Rearrangement of DNA fragments: a branch-and-cut algorithm Abstract. In this paper we consider a problem that arises in the process of reconstruction

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1 Rearrangement of DNA fragments: a branch-and-cut algorithm 1 C. E. Ferreira 1 C. C. de Souza 2 Y. Wakabayashi 1 1 Instituto de Mat. e Estatstica 2 Instituto de Computac~ao Universidade de S~ao Paulo [cef,yw]@ime.usp.br Universidade Estadual de Campinas cid@dcc.unicamp.br Abstract In this paper we consider a problem that arises in the process of reconstruction of DNA fragments. We present a graph theoretical formulation of the problem and mention some extensions. We show this problem to be N P-hard. A 0-1 integer linear programming formulation of the problem is given and some preliminary results of a branch-and-cut algorithm based on this formulation are discussed. Keywords: Integer Programming, Computational Biology, Branch-and-cut, Polyhedral Combinatorics 1 We thank Professor Jo~ao Meidanis (Unicamp) for bringing this problem to our attention. This research has been partially supported by CNPq (Project ProComb of ProTeM{II{CC; Proc /94-6). 1

2 Rearrangement of DNA fragments: a branch-and-cut algorithm Abstract. In this paper we consider a problem that arises in the process of reconstruction of DNA fragments. We present a graph theoretical formulation of the problem and mention some extensions. We show this problem to be N P-hard. A 0-1 integer linear programming formulation of the problem is given and some preliminary results of a branch-and-cut algorithm based on this formulation are discussed. Let be a nite alphabet and ` a positive integer. If s is a string of characters over, then we denote by `-last(s), resp. `-rst(s), the substring of s consisting of the last, resp. rst, ` characters of s. For each positive integer k, we dene the Minimum k-contig Problem, denoted by MkCP, as follows. We are given a collection C of strings of characters over an alphabet and we wish to concatenate the given strings so as to obtain another collection C 0 consisting of a smallest possible number of larger strings, called k-contigs. A k-contig z in C 0 is a string obtained by an overlapping concatenation of a sequence hs 1 ; s 2 ; : : : ; s m i of strings in C, with the property that for any two consecutive strings, say s j and s j+1, there exists an integer `j k such that `j-last(s j )= `j-rst(s j+1 ). We say that a k-contig z is an `1`2 : : : `m overlapping concatenation of hs 1 ; s 2 ; : : : ; s m i if z is a string of the form s 0 1 x 1s 0 2 x 2s 0 3 : : : s0 m?1 x m?1s 0 m where s 0 j x j = s j, x j s 0 j+1 = s j+1 and x j has length `j, for j = 1; : : : ; m? 1. The sequence hs 1 ; s 2 ; : : : ; s m i is called the skeleton of z. We say that each string s j is covered by z. Furthermore, if a string u 2 C is a substring of a string v 2 C, and v is in the skeleton of a k-contig z, we also say that u is covered by z. We require each string in C to be covered exactly once by exactly one k-contig in C 0. Thus the goal is to combine properly the strings in C to form a new collection C 0 with a minimum number of k-contigs. This problem occurs in the reconstruction of DNA fragments. A usual strategy to determine the sequence of the basis in a DNA molecule (which can be seen as a string over the alphabet fa, T, C, Gg) can be described as follows. First, many copies of this molecule are produced and, afterwards, these copies are (by means of chemical substances) broken into several small pieces that we are able to handle. Then, the problem is how to \glue" the small pieces in the correct way to reconstruct the original sequence? (see [M92]). The MkCP arises in this context. The idea is to obtain { for a given positive integer k and a collection C of pieces { a collection C 0 of k-contigs in such a way that the original molecule is completely covered by the k-contigs in C 0. A collection C 0 with the smallest 2

3 possible cardinality gives a best possible approximation to the original molecule. Example 1.1 : Let k = 2 and C consist of the following strings: 1. CCTAATGCTT 5. TGTTTAGCCTG 9. TGCGTTTTGTGC 2. TGTTTAGCCTGCGT 6. CCTGCGTTTTGTGCC 10. GACGTAGACA 3. CTGTGTTTAGCCT 7. TTTTGTCCAT 4. GTAGACAACCCTGTG 8. TTTTGTCCATC An optimal solution for the corresponding minimum 2-contig problem consists of a single 2-contig GACGTAGACAACCCTGTGTTTAGCCT : : : CTAATGCTTTTGTCCATC whose skeleton is the sequence h10; 4; 3; 2; 6; 1; 8i. Observe that string 5 is a substring of 2, string 7 a substring of 8, and string 9 is a substring of 6. 1 A model using graph theory Consider an instance of the MkCP consisting of a collection C of strings and a positive integer k 1. Construct a directed graph G = (V; A) as follows: each node i 2 V corresponds to a string s i 2 C, and there is an arc (i; j) 2 A if and only if there exists ` k such that `-last (s i )= `-rst (s j ). It is easy to see that any directed path in G corresponds to the skeleton of a k-contig. Nodes corresponding to strings that are substrings of others are called Steiner nodes, and need not necessarily be covered by a collection of paths. The other nodes of the graph are called terminals. Thus a smallest set of node-disjoint directed paths covering the terminals of G gives a solution for the given instance of the MkCP. A similar formulation to the problem has been proposed by Kececioglu [K91], [KM95], the main dierence lying in the treatment of the Steiner nodes. In the practical applications we are interested in, many dierent problems occur when the resulting substrings are considered. We know that the DNA-molecule can be seen as two parallel strings over fa; C; G; T g, where the second string is called the reverse complement. After a molecule is broken into pieces one cannot distinguish whether a piece is from the rst string or from the reversed one. We have extended the rst model to handle also this problem { the extended model is shown in the full paper. There are several other versions of the problem which could be of interest. For instance, we may allow that a string in C can be used more than once to form k-contigs. 3 The

4 graph model is the same but now we are looking for a collection of node-disjoint walks (or trails) covering the nodes of the graph. Steiner nodes are also admitted. Other versions allow concatenation of strings which dier by at most characters in their nal and initial positions provided that this occurs in a substring of size at least k. This version is interesting in the practical applications since some errors may occur during the handling of the substrings. A model for this version can also be incorporated in our approach. In this paper we concentrate our attention in both versions, with and without considering reverse complements. 2 N P-completeness of MkCP The version of the problem we are considering here is N P-hard. We prove that, for each xed k 1, the corresponding decision version of the MkCP is N P-complete, even if no Steiner node is allowed. This is shown, by reducing to the MkCP, the problem of nding a minimum covering of the nodes of a directed graph with maximum degree 3 by a collection of node disjoint paths (which is known to be N P-hard (cf [GJ79]). 3 An Integer Programming Formulation for MkCP In this section we show an integer programming formulation for the MkCP. The formulation is based on ow techniques. Consider rst the directed graph G = (V; A) dened in the last section. Now construct a new graph by adding to G two new nodes, a source s and a sink t, and arcs linking s to each node in V and arcs linking each node in V to t. Call A the newly obtained set of arcs, and let Z be the terminal nodes of G. Let us introduce for each arc (i; j) 2 A a variable x ij with the following interpretation: x ij = 1 if the arc (i; j) is in a path and x ij = 0, otherwise. 4

5 Thus, a 0-1 ILP formulation for the problem is given by z = (P ) max e2(s) x e ; j : (j;i)2a j : (i;j)2a j : (i;j)2a e2e(c) x ji = j : (i;j)2a x ij ; for all i 2 V; (1) x ij = 1; for all i 2 Z; (2) x ij 1; for all i 2 V n Z; (3) x e jcj? 1; for all ; 6= C V; (4) x ij 2 f0; 1g; for all (i; j) 2 A; (5) where E(C) := f(i; j) 2 Aj i 2 C and j 2 Cg, and (u) := f(u; v) 2 Ag. The inequalities (2) and (3) guarantee that the terminals must be covered, and the Steiner nodes may be covered by at most one path. The inequalities (4) eliminate the possibility of choosing arcs that \induce" a cycle going through a set of nodes. It is not dicult to check that a 0-1 vector x satises (1) to (5) if and only if the set of arcs a 2 A with x a = 1 induces a collection of node-disjoint paths in G that cover all nodes in Z. Moreover, since we minimize the number of arcs leaving node s, the number of paths is minimized. It is interesting to note that, although the number of inequalities in this formulation is exponential, the separation of these constraints, called subtour elimination constraints, can be solved in polynomial time (cf. [PG85]). It means that an optimum soluiton for the relaxed LP (replacing constraints (4) by 0 x ij 1 for all (i; j) 2 A) can be obtained in polynomial time (cf. [GLS88]). 4 A branch-and-cut algorithm for the MkCP The approach we have used to tackle the problem is to apply the usual linear programming relaxation techniques in a branch-and-cut algorithm. For that we have studied the facial structure of the polyhedron dened as the convex hull of the feasible (integer) solutions of (P). Let us denote this polyhedron by P k (G), that is, P k (G) := conv fx 2 IR A j x satises (1) to (5)g. We were able to derive many results concerning valid and facet-dening inequalities for this polyhedron. Moreover, in some special cases (e.g. the graph is acyclic) we can prove 5

6 that a complete and nonredundant description of the facets of P k (G) can be given. In general, however it is not the case. In the full paper we provide the results and the proofs. The idea of the approach is to start with a relaxation of the polyhedron P k (G) and to solve iteratively better approximations of this polyhedron, obtained by using facet-dening or, at least, valid inequalities that are violated by the optimal solution of the current relaxation. When we are not able to nd any violated valid inequality the procedure xes the value of some variable and proceeds in a branch-and-bound fashion. We have implemented two versions of the algorithm: one for the model without reverse complements, and one that handles it. In both cases we begin with the LP given by constraints (1), (2) and (3), and implemented a separation heuristic [PG85] for the subtour elimination constraints. Whenever no violated inequality is found by the separation routine we perform a branching step. Both versions have been tested with two types of instances. In the rst type the original string is 5000 characters long and has been generated randomly on the alphabet fa, T, C, Gg. Each substring has length between 500 and 700 characters. Each substring is generated by choosing randomly its length and also the position it starts in the original string. For the model allowing reverse complements the sequence is reversed with probability 1. In the second type of instances we use real DNA-molecules. These instances 2 were obtained from the DIMACS challenge 95. For the version without reverse complements we are able to solve instances with up to 200 nodes within 1s (in a Sun Sparc 1000), and in many cases the graph is acyclic, and the rst LP is sucient to provide the optimal solution. We obtained similar results for both the random and real instances. For the version allowing reverse complements, since the number of nodes in the model duplicates, we were able to solve to optimality smaller problems, with up to 100 nodes. The following tables summarize the results for the random instances. The complete results will be given in the full paper. In the rst column are given the number of nodes and arcs in the graph. In the second column the value of k used to construct the graph is given. In column 3 it is shown the value of the optimum solution. Columns 4 and 5 show the number of LPs solved and the number of nodes in the branch-and-bound tree, respectively. Finally, in the last column we present the CPU time spent to solve the problems in a SPARC We use CPLE to solve the LP in each iteration. 6

7 # nodes # arcs k # LP # BB nodes CPU Time (sec.) Table 1: Computational results of the version without reverse complements. # nodes # arcs k # LP # BB nodes CPU Time (sec.) : Table 2: Computational results of the version with reverse complements. 7

8 5 Conclusions and future research The computational results we present in this paper shows that the ILP formulation we suggest for the MkCP can be satisfactorily used to solve medium size instances of the problem, both for the version with and without considering reverse complements. Our aim is to increase the size of the instances we are able to solve to optimality. For that, we need other classes of valid inequalities to improve the lower bounds in each node of the branch-and-cut tree and better primal heuristics. It is still a rather long way until we are able to solve instances with the size of interest for the computational biology community (as considered in the DIMACS challenge 95). The DNA-molecules that they intend to determine have length of millions of bases, and typically these molecules are broken into several hundreds of pieces. Moreover, we aim to extend the model to allow the use of the same piece several times (i.e. covering the graph by node-disjoint walks) and to handle errors. References [GJ79] [GLS88] [K91] [KM95] [M92] [NW88] [PG85] M. R. Garey and D. S. Johnson, Computer and Intractability - A Guide to the Theory of NP-Completeness, W. H. Freeman and Company, New York (1979). M. Grotschel, L. Lovasz and A. Schrijver, Geometric Algorithms and Combinatorial Optimization, Springer-Verlag, Berlin (1988). J.D. Kececioglu, \Exact and Approximation Algorithms for DNA Sequence Reconstruction", PhD. Thesis, University of Arizona, Tucson (1991). J.D. Kececioglu and E.W. Myers, \Combinatorial Algorithms for DNA Sequence Assembly", Algorithmica 13, 7-51 (1995). J. Meidanis, \Algorithms for Problems in Computational Genetics", PhD. Thesis, University of Wisconsin-Madison (1992). G.L. Nemhauser and L.A., \Integer and Combinatorial Optimization", John-Willey and Sons, (1988). M. W. Padberg and M. Grotschel, \Polyhedral Aspects of the Travelling Salesman Problem II : Computations", in E. L. Lawler, J. K. Lenstra and A. H. G. Rinooy Kan eds., The Travelling Salesman Problem, Wiley, New York (1985). 8