Problem 4 [20 pts, (6, 7, 7)]: Protein Folding A major problem in biology is to determine the physical structure of proteins. A protein is a sequence of amino acids, which interact with one another to form a well-defined 3-dimensional structure, the folded protein. The correct three-dimensional structure is critical to the function of the protein; misfolded proteins are believed to be the cause of some diseases. The protein folding problem is to determine the correct 3-dimensional structure of a givein sequence of amino acids. This is determined by various factors, including chemical properties of the amino acids, intramolecular hydrogen bonds, and compactness of the structure. Computational biologists have developed various algorithms and heuristics to compute the most likely structure of a given amino acid sequence. This is, in general, an intractable problem. So often these heuristics have to resort to enumerating a large number of possible protein folds and evaluating their stability. In this exercise, we explore a highly simplified (and somewhat artificial) version of the problem of counting the number of different protein folds. Suppose you are given a protein P, which is a sequence of n amino acids: you can represent this chain as A1-A2 An-I-An where each A; is an amino acid. For example, here is a 10-amino acid (fictional) protein: RHD IRD Figure 1: Some protein folds for the example protein with 10 amino acids A-R-N-D QA-JI-R-D (Amino acids have single-letter abbreviations.) We will consider the number of different two-dimensional folds of a particular kind. A fold of P is obtained by placing the sequence of amino acids left to right on one row, possibly wrapping around at some point and moving right to left on the next row, then possibly wrapping around again, moving left to right on the third row, and so on continuing in a “snake-like” manner until all the amino acids have been placed. Figure 1 depicts 4 (of the many possible) folds for the above 10 amino acid protein. Note that this snake-like form precludes more complex contortions such as spirals and self-intersections Note that the last two folds – (c) and (d) in Figure 1 – are rotationally equivalent in the sense that one can be obtained by rotating another by 90 degrees. But we count them as distinct folds in this exercise Mathematically, a fold of a protein with n amino acids is represented by a sequence ni, n2, …, nk rows, for some k in { 1, 2, . . . ,n), where each ni is a positive integer, and nit n2+ . . . nk equals n. The structure depicted by this fold is as follows: in the first row, the first n amino acids in the sequence are laid left to right; in the second row, the next n2 amino acids are laid right to left starting from the point below the rightmost amino acid on the first row; and so on i. Find the number of different folds for a protein with 8 amino acids The correct protein fold is often a compact structure that occupies little space and leaves few “holes”. Find the number of different folds of a protein with 12 amino acids in which the number of amino acids in every row of the fold is at least 3 iii. We say that a fold is uniform if the fold forms a complete rectangle; i.e., if the number of amino acids in every row is identical. For instance, the above protein with 10 amino acids has 4 uniform folds. Two uniform folds are given in (c) and (d) of Figure 1. The remaining two uniform folds are given in Figure 2. Q (A Figure 2: Two of the four uniform folds for the example protein How many uniform folds does a sequence with 120 amino acids have’?

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