Motivation for Inverse Kinematics in Proteins

One important application of inverse kinematics is in determining missing portions of protein structure. We traditionally depend on experimental techniques to provide us with the picture of an average structure for a protein. X-ray crystallography for example, relies on crystallizing proteins and reporting the structure of the protein crystal within a certain resolution. One of the inherent problems with X-ray crystallography is that mobile protein regions such as loops cause disorder in the crystal and as a consequence, coordinates for the atoms of these mobile regions cannot be reported. Often, in the PDB, crystallographically determined proteins are partially resolved, i.e. a portion of the structure may be missing due to its intrinsic mobility. Even when experimental techniques such as NMR and cryo-EM can report an average picture of the fully resolved protein, the average structure reported is not indicative of the different conformations mobile regions can assume inside our cells at room temperature.

The specific problem of completing a partially resolved protein structure by finding conformations for its missing loop is known as the fragment completion or the loop closure problem. Note that the loop closure problem is actually an inverse kinematics problem. Using sequence information alone, i.e. knowing the aminoacid sequence of the missing loop, one can generate starting loop conformations. The loop closure problem requires these loop conformations to be geometrically constrained by attaching them to the portion of the protein structure that is experimentally determined. Note that, as the picture below indicates, one can generate many loop conformations in space through forward kinematics. One end of the loop can be attached to its counterpart in the protein through translation alone. The other end however, needs to be attached without breaking bonds or stretching bond angles. One way to do this is through inverse kinematics; that is, knowing the goal position in space for the end of the loop, can we solve for the dihedral DOFs of the loop conformation? This question can be answered by Inverse Kinematics techniques.

Figure 1: One "sticky" end of the loop can be attached to its stationary counterpart in the protein through translation. The other end needs to move towards its goal location by solving an Inverse Kinematics problem.

Inverse Kinematics for a Polypeptide Chain

Cyclic Coordinate Descent (CCD)

In this assignment you will complete a loop portion in the 1COA structure of the CI2 protein. X-ray crystallography completely resolves the 1COA structure. However, even though a long loop region from residue 34 to residue 46 is present in the native conformation of 1COA, an interesting exercise is to pretend this loop region cannot be determined. Using an inverse kinematics approach, you will sample many potential loops that can all complete the 1COA structure and compare them to the native conformation obtained through X-ray crystallography. An important question to answer is whether your loop closure algorithm can recover/predict the conformational state of this loop region in CI2; that is, how different are your loops from the one in the native conformation of CI2? You will implement a simple inverse kinematics technique, Cyclic Coordinate Descent (CCD) as presented in [1] . For simplicity, you will work with the native backbone of CI2, whose respective native coordinates you can also obtain from the backbone_native.crd file.

Your task is to generate 10 different loops that all complete the protein structure of CI2 and compare them to the native 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 different conformations 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 [1] to steer these loop conformations to their goal/target location as specified by the coordinates of residue 46 in the native conformation of CI2.

The Cyclic Coordinate Descent, as defined in [1] , allows you to find the optimal dihedral angle for a dihedral bond by whose rotation a desired atom will get 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 [1] does, but only on one atom, the C atom. This means that the equations you need to solve for the optimal dihedral rotation are going to be much simpler, with your distance measure S defined as the Euclidean distance between current position M of atom C and its target position F. Please address Q1 .

CCD for a Chain

You have noticed by now that CCD reports the angle by which to rotate a particular dihedral bond so that the feature atom gets as close as possible to its target position. Consider a chain with the feature atom at its end. As in real polypeptides, this chain does not contain only one dihedral angle, but many. How would you use the CCD routine to solve for all the dihedrals of this chain so that the final end of the chain reaches a target position? Please address Q2 . (Hint: You might consider solving for each dihedral one at a time, after you have imposed an order on the dihedrals you are working with.)

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 different closure 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/initial conformations 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 the respective transformation matrices down the chain through Forward Kinematics, as you have done in Assignment 2. You are not allowed to rotate only one dihedral 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 solve in this assignment.

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 matrix operations. 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 syntax by 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.

Figure 2: Matlab Trigonometry

It is very important that you use the right arctangent implementation for this assignment. The authors in [1] use the atan2 function as implemented in C . There is a similar function in Matlab . Please be careful to understand in what quadrant this function returns its output. You can learn more about this function under the Help pages. Figure 2 captures the help page for this function under the category of Functions By Category , subcategory Elementary Math , as part of the Trigonometric functions.

Figure 3: Matlab atan2 function

For Submission

Please follow this list closely.

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