Finding Motifs in DNA References: 1. Bioinformatics Algorithms, Jones and Pevzner, Chapter Algorithms on Strings, Gusfield, Section Beginning Perl for Bioinformatics, Tisdall, Chapter Wikipedia

Summary Introduce the Motif Finding Problem Explain its significance in bioinformatics Develop a simple model of the problem Design algorithmic solutions: –Brute Force –Branch and Bound –Greedy Compare results of each method.

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The Motif Finding Problem motif noun 1. a recurring subject, theme, idea, etc., esp. in a literary, artistic, or musical work. 2. a distinctive and recurring form, shape, figure, etc., in a design, as in a painting or on wallpaper. 3. a dominant idea or feature: the profit motif of free enterprise.

Example: Fruit Fly Set of immunity genes. DNA pattern: TCGGGGATTTCC Consistently appears upstream of this set of genes. Regulates timing/magnitude of gene expression. “Regulatory Motif” Finding such patterns can be difficult.

Construct an Example: 7 DNA Samples cacgtgaagcgactagctgtactattctgcat cgtccgatctcaggattgtctggggcgacgat gggggcggtgcgggagccagcgctcggcgttt gcaaggcgtcaaattgggaggcgcattctgaa ccacaagcgagcgttcctcgggattggtcacg aggtataatgcgaacagctaaaactccggaaa cccccgcaatttaactagggggcgcttagcgt Pattern acctggcc

Insert Pattern at random locations: cacgtgaacctggccagcgactagctgtactattctgcat cgtccgatctcaggattgtctacctggccggggcgacgat gacctggccggggcggtgcgggagccagcgctcggcgttt gcaaggacctggcccgtcaaattgggaggcgcattctgaa ccacaagcgagcgttcctcgggattggacctggcctcacg aggtataatgcgaaacctggcccagctaaaactccggaaa cccccgcaaacctggcctttaactagggggcgcttagcgt

Add Mutations: cacgtgaacGtggccagcgactagctgtactattctgcat cgtccgatctcaggattgtctacctgAccggggcgacgat gGcctggccggggcggtgcgggagccagcgctcggcgttt gcaaggacctggTccgtcaaattgggaggcgcattctgaa ccacaagcgagcgttcctcgggattggaActggcctcacg aggtataatgcgaaacctTgcccagctaaaactccggaaa cccccgcaaacTtggcctttaactagggggcgcttagcgt

Finally, find the hidden pattern: cacgtgaacgtggccagcgactagctgtactattctgcat cgtccgatctcaggattgtctacctgaccggggcgacgat ggcctggccggggcggtgcgggagccagcgctcggcgttt gcaaggacctggtccgtcaaattgggaggcgcattctgaa ccacaagcgagcgttcctcgggattggaactggcctcacg aggtataatgcgaaaccttgcccagctaaaactccggaaa cccccgcaaacttggcctttaactagggggcgcttagcgt

cacgtgaacgtggccagcgactagctgtactattctgcat cgtccgatctcaggattgtctacctgaccggggcgacgat ggcctggccggggcggtgcgggagccagcgctcggcgttt gcaaggacctggtccgtcaaattgggaggcgcattctgaa ccacaagcgagcgttcctcgggattggaactggcctcacg aggtataatgcgaaaccttgcccagctaaaactccggaaa cccccgcaaacttggcctttaactagggggcgcttagcgt

Three Approachs Brute Force: –check every possible pattern. Branch and Bound: –prune away some of the search space. Greedy: –commit to “nearby” options, never look back.

Brute Force Given that the pattern is of length = L. Generate all DNA patterns of length L. (Called “L-mers”). Match each one to the DNA samples. Keep the L-mer with the best match. “Best” is Based on a scoring function.

Scoring: Hamming Distance accgtaccggtaacaagtaccgtacgggtaacaagtaccgtaggtgtaacaagt gtgtaggt 4 mismatches gtgtaggt 2 mismatches gtgtaggt 8 mismatches dna sequence Try all starting positions Find the position with the fewest mismatches L=8 gtgtaggt an L-mer

Scoring t = 8 DNA samples try all possible L-mers Try each possible L-mer Score is equal to the sum of the mismatches at the locations with fewest mismatches on each string. The L-mer with the lowest such score is the optimal answer total distance = 12 12

Generating all L-mers Systematic enumeration of all DNA strings of length L. DNA has an “alphabet” of 4 letters: { a, c, g, t } Proteins have an alphabet of 20 letters: –one for each of 20 possible amino acids. –{A,B,C,D,E,F,G,H,I,K,L,M,N,P,Q,R,S,T,V,W} Solve problem for any size alphabet (k) and any size L-mer (L).

Definitions k = size of alphabet L = length of strings to be generated a = vector containing a partial or complete L-mer. i = number of entries in a already filled in. Example: k = 4, L = 5, i = 2, a = (2, 4, *, *, * )

Example Alphabet = {1, 2} k = 2, L=4 i = Depth of the Tree (1111) (2222)

NEXT VERTEX i = 3 a = i = 4 a = i = L a = j = Lj = 1 i = L a = NEXTVERTEX(a, i, L, k) if i < L a(i+1) = 1 return (a, i+1) else for j = L to j = 1 if a(j) < k then a(j) = a(j) +1 return(a, j) return (a,0)

i = L a = j = Lj = i = L-1 a = i = L a = i = i = i = i = i = i = i = i = i = i = Example: L = 6 k = 3 alhpabet = {1, 2, 3} When i = L (leaf node)

Brute Force Use NEXTVERTEX to generate nodes in the tree. Translate each numeric value into the corresponding L-mer –(e.g.: 1=a, 2=c, 3=g, 4=t). Score each L-mer (Hamming distance). keep the best L-mer (and where it matched in each dna sample).

Branch and Bound Use same structure as the Brute Force method. Looks for ways to reduce the computation. Prune branches of the tree that cannot produce anything better than what we have so far.

BYPASS BYPASS (a, i, L, k) for j = i to j = 1 –if a(j) < k a(j) = a(j) + 1 return (a, j) return (a, 0)

BRANCHANDBOUND a = (1, 1,..., 1) bestDistance = infinity i = L while (i > 0) –if i < L prefix = translate(a1, a2,..., ai) optimisticDistance = TotalDistance(prefix) if optimisticDistance > bestDistance –(a, i) = BYPASS(a, i) else –(a, i) = NEXTVERTEX( a, i ) –else word = translate (a1, a2,....., aL) if TotalDistance( word, DNA ) < bestDistance –bestDistance = TotalDistance(word, DNA) –bestWord = word (a, i) = NEXTVERTEX( a, i) return bestWord

Greedy Method Picks a “good” solution. Avoids backtracking. Can give good results. Generally, not the best possible solution. But: FAST.

Greedy Method Given t dna samples (each n-long). Find the optimal motif for the first two samples. Lock that choice in place. For the remainder of the samples: –for each dna sample in turn find the L-mer that best fits with the prior choices. never backtrack.

t = 8 DNA samples Step 1: Grab the first two samples and find the optimal alignment (consider all starting points s1 and s2, and keep the largest score). Step 2: Go through each remaining sample, successively finding the starting positions (s3, s4,...., st) that give the best consensus score for all the choices made so far.

atccagct gggcaact atggatct aagcaacc ttggaact atgcatgc Consensus Profile Alignment a g g c a a c t Scoring

Motif Finding Example n=32 t=16 L=5 atgtgaaaaggcccaggctttgttgttctgat aatcagtttgtggctctctactatgtgcgctg catggcgtaagagcaggtgtacaccgatgctg taaatacacagattccttccgactttctgcat caagccttagctttagatctttgtctcccttt gagccatggactgtccgccagtatcttcctag cgccaactgcccgtttcgcagtgccatgttga agttcccagtcccgatcataggaatttgagca tagggatcgaatgagttgtcctagtcaatcct gtagctcctcaagggatacccacctatcgacg agccgcagcgacaacttgctcgctatctaact ccactccctaagcgctgaacaccggagttctg gaagtcttcttgctgacacattacttgctcgc gaatcgtcgtatgttttcgaccttggtggcat tctcaacatgccttcccctccccaggctatgc tgtgtctatcatcccgttagctacctaaatcg

atgtgaaaaggcccaggctttgttgttctgat ***** aatcagtttgtggctctctactatgtgcgctg ***** catggcgtaagagcaggtgtacaccgatgctg ***** taaatacacagattccttccgactttctgcat ***** caagccttagctttagatctttgtctcccttt ***** gagccatggactgtccgccagtatcttcctag ***** cgccaactgcccgtttcgcagtgccatgttga ***** agttcccagtcccgatcataggaatttgagca ***** tagggatcgaatgagttgtcctagtcaatcct ***** gtagctcctcaagggatacccacctatcgacg ***** agccgcagcgacaacttgctcgctatctaact ***** ccactccctaagcgctgaacaccggagttctg ***** gaagtcttcttgctgacacattacttgctcgc ***** gaatcgtcgtatgttttcgaccttggtggcat ***** tctcaacatgccttcccctccccaggctatgc ***** tgtgtctatcatcccgttagctacctaaatcg ***** atgtgaaaaggcccaggctttgttgttctgat ***** aatcagtttgtggctctctactatgtgcgctg ***** catggcgtaagagcaggtgtacaccgatgctg ***** taaatacacagattccttccgactttctgcat ***** caagccttagctttagatctttgtctcccttt ***** gagccatggactgtccgccagtatcttcctag ***** cgccaactgcccgtttcgcagtgccatgttga ***** agttcccagtcccgatcataggaatttgagca ***** tagggatcgaatgagttgtcctagtcaatcct ***** gtagctcctcaagggatacccacctatcgacg ***** agccgcagcgacaacttgctcgctatctaact ***** ccactccctaagcgctgaacaccggagttctg ***** gaagtcttcttgctgacacattacttgctcgc ***** gaatcgtcgtatgttttcgaccttggtggcat ***** tctcaacatgccttcccctccccaggctatgc ***** tgtgtctatcatcccgttagctacctaaatcg ***** consensus_string = ctccc consensus_count = final percent score = consensus_string = atgtg consensus_count = final percent score = Branch and BoundGreedy

ggccc ctctc caccg cttcc ctccc cttcc ctgcc ttccc gtcct ctcct ctcgc ctccc ctcgc cgacc ctccc atccc consensus_string = ctccc count = final percent score = atgtg aggtg ttctg atctt atgga atgtt atttg atgag aaggg acttg aagcg aagtc atgtt acatg gtgtc consensus_string = atgtg count = final percent score = Branch and BoundGreedy

Example 2 n = 64 t = 16 L = 8 gattacttctcgcccccccgctaagtgtatttctctcgctacctactccgctatgcctacaaca tctaccggcattatctatcggcaatgggagcggtggtgatgcacctagcctactcctttgacta tggtccttactggcatcacgcaccgttcttggcggcctgtgcaatatcttgtccctaaataaat aactacggtcattagtgcgtaatcagcacagccgagccggataagcgacttgtaaccatcttcg gagcaagcatgcagtaggtaacgccaagagcggggctttagggagccgcaatcgggacagatct aaaggttctctggatctatagctcacaaatttgcaggggtacgacagagttatagagtgtacca ggcgctttcctcccgagcagagggaacgaacgaccataatgtaagagaatctttatgtccaagc cgtcctgtccatacgtatgttttcaaaactgcgtctagattagtgaggaacagatttaagattc atccagcaacttgtgcattcgtagggagcggacacaaaggacatgatcagacgaaacctatttt cctcaattgaggcccccccccagttgtccgaccgcacgaaccgcttcgcaaaagtgttgcccgc aaccacaccaagtattgctaatgcaccattcttatgtttttgagcagcaaagcgactacgctgt atataggaaaaatcttagtgcaccaagatttaacctgcactttgctttgaaatacaactgtcgg ctttcaataaatgttaattgcgttccctcacttgctcggtcgagtcgtatcgtattcgatcagg tagcgggcacgctcgctcgacgttcatccactcgatagagccggtcatttttcggaactagtaa ggaggaatgagtctacgtcgcgttaagacgaactttacgtgtgtgcaggcttattttcgtccac cctccgggggacgtagactgttcttccacagttctaggcggcgcggtcttggcttgaacaatga

gattacttctcgcccccccgctaagtgtatttctctcgctacctactccgctatgcctacaaca ******** tctaccggcattatctatcggcaatgggagcggtggtgatgcacctagcctactcctttgacta ******** tggtccttactggcatcacgcaccgttcttggcggcctgtgcaatatcttgtccctaaataaat ******** aactacggtcattagtgcgtaatcagcacagccgagccggataagcgacttgtaaccatcttcg ******** gagcaagcatgcagtaggtaacgccaagagcggggctttagggagccgcaatcgggacagatct ******** aaaggttctctggatctatagctcacaaatttgcaggggtacgacagagttatagagtgtacca ******** ggcgctttcctcccgagcagagggaacgaacgaccataatgtaagagaatctttatgtccaagc ******** cgtcctgtccatacgtatgttttcaaaactgcgtctagattagtgaggaacagatttaagattc ******** atccagcaacttgtgcattcgtagggagcggacacaaaggacatgatcagacgaaacctatttt ******** cctcaattgaggcccccccccagttgtccgaccgcacgaaccgcttcgcaaaagtgttgcccgc ******** aaccacaccaagtattgctaatgcaccattcttatgtttttgagcagcaaagcgactacgctgt ******** atataggaaaaatcttagtgcaccaagatttaacctgcactttgctttgaaatacaactgtcgg ******** ctttcaataaatgttaattgcgttccctcacttgctcggtcgagtcgtatcgtattcgatcagg ******** tagcgggcacgctcgctcgacgttcatccactcgatagagccggtcatttttcggaactagtaa ******** ggaggaatgagtctacgtcgcgttaagacgaactttacgtgtgtgcaggcttattttcgtccac ******** cctccgggggacgtagactgttcttccacagttctaggcggcgcggtcttggcttgaacaatga ******** gattacttctcgcccccccgctaagtgtatttctctcgctacctactccgctatgcctacaaca ******** tctaccggcattatctatcggcaatgggagcggtggtgatgcacctagcctactcctttgacta ******** tggtccttactggcatcacgcaccgttcttggcggcctgtgcaatatcttgtccctaaataaat ******** aactacggtcattagtgcgtaatcagcacagccgagccggataagcgacttgtaaccatcttcg ******** gagcaagcatgcagtaggtaacgccaagagcggggctttagggagccgcaatcgggacagatct ******** aaaggttctctggatctatagctcacaaatttgcaggggtacgacagagttatagagtgtacca ******** ggcgctttcctcccgagcagagggaacgaacgaccataatgtaagagaatctttatgtccaagc ******** cgtcctgtccatacgtatgttttcaaaactgcgtctagattagtgaggaacagatttaagattc ******** atccagcaacttgtgcattcgtagggagcggacacaaaggacatgatcagacgaaacctatttt ******** cctcaattgaggcccccccccagttgtccgaccgcacgaaccgcttcgcaaaagtgttgcccgc ******** aaccacaccaagtattgctaatgcaccattcttatgtttttgagcagcaaagcgactacgctgt ******** atataggaaaaatcttagtgcaccaagatttaacctgcactttgctttgaaatacaactgtcgg ******** ctttcaataaatgttaattgcgttccctcacttgctcggtcgagtcgtatcgtattcgatcagg ******** tagcgggcacgctcgctcgacgttcatccactcgatagagccggtcatttttcggaactagtaa ******** ggaggaatgagtctacgtcgcgttaagacgaactttacgtgtgtgcaggcttattttcgtccac ******** cctccgggggacgtagactgttcttccacagttctaggcggcgcggtcttggcttgaacaatga ******** consensus_string = ccatattt count = final percent score = consensus_string = cgtactcc count = final percent score = Branch and BoundGreedy

Summary Introduce the Motif Finding Problem Explain its significance in bioinformatics Develop a simple model of the problem Design algorithmic solutions: –Brute Force –Branch and Bound –Greedy Compare results of each method.

Teaching and Learning

Neural Networks for Optimization Bill Wolfe California State University Channel Islands Reference A Fuzzy Hopfield-Tank TSP Model Wolfe, W. J. INFORMS Journal on Computing, Vol. 11, No. 4, Fall 1999 pp

Neural Models Simple processing units Lots of them Highly interconnected Exchange excitatory and inhibitory signals Variety of connection architectures/strengths “Learning”: changes in connection strengths “Knowledge”: connection architecture No central processor: distributed processing

Simple Neural Model a i Activation e i External input w ij Connection Strength Assume: w ij = w ji (“symmetric” network)  W = (w ij ) is a symmetric matrix

Net Input Vector Format:

Dynamics Basic idea:

Energy

Lower Energy da/dt = net = -grad(E)  seeks lower energy

Problem: Divergence

A Fix: Saturation

Keeps the activation vector inside the hypercube boundaries Encourages convergence to corners

A Neural Model a i Activation e i External Input w ij Connection Strength W (w ij = w ji ) Symmetric

Example: Inhibitory Networks Completely inhibitory –wij = -1 for all i,j –winner take all Inhibitory Grid –neighborhood inhibition –on-center, off-surround

Traveling Salesman Problem Classic combinatorial optimization problem Find the shortest “tour” through n cities n!/2n distinct tours

TSP solution for 15,000 cities in Germany Ref:

TSP 50 City Example

Random Tour

Nearest-City Tour

2-OPT Tour

Centroid Tour

Monotonic Tour

Neural Network Approach neuron

Tours – Permutation Matrices tour: CDBA permutation matrices correspond to the “feasible” states.

Not Allowed

Only one city per time stop Only one time stop per city  Inhibitory rows and columns inhibitory

Distance Connections: Inhibit the neighboring cities in proportion to their distances.

putting it all together:

E = -1/2 { ∑ i ∑ x ∑ j ∑ y a ix a jy w ixjy } = -1/2 { ∑ i ∑ x ∑ y (- d(x,y)) a ix ( a i+1 y + a i-1 y ) + ∑ i ∑ x ∑ j (-1/n) a ix a jx + ∑ i ∑ x ∑ y (-1/n) a ix a iy + ∑ i ∑ x ∑ j ∑ y (1/n 2 ) a ix a jy }

Hopfield JJ, Tank DW. Neural computation of decisions in optimization problems. Biological Cybernetics 1985;52:

typical state of the network before convergence x x x x x x x Fuzzy Tour: GAECBFD

“Fuzzy Readout”

Fuzzy Tour Lengths tour length iteration

Average Results for n=10 to n=70 cities (50 random runs per n) # cities

Conclusions Neurons stimulate intriguing computational models. The models are complex, nonlinear, and difficult to analyze. The interaction of many simple processing units is difficult to visualize. The Neural Model for the TSP mimics some of the properties of the nearest-city heuristic. Much work to be done to understand these models.