Side-chain positioning with integer and linear programming

Transcription

1 Side-chain positioning with integer and linear programming Matt Labrum 1 Introduction One of the components of homology modeling and protein design is side-chain positioning (SCP) In [1], Kingsford, et al, have presented and explored integer linear programming and linear programming (ILP/LP) formulations of this problem For this problem, a fixed backbone is given and for each residue position along the backbone a number of candidate rotamers are considered A pairwise energy function is optimized and an optimal solution, then, is a set of rotamers (one for each position) that minimizes the total energy of the structure In this paper, I summarize the model and results of [1] and present the details of an example native backbone test 2 Model Before discussing the ILP/LP formulations developed to solve these problems, I will summarize the notation used Given a fixed backbone, let p denote its length Then at each residue position, i (i = 1,,p), there will be a set of candidate rotamers, denoted {i r } The SCP problem is then set up in graphtheoretic terms, in which a p-partite graph is developed The node set is V = V 1 V p, where V i corresponds to the set {i r }, and the edge set is D = {(u, v) : u V i, v V j, i j} For each node (rotamer) u V i, E uu is the energy associated with the interaction between the backbone and the chosen rotamer For u V i and v V j, E uv will be the pairwise energy between rotamers u and v Finally, x uv is a binary variable that will equal 1 if nodes u and v are chosen in the solution (otherwise, it will equal 0) x uu indicates whether node u was chosen in the solution Having the notation established, the authors present the following initial integer linear programming formulation: Minimize E = E uu x uu + E uv x uv u V {u,v} D 1

2 subject to x uu = 1 x uv = x vv x uu, x uv {0, 1} for j = 1,,p for j = 1,,p and v V \ V j They note, however, that for many of the uv pairs, E uv = 0 and so this formulation may be simplified by removing all of x uv variables, with u V i and v V j, for which E uv = 0 (given that all of the rotamers in V i and V j are non-positive) The following additional notation is introduced in order to simplify this first formulation Given a set V j, N + (V j ) will denote the union of all V i s for which there is a v V i and u V j with E uv > 0 The edge set will then be modified so that D is the set of {u, v} pairs such that u V j and either of the following conditions holds: (a) v N + (V j ) (b) v / N + (V j ) and E uv < 0 This reduction in variables and constraints leads to the following modified integer linear programming formulation: Minimize E = E uu x uu + E uv x uv u V {u,v} D subject to x uu = 1 for j = 1,, p x uv = x vv for j = 1,, p and v N + (V j ) x uv x vv for j = 1,, p and v / N + (V j ) :E uv<0 x uu, x uv {0, 1} If one desires multiple solutions for the SCP problem, say, to produce a number of candidates for protein design, the following constraints are added: u S k x uu p q for k = 1,,m 1; 1 < q p Here, m is the iteration number of the program run, S k denotes the optimal set of rotamers determined in iteration k, and q is the amount of difference desired between solutions Now the authors note that by replacing the constraints x uu, x uv {0, 1} with 0 x uu, x uv 1, a linear programming formulation is obtained that may 2

3 Problem Type Number Attempted Number Successes w/ LP Native backbone Homology modeling Protein Design 25 6 Table 1: Success rates with the linear programming formulation for each of the three SCP problem types be solved efficiently, and oftentimes results in an optimal solution that is integral (hence, it is an optimal solution for the ILP formulation) Finally, the process of dead-end elimination is presented This allows for a reduction in the number of candidate rotamers in the following way: for u V i, if there exists a v V i such that E uu E vv + j i min w V j (E uw E vw ) > 0, then the node u may be removed from the set, and this removal process continues until a pass through the rotamers fails to yield any such v These formulations and processes lead to the following approach to solving the SCP problem: 1 Attempt to solve the SCP problem with the linear programming formulation If the optimal solution is integral, we are done; if not, proceed to steps 2 and 3 2 Reduce the number of candidate rotamers in each set using dead-end elimination 3 Solve the SCP problem with the integer linear programming formulation 3 Results The approach outlined above for solving the SCP problem was used in the contexts of native backbone tests, homology modeling, and protein design Table 1 summarizes the success rates for the linear programming formulation for each of the three SCP problem types The number of successes with LP indicates the number of proteins for which an optimal (integral) solution was found using only the linear programming formulation An example of a native backbone problem will be presented here The authors have made available the AMPL source code, scripts, and datasets used for solving the SCP problems explored in [1] at The datasets are stored in a format that describes a SCP problem, but they require conversion to the data format needed for processing through AMPL 3

4 To illustrate a native backbone test, I ran the program on the protein 1aac The required information for solving the SCP problem for 1aac is given in the following (abbreviated) scp file provided on the above website: # Written with design on Sat Jul 26 21:12: # PDB file : 1aacnoaltpdb # Design file : ismb/pnas/nativervdw-reducedh/scptux1/1aacdes # VdW paramters : amber_params/amber96-reducedhvdw # HB parameters : # Type names : amber_params/amber94type # Rotamer library: /u/carlk/sdp/rot/pdb/bin/bbdep01julsortlib D 90 K [81 entries omitted] 1496 V 1523 E [1519 entries omitted] [241,310 entries omitted] END This file then needs to be modified for use as an AMPL data file as follows: data; 4

5 # Written with design on Sat Jul 26 21:12: # PDB file : 1aacnoaltpdb # Design file : ismb/pnas/nativervdw-reducedh/scptux1/1aacdes # VdW paramters : amber_params/amber96-reducedhvdw # HB parameters : # Type names : amber_params/amber94type # Rotamer library: /u/carlk/sdp/rot/pdb/bin/bbdep01julsortlib param num_posn := 85; param num_nodes := 1523; param posn_size := [81 entries omitted] ; # nonzeros = param: costv := [1519 entries omitted] ; param: coste := [241,310 entries omitted] ; 5

6 Once the data file is set up, the problem may be solved through AMPL, producing the following output (with the primal vertex variables output rearranged to save space): variables, all linear constraints, all linear; nonzeros 1 linear objective; nonzeros CPLEX 1010: primal dualopt timing=1 Times (seconds): Input = Solve = Output = CPLEX 1010: optimal solution; objective dual simplex iterations (0 in phase I) # energy # problem size # primal vertex variables The primal vertex variables indicate which rotamers provide the optimal solution, whose total energy is given as part of the output The problem size shows the total number of rotamers considered and the number of residue positions on the backbone, respectively 6

7 4 Conclusion The authors have shown that relaxing the integrality constraint on the standard integer linear programming formulation used for solving the SCP problem can greatly improve the efficiency of determining solutions, when the linear programming formulation does produce an integral optimal solution If this is not the case, use of dead-end elimination before proceeding with the integer linear programming formulation will also aid in finding optimal solutions more efficiently References [1] C L Kingsford, B Chazelle, M Singh Solving and analyzing side-chain positioning problems using linear and integer programming Bioinformatics (2005) 21: