BIC I, Week 4 lectures

transcript

Rhys Price Jones and Anne Haake

Rochester Institute of Technology

rpjavp@rit.edu, arh@it.rit.edu

Overview of the need for Dynamic Programming

• Consider Fibonacci• The obvious algorithm is elegant, easily

derived from the definition, and clearly correct.(define fib (lambda (n) (if (<= n 1) 1 (+ (fib (- n 2)) (fib (- n 1))))))

• But it’s hopelessly inefficient• Why?• Because it makes repeated recursive calls

with the same argument

The Traditional Solution

• Change the order in which the computations are performed

• Change the logic of the program– So that it works “bottom up” instead of “top down”– Fill an array with calculated values starting with (fib 0), then

(fib 1) then (fib 2), etc.

• You can do it manually, as in fib.ss• That is dynamic programming!• The main problem is that it requires thought and

programming and hence may introduce error.

It’s not just Fibonacci

• Many programs “write themselves” from the specification of the problem.

• When that happens, we are extremely pleased

• Sadly, the resulting program is often inefficient

• But dynamic programming is a technique to make it efficient again.

Memo-izing

• Redefine the function calling mechanism so that:– We first check to see if we’ve made that calculation before– If no, go ahead and compute it but store the result in a hash

table– If yes, look up the previously computed value in the hash

• Do it once• Inefficient code becomes efficient automatically with

no re-programming memolambda.ss

memofib.ssmemofib.ss

Another Example

• Pascal’s triangle• Each entry is the sum of its parents

– Cn,k = Cn-1,k-1 + Cn-1,k

– C0,k = Cn,0 = 1

• Leading to program• Runs really slowly• Replace lambda by memolambda

badcomb.ss

goodcomb.ss

Review of Pattern Matching

• Does CGGA appear within the sequence ATCGCGTAACGGAGATAGGCTTA ?

• More generally, where does pattern p (length n) appear within text t (length m)

• Boyer-Moore, or Knuth-Morris-Pratt give O(m+n) search

• If p is going to change a lot and t stay the same, suffix tree can be built in O(m), each search is then O(n)

• If p is stable and there are lots of different t, virtual machine can be built in O(n) and then each search is O(m)

Build a Virtual Machine

• CGGA

C G G A

ATCGCGTAACGGAGATAGGCTTA

First Step

• CGGA

C G G A

Second Step

• CGGA

C G G A

Third Step

• CGGA

C G G A

Fourth Step

• CGGA

C G G A

Fifth Step

• CGGA

C G G A

Sixth Step

• CGGA

C G G A

Seventh Step

• CGGA

C G G A

Eighth Step

• CGGA

C G G A

Ninth Step

• CGGA

C G G A

Tenth Step

• CGGA

C G G A

Eleventh Step

• CGGA

C G G A

Twelfth Step

• CGGA

C G G A

Thirteenth Step

• CGGA

C G G A

Fourteenth Step

• CGGA

C G G A

Fifteenth Step

• CGGA

C G G A

Sixteenth Step

• CGGA

C G G A

17th – 23rd Steps

• CGGA

C G G A

Pattern Matching – Conclusion

• Exact pattern matching is easy• Often the naive algorithm is good enough• Fast algorithms are readily available• Sadly, not much use for biological tasks

Why not?

• What’s the difference?• Mutation• Insertion/deletion gaps• We need an inexact way to compare two (or

more) biological sequences

Pattern Matching vs. Sequence Alignment

• In the CS world, we talk of comparing strings, or matching patterns of characters within strings

• For biological applications, we talk of comparing sequences, or aligning sequences of nucleotides (or amino acids) to each other

Evolutionary Relatedness

• Consider ACCGT and CACGT• How likely is it that they are “related”?• Possible alignments:• ACCGT AC-CGTXX||| -|-|||CACGT -CACGT

• Which is better?

It Depends

• ACCGT AC-CGTXX||| -|-|||CACGT -CACGT

• Scoring 2 for a match, -2 for a mismatch, and –1 for a gap, 2 versus 6

• Scoring 2 for a match, 0 for a mismatch and –2 for a gap, 6 versus 4

• And we haven’t even begun to consider experimental evidence that might cause us to rank some mutations better than others!

Distance measure

• Score 0 for a match• 1 for a mismatch or gap• Low score best!• ACCGT AC-CGTXX||| -|-|||CACGT -CACGT

• Now it’s 2 versus 2

Global alignment

• For two sequences• - A C C A C C-ACACC

• Use the scoring scheme to fill in the table, starting with first row and first column

First entries

• Using the distance measure• - A C C A C C- 0 1 2 3 4 5 6A 1 C 2A 3C 4A 5

• Each nucleotide<->gap costs 1 point

Extending inwards

• Extending the distance measure• - A C C A C C- 0 1 2 3 4 5 6A 1 0 1 2 3 4 5 C 2 1 A 3 2 C 4 3 A 5 4

• Extending from North or West costs 1 point, from NW costs 0 (match) or 1 (mismatch)

• Pick cheapest of the three

More extension

• - A C C A C C- 0 1 2 3 4 5 6A 1 0 1 2 3 4 5 C 2 1 0 1 2 3 4 A 3 2 1 1 C 4 3 2 1 A 5 4 3 2

• mi,j = min (mi,j-1+g mi-1,j+g mi-1,j-1+cij)

• where cij = 0 for a match, 1 for a mismatch

Getting there...

• - A C C A C C- 0 1 2 3 4 5 6A 1 0 1 2 3 4 5 C 2 1 0 1 2 3 4A 3 2 1 1 1 2 3C 4 3 2 1 2 A 5 4 3 2 1

• mi,j = min (mi,j-1+1 mi-1,j+1 mi-1,j-1+cij)

Almost done...

• - A C C A C C- 0 1 2 3 4 5 6A 1 0 1 2 3 4 5 C 2 1 0 1 2 3 4A 3 2 1 1 1 2 3C 4 3 2 1 2 1 2A 5 4 3 2 1 2

• mi,j = min (mi,j-1+1 mi-1,j+1 mi-1,j-1+cij)

Finally, we can get a Global alignment

• One of the least-cost routes• - A C C A C C- 0 1 2 3 4 5 6A 1 0 1 2 3 4 5 C 2 1 0 1 2 3 4A 3 2 1 1 1 2 3C 4 3 2 1 2 1 2A 5 4 3 2 1 2 2

• Can you see how this path leads to the alignment• ACCACCAC-ACA

Global alignment program

• Distance measure• Runnable program• Dynamic Programming version

globalig.txt

globalig.ss

globaligm.ss

Global vs Local Alignment

• Global alignment seeks the best alignment between the complete sequence and the complete sequenceA global alignment between GATCCACCA and GTAACACA might be

• G-ATCCACCA|-|X|-||-|GTAAC-AC-A

• A local alignment is the best alignment between subsequences. A local alignment between GATCCACCA and GTAACACA might be

• gATCCACca |X|-||gtAAC-ACa

• Best local alignment depends on scoring scheme

Local Alignment

• For this demo, we will use a different measure– 2 for a match– -1 for a mismatch, -2 for a gap– Find best match withinG C T C T G C G A A T A GC G T T G A G A T A C T C

The solution

• - G C T C T G C G A A T A G

- 0 0 0 0 0 0 0 0 0 0 0 0 0 0

C 0 0 2 0 2 0 0 2 0 0 0 0 0 0

G 0 2 0 1 0 1 2 0 4 2 0 0 0 2

T 0 0 1 2 0 2 0 1 2 3 1 2 0 0

T 0 0 0 3 1 2 1 0 0 1 2 3 1 0

G 0 2 0 1 2 0 4 2 2 0 0 1 2 3

A 0 0 1 0 0 1 2 3 1 4 2 0 3 1

G 0 2 0 0 0 0 3 1 5 3 3 1 1 5

A 0 0 1 0 0 0 1 2 3 7 5 3 3 3

T 0 0 0 3 1 2 0 0 1 5 6 7 5 3

A 0 0 0 1 2 0 1 0 0 3 7 5 9 7

C 0 0 2 0 3 1 0 3 1 1 5 6 7 8

T 0 0 0 4 2 5 3 1 2 0 3 7 5 6

C 0 0 2 2 6 4 4 5 3 1 1 5 6 4

• G C T C T G C G A A T A G

| | x | | X | |

C G T T G A G A - T A C T C

The Program

• Has dynamic programming to make it fast!• This is basically Smith-Waterman• Work has been done on different scoring

schemes, gap penalties, etc.• Runs in time O(mn)

localig.ss

Exercises

• that we will attempt in class:– amend global alignment program to do the

“backtracking” needed for the alignment

• that will be homework– amend local alignment program to do the

“backtracking” needed for the alignment

BIC I, Week 4 lectures

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