Learning Scoring Schemes forSequence Alignment from Partial Examples

Eagu Kim and John Kececioglu

Abstract—When aligning biological sequences, the choice of scoring scheme is critical. Even small changes in gap penalties, for

example, can yield radically different alignments. A rigorous way to learn parameter values that are appropriate for biological

sequences is through inverse parametric sequence alignment. Given a collection of examples of biologically correct referencealignments, this is the problem of finding parameter values that make the scores of the reference alignments be as close as possible to

those of optimal alignments of their sequences. We extend prior work on inverse parametric alignment to partial examples, whichcontain regions where the reference alignment is not specified, and to an improved formulation based on minimizing the average error

between the scores of the reference alignments and the scores of optimal alignments. Experiments on benchmark biologicalalignments show we can learn scoring schemes that generalize across protein families, and that boost the accuracy of multiple

sequence alignment by as much as 25 percent.

Index Terms—Inverse parametric sequence alignment, substitution score matrices, affine gap penalties, linear programming, cutting

plane algorithms.

Ç

1 INTRODUCTION

Afundamental question that arises whenever biologicalsequences are aligned is to decide what parameter

values to use for the alignment scoring scheme. Forexample, the standard scoring function for protein sequencealignment requires determining values for 210 substitutionscores and two gap penalties. In practice, substitution scoresare usually set by convention to a matrix from the PAM [5] orBLOSUM [17] families, while gap penalties are often set byhand through trial and error. Despite this practice, choosingappropriate values for these parameters is critical. Evensmall changes in gap penalties can yield radically differentalignments, while aligning a specific family of sequencesmay require substitution scores that differ substantiallyfrom a conventional matrix.

A rigorous approach to determining these values isthrough inverse parametric sequence alignment [16], [31], whichsets parameters using examples of biologically correctreference alignments. Informally, inverse alignment seeksparameter values for the alignment scoring function thatmake the examples be optimal alignments of their sequences.Put another way, it seeks a parameter choice at which theexamples are optimal solutions to the sequence alignmentproblem. Note that if a parameter choice exists that makesevery example be the unique-optimal alignment of itssequences, then any algorithm that computes optimalsequence alignments will perfectly recover the examples,given the example sequences and this parameter choice asinputs. In general, inverse alignment belongs to the broadarea of inverse parametric optimization [9], where examples

of optimal solutions to a combinatorial optimization problemare used to set parameter values in its objective function.

In practice, a biologically useful parameter choice rarelyexists that makes a collection of example reference align-ments all be optimal [19] (let alone unique-optimal). Conse-quently, the problem becomes one of finding parametervalues that make the examples score as close as possible tooptimal. An important issue in formulating the problem isdeciding what measure of error to minimize between thescores of the examples and the scores of optimal alignments,when assessing closeness to optimality.

Recently, Kececioglu and Kim [19] discovered a newmethod for inverse alignment based on linear programmingthat for the first time could quickly learn best possible valuesfor all 212 parameters in the standard model for globalprotein sequence alignment from hundreds of examples ofcomplete reference alignments. Their approach minimizedthe maximum relative error in scores across the examples. Inthis paper, we extend this work in several directions:

. to partial examples, which contain regions where thereference alignment is not specified,

. to an improved error model involving minimizationof the average absolute error across the examples,

. to a stronger approach for eliminating degeneratesolutions that are not biologically useful, and

. to an experimental study of the recovery rate of thelearned scoring schemes when used for multiplesequence alignment.

The issue of partial versus complete examples is especiallyimportant when the reference alignments are drawn fromsuites of benchmark protein multiple alignments that havebeen determined by aligning the three-dimensional struc-tures of the protein molecules [24], [1], [2], [33]. While suchbenchmarks provide the most accurate protein alignmentscurrently available, the regions where these referencealignments are reliable usually consist of a series of alignedconserved blocks that rarely cover the entire sequences;

IEEE TRANSACTIONS ON COMPUTATIONAL BIOLOGY AND BIOINFORMATICS, VOL. 5, NO. 4, OCTOBER-DECEMBER 2008 1

. The authors are with the Department of Computer Science, The Universityof Arizona, Tucson, AZ 85721. E-mail: {egkim, kece}@cs.arizona.edu.

Manuscript received 7 Dec. 2007; revised 26 Apr. 2008; accepted 21 May2008; published online 3 June 2008.For information on obtaining reprints of this article, please send e-mail to:tcbb@computer.org, and reference IEEECS Log NumberTCBBSI-2007-12-0165.Digital Object Identifier no. 10.1109/TCBB.2008.57.

1545-5963/08/$25.00 ! 2008 IEEE Published by the IEEE Computer Society

between these blocks, the alignment is often unreliable orunspecified.

The improved approach to inverse alignment that resultsfrom these extensions, which minimizes average absoluteerror given partial examples, remains fast and achievesbetter recovery. To briefly summarize the results, therecovery of reference alignments by the learned scoringschemes improves upon conventional amino acid substitu-tion matrices by as much as 25 percent when used formultiple alignment of specific sequence families. Learning ascoring scheme from more than 200 partial examples,whose reference alignments cover about 40 percent of eachsequence on average, takes around 4 hours on a laptop.

1.1 Related WorkInverse parametric sequence alignment was introduced inthe seminal paper of Gusfield and Stelling [16]. Theyconsidered the problem for the case of two parametersand one example, and sought parameter values that madethe example be an optimal scoring alignment of itssequences. For this form of inverse alignment, theypresented an indirect approach that attempted to avoidcomputing the complete parametric decomposition [34],[15], [27] of the parameter space. Sun et al. [31] gave thefirst direct algorithm for inverse alignment for the case ofthree parameters and one example. Their algorithm, whichis quite involved, finds values for the three parametersthat make the example alignment score optimal inO!n2 logn" time, for two strings of length n.

In a significant advance, Kececioglu and Kim [19], using acompletely different approach based on linear program-ming, gave the first polynomial-time algorithm for arbitra-rily many parameters and examples. Their approach is alsoflexible, and solves the unique-optimal and near-optimalvariations of inverse alignment as well. For the near-optimalvariation, their algorithm finds a parameter choice thatmakes the examples score as close as possible to optimal interms of minimizing the maximum relative error of theexamples. As they demonstrated, it is also fast in practice.

The approach to inverse alignment developed in [19]actually solves the more general problem of inverseparametric optimization. For a combinatorial optimizationproblem P, inverse parametric optimization seeks para-meter values for the objective function of P that make agiven example solution to an instance I of P be an optimalsolution to instance I . Their approach solves the inverseparametric problem in polynomial time for any problem Pwhere: 1) the objective function for P is linear in itsparameters, and 2) problem P can be solved in polynomialtime for any fixed choice of its parameters. This solvesinverse parametric optimization for an extremely widerange of problems P, including many classic combinatorialproblems such as sequence alignment, shortest paths,spanning trees, network flow, and maximum matchings.The authors recently learned that Eppstein [9] discovered asimilar approach to general inverse parametric optimiza-tion. Eppstein applied it in the context of minimumspanning trees to find edge weights that make an exampletree be the unique optimal spanning tree. In the context ofbiological sequence alignment, this unique-optimal form ofthe inverse problem rarely has a solution [19].

Alternative approaches for determining alignmentparameters have been recently proposed based on machine

learning. Do et al. [7] use discriminative training onconditional random fields to find parameter values for ahidden Markov model of sequence alignment. Theirapproach requires solving a convex nonlinear numericaloptimization problem that becomes nonconvex in thepresence of examples that are partial alignments, and isnot guaranteed to run in polynomial time. Yu et al. [37]describe a support vector machine approach for learningparameters to align a protein sequence to a protein structure.Their approach involves solving a quadratic numericaloptimization problem with linear constraints, and for thefirst time incorporates a measure of alignment recoverydirectly into the problem formulation. Both of theseapproaches require considerable computational resources,appearing to take days on large instances to compute anapproximate solution.

In contrast to these machine learning approaches, ourmethod for inverse alignment only involves linear pro-gramming, for which an optimal solution can be computedquickly, even for very large instances. We also rigorouslyaddress for the first time the issue of input examples thatare partial alignments.

1.2 OverviewThe next section presents several variations of inversealignment, with relative or absolute error, andwith completeor partial examples. Section 3 reduces these variations tolinear programming, and develops an iterative approach topartial examples. Finally, Section 4 presents results fromexperiments on recovering benchmark protein alignmentswhen using learned parameters for both pairwise andmultiple sequence alignment.

2 INVERSE ALIGNMENT AND ITS VARIATIONS

The conventional sequence alignment problem is: given apair of sequences and a scoring function f on alignments,find an alignment A of the sequences that has an optimalscore under f . The inverse alignment problem turns thisaround: given an example alignment A of a pair ofsequences, find parameter values for scoring function fthat make A be an optimal alignment of its sequences. Tolearn parameter values that are useful in practice forbiologists, this basic form of inverse alignment, which wasfirst studied in [16] and [31], must be generalizedconsiderably: to multiple examples, to suboptimal align-ments, to partial examples, and to handle degeneracy, eachof which we consider in turn.

When function f has many parameters, many examplealignmentsA are needed to determine reliable values for theparameters. Accordingly, we take the input to inversealignment to be a large collection of example alignments. Inpractice, there are usually no biologically useful parametervalues that make all the example alignments have optimalscore. Consequently, we consider finding parameter valuesthat make the examples score as close as possible to optimal,and we examine two optimization criteria for measuring theerror between the example scores and the optimal alignmentscores: minimizing relative error or absolute error. Finally,the type of benchmark reference alignments that areavailable in practice for learning parameters often consistof regions where the alignment is specified, interspersedwith regions where no alignment is specified. We call such a

2 IEEE TRANSACTIONS ON COMPUTATIONAL BIOLOGY AND BIOINFORMATICS, VOL. 5, NO. 4, OCTOBER-DECEMBER 2008

reference alignment a partial example, since it is only a partialalignment of its sequences. When the reference specifies acomplete alignment of its sequences, we call it a completeexample.

Our approach to inverse alignment from partial exam-ples builds upon a solution to the problem with completeexamples, which we discuss first.

2.1 Complete ExamplesInverse alignment from complete examples with arbitrarilymany parameters was first considered by Kececioglu andKim [19]. They examined the relative error criterion, whichwe review below.

Let f be the alignment scoring function, which givesscore f!A" to alignment A. Typically, f is a function ofseveral parameters p1; p2; . . . ; pt, which assign scores orpenalties to various alignment features such as substitu-tions and gaps. (For example, the standard scoring modelfor aligning protein sequences has 210 substitution scoresfor all unordered pairs of amino acids, plus two gappenalties for opening and extending a gap, for a total oft # 212 parameters.) We view the entire set of parametersas a vector p # !p1; . . . ; pt". When we want to emphasizethe dependence of f on its parameters p, we write fp.

The input consists of many example alignments Ai,where each example aligns a corresponding set ofsequences Si. (Typically, the examples Ai are inducedpairwise alignments that come from a structural multiplealignment; in this case, each Si contains two sequences.) Forscoring function f and parameters p, we write f$

p !Si" for thescore of an optimal alignment of sequences Si under fp. Weassume that an optimal alignment minimizes f .

The following defines inverse alignment under therelative error criterion assuming the input contains com-plete examples.

Definition 1 (Inverse Alignment under Relative Error).Inverse Alignment from complete examples under the relativeerror criterion is the following problem. Let the alignmentscoring function be fp with parameter vector p # !p1; . . . ; pt"drawn from domain D. The input is a collection of completealignments A1; . . . ;Ak that respectively align the sets ofsequences S1; . . . ;Sk. The output is parameter vector

x$ :# argminx2D

Erel!x";

where

Erel!x" :# max1%i%k

fx!Ai" & f$x!Si"f$x!Si"

:

In other words, the output vector x$ minimizes the maximumrelative error of the alignment scores of the examples.

An underlying issue in this formulation of inversealignment is that the problem is degenerate. The trivialparameter choice x # !0; ' ' ' ; 0"makes every alignment scorethe same, thus making each example be an optimalalignment. This undesirable solution must be ruled out,though how this is done depends on the particular form ofalignment being considered. Section 3.3 presents a newapproach for eliminating a general family of degeneratesolutions that applies to global sequence alignment [14]. Thisgeneral family includes the trivial solution as a special case.

When scoring function fp is linear in its parameters p,inverse alignment under relative error can be solved inpolynomial time [19] as long as an optimal alignment can becomputed in polynomial time for any fixed parameterchoice. We review the solution in Section 3, which uses areduction to linear programming. The above formulationconsiders the maximum error of the examples, because as wewill see later in Section 3.1.1, minimizing the averagerelative error would lead to an optimization problem withnonlinear constraints.

We also consider here a new model: minimizing theabsolute error. This has the advantage that we can minimizethe average error of the examples and still have a formulationthat is efficiently solvable by linear programming.

The following defines inverse alignment under theabsolute error criterion again assuming complete examples.

Definition 2 (Inverse Alignment under Absolute Error).Inverse Alignment from complete examples under the absoluteerror criterion is the following problem. The input is acollection of complete alignments Ai of sequences Si for1 % i % k. The output is parameter vector

x$ :# argminx2D

Eabs!x";

where

Eabs!x" :# 1

k

X

1%i%k

!fx!Ai" & f$

x!Si"":

Output vector x$ minimizes the average absolute error of thealignment scores of the examples.

Note that this formulation is also degenerate, which isaddressed in Section 3.3.

We next extend to input examples that are partialalignments.

2.2 Partial ExamplesFor inverse alignment of protein sequences, the bestexample alignments that are available come from multiplealignments of protein families that are determined byaligning the three-dimensional structures of family mem-bers. Several suites of such benchmark alignments arecurrently available [24], [1], [2], [33] and are widely used forevaluating the accuracy of software for multiple alignmentof protein sequences. Most all these benchmarks, however,are partial alignments. The benchmark alignment hasregions that are reliable and where the alignment isspecified, but between these regions, the alignment iseffectively left unspecified. These reliable regions areusually the core blocks of the multiple alignment, which aresections of the alignment where structure is conservedacross the family. Core blocks are typically gap free, thoughthey can sometimes contain gaps.

For our purposes, a partial example is an alignment A ofsequences S where each column of A is labeled as beingeither reliable or unreliable. A complete example is a partialexample whose columns are all labeled reliable.

Note that in this definition, a partial example may label agap column as reliable. Consequently, partial examples donot simply specify which pairs of positions to align bysubstitutions, but may also specify which positions shouldbe in gaps.

KIM AND KECECIOGLU: LEARNING SCORING SCHEMES FOR SEQUENCE ALIGNMENT FROM PARTIAL EXAMPLES 3

When learning parameters by inverse alignment frompartial examples, we treat the unreliable columns asmissing information: such columns do not specify thealignment of the example sequences. Given a partialexample A for sequences S, a completion A is a completeexample for S that agrees with the reliable columns of A. Inother words, a completion A can change A on thesubstrings that are in unreliable columns, but must notalter A in reliable columns. Note that in general, a partialexample A does not have a unique completion A.

We define inverse alignment from partial examples asthe problem of finding the optimal parameter choice overall possible completions of the examples.

Definition 3 (Inverse Alignment from Partial Examples).Inverse Alignment from partial examples is the followingproblem. The input is a collection of partial alignments Ai ofsequences Si for 1 % i % k. The output is parameter vector

x$ :# argminx2D

minA1;...;Ak

E!x";

where error function E is either Eabs or Erel, and symbol Ai

varies over all possible completions of Ai. In other words,vector x$ minimizes the error of the example scores over allpossible completions of the partial examples.

We make a few remarks on this definition. First, thisformulation finds parameter values for which optimallyaligning the unreliable regions yields a completion thatscores as close as possible to an optimal unconstrainedalignment of the example sequences. Second, even when thereliable regions are all gap free (which is common whenpartial examples are obtained from the standard suites ofbenchmark protein alignments), this formulation still learnsuseful values for gap penalties, as appropriate values forgaps in the unreliable regions are necessary for the reliableregions to be part of alignments that have a close-to-optimalscore. Third, note that this formulation for partial examplescontains the prior formulations on complete examples asspecial cases.

In the next section, we reduce inverse alignment fromcomplete examples to linear programming, and tacklepartial examples by solving a series of problems oncomplete examples.

3 SOLUTION BY LINEAR PROGRAMMING

When the alignment scoring function fp is linear in itsparameters p, inverse alignment from complete examplesunder relative error can be reduced to linear programming[19], and a similar reduction applies to absolute error. Wedefine a linear scoring function as follows: Suppose f scoresan alignment A by measuring t( 1 features of A throughfunctions g0; g1; . . . ; gt and combines these measures intoone score through a weighted sum involving parametervector p # !p1; . . . ; pt" by

fp!A" :# g0!A" (X

1%i%t

pi gi!A":

Then, we say f is linear in parameters p1; . . . ; pt. (The initialterm given by function g0 usually arises when someparameters in the scoring function are fixed, such as whenlearning gap penalties for a known substitution scoring

matrix [16], [31], [19], in which case g0!A"measures the totalsubstitution score of alignment A. When all parameters arevarying, the term for g0 is typically zero.)

For example, in the standard scoring model for globalalignment of protein sequences, there is a substitutionscore !ab for every unordered pair a, b of amino acids, apenalty " for opening a gap, and a penalty # for extending agap. (A gap in an alignment is a maximal run of eitherinsertion or deletion columns. A gap of ‘ columns has cost" ( #‘, which is often called an affine gap cost.) This gives ascoring function with 212 parameters !ab, ", and # for thealphabet of 20 amino acids. For this scoring model, thefunctions g!;a;b count the number of substitutions of eachtype a; b in A, and functions g" and g# respectively count thenumber of gaps and the total length of all gaps in A.

We first describe the linear programming approach forcomplete examples, and then discuss its extension to partialexamples.

3.1 Complete ExamplesAs described in detail in [19], inverse alignment fromcomplete examples with relative error can be reduced tosolving a series of linear programs. We briefly review thissolution for relative error, and then show how to modify itfor absolute error.

For the standard model of protein sequence alignment,the linear programs have the following form. There is avariable for each parameter !ab, ", and # in the alignmentscoring function f . The domain D of these parameters isdescribed by the following domain inequalities.

When the alignment problem is to minimize the score fof a global alignment, parameter values can always beshifted so they are nonnegative and then scaled so thelargest magnitude is 1, without changing the problem.Thus, domain D has inequalities

!0; 0; 0" % !!ab; ";#" % !1; 1; 1": !1"

The description of D also has the inequalities

!aa % !ab; !2"

since an identity should cost no more than a substitutioninvolving that letter. (As an aside, inequality (2) holds forPAM [5] and BLOSUM [17] similarity scores when they aretransformed into substitution costs.)

The remaining inequalities, which ensure that theexamples score as close as possible to optimal alignments,differ for the relative and absolute error criteria.

3.1.1 Relative ErrorFor the relative error criterion, we first consider the problemassuming a fixed upper bound $ on the relative error. For agiven bound $, we test whether there is a feasible solution xwith relative error at most $ by solving a linear program. Wethen find the smallest $$, to within a specified precision, forwhich there is a feasible solution, using binary search on $.With precision %, this binary search for $$ involves solvingO!log!$$=%"" linear programs. The feasible solution x$ foundat bound $$ is then an optimal solution to inverse alignmentunder relative error.

Beyond the domain inequalities, the remaining inequal-ities in each linear program enforce that the relative error ofall examples is at most $. For each example Ai, and every

4 IEEE TRANSACTIONS ON COMPUTATIONAL BIOLOGY AND BIOINFORMATICS, VOL. 5, NO. 4, OCTOBER-DECEMBER 2008

alignment Bi of sequences Si, the linear program for $ hasan inequality

fx!Ai" % !1( $" fx!Bi": !3"

Notice that for a fixed value of $, this is a linear inequality inparameters x, since function fx is linear in x. Example Ai

satisfies all these inequalities iff the inequalitywithBi # B$ issatisfied, where B$ is an optimal-scoring alignment of Si

under parameters x. In other words, the inequalities are allsatisfied iff the score of Ai has relative error at most $ underfx. Finding the minimum $ for which this system ofinequalities has a feasible solution x corresponds to mini-mizing the maximum relative error of the example scores.

As an aside, one could attempt to minimize the averagerelative error across the examples by modifying thisapproach as follows: Each example Ai would have acorresponding error variable $i, and inequality (3) wouldbe modified by replacing the global constant $ with thevariable $i. The objective function would be to minimize1k

P1%i%k $i. Unfortunately, inequality (3) then becomes

quadratic in variable $i and the vector of variables x, whichis no longer solvable by linear programming.

The above linear program given by inequalities (1), (2),and (3) has an exponential number of inequalities of type (3),since for an exampleAi, the number of alignments Bi of Si isexponential in the lengths of the sequences [11]. Never-theless, this program can be solved in polynomial time [19]using a far-reaching result from linear programming theoryknown as the equivalence of optimization and separation[13]. This equivalence result states that one can solve a linearprogram in polynomial time iff one can solve the separationproblem for the linear program in polynomial time [12], [28],[18]. The separation problem is, given a possibly infeasiblevector ex of parameter values, to report an inequality fromthe linear program that is violated by ex, or to report that exsatisfies the linear program if there is no violated inequality.

We can solve the separation problem in polynomial timefor the above linear program by the following algorithm.Given a vector ex of parameter values, for each example Ai

we compute an optimal-scoring alignment B$ of Si under fusing ex. If inequality (3) is satisfied when Bi # B$, then theinequalities are satisfied for all Bi; on the other hand, ifinequality (3) is not satisfied for B$, this gives the requisiteviolated inequality. For a problem with k examples, solvingthe separation problem involves computing at most oneoptimal alignment for each example, for a total of at most koptimal alignments, which runs in polynomial time.

As a consequence, the equivalence of separation andoptimization implies that the full linear program can besolved in polynomial time. This should mainly be viewed,however, as an existence proof for a polynomial-timealgorithm for inverse alignment. The equivalence theoremrelies on the ellipsoid method for linear programming(which is slow in practice), and does not provide a goodbound on the polynomial for the running time of theresulting algorithm [13].

In practice, a polynomial-time solution to the separationproblem is typically leveraged in the following cuttingplane algorithm [4] for solving a linear program withinequalities L.

1. Start with a small subset P of the inequalities in L.

2. Compute an optimal solution ex to the linear programgiven by subset P. If no such solution exists, halt andreport that L is infeasible.

3. Call the separation algorithm for L on ex. If thealgorithm reports that ex satisfies L, output ex andhalt: ex is an optimal solution for L.

4. Otherwise, add to P the violated inequality returnedby the separation algorithm in step 3, and loop backto step 2.

While such cutting plane algorithms are not guaranteed toterminate in polynomial time, they can be fast in practice [19].

For inverse alignment, we start with subset P initializedto the domain inequalities (1) and (2), together with thenondegeneracy inequality (5) described later in Section 3.3.

3.1.2 Absolute ErrorFor the absolute error criterion,wemodify the linearprogramas follows: For each exampleAi, we have an additional errorvariable &i. Inequality (3) for each example Ai is replaced by

fx!Ai" % fx!Bi" ( &i: !4"

The objective function for the linear program is to minimize

1

k

X

1%i%k

&i:

Notice that minimizing this objective function forces each &ito have the value

&i # fx!Ai" & f$x!Si":

Thus, an optimal solution x$ to this linear program withinequalities (1), (2), and (4) gives a parameter vector thatminimizes the average absolute error of the example scores.Again the program has exponentially many inequalities oftype (4), but the same separation algorithm that computesan optimal alignment B$ solves the separation problem inpolynomial time, so in principle the linear program can besolved in polynomial time. In practice, we use a cuttingplane algorithm as described above.

3.2 Partial ExamplesInverse alignment from partial examples involves optimiz-ing over all possible completions of the examples. While forpartial examples we do not know how to efficiently find anoptimal solution, we present a practical iterative approachwhich as demonstrated in Section 4 finds a good solution.

We start with an initial completion A!0"i for each partial

example Ai. This initial completion may be formed byoptimally aligning each unreliable region of Ai using adefault parameter choice x!0"; we call this the default initialcompletion. (In practice, for x!0" we use a standardsubstitution matrix [17] with appropriate gap penalties.)Alternatively, an initial completion may be obtained bysimply using the complete alignment given by both theunreliable and reliable columns of the partial example; wecall this the trivial initial completion.

We then iterate the following process for j # 0; 1; 2; . . . .

Compute an optimal parameter choice x!j(1" by solving

the inverse alignment problem on the complete examples

KIM AND KECECIOGLU: LEARNING SCORING SCHEMES FOR SEQUENCE ALIGNMENT FROM PARTIAL EXAMPLES 5

A!j"i . Given x!j(1", form a new completion A!j(1"

i of each

example Ai by:

1. computing an alignment of each unreliable regionthat is optimal with respect to parameters x!j(1", and

2. concatenating these alignments of the unreliableregions, alternating with the alignments given by thereliable regions, to form a new complete exampleA!j(1"

i .

Such a completion optimally stitches together the reliableregions of the partial example, using the current estimatefor parameter values.

This iterative scheme alternates a step of finding optimalparameters with a step of finding optimal completions,yielding a sequence of parameters and completions:

A!0" 7! x!1" 7! A!1" 7! x!2" 7! ' ' ' :

As the following shows, the error of successive parameterestimates decreases monotonically.

Theorem 1 (Error Convergence for Partial Examples). Forthe iterative scheme for inverse alignment from partialexamples, denote the error in score for iteration j ) 1 by

ej :# E x!j"; A!j&1"! ";

where the right-hand side measures error criterion Eabs orErel for the given parameters on the given completions. Then,

e1 ) e2 ) e3 ) ' ' ' ) e$;

where e$ is the optimum error for inverse alignment frompartial examples Ai under criterion E.

Proof. Since A!j"i is an optimal-scoring completion of Ai

with respect to parameters x!j",

fx!j" A!j"i

! "% fx!j" A!j&1"

i

! ":

In other words, under parameters x!j", the new

completions A!j"score just as close to optimal as the

old completions A!j&1". This implies that with respect

to error,

E x!j"; A!j"! "% E x!j"; A!j&1"! "

;

whether we consider relative or absolute error. So for the

new completions A!j", error ej is achievable, as witnessed

by x!j". Since the optimum error forA!j", given by the new

parameters x!j(1", cannot be worse,

ej(1 % ej:

Furthermore, e$ lower bounds the error for allcompletions. tuTheorem 1 implies that the error of this iterative scheme

converges, as the error across iterations forms a nonincreas-ing sequence that is bounded from below. (The error mayconverge, however, to a value larger than the optimumerror e$.) As shown in Section 4, choosing a good initialcompletion can reduce the final error. In practice, we iterate

this scheme until the improvement in error becomes toosmall or a bound on the number of iterations is reached. Aserror improves across the iterations, recovery of theexamples generally improves as well.

3.3 Eliminating DegeneracyIn general, the preceding formulations of inverse alignment,for both relative and absolute error, when applied to globalsequence alignment have a family F of degenerate solutions:

!!ab; ";#" 2 F :##!2c; 0; c" : c ) 0

$:

Every parameter choice x 2 F from this family makes allglobal alignments of an example’s sequences have the samescore, namely, c!m( n" for two sequences of lengths mand n. Such an x makes each example be an optimalscoring alignment, hence such an x is an optimal solutionfor every instance of inverse alignment. Of course, thesedegenerate solutions are not of biological interest. Toeliminate them, we use the following approach.

Let nondegeneracy threshold ' be the difference betweenthe expected cost of a random substitution (of differentletters) and the expected cost of a random identity,measured for a default substitution scoring matrix. For asound scoring scheme that distinguishes between substitu-tions and identities, this difference should be positive. Thevalue of ' for the commonly used BLOSUM [17] and PAM [5]matrices for standard amino acid frequencies is around 0.4when substitution scores are translated and scaled to costsin the range [0, 1]. Values of ' for a wide range of BLOSUMmatrices are shown in the next section in Fig. 2.

Given a value for threshold ' , we add to the linearprogram the nondegeneracy inequality

Pa<b papb !abP

a<b papb&

Pa p2a !aaP

a p2a) ' ; !5"

where in the summations a and b range over all aminoacids, and pa is the probability of amino acid a appearing ina random protein sequence. Note that this is a linearinequality in parameters !ab and !aa.

Inequality (5) is equivalent to requiring that for thelearned parameters, the difference between the expectedscore of a random substitution and the expected score of arandom identity is at least ' . When ' is positive, whichholds for standard scoring schemes, this inequality cuts offthe degenerate solution !!ab; ";#" # !2c; 0; c", as the follow-ing theorem states. Furthermore, desirable solutions (suchas the scoring scheme from which ' was measured) are noteliminated.

Theorem 2 (Eliminating Degeneracy). When threshold ' > 0,inequality (5) eliminates all degenerate solutions x 2 F .

Proof. The degenerate solutions have !ab # !aa # 2c, sotheir expected substitution and identity scores are equal.The nondegeneracy inequality then reduces to ' % 0,contradicting ' > 0. In other words, every solution fromF violates inequality (5). tuIn essence, the nondegeneracy inequality forces the

substitution and identity scores in the optimal solution x$

to the linear program to be at least as nondegenerate as thedefault matrix from which ' was measured. In theexperiments described in the next section, we use the valueof ' corresponding to BLOSUM62, namely, ' # 0:4.

6 IEEE TRANSACTIONS ON COMPUTATIONAL BIOLOGY AND BIOINFORMATICS, VOL. 5, NO. 4, OCTOBER-DECEMBER 2008

We emphasize that the nondegeneracy inequality iscrucial. Without it, the inverse alignment algorithm forabsolute error immediately outputs the optimal but trivialsolution x # !0; . . . ; 0".

4 EXPERIMENTAL RESULTS

To evaluate the performance of this approach to inversealignment, we ran several types of experiments on biologicaldata. Our main interest is in comparing the new absoluteerror formulation to the prior relative error formulation, andin investigating whether parameters learned from pairwisealignment examples can improve the recovery of bench-marks when used for multiple sequence alignment. We alsoexamine how the initial completion and the nondegeneracythreshold affect the final recovery.1

To obtain the examples for our experiments, we usedbenchmarks from the PALI [2], [10] suite of structuralmultiple alignments of proteins. PALI contains a benchmarkmultiple alignment for every family from the SCOP [25], [26]classification of proteins, computed by structural alignmentwithout hand curation. In total, PALI has 1,655 benchmarkalignments, from which we selected a subset U consisting ofall PALI alignments on at least seven sequences withnontrivial gap structure. (If an alignment had essentiallyno gaps other than at the terminal ends of its sequences, itwasdeemedtohave trivialgapstructureandwasnot includedin set U .) Set U was further partitioned into two equal-size

subsets P and Q for the purpose of conducting training-set/test-set cross-validation experiments. We also performed adetailed study of a smaller subset S * U containing the25 benchmarks from U with the most sequences. For eachbenchmark in the smaller set S, where the benchmark is amultiple alignment on n sequences, we took for the examplesthe n

2

% &pairwise alignments inducedonall pairs of rowsof the

multiplealignment.For thebenchmarks inthe largersetsU ,P ,andQ, we took as the examples amuch sparser set of inducedpairwise alignments, as described later.

Table 1 summarizes the characteristics of these data sets.Each of the PALI benchmarks in these data sets consists ofpartial examples. The reliable regions in the partialexamples are given by the core blocks of the benchmark.(PALI explicitly identifies the core blocks in each bench-mark, which consist of those columns of the multiplealignment in which all pairs of residues are within 3 !A ontheir C( atoms in the structural multiple alignment.) Forsets U , P , Q, and S, the table reports the number ofbenchmarks in each set and, averaged across the bench-marks in the set, their number of sequences, their sequencelength, and the percent identity of their induced pairwisealignments. Also shown averaged over the core blocks ofthe benchmarks are their percent coverage of the sequencesand their percent identity.

Fig. 1 illustrates the improvement in error for the iterativeapproach to partial examples discussed in Section 3.2. Theplots show the average absolute error and the recovery rateacross the iterations, starting from a given initial comple-tion. The recovery rate is the percentage of columns fromreliable regions that are present in an optimal alignmentcomputed using the estimated parameters. Results areplotted for two initial completions: the default completion,which optimally aligns the unreliable regions using defaultparameters, and the trivial completion, which uses allcolumns of the partial alignment including unreliable ones.

KIM AND KECECIOGLU: LEARNING SCORING SCHEMES FOR SEQUENCE ALIGNMENT FROM PARTIAL EXAMPLES 7

TABLE 1Data Set Characteristics

Fig. 1. Improvement in error and recovery for the iterative approach to partial examples. (a) Plot of the error at each successive iteration. (b) Plot ofthe corresponding recovery for the parameters learned at that iteration. The curves in the plots start from two different initial completions: the trivialcompletion and the default completion (defined in the text). The examples are all induced pairwise alignments from PALI benchmark b.1.8.1.

1. Our experiments do not compare the linear programming approachfor inverse alignment to alternative machine learning approaches [7], [37].The CONTRAlign [6] tool that implements the work of Do et al. [7] does notdirectly produce a substitution matrix and pair of affine gap penalties forthe standard protein alignment scoring model, and reimplementing theirapproach to facilitate a direct comparison requires nonconvex numericaloptimization. The work of Yu et al. [37] is on sequence-to-structure proteinalignment, and reimplementing their approach to apply it to standardsequence-to-sequence alignment requires quadratic programming.

(The default parameters are those used by the multiplealignment tool Opal [35], [36], which are the BLOSUM62

substitution matrix with carefully chosen gap penalties.)The examples for the plot are all induced pairwisealignments of PALI benchmark b.1.8.1, and the errorcriterion is the average absolute error. As the figure shows,the error in alignment scores decreases across the iterations,and the corresponding recovery of the example alignmentstends to improve as well. (Also note that starting from thedefault completion has better recovery and less error thanstarting from the trivial completion.) Generally, smallererror correlates with higher recovery.

Fig. 2 shows that across a very wide range ofconventional substitution matrices, using the correspond-ing value of threshold ' in nondegeneracy inequality (5)does not adversely affect recovery. In other words, thechoice of which substitution matrix is used to establish ' isnot critical. (Nevertheless, enforcing the nondegeneracyinequality is crucial: without it, the inverse alignmentalgorithm immediately outputs an optimal degeneratesolution.) In the plot, the value i along the horizontal axisspecifies the index of a BLOSUM substitution matrix. TheBLOSUM matrices we consider have been transformed intocost matrices with extreme values of 0 and 1. The bottomcurve plots the value of ' measured on matrix BLOSUMi.(Note that BLOSUMi matrices that correspond to a higherpercent identity generally have greater discriminationbetween substitutions and identities, as evidenced by thebottom curve for ' tending to be increasing in i.) The topcurve plots recovery rates when the corresponding valueof ' is used in the nondegeneracy inequality. At everypoint on this curve, parameters are learned using the samecollection of pairwise alignment examples (set eU , which isdescribed later), and the average recovery rate on theseexamples is plotted.2 Across the range of BLOSUM matrices,' varies up to 40 percent, while there is little variation inrecovery. In short, recovery is robust to the particularvalue ' used in the nondegeneracy inequality.

Table 2 shows a detailed comparison of recovery ratesacross several different scenarios for inverse alignment. The

rows of the table correspond to the PALI benchmarks inset S, which are named by their SCOP identifier. (The rowsare sorted in increasing order on column “default (c),”which is described in more detail later.) For each bench-mark, the table reports the average recovery rate across itsexamples, which consist of all induced pairwise alignmentsof the benchmark, for various scenarios. In these scenarios,parameters learned from the examples for a given bench-mark are applied to the sequences in this benchmark, eitherto compute an optimal global pairwise alignment of theexample sequences (corresponding to the first group ofcolumns), or to compute a global multiple alignment of thebenchmark sequences using the tool Opal [35], [36](corresponding to the second group of columns). Opal isa multiple sequence alignment tool, based on optimallyaligning alignments [20], [30], that seeks to optimize thesum-of-pairs objective [3]. In the first group of five columns,the table reports the recovery rates on the examples foroptimal pairwise alignments computed using:

1. the default parameters in Opal, namely, theBLOSUM62 matrix [17] with carefully chosen affinegap penalties [35],

2. the BLOSUM62 matrix with affine gap penaltieslearned using the absolute error criterion,

3. the substitution matrix and gap penalties learnedusing the relative error criterion,

4. the substitution matrix and gap penalties learnedusing absolute error, and

5. the difference between the recovery rates of theabsolute error and default parameters.

In the second group of columns, the table reports therecovery rates of a multiple sequence alignment of thebenchmark sequences computed by Opal using:

6. the default parameters of Opal,7. the substitution matrix and penalties learned under

absolute error, and8. the difference between these two.

In these experiments, the precision of the binary search inthe relative error criterion is 0.01 percent. Parameters foralignment are rounded to integers using a scale of 100.

A key conclusion from examining Table 2 is that theabsolute error criterion substantially outperforms the relativeerror criterion with respect to recovery of example align-ments (seen by comparing the table columns labeledabsolute and relative in the pairwise alignment group). Inaddition, from inspecting the two difference columns in thetable (whose maximum entries are marked by a left arrow),parameters learned under absolute error when used forboth global pairwise and multiple alignment outperformthe default parameters of Opal, which uses the standardBLOSUM62 matrix, by as much as 25 percent in recovery.(This particular matrix had the best recovery among theBLOSUM family.) Also note that the recovery rates ofparameters when used for multiple alignment are generallymuch higher than when used for pairwise alignment. Tosummarize, by performing inverse alignment from partialexamples one can learn parameters for global pairwise ormultiple sequence alignment that are tailored to a givenprotein family and that yield very high recovery.

8 IEEE TRANSACTIONS ON COMPUTATIONAL BIOLOGY AND BIOINFORMATICS, VOL. 5, NO. 4, OCTOBER-DECEMBER 2008

Fig. 2. Robustness of the recovery rate to nondegeneracy threshold ' .The bottom curve plots threshold ' measured on substitution matrixBLOSUMi at each i. The top curve plots the corresponding recovery ratewhen the value of ' from BLOSUMi is used in the nondegeneracyinequality. Parameters are learned and recovery is measured using theexamples in set eU (defined in the text). As shown, recovery is robust tothe particular matrix used to establish ' .

2. The recovery rate in this plot is for computing pairwise alignments ofthe example sequences. Later, Table 3 gives the recovery rate whencomputing multiple alignments of benchmark sequences, which issubstantially higher.

Finally, Table 3 presents recovery results from cross-validation experiments. Parameters learned on sparsetraining sets using the absolute error criterion are appliedto complete test sets. The recovery rate for these parametersis measured on multiple alignments of the benchmarkscomputed by Opal. In the table, parameters learned ontraining sets eU , eP , and eQ using the absolute error criterionare applied to test sets U , P , and Q. For a generic set X ofPALI benchmarks, the examples in training set eX are asubset of the induced pairwise alignments of the bench-marks in X. Set eX contains pairwise alignments selectedaccording to their recovery rate in an initial multiplealignment of the benchmark computed by Opal usingits default parameters. For eP and eQ, the subset contains theinduced pairwise alignments from each benchmark whoseinitial recovery rates rank at the 25th, 50th, and75th percentiles; for eU , the examples rank at the 33rd and66th percentiles. Parameters from a given training set wereused in Opal to compute multiple alignments of thebenchmarks in the test set, and the table reports the averagerecovery.

Note that there is only a small difference in recoverywhenparameters are applied for multiple sequence alignment to

disjoint test sets, compared to their recovery on their trainingset. This suggests that the absolute error method is notoverfitting the parameters to the training data.

To give a sense of running time, performing inversealignment on training set eU of more than 200 examplesinvolved around 6 iterations for completing partial exam-ples and took about 4 hours total on a 3.2 GHz Pentium 4processor with 1 Gbyte of RAM, using GLPK [23] to solvethe linear programs. An iteration took roughly 40 minutesand required around 4,000 cutting planes.

5 CONCLUSION

We have explored a new approach to inverse parametricsequence alignment that for the first time carefully treatspartial examples. This new approach minimizes the averageabsolute error of alignment scores, and iterates overcompletions of partial examples. We also studied for thefirst time the performance of the resulting scoring schemeswhen used for multiple sequence alignment. A keyconclusion from this study is that the new absolute errorcriterion outperforms the prior relative error criterion interms of recovery rate. Our results also indicate that asubstantial improvement in alignment accuracy can beachieved on individual protein families, and that scoringschemes learned from a sample of families generalize wellto other protein families.

5.1 Further ResearchThere are many directions for further research. Given thatwe can solve instances of inverse alignment with hundredsof parameters, is it possible to significantly increase theaccuracy of protein sequence alignment through a moresophisticated scoring model with additional parametersthat incorporate features such as amino acid hydrophobicity

KIM AND KECECIOGLU: LEARNING SCORING SCHEMES FOR SEQUENCE ALIGNMENT FROM PARTIAL EXAMPLES 9

TABLE 2Recovery Rates on Data Set S for Variations of Inverse Alignment

TABLE 3Recovery Rates for Cross-Validation Experiments

[7], [32], [8] and predicted secondary structure [38], [29],

[22]? Can accuracy be boosted by modifying inverse

alignment to directly incorporate example recovery [37] into

the formulation? Is parameter generalization benefited by

including a regularization term in the objective function [7],

[37], for instance by penalizing parameter overfitting

through the L1 norm to retain a linear programming

formulation? There remains much to investigate.

ACKNOWLEDGMENTS

The authors wish to thank Chuong Do for helpful discus-

sions, and Travis Wheeler for assistance with using Opal

[36]. We also thank the anonymous referees for suggestions

that improved the paper. This researchwas supported by the

US National Science Foundation through Grant DBI-

0317498. An earlier version of this paper appeared in [21].

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10 IEEE TRANSACTIONS ON COMPUTATIONAL BIOLOGY AND BIOINFORMATICS, VOL. 5, NO. 4, OCTOBER-DECEMBER 2008

Eagu Kim received the MS degree in computerscience from Yonsei University, Korea, in 1998,with a thesis in cryptography. He is a PhDstudent in the area of algorithms for computa-tional biology in the Department of ComputerScience, University of Arizona.

John Kececioglu received the PhD degree incomputer science from the University of Arizonain 1991. He did postdoctoral study at theUniversite de Montreal and the University ofCalifornia, Davis, then taught at the University ofGeorgia, before joining the Department ofComputer Science, University of Arizona in2000, where he is an associate professor ofcomputer science. He is the recipient of a USNational Science Foundation CAREER Award,

has served on the Scientific Advisory Board of the Max-Planck-Institutfur Informatik, and serves on the Editorial Board of Algorithms forMolecular Biology. His research is in the design, analysis, andimplementation of algorithms for computational biology.

. For more information on this or any other computing topic,please visit our Digital Library at www.computer.org/publications/dlib.

KIM AND KECECIOGLU: LEARNING SCORING SCHEMES FOR SEQUENCE ALIGNMENT FROM PARTIAL EXAMPLES 11