This lecture ¾Restriction enzymes and the partial digest problem Exhaustive search ¾Finding regulatory motifs in DNA Sequences Torgeir R. Hvidsten ¾Exhaustive search methods T.R. Hvidsten: 1MB304: Discrete structures for bioinformatics II 1 T.R. Hvidsten: 1MB304: Discrete structures for bioinformatics II 2 Restriction enzymes ¾ HindII - first restriction enzyme – Restriction enzymes and the was discovered accidentally in 1970 while studying how the ppgpartial digest problem bacterium Haemophilus influenzae takes up DNA from the virus ¾ Restriction enzymes are used as a defense mechanism by bacteria to break down the DNA of attacking viruses T.R. Hvidsten: 1MB304: Discrete structures for bioinformatics II 3 T.R. Hvidsten: 1MB304: Discrete structures for bioinformatics II 4 Uses of restriction enzymes Restriction maps ¾Recombinant DNA technology (i.e. combining DNA ¾ A map showing the positions of restriction sites in a DNA sequences that would not normally occur together) sequence ¾Cloning ¾ If the DNA sequence is known then constructing a ¾cDNA/genomic library construction restriiiction map is tr iilivial ¾DNA mapping ¾ In early days of molecular biology, DNA sequences were often unknown ¾ Biologists had to solve the problem of constructing restriction maps without knowing DNA sequences T.R. Hvidsten: 1MB304: Discrete structures for bioinformatics II 5 T.R. Hvidsten: 1MB304: Discrete structures for bioinformatics II 6 1 Measuring length of restriction fragments Partial restriction digest ¾ The sample of DNA is exposed to the restriction enzyme for only a limited amount of time to prevent it from being cut at all restriction sites ¾ Restriction enzymes break DNA into restriction ¾ This experiment generates the set of all possible restriction fragments fragments between every two (not necessarily consecutive) cuts ¾ Gel electrophoresis is a process for separating DNA by size and measuring sizes of restriction fragments ¾ This set of fragment sizes is used to ¾ Can separate DNA fragments that differ in length in determine the only 1 nucleotide for fragments up to 500 nucleotides positions of the long restriction sites in the DNA sequence T.R. Hvidsten: 1MB304: Discrete structures for bioinformatics II 7 T.R. Hvidsten: 1MB304: Discrete structures for bioinformatics II 8 Multiset of restriction fragments Partial Digest Problem (PDP) ¾ We assume that multiplicity of a fragment can be detected, i.e., X the set of n integers representing the location of all the number of restriction fragments of the same length can be cuts in the restriction map, including the start and determined end i.e. X = {x1 = 0, x2, …, xn}. ¾ Restriction sites: X = {0, 5, 14, 19, 22} n the total number of cuts ¾ Multiset: ΔX = {3, 5, 5, 8, 9, 14, 14, 17, 19, 22} ΔX thlifihe multiset of integers represent ilhfing lengths of each of the n(n-1)/2 fragments produced from a partial digest i.e. ΔX = {xj –xi | 1 ≤ i < j ≤ n} Problem: Given the multiset L, find a set X such that ΔX = L T.R. Hvidsten: 1MB304: Discrete structures for bioinformatics II 9 T.R. Hvidsten: 1MB304: Discrete structures for bioinformatics II 10 Partial Digest: Multiple Solutions (I) Partial Digest: Multiple Solutions (II) It is not always possible to uniquely reconstruct a set X based only on ΔX. 01257912 015781012 For example, the set 0 1 2 5 7 912 0 1 5 7 81012 X = {0, 2, 5} 1 1 4 6811 1 4 6 7 911 and 2 3 5 7 10 5 2 3 5 7 (X + 10) = {10, 12, 15} both produce ΔX={2, 3, 5} as their partial digest set (they are 5 2 4 7 7 1 3 5 homometric). 7 2 5 8 2 4 The sets {0,1,2,5,7,9,12} and {0,1,5,7,8,10,12} present a less trivial 9 3 10 2 example of non-uniqueness. They both digest into: {1, 1, 2, 2, 2, 3, 3, 4, 4, 5, 5, 5, 6, 7, 7, 7, 8, 9, 10, 11, 12}. 12 12 T.R. Hvidsten: 1MB304: Discrete structures for bioinformatics II 11 T.R. Hvidsten: 1MB304: Discrete structures for bioinformatics II 12 2 BruteForcePDP Efficiency of BruteForcePDP BruteForcePDP(L, n) ¾The running time of BruteForcePDP is O(|L|n-2) 1 M ← Maximum element in L or, since |L| = n(n-1)/2, O(n2n-4) 2 for every set of n – 2 integers 0 < x2 < … xn-1 < M ¾ Note that without restricting the elements of X to n-2 such that xi ε L for 1 < i < n elements in L, the running time would be O(M ) 3 X ← {0,x2,…,xn-1,M } ¾If L = {2, 998, 1000} (n = 3, M = 1000), the 4Form ΔX from X running time would be very different 5 if ΔX = L ¾Nonetheless, both algorithms have a exponential 6 return X running time 7 output “No solution” T.R. Hvidsten: 1MB304: Discrete structures for bioinformatics II 13 T.R. Hvidsten: 1MB304: Discrete structures for bioinformatics II 14 Branch and Bound Algorithm for PDP Definitions 1. Initiation: Add the start (0) and end (width) point of the Before describing PartialDigest, first define sequence to X (and remove the end point from L) Δ(y, X) 2. Find the largest element y in L as the multiset of all distances between point y and all 3. See if y fits o n t he rig ht o r le ft s ide o f t he rest rict io n map other points in the set X by checking whether the other lengths (fragments) it Δ(y, X) = {|y – x1|, |y – x2|, …, |y – xn|} creates are in L 4. If it fits, remove these lengths from L and add y (or width for X = {x1, x2, …, xn} –y) to X (if not, backtrack) 5. Go back to step 2 until L is empty T.R. Hvidsten: 1MB304: Discrete structures for bioinformatics II 15 T.R. Hvidsten: 1MB304: Discrete structures for bioinformatics II 16 PartialDigest(L) 1 width ← Maximum element in L 2 Remove width from L PartialDigest An Example 3 X ← {0, width} 4 Place(L, X) L = {2, 2, 3, 3, 4, 5, 6, 7, 8, 10} Place(L,X) 1 if L is empty Inititation: 2 output X width = 10, so delete 10 from L and add 0 and 10 to X 3 return 4 y ← MiMaximum e lement in L 5 if Δ(y, X ) ⊆ L L = {2, 2, 3, 3, 4, 5, 6, 7, 8} 6 Remove lengths Δ(y, X) from L and add y to X X = {0, 10} 7 Place(L, X ) 8 Remove y from X and add lengths Δ(y, X) to L 9 if Δ(width – y, X ) ⊆ L 10 Remove lengths Δ(width – y, X) from L and add width – y to X and 11 Place(L, X ) 12 Remove width – y from X and add lengths Δ(width – y, X ) to L 13 return T.R. Hvidsten: 1MB304: Discrete structures for bioinformatics II 17 T.R. Hvidsten: 1MB304: Discrete structures for bioinformatics II 18 3 An Example An Example L = {2, 2, 3, 3, 4, 5, 6, 7, 8} L = {2, 3, 3, 4, 5, 6, 7} X = {0, 10} X = {0, 8, 10} y = 8 y = 7 Δ(y, X) = { 8, 2} , which is a subset of L Δ(y, X) = { 7, 1, 3} , which is not a subset of L Δ(width – y, X) = {3, 5, 7}, which is a subset of L L = {2, 3, 3, 4, 5, 6, 7} X = {0, 8, 10} L = {2, 3, 4, 6} X = {0, 3, 8, 10} T.R. Hvidsten: 1MB304: Discrete structures for bioinformatics II 19 T.R. Hvidsten: 1MB304: Discrete structures for bioinformatics II 20 An Example An Example L = {2, 3, 4, 6} L = {} X = {0, 3, 8, 10} X = {0, 3, 6, 8, 10} y = 6 L is empty! Δ(y, X) = { 6, 3, 2, 4} , which is a subset of L Output X L = {} And continue searching for more solutions … X = {0, 3, 6, 8, 10} T.R. Hvidsten: 1MB304: Discrete structures for bioinformatics II 21 T.R. Hvidsten: 1MB304: Discrete structures for bioinformatics II 22 An Example An Example L = {2, 3, 4, 6} L = {2, 3, 3, 4, 5, 6, 7} X = {0, 3, 8, 10} X = {0, 8, 10} y = 6 y = 7 Δ(y, X) = { 6, 3, 2, 4} , which is a subset of L Δ(y, X) = { 7, 1, 3} , which is not a subset of L Δ(width – y, X) = {4, 1, 4, 6}, which is not a subset of L Δ(width – y, X) = {3, 5, 7}, which is a subset of L Backtrack! Backtrack! T.R. Hvidsten: 1MB304: Discrete structures for bioinformatics II 23 T.R. Hvidsten: 1MB304: Discrete structures for bioinformatics II 24 4 An Example An Example L = {2, 2, 3, 3, 4, 5, 6, 7, 8} L = {2, 3, 3, 4, 5, 6, 7} X = {0, 10} X = {0, 2, 10} y = 8 y = 7 Δ(y, X) = { 8, 2} , which is a subset of L Δ(y, X) = { 7, 5, 3} , which is a subset of L Δ(width – y, X) = {2, 8}, which is a subset of L L = {2, 3, 4, 6} L = {2, 3, 3, 4, 5, 6, 7} X = {0, 2, 7, 10} X = {0, 2, 10} T.R. Hvidsten: 1MB304: Discrete structures for bioinformatics II 25 T.R. Hvidsten: 1MB304: Discrete structures for bioinformatics II 26 An Example An Example L = {2, 3, 4, 6} L = {} X = {0, 2, 7, 10} X = {0, 2, 4, 7, 10} y = 6 L is empty! Δ(y, X) = { 6, 4, 1, 4} , which is not a subset of L Output X Δ(width – y, X) = {4, 2, 3, 6}, which is a subset of L And continue searching for more solutions … L = {} X = {0, 2, 4, 7, 10} T.R.