1.2 Assignment 3: inverse kinematics  (Page 2/2)

Your task is to generate 10 different loops that all complete the protein structure of CI2 and compare them to thenative loop conformation. To do so, even though the native loop conformation is given to you in the backbone.pdb file, you will pretend that the loop region from residue 34 to residue 46 is not resolved. You will generate differentconformations for this loop region by randomly sampling values for the dihedral of the loop, uniformly in the range [-PI, PI]. Then you will use the CCD algorithm described in to steer these loop conformations to their goal/target location asspecified by the coordinates of residue 46 in the native conformation of CI2.

The Cyclic Coordinate Descent, as defined in , allows you to find the optimal dihedral angle for a dihedral bond by whose rotation a desired atom willget as close as possible to a target position. To simplify this problem, you will not have target positions for the three atoms N, CA, C as does, but only on one atom, the C atom. This means that the equations you need to solve for the optimaldihedral rotation are going to be much simpler, with your distance measure S defined as the Euclidean distance between current position M of atom Cand its target position F. Please address Q1 .

Ccd for a chain

For convenience, we will refer to the conformation of the chain where the atom reaches its target position as the closure conformation . The problem you will address in this assignment is how to obtain differentclosure conformations by solving for the dihedral DOFs of a chain in order to go from a starting/initial/current conformation to a closure conformation.In order to generate closure conformations, before solving for the dihedral DOFs through CCD for a chain, you need to generate starting/initialconformations for the loop region of CI2. One way to do so is to generate random angles for the dihedrals of the chain and then accumulate therespective transformation matrices down the chain through Forward Kinematics, as you have done in Assignment 2. You are not allowed to rotate only onedihedral bond to generate a random initial transformation . You have to rotate all dihedral bonds of a chain by random angles in the interval[-PI, PI]. Please answer Q3 .

Now it is time for you to put your pseudocode to the test. We first list the particular specifications for the inverse kinematics problem you will solvein this assignment.

• For this problem, you will steer the C atom of residue 46 of the loop region towards its target position. We will refer to this atom as the feature atom .
• We need to define the DOFs. As you will only generate closure conformations for the loop region from residue 34 to residue 46, theonly DOFs you can sample in the intial loop conformations and solve for in the closure conformations are those of the loop region. Everyother dihedral you cannot change.
• The CCD algorithm needs to have a measure of distance between the current position of the feature atom and its target position. Afteryou have generated initial conformations for the loop region, the current position of the feature atom is position in space of the Catom of residue 46 in this initial conformation.
• The target position of the feature atom will be its position as specified in the original (native) conformation contained in the backbone_native.crd file.
• Now we can put the pieces together: Generate an initial conformation by rotating the dihedrals of the 34-46 loop region of CI2 with randomangles sampled uniformly in the [-PI, PI] interval. Solve forthe dihedrals of the 34-46 region so that the C atom of aminoacid 46 in the initial conformation you have just generated reaches itstarget position as specified by its position in the native conformation in the backbone_native.crd file.Note that depending on the random conformation you start with, itmight be hard to steer atom C to reach its target position. After applying the dihedrals you have solved for, you might need to doadditional iterations to solve for the dihedrals of the updated conformation, apply those and so on until the spatial constraint foratom C is satisfied. To have a quantitative criterion for when atom C satisfies its spatial constraint, you need to define a thresholdparameter for S, as in . Set this threshold criterion to 0.01 for this assignment.Please answer Q4 .

Setup with matlab

Even though you are welcome to use any programming language to complete this assignment, we suggest that you use Matlab due to its convenient matrixoperations. For this assignment you will need some basic trigonometry functions such as sine , cosine , tangent , and inverse tangent/arctangent . You can find all these functions already implemented in Matlab. Please get familiar with their syntaxby using the Help pages. Figure 1 captures the Help page for the category of Functions By Category , subcategory Elementary Math . You need to read this category to learn more on how to use these functions.

For submission

Please follow this list closely.

Deliverables

• Q1

  Please repeat the derivation in with the simplification we made that only one atom is the one we want to tomove to its target position. Typeset these derivations for submission.

• Q2

  Please typeset your pseudocode for how you would iterate over a chain until the end of the chain reaches its target position.

• Q3

  Please provide pseudocode for how you would generate a random conformation for the 34-46 region of the protein. Please generate 10 conformations inthis way and visualize these conformations through VMD, all superimposed over the native. Submit this image.

• Q4

  Please plot all the final 10 conformations where the C atom of aminoacid 46 reaches its target position. Plot these conformations over the native.Indicate where the C atom lies with a VDW representation in red. Submit this image. (Note: remember that VMD requires a dummy first line in .crd files, so add one if your code does not).

• Q5

  Please compute the RMSD of the 10 closure conformations from the native. Please plot these values and submit this plot. Please check the 10 closureconformations and the native conformation for collisions with the energy function you implemented in assignment 2. Please report the energetic scoresof each conformation. Does the native conformation contain any collisions? How do the other closure conformations compare to the native in terms ofenergy? What can you conclude about your implementation of CCD to get a closure conformation with regard to the energy of this closure conformation?

• Q6

  The CCD algorithm does not take into consideration energetic constraints due to unfavorable steric clashes. One way to impose energetic considerations isto discriminate against closure conformations that contain steric clashes. You have already implemented a simplistic energy function that can report thepresence of collisions. Please provide pseudo-code on how you would couple CCD with your energy function to report only closure conformations that donot contain collisions.

Note: While you are welcome to use any programming language, please typeset your deliverables. We encourage you to use Matlab. You may use anyvisualization software to produce your images.