by Guang Song and Nancy M. Amato Journal of Computational Biology, April 1, 2002 Presentation by Athina Ropodi

 Introduction- Definition Motion planning Probabilistic Roadmap Method Denaturation C-space  Probabilistic Roadmap Method Basic steps Degrees of freedom/C-space Node Generation Sampling strategy Potential Energy Computations  Results

 Motion planning is a term used in robotics for the process of detailing a task into atomic robotic motions.  This issue is also known as the “navigation problem”:  Given an environment, a start and a goal position of an object, the objective is to find a valid path (continuous sequence of valid configurations) from start to goal.

 Probabilistic Roadmap (PRM) motion planning techniques are applied to “small” proteins (up to 60 residues) in order to compute folding pathways from a denaturated state to its native fold.  Denaturate: To cause the structure to unfold, so that some of its original properties, especially its biological activity, are diminished or eliminated. Usually caused by extreme conditions, e.g. high temperature.

 In quaternary structure denaturation, protein sub-units are dissociated and/or the spatial arrangement of protein subunits is disrupted.  Tertiary structure denaturation involves the disruption of:  Covalent interactions between amino acid side chains (such as disulfide bridges between cysteine groups)  Noncovalent dipole-dipole interactions between polar amino acid side chains (and the surrounding solvent)  Van der Waals (induced dipole) interactions between nonpolar amino acid side chains.  In secondary structure denaturation, proteins lose all regular repeating patterns such as alpha-helices and beta- pleated sheets, and adopt a random coil configuration.  Primary structure, such as the sequence of amino acids held together by covalent peptide bonds, is not disrupted by denaturation

 The article investigates the folding mechanisms of a protein assuming we know its native fold.  Results are validated by comparing the formation order to pulse-labeling experimental results.  The configuration space (C-Space) of a movable object is the space consisting of all positions and orientations of that object.  Pulse labeling is a biochemistry technique of identifying the target molecule presence by inclusion of a pulse of a radioactive compound.

 Introduction- Definition Motion planning Probabilistic Roadmap Method Denaturation C-space  Probabilistic Roadmap Method Basic steps Degrees of freedom/C-space Node Generation Sampling strategy Potential Energy Computations  Results

 Any complete motion planner would require time exponential in the number of degrees of freedom (dof).  Several methods such as energy minimization, molecular dynamics, Monte-Carlo and genetic algorithms have been used.  This method tries to simulate the true dynamics of the folding process using the classical Newton’s motion equations.  However, an exact simulation would depend on the start conformation and could result in local minima.

The basic steps are:  First of all, PRM samples points randomly from C-space and retains those that satisfy certain requirements.  Then, the points are connected and form a graph using a simple planning method.  Finally, paths connecting the start and goal configurations are extracted using standard graph search techniques.  [Kavraki, Svestka, Latombe,Overmars 1996] In this case, low-energy conformations are preferable.

 All atomic bond lengths and angles are considered to be constants.  We only consider 2 dofs, φ and ψ angles.  Side-chains are modeled as spheres with no dof.  Fold positions (atomic bonds) correspond to joints and atoms correspond to links. Thus, for k residues our model has 2k links and 2k revolute joints…

 Different configurations are produced by assigning possible angle values.  But, the nodes are accepted or rejected based on their potential energy: where E min=50000 and Emax=89000 KJouls/mol A configuration with higher potential is more likely to be rejected.

 Due to the high dimensionality of the problem, a very dense uniform sampling is required.  Since prior knowledge of the native fold is assumed, a sampling strategy biased to the native fold is applied:  Sampling is performed from a set of normal distributions around the native fold. The standard deviations used are {5, 10, 20, 40, 80,160 degrees}. Small STDs capture the detail around the goal, and larger ensure adequate roadmap coverage

 For each node, the k-nearest neighbors are found(k=20 and the metric is Euclidean).  For each connection a feasibility check is performed. (two nodes are connected by a straight line)  For 2 consecutive intermediate conformations, i and i+1, we first check their potential energies and then the probability of moving from i to i+1: RMSD metric proved inferior

 The total weight of the edge is:  Dijkstra’s algorithm is then used to find the smallest weight path.  Path optimization: resampling is performed around the nodes of paths with high potential.

 To reduce the cost of calculations, an approximation function is used. We only consider contribution from side chains and those are modeled as spheres.  The cost is then reduced by 2 orders of magnitude. The 1 st term represents constraints that favor secondary structure, hydrogen and disulphide bonds The 2 nd the van der Waals interactions among atoms.

 Introduction- Definition Motion planning Probabilistic Roadmap Method Denaturation C-space  Probabilistic Roadmap Method Basic steps Degrees of freedom/C-space Node Generation Sampling strategy Potential Energy Computations  Results

 Folding process of GB1:

 Protein GB1(56 residues): 1 α-helix and 4 β-strands  Protein A (60 residues): 3 α-helices

Peaks show where atoms are close and Van der Waals interactions dominate. Bigger roadmaps have smoother paths.

More samples around the peaks improve the path.

 Different pathways tend to come together and appear to have some common channels, as they approach the native fold.

Bibliography: 1. Probabilistic roadmaps for path planning in high-dimensional configuration spaces L. Kavraki, P. Svestka, J-C. Latombe, M. H. Overmars, Protein Folding by Restrained Energy Minimization and Molecular Dynamics Michael Levitt, esearch/folding/index.shtml.OLD2