threading.


In order to study the energy-similarity relationshipfor each case we perform a long MC run at thetransition temperature. Alignments sampled every1000 MC steps constitute the equilibrium ensemble.We compute the number of alignments in theensemble which have the energy in the interval[E, 2 E � dE} and similarity to the optimalalignment Qalg 2 {Qalg,Qalg � Qdalg].

Structure-structure alignment

Figure 7A, B and C present logarithms of thenumber of alignments in each energy-similarityinterval for different cases.

These results clearly demonstrate the fundamen-tal difference between the cases which exhibit ®rst-order transition (self-threading and Ubi-Gua) andthose which do not (Ubi-Igd and Ubi-Frr). In thecase of the ®rst-order transition (see Figure 7A andB) we observe a well-focusing landscape, i.e. apronounced correlation between the degree ofsimilarity Qalg and energy. Two states are predomi-nantly populated: ``folded'' low E, high Qalg; and`ùnfolded'', high E, low Qalg. In contrast, when no®rst-order transition is observed, the landscape ismuch less focusing, i.e. there are several lowenergy alignments different from the optimal one

(low E, high Qalg). These decoy low-energy align-ments constitute a serious danger for the success ofa threading procedure. As explained above struc-ture-structure comparison is equivalent to thread-ing with an `ìdeal'' potential. Then, a potentialdifferent from an `ìdeal'' one can decrease analready small energy difference between the opti-mal and the decoy alignments and, hence, makeone of the decoys the optimal one. As shownabove, that is exactly the case when the residue-residue potential is used for threading.

Sequence-structure alignment

Figure 7D, E and F present logarithms of thenumber of alignments in each energy-distance forthreading with the ``best'' residue-residue poten-tial. Comparing these with corresponding ones forstructure-structure alignment (A, B and C) weobserve the following. (i) Both Ubi-Ubi (self-align-ment) and Ubi-Gua energy landscapes sustain thechange of the potential and both have focusinglandscape. (ii) The threading of Ubi-Igd, however,suffers a lot from the application of a non-idealpotential (compare Figure 7C and F). The optimalalignment becomes very different from the struc-tural one. As expected, one of the former low-

Figure 7. Energy landscapes of threading. The gray level shows the number of alignments in each E, Qalg interval.A, B and C, Structure-structure alignments. D, E and F, Sequence-structure alignments. Pairs of proteins used areshown as the title of each column. All for the equilibrium ensembles of alignments obtained at Tf .


energy decoys becomes the optimal alignment,while the structural alignment obtains a ratherhigh energy. (iii) The nicely focusing landscape ofUbi-Gua is deformed and several low-energydecoys with Qalg � 60% appear (compare Figure 7Band E). We expect that further changes in thepotentials can make one of the decoys the optimalalignment.

These results bring us to the conclusion that theoptimal alignment may be sensitive to the noise inthe potential. The degree of this sensitivitydepends on the degree of structural similaritybetween the native structure and the templatestructure used for threading. The closer the tem-plate structure is to the native structure, the lesssensitive the optimal alignment. It is crucial toestablish the generality of this conclusion andmake it as quantitative as possible. To achieve that,other threading examples should be considered. Inthe following subsection we brie¯y discuss thread-ing experiments for two more proteins folds:immunoglobulins (all b) and globins (all a).

Examples of threading with other proteins

In order to generalize conclusions drawn fromthe study of proteins with a ubiquitin-like fold (abclass) we repeated the analysis for proteins ofimmunoglobulin (b class) and globin (a class)folds. For each fold we make structural alignmentsand threading alignments with both ``best'' andMJ96 potentials. Tables 4 and 5 summarize theresults of these experiments.

Consistent with our results for ubiquitin, thefurther the template is from the native structure,the more sensitive the alignments are to the noisein potential.

By choosing templates of increasing distance(RMS) from the native structure we can see howalignment accuracy depends on the distance. Thestructure of ®bronectin (Fnf) is very close to thestructure of tenascin with RMS < 2 AÊ . Threading oftenascin's sequence through the structure of ®bro-nectin gives very accurate alignment for both``best'' and MJ96 potential. A template that is closeto the native structure can provide good structurepredictions, even with an available knowledge-based contact potential (of course, if the threadingalgorithm is good enough to ®nd alignments of thelowest energy!). A bit more distant structure ofhuman growth hormone (HhrB) provides a goodalignment with the ``best'' potential (Qalg > 0.75,RMS < 3 AÊ ), but it misses the right alignment witha more ``noisy'' MJ96 potential (Qalg < 0.5,RMS > 4 AÊ ). More distant from HhrB templatestructure of tenascin provide RMS > 3 AÊ even forthe ``best'' potential and yields totally wrong align-ments with MJ96 (see Table 5).

For a single domain of hemoglobin (Ash) eventhe closest available structure (Fal) gives a pooralignment with RMS > 4 AÊ when the ``best'' poten-tial is used. Colicin A, a well-known structural ana-log of hemoglobin yields completely incorrectalignments even for the ``best'' potential. We seethat as in the other cases, the templates that aredistant from the native structure can not allow suc-cessful in threading with any potential.

Table 4. Comparison of the optimal structure-structure and optimal sequence-structure alignments for the immuno-globulin fold


Proteins Lalg RMS dRMS Lalg RMS dRMS Qalg Lalg RMS dRMS Qalg

Ten-Ten 89 0.00 0.00 84 0.00 0.00 0.94 84 3.27 3.24 0.60Ten-Fnf 89 0.97 1.00 82 0.94 0.90 0.93 88 2.64 2.86 0.57Ten-HhrB 80 4.41 3.25 82 2.74 2.26 0.77 87 4.26 3.98 0.47Ten-Hnf 73 2.88 3.02 73 3.95 3.73 0.53 84 16.89 11.75 0.13Ten-Cid 74 2.73 2.78 69 3.67 3.31 0.23 82 13.96 8.66 0.17Ten-Tit 80 5.34 3.52 75 4.83 4.42 0.34 80 5.39 5.05 0.34

Notation as for Table 3. Proteins codes used in the Table and sequence identity with 1ten: Ten, tenascin (100%); Fnf3, ®bronectin(3-d domain) (22%); HhrB, human growth hormone (19%); Hnf, CD2 (8%); Cid, CD4 (11%); Tit, titin (9%).

Table 5. Comparison of the optimal structure-structure and optimal sequence-structure alignments for globin fold


Proteins Lalg RMS dRMS Lalg RMS dRMS Qalg Lalg RMS dRMS Qalg

Ash-Ash 147 0.00 0.00 141 0.00 0.00 0.93 146 0.77 0.82 0.95Ash-Fal 140 1.91 1.76 123 4.19 3.44 0.62 143 3.44 3.02 0.81Ash-Hbg 137 3.16 2.60 115 5.77 4.36 0.51 138 5.87 4.62 0.31Ash-ColA 114 4.24 3.43 102 7.44 6.05 0.42 146 15.30 8.68 0.00

Notation as for Table 3. Protein codes used in the table and sequence identity with 1ash; Ash, hemoglobin (domain one) (100%);Fal, myoglobin (12%); Hbg, hemoglobin(deoxy) (14%); ColA, colicin a (6%).


MC threading versus ``frozen'' approximation

Finally we compare the performance of MonteCarlo threading with the widely used ``frozen''approximation (Flockner et al., 1997). Under thisapproximation the energy of a target sequenceaj, j � 1, . . . ,J of protein J threaded into a structureof protein I is assumed to be a sum of energies ofsingle mutations needed to make sequence J outof the sequence bi, i � 1, . . . ,I of protein I . If theenergy of protein I after a single mutation bk! ak

is:

e�bk ! ak� �X

i

U�bi; ak��ik

then the energy of the whole alignment pi under``frozen'' approximation is:

E �XI

i0�1

e�bi0 ! ap0i� �

XI

i<i0�1

U�bi; ap0i��ii0

The ``frozen'' approximation essentially turns aproblem of sequence-structure alignment into asequence-sequence alignment with a position-dependent substitution matrix, allowing appli-cation of fast and exact sequence alignment algor-ithms (Smith & Waterman, 1981; Needleman &Wunsch, 1970). This approximation, however, isvery crude, since all interactions between residuesof the target sequence J are replaced by an exter-nal ®eld acting on the sequence from the residuesof protein I .

We implemented the ``frozen'' approximationusing local sequence alignment (Smith &Waterman, 1981) and linear gap penalty function.To get the best results from the ``frozen'' approxi-mation we varied the values of the two gap penal-ties trying 500 different values for each. Analignment which produced the lowest dRMS wasselected for further comparison. The results for the

``frozen'' approximation applied to ubiquitinthreaded through its three analogs with the MJ96potential are presented in Table 6.

Clearly, the alignments obtained with the ``fro-zen'' approximation are very far from the structur-al ones. The only example where alignment wasclose to the optimal is tenascin threaded throughthe third domain of ®bronectin. This result comesas no surprise, since the two proteins haveextremely close structures (RMS < 1.0 AÊ ) andsome sequence and evolutionary homology(SeqID � 20%). For all other cases the alignmentsobtained by MC threading with the same potentialwere much more accurate than alignmentsobtained using the ``frozen'' approximation.

Discussion

Here we presented a systematic approach to theproblem of protein structure prediction by thread-ing. As in folding, the problem of threading hastwo components: (i) a search strategy, which isable to ®nd the optimal alignment of sequence anda structure; and (ii) a potential, which provides thelowest energy to the native and similar structuresof a protein. First, we developed a Monte Carloalgorithm to search through the space of align-ments for the optimal one. To test the algorithmwe applied it to structure-structure alignmentsusing a Dali potential. For each case studied ouralgorithm successfully found the optimal align-ments which another heuristic algorithm (Holm &Sander, 1993) failed to ®nd. We also showed thatfor proteins with similar structures, the optimalalignment is not substantially changed when thedistance-dependent potential is replaced by amuch simpler contact potential. Then we con-sidered structure-structure alignments as sequence-structure threading with an hypothesized `ìdeal''potential, which makes all native and only nativecontacts attractive (so-called ``Go-model'' in pro-tein folding).

Although no residue-residue potential can pro-vide this `ìdeal'' pattern of interaction, one canbuild a potential which provides an interactionpattern as close to the `ìdeal'' one as possible(Mirny & Shakhnovich, 1996). Using an optimiz-ation technique (Mirny & Shakhnovich, 1996) webuilt this potential for a single protein, ubiquitin,and then threaded its sequence through differentprotein structures. Comparison of the optimalalignments obtained by threading and correspond-ing structural alignments brought us to the follow-ing conclusions. (i) Although the used residue-residue potential reconstitutes an `ìdeal'' patternof interactions in the best possible way, optimalthreading alignments are different from optimalstructural alignments. (ii) The degree to which anoptimal alignment changes depends strongly onthe degree of similarity between the native struc-ture and the template structure used for threading.Particularly, threading through the native structure

Table 6. Results of threading with ``frozen'' approxi-mation using the MJ96 potential

Proteins SeqID (%) Lalg dRMS Cb Qalg

Ubi-Ubi 100 76 0.00 1.00Ubi-Gua 14 76 6.00 0.59Ubi-Igd 4 61 7.11 0.04Ubi-Frr 6 76 6.61 0.13Ten-Ten 100 89 0.00 1.00Ten-Fnf 22 89 1.79 0.90Ten-HhrB 19 80 7.23 0.11Ten-Hnf 9 89 18.21 0.08Ten-Cid 11 89 8.87 0.00Ten-Tit 9 89 6.06 0.07Ash-Ash 100 147 0.00 1.00Ash-Fal 12 146 4.87 0.01Ash-Hbg 14 147 6.49 0.17Ash-ColA 6 147 8.49 0.00

Gap penalties were chosen to minimize the dRMS. Asexpected, dRMS values are much higher for all proteins withno distinct sequence homology (compare dRMS with Tables 3to 5). The only pair that yields an accurate alignment (Ten-Fnf)has a distinct sequence homology.


and a highly similar one yields an optimal align-ment very similar to the structural one. In contrast,for proteins sharing a moderate similarity with thenative one, the optimal threading alignment is verydifferent from the structural one and provides apoor model of the native structure (RMS Ca > 4 AÊ ).These results and detailed analysis of the threadinglandscape brought us to the major conclusion thatin order to get an accurate threading alignmentone needs, most importantly, a very similar tem-plate structure. If this is not present, even anincredibly accurate residue-residue potential is use-less.

Threading versus folding

In folding, the problem is to ®nd the native con-formation, whereas threading aims at ®nding aconformation similar to the native one. The funda-mental difference is that in threading the nativeconformation is not present in the space of possibleconformations and only an approximate (``native-like'') conformation can be found. The native con-formation of a protein has an energy much lowerthan the energy of other (random) conformations,where a ``native-like'' conformations is less stableand not that much below other conformations inenergy. This makes threading a much morechallenging problem as one needs to ®nd a confor-mation that is not as distinct in energy from othersas the native. When one threads a sequencethrough its native structure, the optimal alignmentis separated from dissimilar decoys by a largeenergy gap, whereas when one uses a structuresimilar to the native as a template for threading,the gap is much smaller. The further the templatestructure is from the native one, the smaller theenergy gap between the optimal alignment and therandom ones. Figure 8 presents a schematic ofenergy spectra for folding and threading.

From the physical point of view, threading usinga template close to the native structure is similar to

folding to the structure which has a moderateenergy gap. This situation was studied extensivelywith model proteins (Mirny et al., 1996; Abkevichet al., 1996), where different energy gaps betweenthe native conformation and the bulk of unfoldedones are responsible for different regimes of fold-ing. When the energy gap is large (folding ofdesigned, evolved sequences), the native confor-mation is stable relative to changes in temperatureand to mutations (Pande et al., 1995; Vendruscoloet al., 1997), and folding is fast and exhibits a coop-erative, ®rst-order-like transition. When the energygap is small (poorly designed, random sequences;Mirny et al., 1996; Abkevich et al., 1996) the nativeconformation is much less stable with respect totemperature and, importantly, is very unstableto mutations (Shakhnovich & Gutin, 1991) and toerrors in the potential (Vendruscolo et al., 1997;Pande et al., 1995; Bryngelson, 1994). Folding inthis case is slow because thermodynamic stabilityrequires a low temperature at which a protein getstrapped in several misfolded low-energy confor-mations. Stability with respect to changes in poten-tial and mutations have the same nature: both raisethe energy of the ground state, decreasing the gap.If the gap is large enough it sustains destabilizationof the native state and it remains the ground state.If the gap is small, the destabilizing decrease in thegap can eliminate the gap completely and makeanother (random!) conformation the state with thelowest energy (see the cartoon in Figure 8).

In this study we observed similar regimes forthreading. When the template is close to the nativestructure (self-threading, Ubi-Igd) the optimalalignment is separated by a large energy gap andwe observe: (i) cooperative transition; (ii) focusingenergy landscape; (iii) high reliability of alignment(low Si, similar suboptimal alignments); and, mostimportantly (iv) stability of the optimal alignmentto the changes in potential. Alternatively, when thetemplate is moderately close to the native structure(Ubi-Gua, Ubi-Frr) the gap is small and we

Figure 8. Schematic cartoon of the density of states for folding and threading. See Discussion.


observe: (i) smooth transition; (ii) poorly focusingenergy landscape; (iii) low reliability of alignment(high Si, dissimilar suboptimal alignments); and,most importantly (iv) low stability of the optimalalignment to the changes in potential.

Factors affecting the accuracy of alignment

These results suggest that there are two majorfactors affecting the accuracy of threading align-ment: (i) the degree of similarity between the tem-plate structure and the native one; (ii) the accuracyof the potential. Both factors are responsible forproviding a large energy gap between the rightalignment and the bulk of decoys. The gap isessential to guarantee the lowest energy and highreliability for the right alignment.

The crucial role of the degree of similarity forthe success of threading became evident after therecent experiment in ``blind'' structure prediction,CASP2 (Moult et al., 1997; Levitt, 1997; Marchler-Bauer & Bryant, 1997; Eisenberg, 1997; Shortle,1997; Dunbrack et al., 1997). Several predictorsJones, 1997; Marchler-Bauer & Bryant, 1997) notedthat success in threading could be attributed moreto target proteins rather than methods. Only thoseproteins that have a globally similar (more than60% of residues superimposable) structure in thedatabase were modeled accurately (Marchler-Bauer& Bryant, 1997; Levitt, 1997). The degree of simi-larity between the two structures (dRMS, fractionof matched residues, fraction of common contacts,etc.) required to provide high accuracy of thread-ing using currently available potentials is still to beestimated.

Earlier we showed (Mirny et al., 1996) than an`ìndividual'' potential optimized for a single pro-tein is not transferable to others. On the otherhand, a `ùniversal'' potential optimized simul-taneously for all proteins in the database worksworse for every single individual protein than the`ìndividual'' potential. Hence, alignments obtainedwith any realistic `ùniversal'' potential should bemuch worse than those obtained with the `ìndivid-ual'' one used in our threading tests. This is indeedthe case as seen in Tables 3 and 5.

While our analysis is carried out only on a lim-ited set of examples (three folds) the consistency ofthe results between them makes it possible tomake a preliminary quantitative conclusion ofwhat degree of success in sequence-structure align-ment can one expect from threading calculations.

When the RMS between the template and thenative structure is less than 2 AÊ the sequence-struc-ture alignment is likely to be accurate with anyavailable and reasonably good potential function.

When the RMS between the template and thenative structure is greater than 3.5 AÊ , the sequence-structure alignment is expected to be grossly inac-curate, independent of the potential function used.

The RMS between 2 AÊ and 3.5 AÊ represents a``twilight zone'' where the results may depend

strongly on the potential used as well as particularproteins.

The analysis of more examples using the MCthreading procedure will allow us to make thequantitative estimates of expected threading accu-racy more precise.

We are planning to carry our this research in thenear future.

Fold recognition versus alignment recognition

How does the accuracy of threading alignmentaffect the accuracy of fold recognition? Why arecurrent threading algorithms able to ®nd the righttemplate structure, yet fail to produce accuratealignments (Marchler-Bauer & Bryant, 1997; Levitt,1997; Jones, 1997)?

The major difference between fold recognitionand sequence-structure alignments is in the num-ber of alternatives to choose from. Fold recognitionis aimed at ®nding a structure in a representativefold database which contains about 1000 folds(Holm & Sander, 19970. In contrast, the threadingalgorithm applied to two proteins of length 100needs to ®nd the optimal in the space of 2100 � 1030

alignments (Waterman, 1995). Clearly, fold recog-nition is much less demanding of the accuracy ofpotential and the power of a search strategy. Foldrecognition can tolerate errors in threading align-ment. In fact, to distinguish a structure close to thenative one from a distant structure one needs to®nd any low energy alignment of the close struc-ture. Only a tiny fraction of alignments has theenergy low enough to distinguish the close struc-ture, but because of the huge alignment space thisfraction numbers in the millions of alignments andany of them is equally good for the purpose of foldrecognition. In contrast, here we show that align-ment recognition requires much more accuratepotential and a more powerful search strategy.

To be successful in fold recognition, one neednot send the arrow right to the apple; one justneeds to shoot in the right direction.

Future directions

One of the possible ways to overcome this fun-damental problem in threading is to use someextra information about the query sequence. Evol-utionary information can be of great importancefor structure prediction and has already been takeninto account (explicitly or implicitly) in differentways (Fischer & Eisenberg, 1996; Gerloff et al.,1997; Russell et al., 1996; Defay & Cohen, 1996;Goldman et al., 1996; Ortiz et al., 1998). A popularway of using evolutionary information is to predicta secondary structure or the degree of solventaccessibility from multiple sequence alignments(Rost & Sander, 1995) and then use this predictionto re®ne threading. Use of intrinsically inaccuratepredictions, however, should not necessarilyimprove folding or threading, (Rost et al., 1997) butinstead can derail them. Rational utilization of


evolutionary information clearly requires deeperstructural understanding of the evolutionary tracesobserved in homologous proteins (Shakhnovichet al., 1996b; Mirny et al., 1998).

Another substantial improvement may comefrom building more complicated (but still compu-tationally tractable!) models of proteins. Takinginto account side-chains, their sizes and shapes,can eliminate from the search some threadingalignments that produce unfeasible side-chainpacking and thus focus the landscape to the nativestate.

Methods

Alignment representation

An alignment between two proteins of length I and Jis represented by a matrix Aij, where i � 1, . . . ,I andj � 1, . . . ,J:

Aij � 1 if i is aligned with j0 otherwise

��10�

Another way of presenting an alignment is by a pointerpi:

pi � j if i is aligned with j0 if i is not aligned to any residue

��11�

In this study we do not allow double matches(i.e. �i�1, . . . ,I Aij 4 1). The reverse of any fragment in thealignment is also forbidden, i.e. if Aij � 1, then for anyi0 > i and j0 < j Ai0 j0 � 0. (In general, reverse of proteinfragments in sequence structure alignment might makesense and improve performance (Finkelstein, 1997).)Under these constraints, matrix Aij should have the formshown in Figure 9A, i.e. an alignment composed ofruns of aligned residues separated by gaps in either or inboth proteins. These runs are referred to below as frag-ments of alignment. Each fragment is a set of matches

(Aij � 1 for (ij) � (i0, j0), (i0 � 1, j0 � 1), . . . ,(i0 � L, j0 � L))framed into gaps Ai0 ÿ 1,j0 ÿ 1 � Ai � L � 1,j � L � 1 � 0. Thesefragments are used in the move set as elementary build-ing blocks.

Move set

Each move in the move set is designed to change thealignment preserving most of the matches and, hence,leading to a small change in energy. Another importantfeature of the proposed move set is that it allows easyintroduction of constraints on the minimum length of afragment or maximum length of a gap. This ¯exibility isachieved by making moves on fragments, rather thancreating and destroying single matches. Here we con-strain the fragment length to be greater or equal toLmin � six residues. Moves used in the current implemen-tation are shown in Figure 10.

Shift

A whole fragment is shifted in any of four directions.The distance fragment shifted is chosen randomly anduniformly between 1 and M, where M is the space avail-able to the next fragment in the direction of the shift.

Shrink/expand

A fragment is shrunk (expanded) on either end on nresidues. The value of n is chosen randomly from theexponential distribution, i.e. P(n) � a � b � exp(ÿn/5).Parameters a and b set the limits on n. Limits are chosento provide the length of the obtained fragment L 5 Lmin

and not overlap with another fragment.

Figure 9. Representation of a protein-protein align-ments. A, Example of the alignment; B, its matrixrepresentation.

Figure 10. Move set. A, Shift; B, shrink/expand; C,split/merge; D, jump.


Split/merge

Fragments which happen to be located head-to-tail ofeach other are merged into a single fragment. Fragmentsare merged every ten steps (starting from step 5), a ran-domly chosen fragment is split into two fragments at arandom point, providing that lengths of both fragmentsL1, L2 5 Lmin.

These moves are suf®cient for successful MC align-ment. However, we use one extra move which makessampling more ef®cient.

Jump

A piece of one fragment is deleted and joined to thenext fragment. The length of the piece is chosen ran-domly and uniformly so that the length of the remainingpart of the fragment L 5 Lmin.

Although we do not have an explicit move for the cre-ation (destruction) of a fragment, new fragments canappear (disappear). In fact, a new fragment can appearby expansion of an existing one and further splitting.Similarly, a fragment can disappear by merging with anexisting one followed by shrinking. At least one frag-ment is present in the alignment. These moves can ef®-ciently sample all possible alignments between twoproteins. Note that in contrast to other work (Lathrop &Smith, 1996) the lengths and number (Bryant, 1996) offragments can vary.

This move set provides an acceptance ratio forMonte Carlo steps of about 0.6 at transition tempera-ture Tf . A typical simulated annealing protocol requires20-50 temperature levels with 106 steps on each. Ittakes about four minutes of CPU time on a Pentium233 for each 106 MC steps for proteins of approxi-mately 100 residues.

Measures of similarity

Distance between protein conformations

We use three different measure of distance betweenprotein structures. Consider proteins I and J whichhave length I and J residues, respectively, and alignmentbetween them given by pi (see above). Then:

dRMS ��

1

�Nalg ÿ 1��Nalg ÿ 2�XI

i;i0>i�2

�rJpi;p0iÿ rIii0 �2

vuut �12�

where rIii0 and rJjj0 are distances between residues in eachprotein. In this study we use distances between Cb atomsas distances between corresponding residues.

Another measure based on distance matrices rij0 wasintroduced by Holm & Sander, 1993) and is used in Dalistructure comparison program:

Dali �X

i;i0>i�2

0:2ÿrIii0 ÿ rJri;p0i

d

!exp�ÿd=20� �13�

where:

d �rIii0 � rJpi;p0i

2

The Dali measure is much more useful for structure com-parison as it ``weights'' matches between closely locatedresidues much more than between distant ones (seeModel for details).

A much simpler measure of protein similarity is theoverlap of contact maps to the two proteins:

Q � 1

min�NI ;NJ �XI

i;i0>i�2

�Jpi;p0i��Iii0 �14�

where:

�ii0 �1 if i and i0 are in contact: rii0 < Rcut

0 otherwise

(�15�

Nw is the number of contacts in protein w. In this studywe use contact cutoff Rcut � 8 AÊ between Cb atoms. Con-tact overlap Q, in contrast to dRMS and Dali, is focusedon contacts between residues and, hence, it completelyignores residues that are far from each other and mightnot contribute to the energy of the conformation.

Distance between alignments

To measure distance between alignments we intro-duce overlap Qalg between alignment pi and qi:

Qalg � 1

Nalg

XI

i�1

sign�pi � qi� � jpi ÿ qij �16�

where sign(0) � 0 and sign(x > 0) � 1, (qi, pi 5 0 for all i).

Random alignments

To generate a random alignment we make 104 steps ofthe MC procedure at T �1, i.e. accepting all moves.Alignments randomized in this way are used as startingpoints for optimization runs.

Optimization of potential

The aim of this procedure is to ®nd a potentialU(x, Z), x, Z � 1, . . . ,20 that provides the pattern of inter-actions between residues Bjj0 � U(aj, aj0) as close to an``ideal'' pattern Bideal

jj0 � ÿ�jj0 as possible. This is achievedby maximizing the correlation between Bjj0 and ÿ�jj0:

r�U� � r�B�U�;ÿ�� ÿ hB�i ÿ hBih�i��s2�B�s2��

p �17�

as a function of potential U. Averages h � i are taken overall j < j0 elements. Optimization is performed by a MonteCarlo procedure, where at each step a randomly chosenelement U(x,Z) is increased (decreased) by d � 0.01 andthe increase is accepted or rejected according to theMetropolis scheme (Metropolis et al., 1953). Sinceequation (17) does not change upon linear transform-ation of U, we set hUi � 0 and s(U) � 1 (see Mirny &Shakhnovich (1996) for details).

Importantly, r(U) de®ned above is a linear function ofthe Z-score, which measures the energy gap between thenative conformation and the bulk of alternativeconformations:

Z � EN ÿ hEiconf

s�E�conf

�18�

where EN is the energy of the native conformation;hEiconf and s(E)conf are the mean and the variance ofenergy of alternative conformations. When hEiconf andsconf(E) are computed as described by Mirny &Shakhnovich (1996):


r � Z��ntotal ÿ np �19�

where n is the number of native contacts in J and ntotal

is the total number of possible contacts J �J ÿ 2�=2.Then, by maximizing r we maximize Z and make thenative conformation of the proteins a pronounced energyminimum (Mirny & Shakhnovich, 1996).

Acknowledgments

We are grateful to Victor Abkevich and CeciliaClementi for fruitful discussions. This work was sup-ported by NIH grant GM52126.

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Edited by F. Cohen

(Received 13 February 1998; received in revised form 15 July 1998; accepted 16 July 1998)


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