Texas A&M University

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1 Texas A&M University Electrical & Computer Engineering Department Graphical Modeling Course Project Author: Mostafa Karimi UIN: Prof Krishna Narayanan May 10

2 1 Introduction Proteins are made by linking amino acids which have a specific structure shown in figure 1. Every Amino acid has 4 important parts. Central carbon atom c α amino group (NH 2 ) carboxyl group (COOH) Side chain Figure 1: amino acid structure There are 20 natural amino acids in nature which are distinguished by their side chains. During protein synthesis, 2 amino acids will be linked end to end exactly like figure 2 and make a peptide. If several amino acids join, they will make a protein. An amino acid in protein structure is an unit which is called a residue. The formation of succession of peptide bonds make a backbone, upon which side chains are hanged on. Figure 2: Process of joining 2 amino acids to make a peptide 1

3 Conformation of every side chain is represented by at most 4 dihedral angles χ 1, χ 2, χ 3 and χ 4 shown in figure 3. Every assignment of χ angles for all of residues, is a valid conformation in 3D dimension. Figure 3: Dihedral angles in an amino acid Problem is to find the best assignment χ angles for all of the side chains which is nonlinear continuous energy minimization with non convex energy function. To reduce the size of problem, discretizing the χ angles with highly favorable energy will produce set of rotamers for every residue. Now problem will reduced to combinatorial optimization problem which is called Side chain Packing (SCP). SCP problem is a NP-hard problem which there are multiple exact algorithms with worst case of exponential time complexity. In following sections I will explain how we can exploit graphical modeling to solve the problem. 2 Graphical modeling representation One of the famous and simple model in statistical physics area is Ising model which exploits singleton and pairwise factor nodes. By exploiting Ising Model for protein energy we can at first derive a simple but very 2

4 useful energy function for every conformation: E(r) = E template + which consists of 3 terms: n E(i r ) + i=1 n n i=1 j=i+1 E(i r, j s ) (1) Template energy: correspond to energy of backbone and fixed residues Singleton energy: correspond to self energy of every rotamer and energy of every rotamer to backbone Pairwise energy: correspond to energy of a rotamer of one residue to a rotamer of another residue Based on Ising model we can derive the factor nodes: Singleton factor node: ψ i (i r ) = e E(ir) T (2) Pairwise factor node: ψ i,j (i r, j s ) = e E(ir,js) T (3) Therefore, based on Hammersley Clifford theorem, we can derive the probability of a conformation from factor nodes which are the cliques in markov graph and Z correspond to the partition function. P (r) = 1 ψ i (i r ) ψ i,j (i r, j s ) (4) Z i i,j 3 Approximate Algorithms As we know, Message passing algorithm is exact algorithm for inference in tree graphs. Therefore, if the problem is tree, we can use max-product belief propagation to find the most probable conformation or sum-product belief propagation to find marginal probability for every residue. However, in Ising model of proteins there are hundreds of small cycles. Therefore, if we use these algorithms in non-tree graphs due to positive feedback and other reasons, they may not converge or if they converge they may converge to sub-optimal solution. If we use belief propagation algorithms on loopy graphs, it is called loopy 3

5 belief propagation which is an approximate algorithm. There are two popular belief propagation that can be used for this combinatorial optimization problem. Max product algorithm is mainly used for finding the most probable conformation. Max product belief propagation update is: ( ) m i j (j s ) = max i r e E i (ir) E i,j (ir,js) T k N(i)\j m k i (i r ) Sum product algorithm is mainly used for finding marginal probability for every rotamer in every residue. Sum product belief propagation update is: m i j (j s ) = ( ) e E i (ir) E i,j (ir,js) T m k i (i r ) (6) i r k N(i)\j After convergence or a pre-determined number of iteration, we can derive the beliefs for every rotamer of every residue by multiplying every messages that is coming from neighbors and multiplying to its singleton factor node. Finally beliefs will be: B(i r ) = e E i (ir) T k N(i) (5) m k i (i r ) (7) After calculating beliefs, we can find the most probable conformation, by choosing the most probable rotamer for every residue, which will be: 4 Simulation setup r = arg max B(i r ) (8) i r I used the Data which was prepared from [1]. At first I write a c++ code to convert the text file to the correct format that our existing protein design code use. Then, I add 2 c code scripts, for Message passing algorithms such as max-product and sum-product. For temperature I used T=1, and for finding neighbors for every residue, I use the threshold = 0.01 based on pairwise energy terms which means that if two residues have absolute pairwise energy for all pair of their rotamer less than threshold then they are not neighbors. For convergence, I used threshold = and pre-determined maximum 4

6 iteration = iteration. Due to underflow issue, I used normalizing in every 5 Results & Discussion At first I will explain about the result from [5]. In this paper, at first they discussed the convergence problem in loopy belief propagation and showed that they have pretty good convergence in real world data which shows that all of these algorithms have convergence above 90%. Figure 4: Comparison of convergence Then, they compare the best solution found by max-product, min-sum, mean field algorithm. Since max-product always find better solution compare to others, they calculated histogram of difference between sum product and max product which is: Figure 5: Sum product Vs Max product minimum energy found 5

7 And, they calculated the histogram of difference between mean field and max product, which is: Figure 6: Mean Field Vs Max product minimum energy found There is another plot from [1] which compare the running time of different algorithm with respect to size of rotameric conformation in protein design problem. Figure 7: Comparison of different Algorithm Finally I implemented sum product and max product belief propagation algorithms and compare their sub-optimal conformation result to the optimal result from exact algorithm like A or Branch and Bound algorithm. As we can see most of the time max product algorithm is better than sum product in finding better solution. 6

8 6 Further Research We can use protein 3 dimensional structure to form the neighborhood between two residue not just based on pairwise energies between two residues. Therefore, we can have much sparser graph than the one in this report. Furthermore, we can use smooth version message passing which may avoid positive feedback more than before, which may increase the performance. References [1] D. Allouche, I. Andr, S. Barbe, J. Davies, S. de Givry, G. Katsirelos, B. O Sullivan, S. Prestwich, T. Schiex, and S. Traor. Computational protein design as an optimization problem. Artificial Intelligence, 212:59 79, July [2] M. Fromer and C. Yanover. Accurate prediction for atomic-level protein design and its application in diversifying the near-optimal sequence space. Proteins, 75(3): , May [3] A. R. Leach and A. P. Lemon. Exploring the conformational space of protein side chains using dead-end elimination and the A* algorithm. Proteins, 33(2): , Nov [4] C. Yanover and M. Fromer. Prediction of Low Energy Protein Side Chain Configurations Using Markov Random Fields. In T. Hamelryck, K. Mardia, and J. Ferkinghoff-Borg, editors, Bayesian Methods in Structural Bioinformatics, Statistics for Biology and Health, pages Springer Berlin Heidelberg, DOI: / [5] C. Yanover and Y. Weiss. Approximate inference and protein folding. In Advances in Neural Information Processing Systems, pages MIT Press,

9 size of rotameric conformation max product sum product A* 1BK2 1.18e BRS 1.67e C9O 3.77e CDL 5.68e CM1 3.73e CSP 5.02e CTF 3.95e DKT 3.94e CSE 8.35e CSK 4.09e MJC 4.36e NXB 2.61e FNA 3.02e GVP 1.51e HNG 3.77e L e LZ1 1.04e PIN 5.32e POH 8.02e SHF 1.05e SHG 2.13e TEN 6.17e UBI 2.43e DRI 7.26e PCY 2.34e RN2 3.68e TRX 9.02e Table 1: The lowest conformation found in different algorithms 8