Physical Mapping II + Perl

CIS 667 March 2, 2004

Restriction Site Models

• Let each fragment in the Double Digest Problem be represented by its length No measurement errors All fragments present

• Digesting the target DNA by the first enzyme gives the multiset A = {a1, a2, …, an}

• The second enzyme gives B = {b1, b2, …, bn}• Digestion with both gives O = {o1, o2, …, on}

Restriction Site Models

• We want to find a permutation A of the elements of A and B of the elements of B Plot lengths A from on a line in the order of A

Plot lengths B from on a line in the order of B

on top of previous plot Several new subintervals may be produced

We need a one-to-one correspondence between each resulting subinterval and each element of O

Restriction Site Models

• This problem is NP-complete It is a generalization of the set partition

problem The number of solutions is exponential

• Partial Digest problem has not been proven to be NP-complete The number of solutions is much smaller

than for DDP

Interval Graph Models

• We model hybridization mapping using interval graphs Much simpler than the real problem, but

still NP-complete Uses graphs

Vertices represent clones Edges represent overlap information

between clones

First Interval Graph Model

• Uses two graphs Gr = (V, Er)

(i, j) Er means we know clones i, j overlap

Gt = (V, Et) Et represents known and unknown overlap

information If we know for sure that two clones don’t

overlap, the corresponding edge is left out of the graph Gt

First Interval Graph Model

• Does there exist a graph Gs = (V, Es) such that Er Es Et such that Gs is an interval graph? An interval graph G = (V, E) is an

undirected graph obtained from a collection C of intervals on the real line To each interval in C there corresponds a

vertex in G There is an edge between u and v only if

their intervals have a non-empty intersection

First Interval Graph Model

a

bc

d e

a

b

c

d

e

Non-Interval Graphs

a

b

c

d

e

a

b

c

d

e

Second Interval Graph Model

• Don’t assume that known overlap information is reliable Construct a graph G = (V, E) using that

information Does there exist a graph G’ = (V, E’)

such that E’ E, G’ is an interval graph and |E’| is maximum? We have discarded some false positives The solution is the interpretation that

contains the minimum number of false positives

Third Interval Graph Model

• Use overlap information along with information about each clone Different clones come from different

copies of the same molecule Label each clone with the identification

of the molecule copy it came from Assume we had k copies of the target

DNA and different restriction enzymes were used to break up each copy

Third Interval Graph Model

• Build a graph G = (V, E) with known overlap information between clones Use k colors to color the vertices No edges between vertices of the same color

since they come from the same clone and hence cannot overlap We say that such a graph has a valid coloring Does there exist graph G’ = (V’, E) such that , G’ is an

interval graph, and the coloring of G is valid for G’? I.e., Can we add edges to G transforming it into an

interval graph without violating the coloring?

Consecutive Ones Property

• We can apply the previous models in any situation where we can obtain some type of fingerprint for each fragment Now we use as a clone fingerprint the set of

probes that hybridize to it Assumptions

Reverse complement of each probe’s sequence occurs only once in the target DNA (“probes are unique”

There are no errors All “clones X probes” hybridization experiments have

been done

Consecutive Ones Property

• If we have n clones and m probes we will build an n m binary matrix M, where each entry Mij tells us whether probe j hybridized to clone i or not Then obtaining a physical map from the matrix

becomes the problem of finding a permutation of the columns (probes) such that all 1s in each row (clone) are consecutive Such a matrix is said to have the consecutive 1s

property for rows (C1P)

Consecutive Ones Property

• There exist polynomial algorithms for the C1P property Works only for data with no errors Realistic algorithms should try to find matrixes

which approximate the C1P property, while minimizing the number of errors which must have been present to lead to such a solution Allow 2 or 3 runs of 1s in a row Minimize the number of runs of 1s in the matrix

• Problem is now NP-hard

Now we will look at some Perl in

preparation for assignment 1

Perl substitution operator

• Example of Perl substitution operator

$RNA =~ s/T/U/g;

variable binding operator

substitute operator

PATTERN regular expressionTo be replaced by REPLACEMENT

delimiter

REPLACEMENTtext to replace PATTERN

Pattern modifier: g meansglobally, throughout thestring. Others:i case insensitivem multilines single line

Example 1

• Let’s use the substitution operator to calculate the reverse complement of a strand of DNA

Example 2

• One common task in bioinformatics is to look for motifs, short segments of DNA or protein of interest For example, regulatory elements of DNA

• Let’s see a program to Read in protein sequence data from a file Put all the sequence data into one string for

easy searching Look for motifs the user types in at the

keyboard

Turning arrays into Scalars

• We often find sequence data broken into short segments of 80 or so characters This is inconvenient for the Perl program

Have to deal with motifs on more than one line

Collapse an array into a scalar with join $protein = join( ‘’, @protein)

Regular expressions

• Regular expressions are ways of matching one or more strings using special wildcard-like operators $protein =~ s/\s//g

\s matches whitespace Can also be written [ \t\n\f\r]

if ($motif =~ /^\s*$/ ) { ^ - beginning of line; $ - end of line * repeated zero or more times

Hashes

• There are three main data types in Perl: scalar variables, arrays and hashes (also called associative arrays) A hash provides a fast lookup of the

value associated with a key Initialized like this:%classification = (

‘dog’ => ‘mammal’,‘robin’ => ‘bird’‘asp’ => ‘reptile’

);

Example 3

• Let’s look at the use of a hash by a subroutine to translate a codon to an amino acid using hash lookup codon2aa

Example 3

• The arguments to the subroutine are in the @_ array

• Declare a local variable as a my variable

• my($dna) = @_;

Example 4

• We can use that subroutine to translate DNA into protein

• Note the use of a module (library)• Note the use of .= to concatenate