0
DNA Mapping and Brute
Force Algorithms
1. Restriction Enzymes
2. Gel Electrophoresis
3. Partial Digest Problem
4. Brute Force Algorithm for Partial Digest Proble...
Section 1:
Restriction Enzymes
Discovery of Restriction Enzymes
• HindII: First restriction enzyme.
• Was discovered accidentally in 1970 while scientist...
Discovering Restriction Enzymes
Werner Arber
My father has discovered a
servant who serves as a pair of
scissors. If a for...
Molecular Cell Biology, 4thEdition
Molecular Scissors
Restriction Enzymes: Common Recognition Sites
Molecular Cell Biology, 4thEdition
• Recombinant DNA technology
• Cloning
• cDNA/genomic library construction
• DNA mapping
Uses of Restriction Enzymes
Recombinant DNA technology
cDNA Library
• A restriction map is a map showing positions of restriction sites
in a DNA sequence.
• If DNA sequence is known then con...
Full Restriction Digest
• Cutting DNA at each restriction site creates multiple
restriction fragments:
• Full Restriction ...
Full Restriction Digest: Multiple Solutions
• For the set of fragment lengths {3, 5, 5, 9} we have the
original segment as...
Section 2:
Gel Electrophoresis
• Restriction enzymes break DNA into restriction fragments.
• Gel electrophoresis:A process for separating DNA by size
and...
• DNA fragments are injected into a gel positioned in an
electric field.
• DNA are negatively charged near neutral pH.
• T...
Gel Electrophoresis
• DNA fragments of different lengths are separated according to
size.
• Smaller molecules move through...
Gel Electrophoresis
Direction of DNA
movement
Detecting DNA: Autoradiography
• Separated DNA bands on a
gel can be viewed via
autoradiography:...
• Another way to visualize DNA bands in gel is through
fluorescence:
• The gel is incubated with a solution containing the...
Section 3:
Partial Digest Problem
• The sample of DNA is exposed to the restriction enzyme for
only a limited amount of time to prevent it from being cut at...
• Partial Digest results in the following 10 restriction fragments:
Partial Digest: Example
• We assume that multiplicity of a fragment can be detected, i.e.,
the number of restriction fragments of the same length ...
Multiset: {3, 5, 5, 8, 9, 14, 14, 17, 19, 22}
Partial Digest: Example
• We therefore have a multiset of fragment lengths.
• We now provide a basic mathematical framework for the
partial digest process.
The multiset of integers representing leng...
Return to Partial Digest Example
Return to Partial Digest Example
0 5 14 19 22
• n = 5
Return to Partial Digest Example
0 5 14 19 22
• n = 5
• X = {0, 5, 14, 19, 22}
Return to Partial Digest Example
0 5 14 19 22
Return to Partial Digest Example
0 5 14 19 22
• n = 5
• X = {0, 5, 14, 19, 22}
• DX = {3,5,5,8,9,14,14,17,19,22}
• n = 5
• X = {0, 5, 14, 19, 22}
• DX = {3,5,5,8,9,14,14,17,19,22}
• Represent DX as a table, with
elements of X along bot...
• n = 5
• X = {0, 5, 14, 19, 22}
• DX = {3,5,5,8,9,14,14,17,19,22}
• Represent DX as a table, with
elements of X along bot...
• n = 5
• X = {0, 5, 14, 19, 22}
• DX = {3,5,5,8,9,14,14,17,19,22}
• Represent DX as a table, with
elements of X along bot...
• Goal: Given all pairwise distances between points on a line,
reconstruct the positions of those points.
• Input: The mul...
• It is not always possible to uniquely reconstruct a set X based
only on ∆X.
• Example:The sets
X = {0, 2, 5} (X + 10) = ...
X = {0,1,2,5,7,9,12}
Homometric Sets: Example
0 1 2 5 7 9 12
0 1 2 5 7 9 12
1 1 4 6 8 11
2 3 5 7 10
5 2 4 7
7 2 5
9 3
12
X = {0,1,2,5,7,9,12}
Homometric Sets: Example
0 1 2 5 7 9 12
0 1 2 5 7 9 12
1 1 4 6 8 11
2 3 5 7 10
5 2 4 7
7 2 5
9 3
12
X = {0,1,2,5,7,9,12} Y = {0,1,5,7,8,10,12}
Homo...
0 1 2 5 7 9 12
0 1 2 5 7 9 12
1 1 4 6 8 11
2 3 5 7 10
5 2 4 7
7 2 5
9 3
12
0 1 5 7 8 10 12
0 1 5 7 8 10 12
1 4 6 7 9 11
5 ...
0 1 2 5 7 9 12
0 1 2 5 7 9 12
1 1 4 6 8 11
2 3 5 7 10
5 2 4 7
7 2 5
9 3
12
0 1 5 7 8 10 12
0 1 5 7 8 10 12
1 4 6 7 9 11
5 ...
0 1 2 5 7 9 12
0 1 2 5 7 9 12
1 1 4 6 8 11
2 3 5 7 10
5 2 4 7
7 2 5
9 3
12
0 1 5 7 8 10 12
0 1 5 7 8 10 12
1 4 6 7 9 11
5 ...
0 1 2 5 7 9 12
0 1 2 5 7 9 12
1 1 4 6 8 11
2 3 5 7 10
5 2 4 7
7 2 5
9 3
12
0 1 5 7 8 10 12
0 1 5 7 8 10 12
1 4 6 7 9 11
5 ...
0 1 2 5 7 9 12
0 1 2 5 7 9 12
1 1 4 6 8 11
2 3 5 7 10
5 2 4 7
7 2 5
9 3
12
0 1 5 7 8 10 12
0 1 5 7 8 10 12
1 4 6 7 9 11
5 ...
0 1 2 5 7 9 12
0 1 2 5 7 9 12
1 1 4 6 8 11
2 3 5 7 10
5 2 4 7
7 2 5
9 3
12
0 1 5 7 8 10 12
0 1 5 7 8 10 12
1 4 6 7 9 11
5 ...
0 1 2 5 7 9 12
0 1 2 5 7 9 12
1 1 4 6 8 11
2 3 5 7 10
5 2 4 7
7 2 5
9 3
12
0 1 5 7 8 10 12
0 1 5 7 8 10 12
1 4 6 7 9 11
5 ...
0 1 2 5 7 9 12
0 1 2 5 7 9 12
1 1 4 6 8 11
2 3 5 7 10
5 2 4 7
7 2 5
9 3
12
0 1 5 7 8 10 12
0 1 5 7 8 10 12
1 4 6 7 9 11
5 ...
0 1 2 5 7 9 12
0 1 2 5 7 9 12
1 1 4 6 8 11
2 3 5 7 10
5 2 4 7
7 2 5
9 3
12
0 1 5 7 8 10 12
0 1 5 7 8 10 12
1 4 6 7 9 11
5 ...
0 1 2 5 7 9 12
0 1 2 5 7 9 12
1 1 4 6 8 11
2 3 5 7 10
5 2 4 7
7 2 5
9 3
12
0 1 5 7 8 10 12
0 1 5 7 8 10 12
1 4 6 7 9 11
5 ...
0 1 2 5 7 9 12
0 1 2 5 7 9 12
1 1 4 6 8 11
2 3 5 7 10
5 2 4 7
7 2 5
9 3
12
0 1 5 7 8 10 12
0 1 5 7 8 10 12
1 4 6 7 9 11
5 ...
Section 4:
Brute Force Algorithm for
Partial Digest Problem
• Brute force algorithms, also known as exhaustive search
algorithms, examine every possible variant to find a solution.
•...
1. Find the restriction fragment of maximum length M. Note: M
is the length of the DNA sequence.
2. For every possible set...
Partial Digest: Brute Force
1. AnotherBruteForcePDP(L, n)
2. M  maximum element in L
3. for every set of n – 2 integers 0...
• BruteForcePDP takes O(M n-2) time since it must examine all
possible sets of positions. Note: the number of such sets is...
Another BruteForcePDP
• Limiting the members of X to those contained in L is almost
identical to BruteForcePDP, except for...
Another BruteForcePDP
• Limiting the members of X to those contained in L is almost
identical to BruteForcePDP, except for...
• More efficient than BruteForce PDP, but still slow.
• If L = {2, 998, 1000} (n = 3, M= 1000), BruteForcePDP will
be extr...
Section 5:
Branch and Bound
Algorithm for Partial
Digest Problem
1. Begin with X = {0}.
Branch and Bound Algorithm for PDP
1. Begin with X = {0}.
2. Remove the largest element in L and place it in X.
Branch and Bound Algorithm for PDP
1. Begin with X = {0}.
2. Remove the largest element in L and place it in X.
3. See if the element fits on the right or le...
1. Begin with X = {0}.
2. Remove the largest element in L and place it in X.
3. See if the element fits on the right or le...
1. Begin with X = {0}.
2. Remove the largest element in L and place it in X.
3. See if the element fits on the right or le...
1. Begin with X = {0}.
2. Remove the largest element in L and place it in X.
3. See if the element fits on the right or le...
• Before describing PartialDigest, first define D(y, X) as the
multiset of all distances between point y and all other poi...
return15
to),(lengthsaddandfromRemove14
),PLACE(13
from),(lengthsremoveandtoAdd12
),(if11
to),(lengthsaddandfromRemove10
)...
L = { 2, 2, 3, 3, 4, 5, 6, 7, 8, 10 }
X = { }
PartialDigest: Example
L = { 2, 2, 3, 3, 4, 5, 6, 7, 8, 10 }
X = { 0 }
• Remove 10 from L and insert it (along with 0) into X. We know
this must ...
L = { 2, 2, 3, 3, 4, 5, 6, 7, 8, 10 }
X = { 0, 10 }
PartialDigest: Example
L = { 2, 2, 3, 3, 4, 5, 6, 7, 8, 10 }
X = { 0, 10 }
• Take 8 from L and make y = 2 or 8. But since the two cases are
symme...
L = { 2, 2, 3, 3, 4, 5, 6, 7, 8, 10 }
X = { 0, 10 }
• We find that the distances from y=2 to other elements in X are
D(y, ...
L = { 2, 2, 3, 3, 4, 5, 6, 7, 8, 10 }
X = { 0, 2, 10 }
PartialDigest: Example
L = { 2, 2, 3, 3, 4, 5, 6, 7, 8, 10 }
X = { 0, 2, 10 }
• Take 7 from L and make y = 7 or y = 10 – 7 = 3. We will
explore y...
L = { 2, 2, 3, 3, 4, 5, 6, 7, 8, 10 }
X = { 0, 2, 10 }
• For y = 7 first, D(y, X ) = {7, 5, 3}. Therefore we remove {7, 5
...
L = { 2, 2, 3, 3, 4, 5, 6, 7, 8, 10 }
X = { 0, 2, 7, 10 }
PartialDigest: Example
L = { 2, 2, 3, 3, 4, 5, 6, 7, 8, 10 }
X = { 0, 2, 7, 10 }
• Take 6 from L and make y = 6.
• Unfortunately D(y, X) = {6, 4,...
L = { 2, 2, 3, 3, 4, 5, 6, 7, 8, 10 }
X = { 0, 2, 7, 10 }
• This time make y = 4. D(y, X) = {4, 2, 3 ,6}, which is a subse...
L = { 2, 2, 3, 3, 4, 5, 6, 7, 8, 10 }
X = { 0, 2, 4, 7, 10 }
PartialDigest: Example
L = { 2, 2, 3, 3, 4, 5, 6, 7, 8, 10 }
X = { 0, 2, 4, 7, 10 }
• L is now empty, so we have a solution, which is X.
PartialD...
L = { 2, 2, 3, 3, 4, 5, 6, 7, 8, 10 }
X = { 0, 2, 7, 10 }
• To find other solutions, we backtrack.
PartialDigest: Example
L = { 2, 2, 3, 3, 4, 5, 6, 7, 8, 10 }
X = { 0, 2, 10 }
• More backtrack.
PartialDigest: Example
L = { 2, 2, 3, 3, 4, 5, 6, 7, 8, 10 }
X = { 0, 2, 10 }
• This time we will explore y = 3. D(y, X) = {3, 1, 7}, which is
no...
L = { 2, 2, 3, 3, 4, 5, 6, 7, 8, 10 }
X = { 0, 10 }
• We backtracked back to the root. Therefore we have found all
the sol...
• Still exponential in worst case, but is very fast on average.
• Informally, let T(n) be time PartialDigest takes to plac...
Section 6:
Double Digest Problem
• Double Digest is yet another experimentally method to
construct restriction maps
• Use two restriction enzymes; three fu...
Double Digest: Example
• Without the information about X (i.e. A+B), it is impossible to
solve the double digest problem as this diagram illustra...
• Input:
• dA – fragment lengths from the digest with enzyme A.
• dB – fragment lengths from the digest with enzyme B.
• d...
Double Digest: Multiple Solutions
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Transcript of "Ch04 dna mapping"

  1. 1. DNA Mapping and Brute Force Algorithms
  2. 2. 1. Restriction Enzymes 2. Gel Electrophoresis 3. Partial Digest Problem 4. Brute Force Algorithm for Partial Digest Problem 5. Branch and Bound Algorithm for Partial Digest Problem 6. Double Digest Problem Outline
  3. 3. Section 1: Restriction Enzymes
  4. 4. Discovery of Restriction Enzymes • HindII: First restriction enzyme. • Was discovered accidentally in 1970 while scientists were studying how the bacterium Haemophilus influenzae takes up DNA from the virus. • Recognizes and cuts DNA at sequences: • GTGCAC • GTTAAC
  5. 5. Discovering Restriction Enzymes Werner Arber My father has discovered a servant who serves as a pair of scissors. If a foreign king invades a bacterium, this servant can cut him in small fragments, but he does not do any harm to his own king. Clever people use the servant with the scissors to find out the secrets of the kings. For this reason my father received the Nobel Prize for the discovery of the servant with the scissors.” Daniel Nathans’ daughter (from Nobel lecture) Hamilton SmithDaniel Nathans Werner Arber – discovered restriction enzymes Daniel Nathans - pioneered the application of restriction for the construction of genetic maps Hamilton Smith - showed that restriction enzyme cuts DNA in the middle of a specific sequence
  6. 6. Molecular Cell Biology, 4thEdition Molecular Scissors
  7. 7. Restriction Enzymes: Common Recognition Sites Molecular Cell Biology, 4thEdition
  8. 8. • Recombinant DNA technology • Cloning • cDNA/genomic library construction • DNA mapping Uses of Restriction Enzymes
  9. 9. Recombinant DNA technology
  10. 10. cDNA Library
  11. 11. • A restriction map is a map showing positions of restriction sites in a DNA sequence. • If DNA sequence is known then construction of restriction map is trivial exercise. • In early days of molecular biology DNA sequences were often unknown. • Biologists had to solve the problem of constructing restriction maps without knowing DNA sequences. Restriction Maps
  12. 12. Full Restriction Digest • Cutting DNA at each restriction site creates multiple restriction fragments: • Full Restriction Digest: Is it possible to reconstruct the order of the fragments from the sizes of the fragments? • Example: Say the fragments have lengths {3,5,5,9} as in the above sequence.
  13. 13. Full Restriction Digest: Multiple Solutions • For the set of fragment lengths {3, 5, 5, 9} we have the original segment as a possible solution: • However, we could also have the following segment:
  14. 14. Section 2: Gel Electrophoresis
  15. 15. • Restriction enzymes break DNA into restriction fragments. • Gel electrophoresis:A process for separating DNA by size and measuring sizes of restriction fragments. • Modern electrophoresis machines can separate DNA fragments that differ in length by 1 nucleotide for fragments up to 500 nucleotides long. Gel Electrophoresis: Measure Segment Lengths
  16. 16. • DNA fragments are injected into a gel positioned in an electric field. • DNA are negatively charged near neutral pH. • The ribose phosphate backbone of each nucleotide is acidic; DNA has an overall negative charge. • Thus DNA molecules move towards the positive electrode. Gel Electrophoresis: How It Works
  17. 17. Gel Electrophoresis • DNA fragments of different lengths are separated according to size. • Smaller molecules move through the gel matrix more readily than larger molecules. • The gel matrix restricts random diffusion so molecules of different lengths separate into different bands.
  18. 18. Gel Electrophoresis
  19. 19. Direction of DNA movement Detecting DNA: Autoradiography • Separated DNA bands on a gel can be viewed via autoradiography: 1. DNA is radioactively labeled. 2. The gel is laid against a sheet of photographic film in the dark, exposing the film at the positions where the DNA is present. Molecular Cell Biology, 4th edition
  20. 20. • Another way to visualize DNA bands in gel is through fluorescence: • The gel is incubated with a solution containing the fluorescent dye ethidium. • Ethidium binds to the DNA. • The DNA lights up when the gel is exposed to ultraviolet light. Detecting DNA: Fluorescence
  21. 21. Section 3: Partial Digest Problem
  22. 22. • 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; this procedure is called partial (restriction) digest. • This experiment generates the set of all possible restriction fragments between every two (not necessarily consecutive) cuts. • This set of fragment sizes is used to determine the positions of the restriction sites in the DNA sequence. Partial Restriction Digest
  23. 23. • Partial Digest results in the following 10 restriction fragments: Partial Digest: Example
  24. 24. • We assume that multiplicity of a fragment can be detected, i.e., the number of restriction fragments of the same length can be determined. • Here we would detect two fragments of length 5 and two of length 14. Partial Digest: Example
  25. 25. Multiset: {3, 5, 5, 8, 9, 14, 14, 17, 19, 22} Partial Digest: Example • We therefore have a multiset of fragment lengths.
  26. 26. • We now provide a basic mathematical framework for the partial digest process. The multiset of integers representing lengths of each of the DNA fragments produced from a partial digest; formed from X by taking all pairwise differences. The set of n integers representing the location of all cuts in the restriction map, including the start and end. X: DX: Partial Digest: Mathematical Framework
  27. 27. Return to Partial Digest Example
  28. 28. Return to Partial Digest Example 0 5 14 19 22
  29. 29. • n = 5 Return to Partial Digest Example 0 5 14 19 22
  30. 30. • n = 5 • X = {0, 5, 14, 19, 22} Return to Partial Digest Example 0 5 14 19 22
  31. 31. Return to Partial Digest Example 0 5 14 19 22 • n = 5 • X = {0, 5, 14, 19, 22} • DX = {3,5,5,8,9,14,14,17,19,22}
  32. 32. • n = 5 • X = {0, 5, 14, 19, 22} • DX = {3,5,5,8,9,14,14,17,19,22} • Represent DX as a table, with elements of X along both the top and left sides. Return to Partial Digest Example
  33. 33. • n = 5 • X = {0, 5, 14, 19, 22} • DX = {3,5,5,8,9,14,14,17,19,22} • Represent DX as a table, with elements of X along both the top and left sides. X 0 5 14 19 22 0 5 14 19 22 Return to Partial Digest Example
  34. 34. • n = 5 • X = {0, 5, 14, 19, 22} • DX = {3,5,5,8,9,14,14,17,19,22} • Represent DX as a table, with elements of X along both the top and left sides. • We place xj – xi into entry (i,j) for all 1 ≤ i<j ≤ n X 0 5 14 19 22 0 5 14 19 22 5 9 14 17 14 5 8 19 3 22 Return to Partial Digest Example
  35. 35. • Goal: Given all pairwise distances between points on a line, reconstruct the positions of those points. • Input: The multiset of pairwise distances L, containing n(n- 1)/2 integers. • Output: A set X, of n integers, such that ∆X = L. Partial Digest Problem (PDP): Formulation
  36. 36. • It is not always possible to uniquely reconstruct a set X based only on ∆X. • Example:The sets X = {0, 2, 5} (X + 10) = {10, 12, 15} both produce ∆X = ∆(X + 10) ={2, 3, 5} as their partial digest. • Two sets X and Y are homometric if ∆X = ∆Y. • The sets {0,1,2,5,7,9,12} and {0,1,5,7,8,10,12} present a less trivial example of homometric sets. 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} Multiple Solutions to the PDP
  37. 37. X = {0,1,2,5,7,9,12} Homometric Sets: Example
  38. 38. 0 1 2 5 7 9 12 0 1 2 5 7 9 12 1 1 4 6 8 11 2 3 5 7 10 5 2 4 7 7 2 5 9 3 12 X = {0,1,2,5,7,9,12} Homometric Sets: Example
  39. 39. 0 1 2 5 7 9 12 0 1 2 5 7 9 12 1 1 4 6 8 11 2 3 5 7 10 5 2 4 7 7 2 5 9 3 12 X = {0,1,2,5,7,9,12} Y = {0,1,5,7,8,10,12} Homometric Sets: Example
  40. 40. 0 1 2 5 7 9 12 0 1 2 5 7 9 12 1 1 4 6 8 11 2 3 5 7 10 5 2 4 7 7 2 5 9 3 12 0 1 5 7 8 10 12 0 1 5 7 8 10 12 1 4 6 7 9 11 5 2 3 5 7 7 1 3 5 8 2 4 10 2 12 X = {0,1,2,5,7,9,12} Y = {0,1,5,7,8,10,12} Homometric Sets: Example
  41. 41. 0 1 2 5 7 9 12 0 1 2 5 7 9 12 1 1 4 6 8 11 2 3 5 7 10 5 2 4 7 7 2 5 9 3 12 0 1 5 7 8 10 12 0 1 5 7 8 10 12 1 4 6 7 9 11 5 2 3 5 7 7 1 3 5 8 2 4 10 2 12 X = {0,1,2,5,7,9,12} Y = {0,1,5,7,8,10,12} Homometric Sets: Example
  42. 42. 0 1 2 5 7 9 12 0 1 2 5 7 9 12 1 1 4 6 8 11 2 3 5 7 10 5 2 4 7 7 2 5 9 3 12 0 1 5 7 8 10 12 0 1 5 7 8 10 12 1 4 6 7 9 11 5 2 3 5 7 7 1 3 5 8 2 4 10 2 12 X = {0,1,2,5,7,9,12} Y = {0,1,5,7,8,10,12} Homometric Sets: Example
  43. 43. 0 1 2 5 7 9 12 0 1 2 5 7 9 12 1 1 4 6 8 11 2 3 5 7 10 5 2 4 7 7 2 5 9 3 12 0 1 5 7 8 10 12 0 1 5 7 8 10 12 1 4 6 7 9 11 5 2 3 5 7 7 1 3 5 8 2 4 10 2 12 X = {0,1,2,5,7,9,12} Y = {0,1,5,7,8,10,12} Homometric Sets: Example
  44. 44. 0 1 2 5 7 9 12 0 1 2 5 7 9 12 1 1 4 6 8 11 2 3 5 7 10 5 2 4 7 7 2 5 9 3 12 0 1 5 7 8 10 12 0 1 5 7 8 10 12 1 4 6 7 9 11 5 2 3 5 7 7 1 3 5 8 2 4 10 2 12 X = {0,1,2,5,7,9,12} Y = {0,1,5,7,8,10,12} Homometric Sets: Example
  45. 45. 0 1 2 5 7 9 12 0 1 2 5 7 9 12 1 1 4 6 8 11 2 3 5 7 10 5 2 4 7 7 2 5 9 3 12 0 1 5 7 8 10 12 0 1 5 7 8 10 12 1 4 6 7 9 11 5 2 3 5 7 7 1 3 5 8 2 4 10 2 12 X = {0,1,2,5,7,9,12} Y = {0,1,5,7,8,10,12} Homometric Sets: Example
  46. 46. 0 1 2 5 7 9 12 0 1 2 5 7 9 12 1 1 4 6 8 11 2 3 5 7 10 5 2 4 7 7 2 5 9 3 12 0 1 5 7 8 10 12 0 1 5 7 8 10 12 1 4 6 7 9 11 5 2 3 5 7 7 1 3 5 8 2 4 10 2 12 X = {0,1,2,5,7,9,12} Y = {0,1,5,7,8,10,12} Homometric Sets: Example
  47. 47. 0 1 2 5 7 9 12 0 1 2 5 7 9 12 1 1 4 6 8 11 2 3 5 7 10 5 2 4 7 7 2 5 9 3 12 0 1 5 7 8 10 12 0 1 5 7 8 10 12 1 4 6 7 9 11 5 2 3 5 7 7 1 3 5 8 2 4 10 2 12 X = {0,1,2,5,7,9,12} Y = {0,1,5,7,8,10,12} Homometric Sets: Example
  48. 48. 0 1 2 5 7 9 12 0 1 2 5 7 9 12 1 1 4 6 8 11 2 3 5 7 10 5 2 4 7 7 2 5 9 3 12 0 1 5 7 8 10 12 0 1 5 7 8 10 12 1 4 6 7 9 11 5 2 3 5 7 7 1 3 5 8 2 4 10 2 12 X = {0,1,2,5,7,9,12} Y = {0,1,5,7,8,10,12} Homometric Sets: Example
  49. 49. 0 1 2 5 7 9 12 0 1 2 5 7 9 12 1 1 4 6 8 11 2 3 5 7 10 5 2 4 7 7 2 5 9 3 12 0 1 5 7 8 10 12 0 1 5 7 8 10 12 1 4 6 7 9 11 5 2 3 5 7 7 1 3 5 8 2 4 10 2 12 X = {0,1,2,5,7,9,12} Y = {0,1,5,7,8,10,12} Homometric Sets: Example
  50. 50. 0 1 2 5 7 9 12 0 1 2 5 7 9 12 1 1 4 6 8 11 2 3 5 7 10 5 2 4 7 7 2 5 9 3 12 0 1 5 7 8 10 12 0 1 5 7 8 10 12 1 4 6 7 9 11 5 2 3 5 7 7 1 3 5 8 2 4 10 2 12 X = {0,1,2,5,7,9,12} Y = {0,1,5,7,8,10,12} Homometric Sets: Example
  51. 51. Section 4: Brute Force Algorithm for Partial Digest Problem
  52. 52. • Brute force algorithms, also known as exhaustive search algorithms, examine every possible variant to find a solution. • Efficient only in rare cases; usually impractical. Brute Force Algorithms
  53. 53. 1. Find the restriction fragment of maximum length M. Note: M is the length of the DNA sequence. 2. For every possible set X={0, x2, … ,xn-1, M} compute the corresponding DX. 3. If DX is equal to the experimental partial digest L, then X is a possible restriction map. Partial Digest: Brute Force
  54. 54. Partial Digest: Brute Force 1. AnotherBruteForcePDP(L, n) 2. M  maximum element in L 3. for every set of n – 2 integers 0 < x2 < … xn-1 < M 4. X { 0,x2,…,xn-1,M } 5. Form DX from X 6. if DX = L 7. return X 8. output “no solution”
  55. 55. • BruteForcePDP takes O(M n-2) time since it must examine all possible sets of positions. Note: the number of such sets is • One way to improve the algorithm is to limit the values of xi to only those values which occur in L, because we are assuming for the sake of simplicity that 0 is contained in X. )()2()2)(1( )!2( 1 )!1()!2( )!1( 2 1 2              n MOnMMM nnMn M n M  Efficiency of BruteForcePDP
  56. 56. Another BruteForcePDP • Limiting the members of X to those contained in L is almost identical to BruteForcePDP, except for line 3: 1. AnotherBruteForcePDP(L, n) 2. M  maximum element in L 3. for every set of n – 2 integers 0 < x2 < … xn-1 < M 4. X  { 0,x2,…,xn-1,M } 5. Form DX from X 6. if DX = L 7. return X 8. output “no solution”
  57. 57. Another BruteForcePDP • Limiting the members of X to those contained in L is almost identical to BruteForcePDP, except for line 3: 1. AnotherBruteForcePDP(L, n) 2. M  maximum element in L 3. for every set of n – 2 integers 0 < x2 < … xn-1 < M from L 4. X  { 0,x2,…,xn-1,M } 5. Form DX from X 6. if DX = L 7. return X 8. output “no solution”
  58. 58. • More efficient than BruteForce PDP, but still slow. • If L = {2, 998, 1000} (n = 3, M= 1000), BruteForcePDP will be extremely slow, but AnotherBruteForcePDP will be quite fast. • Fewer sets are examined, but runtime is still exponential: O(n2n-4). Another BruteForcePDP: Efficiency
  59. 59. Section 5: Branch and Bound Algorithm for Partial Digest Problem
  60. 60. 1. Begin with X = {0}. Branch and Bound Algorithm for PDP
  61. 61. 1. Begin with X = {0}. 2. Remove the largest element in L and place it in X. Branch and Bound Algorithm for PDP
  62. 62. 1. Begin with X = {0}. 2. Remove the largest element in L and place it in X. 3. See if the element fits on the right or left side of the restriction map. Branch and Bound Algorithm for PDP
  63. 63. 1. Begin with X = {0}. 2. Remove the largest element in L and place it in X. 3. See if the element fits on the right or left side of the restriction map. 4. When if fits, find the other lengths it creates and remove those from L. Branch and Bound Algorithm for PDP
  64. 64. 1. Begin with X = {0}. 2. Remove the largest element in L and place it in X. 3. See if the element fits on the right or left side of the restriction map. 4. When if fits, find the other lengths it creates and remove those from L. 5. Go back to step 1 until L is empty. Branch and Bound Algorithm for PDP
  65. 65. 1. Begin with X = {0}. 2. Remove the largest element in L and place it in X. 3. See if the element fits on the right or left side of the restriction map. 4. When if fits, find the other lengths it creates and remove those from L. 5. Go back to step 1 until L is empty. Branch and Bound Algorithm for PDP WRONG ALGORITHM
  66. 66. • Before describing PartialDigest, first define D(y, X) as the multiset of all distances between point y and all other points in the set X.    n n xxxX xyxyxyXyD ,,,for ,,,),( 21 21     Defining D(y, X)
  67. 67. return15 to),(lengthsaddandfromRemove14 ),PLACE(13 from),(lengthsremoveandtoAdd12 ),(if11 to),(lengthsaddandfromRemove10 ),PLACE(9 from),(lengthsremoveandtoAdd8 ),(if7 ),DELETE(6 inelementmaximumy5 return4 output3 emptyisif2 ),PLACE(1 LXywidthDXywidth XL LXywidthDXywidth LXywidthD LXyDXy XL LXyDXy LXyD Ly L X L XL        ),PLACE(4 ,0X3 ),DELETE(2 inelementMaximum1 )est(PartialDig XL width Lwidth Lwidth L   Simply deletes width from L PartialDigest: Pseudocode
  68. 68. L = { 2, 2, 3, 3, 4, 5, 6, 7, 8, 10 } X = { } PartialDigest: Example
  69. 69. L = { 2, 2, 3, 3, 4, 5, 6, 7, 8, 10 } X = { 0 } • Remove 10 from L and insert it (along with 0) into X. We know this must be the length of the DNA sequence because it is the largest fragment. PartialDigest: Example
  70. 70. L = { 2, 2, 3, 3, 4, 5, 6, 7, 8, 10 } X = { 0, 10 } PartialDigest: Example
  71. 71. L = { 2, 2, 3, 3, 4, 5, 6, 7, 8, 10 } X = { 0, 10 } • Take 8 from L and make y = 2 or 8. But since the two cases are symmetric, we can assume y = 2. PartialDigest: Example
  72. 72. L = { 2, 2, 3, 3, 4, 5, 6, 7, 8, 10 } X = { 0, 10 } • We find that the distances from y=2 to other elements in X are D(y, X) = {8, 2}, so we remove {8, 2} from L and add 2 to X. PartialDigest: Example
  73. 73. L = { 2, 2, 3, 3, 4, 5, 6, 7, 8, 10 } X = { 0, 2, 10 } PartialDigest: Example
  74. 74. L = { 2, 2, 3, 3, 4, 5, 6, 7, 8, 10 } X = { 0, 2, 10 } • Take 7 from L and make y = 7 or y = 10 – 7 = 3. We will explore y = 7 first, so D(y, X ) = {7, 5, 3}. PartialDigest: Example
  75. 75. L = { 2, 2, 3, 3, 4, 5, 6, 7, 8, 10 } X = { 0, 2, 10 } • For y = 7 first, D(y, X ) = {7, 5, 3}. Therefore we remove {7, 5 ,3} from L and add 7 to X. D(y, X) = {7, 5, 3} = {½7 – 0½, ½7 – 2½, ½7 – 10½} PartialDigest: Example
  76. 76. L = { 2, 2, 3, 3, 4, 5, 6, 7, 8, 10 } X = { 0, 2, 7, 10 } PartialDigest: Example
  77. 77. L = { 2, 2, 3, 3, 4, 5, 6, 7, 8, 10 } X = { 0, 2, 7, 10 } • Take 6 from L and make y = 6. • Unfortunately D(y, X) = {6, 4, 1 ,4}, which is not a subset of L. Therefore we won’t explore this branch. 6 PartialDigest: Example
  78. 78. L = { 2, 2, 3, 3, 4, 5, 6, 7, 8, 10 } X = { 0, 2, 7, 10 } • This time make y = 4. D(y, X) = {4, 2, 3 ,6}, which is a subset of L so we will explore this branch. We remove {4, 2, 3 ,6} from L and add 4 to X. PartialDigest: Example
  79. 79. L = { 2, 2, 3, 3, 4, 5, 6, 7, 8, 10 } X = { 0, 2, 4, 7, 10 } PartialDigest: Example
  80. 80. L = { 2, 2, 3, 3, 4, 5, 6, 7, 8, 10 } X = { 0, 2, 4, 7, 10 } • L is now empty, so we have a solution, which is X. PartialDigest: Example
  81. 81. L = { 2, 2, 3, 3, 4, 5, 6, 7, 8, 10 } X = { 0, 2, 7, 10 } • To find other solutions, we backtrack. PartialDigest: Example
  82. 82. L = { 2, 2, 3, 3, 4, 5, 6, 7, 8, 10 } X = { 0, 2, 10 } • More backtrack. PartialDigest: Example
  83. 83. L = { 2, 2, 3, 3, 4, 5, 6, 7, 8, 10 } X = { 0, 2, 10 } • This time we will explore y = 3. D(y, X) = {3, 1, 7}, which is not a subset of L, so we won’t explore this branch. PartialDigest: Example
  84. 84. L = { 2, 2, 3, 3, 4, 5, 6, 7, 8, 10 } X = { 0, 10 } • We backtracked back to the root. Therefore we have found all the solutions. PartialDigest: Example
  85. 85. • Still exponential in worst case, but is very fast on average. • Informally, let T(n) be time PartialDigest takes to place n cuts. • No branching case: T(n) < T(n-1) + O(n) • Quadratic • Branching case: T(n) < 2T(n-1) + O(n) • Exponential Analyzing the PartialDigest Algorithm
  86. 86. Section 6: Double Digest Problem
  87. 87. • Double Digest is yet another experimentally method to construct restriction maps • Use two restriction enzymes; three full digests: 1. One with only first enzyme 2. One with only second enzyme 3. One with both enzymes • Computationally, Double Digest problem is more complex than Partial Digest problem Double Digest Mapping
  88. 88. Double Digest: Example
  89. 89. • Without the information about X (i.e. A+B), it is impossible to solve the double digest problem as this diagram illustrates Double Digest: Example
  90. 90. • Input: • dA – fragment lengths from the digest with enzyme A. • dB – fragment lengths from the digest with enzyme B. • dX – fragment lengths from the digest with both A and B. • Output: • A – location of the cuts in the restriction map for the enzyme A. • B – location of the cuts in the restriction map for the enzyme B. Double Digest Problem
  91. 91. Double Digest: Multiple Solutions
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