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Application of evolutionary algorithms to protein folding prediction A. Piccolboni and G. Mauri Dipartimento di Scienze dell'Informazione Universita di Milano Via Comelico 39/41, 20135 Milano, Italy

Abstract. The aim of this paper is to show how evolutionary algorithms

can be applied to protein folding prediction. We start reviewing previous similar approaches, that we criticize emphasizing the key issue of representation. A new evolutionary algorithm is described, based on the notion of distance matrix representation, together with a software package that implements it. Finally, experimental results are discussed.

1 Protein folding prediction Proteins are molecules of extraordinary relevance for live beings. They are chains of aminoacids (called also residues) that assume very rich and involved shapes in vivo. The prediction of protein tertiary structure (3D shape) from primary structure (sequence of aminoacids) is a daunting as well as a fundamental task in molecular biology. A large amount of experimental data is available as far as it concerns sequence, and large projects are creating huge amounts of sequence data. But to infer the biological function (the ultimate goal for molecular biology) from sequence we have to pass through tertiary structure, and how to accomplish this is still an open problem (ab initio prediction). Indeed, experimental resolution of structures is a dicult, costly and error-prone process. The prediction problem can be recast as the optimization of the energy function of a protein, under the assumption that an accurate enough approximation of this function is available. Indeed, according to the so-called An nsen hypothesis [AHSJ61], native conformations correspond, at a rst approximation, to global minima of this function.

2 Evolutionary algorithms Evolutionary algorithms (EAs) [Hol75, De 75, FOW66, BS93] are optimization methods based on an evolutionary metaphor that showed eective in solving dicult problems. Distinctive features of EAs are:

{ a set of candidate solutions is considered at each time step instead of a single

one (population); { candidate solutions are combined to form new ones (mating operator);

{ solutions can be randomly slightly modi ed (mutation operator); { better solutions according to the optimization criterion ( tness) are given more reproductive trials.

These basic principles result in an overall population dynamics that can be roughly described as the spreading of good features throughout the population. This naive idea is made more precise in the so called \schema theorem" [Hol75]. According to [Gol90], the best performances can be obtained when hyperplanes in the solution space with under average energy exist (building blocks). According to the more general de nition found in [Vos91], a building block is a property of solutions which is inherited with high probability by osprings, i.e. which is (almost) preserved after crossover and mutation and is rewarding from the point of view of tness. In the following we will refer also to genetic algorithms (GAs) which are a special case of EAs.

3 Previous work The in uence of representation on the dynamics and eectiveness of EAs has been already recognized [BBM94]. Three main representation techniques have been proposed for protein structures:

Cartesian coordinates are unsuitable for a population based algorithm, since

basically identical structures (up to a roto-translation) can have completely dierent coordinates; internal coordinates de ne aminoacid positions w.r.t. neighboring aminoacids, specifying distances and angles; this is the choice of all genetic approaches to protein folding so far; distance geometry describes a structure by means of the matrix of all the distances between every couple of points and has been proposed as a tool for energy minimization since [NS77]; our main contribution is its joint use together with EAs. To the best of our knowledge, all evolutionary approaches to folding prediction so far have been based on an internal coordinate representation. It is straightforward to show that relevant structural features can not be described as hyperplanes under this approach. Schulze-Kremer [SK93, SK95] de nes a real-coded GA in internal coordinate space and a simpli ed energy function, but fails ab initio prediction for a test protein. His algorithm proves useful for side chain placement. Unger and Moult [UM93] compare a GA against Monte Carlo methods using an idealized 2D lattice model and simpli ed internal coordinates. Large performance gains over Monte-Carlo are achieved but no comparison is possible with real proteins (Patton et al. [PPG95] report an improvement to this approach and we will compare our results to theirs).

Dandekar and Argos [DA94] use a standard GA with an heuristic, heavily tailored tness and an internal coordinate discretized representation. Results are encouraging, but the generality of the method is questionable. Herrmann and Suhai [HS95] use a standard genetic algorithm in internal coordinate space together with local search and a detailed model. It proved interesting only for very small structures. A simple observation against internal coordinate representation is the following: typical force elds are a sum of terms like relaxation distances or relaxation angles that are convex functions of pairwise distances. It is likely that structures minimizing these relaxation terms will have a lower energy and optimal structures have a lot of relaxation terms close to zero. So distances and angles can act as building blocks for genetic search. But internal coordinates include only distances between neighboring residues in the sequence. The distance between an arbitrary pair of residues can be calculated only using a complex formula involving coordinates of all the residues that appear between the two in the sequence, i.e., in the worst case, the whole representation. The same is true for angles. It is very dicult, thus, to describe useful schemas in internal coordinate space and guarantee some minimal properties such as stability [Vos91], low epistasis [BBM93] and others that can not guarantee the success of a GA but are believed to be key ingredients of it [Gol90]. On the contrary, we show that some of these properties hold for suitable genetic operators in distance matrix space.

4 Energy function We tried to keep our model as simple as possible. We model each residue as a unique point and according to [Dil85] we consider the hydrophobic/hydrophilic interaction as the dominant force that drives folding. Under this assumption, protein instances can be seen as sequences of either hydrophobic or hydrophilic aminoacids. Thus our energy function Etot is made up of three terms only Etot = Erep + Echn + Ehyd that we will describe brie y. Erep is a \solid body" constraint penalty term, that prevents two residues from occupying the same place. Namely we have NX ?2 X N

g(dij ) i=1 j =i+2 is the length of the protein, krep a suitable constant, dij is the distance Erep = krep

where N between residues i and j and g is de ned as  1 g(x) = 0x ? 2 + x x0 x1  1 Echn is a chain constraint penalty term, that forces neighboring residues in the sequence to lay spatially close, whose form is

Echn = kchn

NX ?1 i=1

(di;i+1 ? 1)2

where kchn is a constant. Finally, Ehyd is an hydrophobic interaction term, rewarding closeness between hydrophobic aminoacids and is de ned as

Ehyd = khyd

NX ?2 X N i=1 j =i+2

h(dij )

where khyd is a constant and 1 2 residue i and j are both hydrophobic h(x) = 0log((x ? 1) ) + e ) ifotherwise



Although the exact form of the energy is subject to wide variations [Neu93], this energy function models at least qualitatively the most important structural properties of proteins. From an evolutionary algorithm point of view this function satis es the ideal condition of no epistasis [BBM93], since a change in a variable always produces the same change in energy, despite the values of all other variables. This is the main advantage of a distance based representation.

5 Distance matrix representation We describe a dierent approach to folding prediction with EAs. The solution space is the set of aminoacid distance matrices, that is the set of N  N symmetric, zero diagonal matrices with positive entries representing pairwise distances between aminoacids. Unfortunately, it turns out that such a representation describes a superset of possible con gurations. To cope with this fact and, in general, to deal with such a kind of representation we need some concepts belonging to distance geometry [HKC83].

De nition 1 A distance matrix D = fdij g is embeddable in Rn if there exist a set of N points in Rn , C = fci g s.t. jjci ? cj jj = dij and such a C is called an embedding of D. We use the same notation for the set of points C = fci g and for the n  N matrix with fci g as columns. De nition 2 Given a matrix D, its Gram matrix is M = fmij g, with mij = di + dj ? dij De nition 3 Given a set of points C, its metric matrix is CT C. Theorem 1. In an inner product space, if C is an embedding of D and M is the Gram matrix of D, then M is equal to the metric matrix of C. 2 1

2 1

2

In our context this equality will always hold. From an algorithmic point of view, C is computable from M by Cholesky factorization [Van92]. Theorem 2. A matrix D is embeddable in Rn i its Gram matrix M is positive semide nite of rank at most n. We observe that, since M is N  N its rank can be at most N . We have obtained thus a very elegant way to de ne our solution space: it is the set of symmetric, zero diagonal matrices with positive entries whose metric matrix is positive semide nite of rank at most 3.

6 EA speci cation According to [MS96], among dierent constraint handling strategies for EAs we explored these two possibilities: { whenever an unfeasible solution is produced \repair" it, i.e. nd a feasible solution "close" to the unfeasible one (repair strategy ); { admit unfeasible individuals in the population, but penalize them adding a suitable term to the energy (penalize strategy ).

6.1 The repair algorithm

We are given a distance matrix D which is, for some reason, unfeasible and we want to nd a feasible solution \close" to it, where \close" is to be speci ed. First of all, we may safely assume that D is symmetric, zero diagonal with positive entries, since, as we will see it, is easy to enforce such properties in the population, whatever kind of GA we are de ning. Next, we turn our attention to positive semide niteness. We can test this property in polynomial time by evaluating the smallest eigenvalue (it is positive i the matrix is positive semide nite), but in case it isn't veri ed there's nothing we can do (apart from rejecting the solution). This shortcoming is common to other similar algorithms in the literature [HKC83] and we are currently investigating this issue. Our guess is that without positive semide niteness the problem of embedding a distance matrix could be much harder. Let us suppose we are given a symmetric, zero diagonal matrix D with positive entries whose metric matrix M is positive semide nite of rank n > 3, so that the condition on rank only is not satis ed. The repair algorithm proceeds as follows (it is a modi cation and generalization of the one in [LLR95]): 1. nd a coordinate set C in Rn, n  N by Cholesky factorization; 2. compute the projection of C (with distance matrix D0 = fd0ij g) onto a random hyperplane P through the origin; call it C0 ; p 3. multiply C0 element-wise by n=3 (in the following we will consider, for the sake of generality, an m-dimensional hyperplane and a multiplicative factor p of n=m).

We have the following Theorem 3. With probability greater than 1=e, for each i; j 2



p dij ? d0ij2 n log N )  O( 2 dij

Proof. (It is a generalization of the one in [Tre96]) Let us introduce the random variable (call it distortion ) 2

and the quantities



d ? d02 X = ij d2 ij ij

yk = jjc0ik ? c0jk jj2

(we can temporarily forget about the dependence of yk from i and j since what we are going to prove with yk 's is true for every i; j ). Without loss of generality we may assume that P is a coordinate hyperplane (whose dimension is m to be more general, but we are interested in m = 3). This is because we are interested in properties of distances, which are invariant under rotation. Since

c0ik =

r

n m cih

for some h, depending only on P , we have that m nX d0ij2 = m yI

whereas

k=1

d2ij =

n

X

k=1

k

yk

where Ik are the indices of dimensions parallel to P , so that d0ij2 is a random variable which is the sum of a subset of size m of fyk g multiplied by n=m. Let so that

n yI ? 1 Pn yi m Yk = k Pnm y i=1 y=1 i

X=

X

K

= 1m Yk



Straightforward calculations show that E [Yk ] = 0 and ?1  Yk  1. We are ready to apply Hoeding inequality [Hoe63] to obtain   p P X  21 n(log 2 + log(N 2 + 1))  1 ? N 21+ 1 The probability of the same bound to hold for all N 2 distances at once is greater or equal than (1 ? N 21+1 )N 2 which is greater than 1=e. Q.E.D.

6.2 The penalization term

We would like to be able to measure the \level of unfeasibility" of a solution D. As with the repair algorithm, we take for granted that D has positive entries and zero diagonal. Next, in case D metric matrix is not positive semide nite, we assign D a conventional, very high penalization. Thus we have to deal only with positive semide nite matrices. If we measure the dierence between two con gurations C and C0 (the rst one of higher dimension than the second) by the Frobenius norm of their dierence

jjC ? C0 jjF =

v u uX t

N

ij

cij ? c0ij

we have the following theorem (adapted from [HKC83]) Theorem 4. If C has rank n the matrix C0 of rank m  n that minimizes

jjC ? C0 jjF

is obtained as follows 1. compute C metric matrix M; 2. decompose M as M = YT Y, where Y is unitary and  is Diag(1 ; : : : ; N ) with 1 ; : : : ; N the eigenvalues of M in decreasing order; 3. nally, C0 = Diag(12 ; : : : ; m2 ; 0; : : : ; 0)Y Moreover, we observe that the minimum of jjC ? C0 jjF so attained is 1

1

v u u X t

N

i=m+1

i

Therefore, this is (with m = 3) a good candidate as a penalization term. Since we would like it to be invariant to scale changes, we normalized it to obtain q P

k

N i=m+1 i PN i=1 i

q

where k is a weighting factor to be described later on.

6.3 Genetic operators

The EA can be further speci ed with the de nition of genetic operators. We de ned and experimented with a number of dierent recombination operators and we studied the stability of schemata, the feasibility of solutions produced by each of them and their behavior w.r.t. the repair algorithm. The following de nitions are a generalization of the ones in [Vos91].

De nition 4 A schema is a subset of the solution space. De nition 5 A schema H is said to be stable under some genetic operator G if whenever G is applied to individuals all belonging to H then every generated ospring belongs to H .

Every operator is supposed to have as input one or two zero diagonal symmetric matrices (parents) with positive entries and outputs, as it is easy to check, a matrix with the same properties. The rst genetic operator is the customary uniform crossover, i.e. let D0 and 00 D be parents of D, then for each i; j independently P (dij = d0ij ) = 21 P (dij = d00ij ) = 21 It easy to show that every coordinate hyperplane in matrix space is stable under this operator. The resulting matrix is not guaranteed to have a positive semidefinite metric matrix M, so that the new individual has to be rejected or penalized, depending on constraint handling strategy. Moreover, even when M is positive semide nite, it can have full rank also when parents have rank 3, so that the upper bound on distortion in theorem 3 can be rewritten as 2



p dij ? d0ij2 N log N )  O( 2 dij

This can be better evaluated considering that, in general, in a feasible structure D 1  dij  N and in a maximally compact 1structure (and low energy structures are usually very compact) 1  dij  O(N 3 ). We found experimentally that using a similar operator but with probability biased toward one of the parents (19=20) positive semide niteness is often preserved and distortion is lower. We call this biased operator graft crossover. See table 1 for an experimental analysis of this operator. The third operator (called arithmetic crossover ) is a convex combination of element-wise squared matrices i.e.

d2ij = d0ij2 + (1 ? )d00ij2 with  uniformly distributed between 0 and 1. It is straightforward to show that every convex set in matrix space is stable under this operator, so that it has a very rich set of stable schemata, that includes, for example, coordinate hyperplanes and general hyperplanes. As to what concerns feasibility, we have the following Theorem 5. [HKC83] The set of all matrices of squared distances is convex in zero-diagonal matrix space. Furthermore, the convex combination of two matrices of squared distances embeddable in k an l dimensions respectively is embeddable in at most k + l dimensions.

This means that when adopting the repair strategy we are guaranteed that the generated individual has a positive semide nite metric matrix of rank at most 6, since parents are guaranteed to be feasible. This turns out in a much lower distortion, as we can see specializing theorem 3 to this case as follows 2



p dij ? d0ij2 log N )  O( 2 dij

A fourth operator (block crossover ) tries to emulate the eects of multi-point crossover applied to internal coordinates, as in previous approaches cited before. It is akin to uniform crossover with submatrices instead of elements as basic entities. It works as follows 1. The set of aminoacids is partitioned in intervals I1 ; :::; Ik , according to their order in the aminoacidic sequence; 2. Both parents' distance matrices (D0 and D00 ) are partitioned in submatrices D0ij = fd0hk gh2I ;k2I D00ij = fd00hk gh2I ;k2I i

i

j

j

3. For each i; j choose a parent p(i; j ) with probability 1=2; 4. De ne a new solution as p(ij ) g dhk = fdhk h2I ;k2I i

j

The analysis of this operator is basically the same as the one for uniform crossover. In table 1 the four crossover operators are analyzed experimentally on a sample population of 27 residue long individuals and size 1000, generated according to the strategy explained in 6.3. For 1000 individuals generated by crossover we report the ratio of individuals with positive semide nite metric matrix, the ratio of individuals with energy lower, intermediate or higher than the two parents. crossover pos. sem. arithmetic 100.0 block 45.4 graft 65.6 uniform 33.1

lower 23.3 7.3 6.3 1.2

interm. 49.5 19.2 17.6 5.3

higher 27.2 73.5 76.1 93.5

Table 1. Statistical analysis of crossover operators The mutation operator is a classic creep operator that randomly choices a variable and modi es it with the addition of a small gaussianly distributed

amount ", preserving distance symmetry. The positivity of entries is preserved by suitable truncation of the probability distribution. Since the corresponding creep in Gram matrix eigenvalues is O(") [Van92], mutation can be tuned so that feasibility is almost preserved. Small negative eigenvalues are rounded to zero. Further details of the EA can be summarized as follows: { we use a steady-state algorithm (osprings are inserted in the population as soon as they are created); { reproduction and replacement candidates are selected out of a small pool (tournament selection ), according to tness and diversity criteria (individual less t and more similar to a new ospring are more likely to be replaced). { each individual in the initial population is generated in the following way: the rst aminoacid is at the origin; the position of the ith aminoacid is chosen uniformly at random on the unit sphere centered around the (i ? 1)th aminoacid.

7 Implementation issues The simulation program has been coded in C++, using the mathematical library Lapack++ and the genetic algorithm library GAlib. A graphical front end for simulation control and basic data analysis has been written in Perl and Tcl/Tk and using the visualization program Rasmol. Simulation have been run on a IBM-compatible with Pentium processor under Linux operating system, but a porting to a four processor Silicon Graphics Origin 2000 server is under way.

8 Simulation results We compare our results to those found in [PPG95], but the reader should be warned that the comparison is somewhat unfair, since in that paper a lattice model is used, so that a residue neighborhood can accommodate only 6 residues, instead of 12 [CS92] as in our continuous model. Moreover, in the same paper the energy is the opposite of the number of hydrophobic residue pairs at unit distance, so we tried to evaluate our structures using the same criterion, but introducing a little tolerance. In table 2 we summarize our results on three test cases taken from [PPG95] using the repair strategy, comparing dierent crossover operators and tournament sizes. Column C1 reports the number of hydrophobic pairs at distance 1  0:2, C2 at distance 1  0:1. The corresponding results in [PPG95] are 15 for P27-4, 11 for P27-6 and 13 for P27-7. In gure 1 the best structure obtained for P27-4 is shown. The compact hydrophobic core is very protein-like. The simulations ran with a population of 1000 individuals and the algorithm stopped after the generation of 200,000

sequence crossover arithmetic P27-4 block graft uniform arithmetic P27-6 block graft uniform arithmetic P27-7 block graft uniform

Tournament size 10 Tournament size 4 C1 C2 energy C1 C2 energy 20 20 24.211 18 15 19.655 35 28 -23.438 31 27 -19.484 35 31 -23.051 36 30 -28.068 22 19 22.071 25 22 5.267 10 7 16.592 10 8 15.730 17 15 -4.172 17 15 -4.511 13 10 0.961 16 11 0.459 12 8 6.580 13 9 8.270 20 15 -2.190 16 12 12.719 25 21 -5.368 21 19 -4.231 23 19 -1.343 21 17 2.959 22 19 -4.640 15 12 13.631

Table 2. Simulation results for \repair" strategy

Fig. 1. Best structure for P27-4 osprings, taking about 21 minutes. The probability of applying the crossover and mutation operators were set to 0.9 and 0.5 respectively. Simulation with the penalize strategy gave unsatisfactory results. We tried dierent penalization strategies, with xed or time varying weight factor. No run stopped producing a feasible solution. This can be explained noting that all genetic operators described tend to increase the rank of the metric matrix of new individuals.

9 Discussion and further work The ultimate test for any protein folding prediction algorithm is the prediction of experimental structures of biological sequences. Of course the EA described in the present work is not ready for such a task, at least because the very simple model prevents accurate predictions. Anyway it compares favorably with previous approaches and can be straightforwardly extended to more complex and realistic models, something that lattice model oriented algorithms can't. The experimental analysis of this algorithm is clearly in its preliminary stage and we plan to test longer sequences as soon as possible. Moreover we would like to clarify the relative strengths and weaknesses of dierent crossover operators. A rst distinction can be made between arithmetic crossover and the others, since this operator has some nice property w.r.t rank constraints but doesn't explore thoroughly the search space. This observation should be made more precise and supported with analytic or experimental evidence.

Acknowledgments This work has been partly supported by MURST project \Ecienza di Algoritmi e Progetto di Sistemi Informativi" and by CNR grant 97.02399.CT12.

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