Protein Folding Pathways

Yogesh A. girdhar

girdhy(AT)cs(DOT)rpi(DOT)edu

I developed the following simulator to simulate protein folding pathways. Given a protein in its native configuration, the simulator generates thousands of possible folding pathways(from different initial configuration) which the protein takes while it is folding. This program has applications in molecular drug design and studying diseases which are caused due to protein mis-folding.

The simulator is based on Probabilistic Roadmaps (PRM) method of motion planning. Its working can be divided into the following 3 steps:

• Sample Generation - Generating c-space

• Graph Generation - Connecting c-space

• Pathways Generation - Finding pathways.

Sample Generation

Samples are generated by the following procedure:

• Input: native PDB file for the protein.

• Approximate the side chains with a CB carbon atom.(this is done for speed)

• Build a atomgroup[3] model of this molecule so that we can change the torsion angles and get new configurations.

• Generate a sample by randomly changing the torsion angle using a normal random number distribution biased around native configuration.

• If any of the CB atoms in the sample are closer than 2.4A then discard the sample.

• Calculate the energy of the sample[1,2].

• Repeat till required number of samples have been generated.

Graph Generation

Once we have the samples (c-space), we now try to connect them as following:

• For each samples s in c-space (this is the source vertex in the edge)

  • find K-nearest neighbors.

  • For each neighbor n (this is the target vertex in the edge)

    • calculate deltaE = Energy(n) - Energy(s)

    • if(deltaE < 0) deltaE = 0 (deltaE is the edge weight)

    • output Edge(s,n,deltaE)

Distance Matrix Error is used as the distance function above. Another possible function could be euclidean distance in c-space.

DME is calculate as:

DME = Sum( | a_ij - b-ij | ) / (N*(N-1)/2)

a_ij is the distance between atom i and atom j in configuration a.
b_ij is the distance between atom i and atom j in configuration b.
N is the number of atoms

Path Generation

One we have the Graph(list of edges and vertices), we simply run Dijkstra's single source shortest path algorithm to get all possible paths leading to the native. Dijkstra's algorithms minimizes the edge weights; as a result we get paths with lowest energy climb.

Some Results

I ran the simulator on 2GB1 protein fragment with the following parameters:

• Residues 39-56
• 9000 samples with stddev. of 1, 2, 4, 8, 16, 32, 64, 128, 180 degrees(1000 samples from each class)
• For nearest neighbor, K=20
• Energy cutoff 1000000 KJ/mol

Here are some randomly chosen output paths(you need VMD to see these):

References

• Using Motion Planning to Map Protein Folding Landscapes and Analyze Folding Kinetics of Known Native Structures , Nancy M. Amato, Ken A. Dill, and Guang Song, Journal of Computational Biology, 10(3), 2003, pp. 239-255. Preliminary version appeared in Proceedings of the 6th International Conference on Computational Molecular Biology (RECOMB), April 2002, pages 2-11. Also, Technical Report TR01-001, Parasol LAB, Department of Computer Science, Texas A&M University, October 2001.
• Using Motion Planning to Study Protein Folding Pathways, Guang Song and Nancy M. Amato, Journal of Computational Biology, 9(2), 2002, pp. 149-168. Preliminary version appeared in Proceedings the 5th International Conference on Computational Molecular Biology (RECOMB), April 2001, pp. 287-296. Also, Technical Report TR00-026, Department of Computer Science, Texas A&M University, October 2000.
• Zhang, M. and Kavraki, L. E.. A New Method for Fast and Accurate Derivation of Molecular Conformations. Journal of Chemical Information and Computer Sciences, 42(1):64-70, 2002