DNA Mapping and Brute Force Algorithms : DNA Mapping and Brute Force Algorithms
Outline : Outline Restriction Enzymes
Gel Electrophoresis
Partial Digest Problem
Brute Force Algorithm for Partial Digest Problem
Branch and Bound Algorithm for Partial Digest Problem
Double Digest Problem
Molecular Scissors : Molecular Scissors Molecular Cell Biology, 4th edition
Discovering Restriction Enzymes : Discovering Restriction Enzymes HindII - first restriction enzyme – was discovered accidentally in 1970 while studying how the bacterium Haemophilus influenzae takes up DNA from the virus
Recognizes and cuts DNA at sequences:
GTGCAC
GTTAAC
Discovering Restriction Enzymes : Discovering Restriction Enzymes Werner Arber Daniel Nathans Hamilton Smith 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
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)
Recognition Sites of Restriction Enzymes : Recognition Sites of Restriction Enzymes Molecular Cell Biology, 4th edition
Uses of Restriction Enzymes : Uses of Restriction Enzymes Recombinant DNA technology
Cloning
cDNA/genomic library construction
DNA mapping
Restriction Maps : Restriction Maps • A map showing positions of restriction sites in a DNA sequence
• If DNA sequence is known then construction of restriction map is a 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
Full Restriction Digest : Full Restriction Digest
• Cutting DNA at each restriction site creates
multiple restriction fragments: • Is it possible to reconstruct the order of
the fragments from the sizes of the fragments
{3,5,5,9} ?
Full Restriction Digest: Multiple Solutions : Full Restriction Digest: Multiple Solutions • Alternative ordering of restriction fragments: vs
Measuring Length of Restriction Fragments : Measuring Length of Restriction Fragments Restriction enzymes break DNA into restriction fragments.
Gel electrophoresis is a process for separating DNA by size and measuring sizes of restriction fragments
Can separate DNA fragments that differ in length in only 1 nucleotide for fragments up to 500 nucleotides long
Gel Electrophoresis : Gel Electrophoresis 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
DNA molecules move towards the positive electrode
Gel Electrophoresis (cont’d) : Gel Electrophoresis (cont’d) 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
Gel Electrophoresis: Example : Gel Electrophoresis: Example Direction of DNA movement Smaller fragments travel farther Molecular Cell Biology, 4th edition
Detecting DNA: Autoradiography : Detecting DNA: Autoradiography One way to visualize separated DNA bands on a gel is autoradiography:
The DNA is radioactively labeled
The gel is laid against a sheet of photographic film in the dark, exposing the film at the positions where the DNA is present.
Detecting DNA: Fluorescence : Detecting DNA: Fluorescence Another way to visualize DNA bands in gel is 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.
Partial Restriction Digest : Partial Restriction Digest 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 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 Digest Example : Partial Digest Example Partial Digest results in the following 10 restriction fragments:
Multiset of Restriction Fragments : Multiset of Restriction Fragments We assume that multiplicity of a fragment can be detected, i.e., the number of restriction fragments of the same length can be determined (e.g., by observing twice as much fluorescence intensity for a double fragment than for a single fragment) Multiset: {3, 5, 5, 8, 9, 14, 14, 17, 19, 22}
Partial Digest Fundamentals : Partial Digest Fundamentals the set of n integers representing the location of all cuts in the restriction map, including the start and end
the multiset of integers representing lengths of each of the fragments produced from a partial digest the total number of cuts
X:
n: DX:
One More Partial Digest Example : One More Partial Digest Example Representation of DX = {2, 2, 3, 3, 4, 5, 6, 7, 8, 10} as a two dimensional table, with elements of
X = {0, 2, 4, 7, 10}
along both the top and left side. The elements at (i, j) in the table is xj – xi for 1 ≤ i < j ≤ n.
Partial Digest Problem: Formulation : Partial Digest Problem: Formulation 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 DX = L
Partial Digest: Multiple Solutions : Partial Digest: Multiple Solutions It is not always possible to uniquely reconstruct a set X based only on DX.
For example, the set
X = {0, 2, 5}
and (X + 10) = {10, 12, 15}
both produce DX={2, 3, 5} as their partial digest set.
The sets {0,1,2,5,7,9,12} and {0,1,5,7,8,10,12} present a less trivial 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}
Homometric Sets : Homometric Sets
Brute Force Algorithms : Brute Force Algorithms Also known as exhaustive search algorithms; examine every possible variant to find a solution
Efficient in rare cases; usually impractical
Partial Digest: Brute Force : Partial Digest: Brute Force Find the restriction fragment of maximum length M. M is the length of the DNA sequence.
For every possible set
X={0, x2, … ,xn-1, M}
compute the corresponding DX
If DX is equal to the experimental partial digest L, then X is the correct restriction map
BruteForcePDP : BruteForcePDP BruteForcePDP(L, n):
M <- maximum element in L
for every set of n – 2 integers 0 < x2 < … xn-1 < M
X <- {0,x2,…,xn-1,M}
Form DX from X
if DX = L
return X
output “no solution”
Efficiency of BruteForcePDP : Efficiency of BruteForcePDP BruteForcePDP takes O(M n-2) time since it must examine all possible sets of positions.
One way to improve the algorithm is to limit the values of xi to only those values which occur in L.
AnotherBruteForcePDP : AnotherBruteForcePDP AnotherBruteForcePDP(L, n)
M <- maximum element in L
for every set of n – 2 integers 0 < x2 < … xn-1 < M
X <- { 0,x2,…,xn-1,M }
Form DX from X
if DX = L
return X
output “no solution”
AnotherBruteForcePDP : AnotherBruteForcePDP AnotherBruteForcePDP(L, n)
M <- maximum element in L
for every set of n – 2 integers 0 < x2 < … xn-1 < M from L
X <- { 0,x2,…,xn-1,M }
Form DX from X
if DX = L
return X
output “no solution”
Efficiency of AnotherBruteForcePDP : Efficiency of AnotherBruteForcePDP
It’s more efficient, 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)
Branch and Bound Algorithm for PDP : Branch and Bound Algorithm for PDP Begin with X = {0}
Remove the largest element in L and place it in X
See if the element fits on the right or left side of the restriction map
When it fits, find the other lengths it creates and remove those from L
Go back to step 1 until L is empty
Branch and Bound Algorithm for PDP : Branch and Bound Algorithm for PDP Begin with X = {0}
Remove the largest element in L and place it in X
See if the element fits on the right or left side of the restriction map
When it fits, find the other lengths it creates and remove those from L
Go back to step 1 until L is empty WRONG ALGORITHM
Defining D(y, X) : Defining D(y, X) 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
D(y, X) = {|y – x1|, |y – x2|, …, |y – xn|}
for X = {x1, x2, …, xn}
PartialDigest Algorithm : PartialDigest Algorithm PartialDigest(L):
width <- Maximum element in L
DELETE(width, L)
X <- {0, width}
PLACE(L, X)
PartialDigest Algorithm (cont’d) : PartialDigest Algorithm (cont’d) PLACE(L, X)
if L is empty
output X
return
y <- maximum element in L
Delete(y,L)
if D(y, X ) Í L
Add y to X and remove lengths D(y, X) from L
PLACE(L,X )
Remove y from X and add lengths D(y, X) to L
if D(width-y, X ) Í L
Add width-y to X and remove lengths D(width-y, X) from L
PLACE(L,X )
Remove width-y from X and add lengths D(width-y, X ) to L
return
An Example : An Example L = { 2, 2, 3, 3, 4, 5, 6, 7, 8, 10 }
X = { 0 }
An Example : An Example L = { 2, 2, 3, 3, 4, 5, 6, 7, 8, 10 }
X = { 0 }
Remove 10 from L and insert it into X. We know this must be
the length of the DNA sequence because it is the largest
fragment.
An Example : An Example L = { 2, 2, 3, 3, 4, 5, 6, 7, 8, 10 }
X = { 0, 10 }
An Example : An 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 symmetric, we can assume y = 2.
An Example : An Example 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.
An Example : An Example L = { 2, 2, 3, 3, 4, 5, 6, 7, 8, 10 }
X = { 0, 2, 10 }
An Example : An 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 = 7 first, so D(y, X ) = {7, 5, 3}.
An Example : An Example 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}
An Example : An Example L = { 2, 2, 3, 3, 4, 5, 6, 7, 8, 10 }
X = { 0, 2, 7, 10 }
An Example : An 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, 1 ,4}, which is not a subset of L. Therefore we won’t explore this branch.
An Example : An Example 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.
An Example : An Example L = { 2, 2, 3, 3, 4, 5, 6, 7, 8, 10 }
X = { 0, 2, 4, 7, 10 }
An Example : An 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.
An Example : An Example L = { 2, 2, 3, 3, 4, 5, 6, 7, 8, 10 }
X = { 0, 2, 7, 10 }
To find other solutions, we backtrack.
An Example : An Example L = { 2, 2, 3, 3, 4, 5, 6, 7, 8, 10 }
X = { 0, 2, 10 }
More backtrack.
An Example : An 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
not a subset of L, so we won’t explore this branch.
An Example : An Example 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.
Analyzing PartialDigest Algorithm : Analyzing PartialDigest Algorithm 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
Double Digest Mapping : Double Digest Mapping Double Digest is yet another experimentally method to construct restriction maps
Use two restriction enzymes; three full digests:
One with only first enzyme
One with only second enzyme
One with both enzymes
Computationally, Double Digest problem is more complex than Partial Digest problem
Double Digest: Example : Double Digest: Example
Double Digest: Example : Double Digest: Example Without the information about X (i.e. A+B), it is impossible to solve the double digest problem as this diagram illustrates
Double Digest Problem : Double Digest Problem 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: Multiple Solutions : Double Digest: Multiple Solutions