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genesonoroﬀ. Thesemoleculesbindpreferentially tospeciﬁcDNA sequences,

andthisbindingpreferencepatterncanberepresentedbyatableof frequencies

ofgiven symbolsateachpositionofthe pattern. More precisely, eachrowin

thetable correspondsto thebasesA,C,G,andT,while columnjreﬂectshow

many timesthebase appearsinpositionjinthe DNAsequence.

Forexample, ifoursetofDNAsequencesare TAG,GGT, andGGG, the

table becomes

base 0 1 2

0 1 0

0 0 0

2 2 2

1 0 1

FromthistablewecanreadthatbaseAappearsonce inindex1intheDNA

strings,baseCdoesnotappearatall,baseGappearstwiceinallpositions,and

baseTappearsonceinthe beginningandendofthestrings.

Inthefollowingweshallpresentdiﬀerentdatastructurestoholdsuchatable

anddiﬀerentwaysof computingthem. Thetableisknownasafrequencymatrix

inbioinformaticsandthisisthe termusedheretoo.

Separate FrequencyLists. Sinceweknowthatthere areonlyfourrowsin

thefrequencymatrix,anobviousdatastructurewouldbefourlists,eachholding

arow. A functioncomputingtheselistsmaylooklike

def freq_lists(dna_list):

n = len(dna_list[0])

A = [0]*n

T = [0]*n

G = [0]*n

C = [0]*n

for dna in dna_list:

for index, base in enumerate(dna):

if base == ’A’:

A[index] += 1

elif base == ’C’:

C[index] += 1

elif base == ’G’:

G[index] += 1

elif base == ’T’:

T[index] += 1

return A, C, G, T

Weneedtoinitializethelistswiththerightlengthandazeroforeachelement,

since each listelementisto be usedasa counter. Creating a listoflengthn

withobjectx in all positionsisdone by[x]*n. Finding the proper lengthis

here carriedoutbyinspecting the lengthoftheﬁrstelementindna_list,the

listofallDNAstringstobecounted,assumingthatallelementsinthislisthave

thesame length.

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Intheforloopwe applytheenumeratefunction,whichisusedtoextract

boththeelementvalue and the elementindexwheniterating overa sequence.

Forexample,

>>> for index, base in enumerate([’t’, ’e’, ’s’, ’t’]):

...

print index, base

...

0 t

1 e

2 s

3 t

Here isatest,

dna_list = [’GGTAG’, ’GGTAC’, ’GGTGC’]

A, C, G, T = freq_lists(dna_list)

print A

print C

print G

print T

withoutput

[0, 0, 0, 2, 0]

[0, 0, 0, 0, 2]

[3, 3, 0, 1, 1]

[0, 0, 3, 0, 0]

Nested List. . ThefrequencymatrixcanalsoberepresentedasanestedlistM

suchthatM[i][j]isthe frequencyofbasei in positionjin the set ofDNA

strings. Herei isaninteger, where 0correspondsto A,1toT,2to G,and3

to C.Thefrequencyis the numberoftimesbasei appearsinpositionj in a

setofDNAstrings. SometimesthisnumberisdividedbythenumberofDNA

stringsinthe setso thatthe frequencyisbetween 0and 1. Note thatallthe

DNA stringsmusthavethesamelength.

The simplestwayto make a nestedlististo inserttheA,C,G, andT lists

intoanotherlist:

>>> frequency_matrix = [A, C, G, T]

>>> frequency_matrix[2][3]

>>> G[3] # same element

Alternatively, we can illustrate how to compute this type of nested list

directly:

def freq_list_of_lists_v1(dna_list):

# Create empty frequency_matrix[i][j] = 0

# i=0,1,2,3 corresponds to A,T,G,C

# j=0,...,length of dna_list[0]

frequency_matrix = [[0 for v in dna_list[0]] for x in ’ACGT’]

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for dna in dna_list:

for index, base in enumerate(dna):

if base == ’A’:

frequency_matrix[0][index] +=1

elif base == ’C’:

frequency_matrix[1][index] +=1

elif base == ’G’:

frequency_matrix[2][index] +=1

elif base == ’T’:

frequency_matrix[3][index] +=1

return frequency_matrix

Asinthecasewithindividuallistsweneedtoinitializeallelementsinthe

nestedlisttozero.

Acallandprintout,

dna_list = [’GGTAG’, ’GGTAC’, ’GGTGC’]

frequency_matrix = freq_list_of_lists_v1(dna_list)

print frequency_matrix

resultsin

[[0, 0, 0, 2, 0], [0, 0, 0, 0, 2], [3, 3, 0, 1, 1], [0, 0, 3, 0, 0]]

DictionaryforMore Convenient Indexing. . Theseriesofiftestsinthe

Pythonfunctionfreq_list_of_lists_v1aresomewhatcumbersome,especially

ifwewanttoextendthecodetootherbioinformaticsproblemswherethealphabet

islarger. Whatwewantisamappingfrombase,whichisa character,tothe

corresponding index 0, 1, 2, or 3. A Python dictionarymay representsuch

mappings:

>>> base2index = {’A’: 0, ’C’: 1, ’G’: 2, ’T’: 3}

>>> base2index[’G’]

Withthebase2index dictionarywedonotneedthe seriesof f if testsandthe

alphabet’ATGC’couldbe muchlargerwithoutaﬀectingthelengthofthecode:

def freq_list_of_lists_v2(dna_list):

frequency_matrix = [[0 for v in dna_list[0]] for x in ’ACGT’]

base2index = {’A’: 0, ’C’: 1, ’G’: 2, ’T’: 3}

for dna in dna_list:

for index, base in enumerate(dna):

frequency_matrix[base2index[base]][index] += 1

return frequency_matrix

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NumericalPython Array. . Aslongaseachsublistinalistoflistshasthe

samelength,alistoflistscanbereplacedbyaNumericalPython(numpy)array.

Processing ofsucharrays is oftenmuch more eﬃcient than processingofthe

nestedlistdatastructure. Toinitializeatwo-dimensionalnumpyarrayweneedto

knowitssize,here4timeslen(dna_list[0]). Onlytheﬁrstlineinthefunction

freq_list_of_lists_v2needstobechangedinordertoutilizeanumpyarray:

import numpy as np

def freq_numpy(dna_list):

frequency_matrix = np.zeros((4, len(dna_list[0])), dtype=np.int)

base2index = {’A’: 0, ’C’: 1, ’G’: 2, ’T’: 3}

for dna in dna_list:

for index, base in enumerate(dna):

frequency_matrix[base2index[base]][index] += 1

return frequency_matrix

Theresultingfrequency_matrixobjectcanbeindexedas[b][i]or[b,i],

withintegersb andi. Typically,b willbe somethinglinebase2index[’C’].

Dictionaryof Lists. . Insteadofgoingfromacharactertoanintegerindex

viabase2index,we mayprefertoindexfrequency_matrix bythebasename

andthe position indexdirectly, like in[’C’][14]. This isthe mostnatural

syntaxforauserofthefrequencymatrix. TherelevantPythondatastructureis

thenadictionary of lists. Thatis,frequency_matrixisadictionary withkeys

’A’,’C’,’G’,and ’T’.Thevalueforeachkeyisalist. Letusnowalsoextend

the ﬂexibility such thatdna_list can have DNAstrings of diﬀerent lengths.

Thelistsinfrequency_listwillhave lengthsequaltothelongestDNAstring.

Arelevantfunctionis

def freq_dict_of_lists_v1(dna_list):

n = max([len(dna) for dna in dna_list])

frequency_matrix = {

’A’: [0]*n,

’C’: [0]*n,

’G’: [0]*n,

’T’: [0]*n,

}

for dna in dna_list:

for index, base in enumerate(dna):

frequency_matrix[base][index] += 1

return frequency_matrix

Runningthe testcode

frequency_matrix = freq_dict_of_lists_v1(dna_list)

import pprint

# for nice printout of nested data structures

pprint.pprint(frequency_matrix)

resultsinthe output

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{’A’: [0, 0, 0, 2, 0],

’C’: [0, 0, 0, 0, 2],

’G’: [3, 3, 0, 1, 1],

’T’: [0, 0, 3, 0, 0]}

Theinitializationoffrequency_matrixintheabovecodecanbemademore

compactbyusingadictionary comprehension:

dict = {key: value for key in some_sequence}

Inourexample weset

frequency_matrix = {base: [0]*n for base in ’ACGT’}

Adoptingthisconstructioninthefreq_dict_of_lists_v1functionleadstoa

slightlymorecompactversion:

def freq_dict_of_lists_v2(dna_list):

n = max([len(dna) for dna in dna_list])

frequency_matrix = {base: [0]*n for base in ’ACGT’}

for dna in dna_list:

for index, base in enumerate(dna):

frequency_matrix[base][index] += 1

return frequency_matrix

Asanadditionalcommentoncomputingthemaximumlengthofthe DNA

strings canbe made asthere are severalalternative waysofdoing this. The

classical useof maxistoapplyittoalistasdoneabove:

n = max([len(dna) for dna in dna_list])

However,forverylonglistsitispossibletoavoidthememorydemandsofstoring

the resultofthe listcomprehension,i.e., the listoflengths. Insteadmax can

workwiththe lengthsasthey are computed:

n = max(len(dna) for dna in dna_list)

Itisalsopossible towrite

n = max(dna_list, key=len)

Here,len is appliedto each element indna_list, and the maximum of the

resultingvaluesisreturned.

DictionaryofDictionaries. Thedictionaryoflistsdatastructurecanalter-

nativelybe replacedbya dictionaryofdictionariesobject, oftenjustcalleda

dictofdictsobject. Thatis,frequency_matrix[base]isadictionarywithkey

iandvalueequaltotheaddednumberofoccurrencesof basein dna[i]for

alldnastringsinthelistdna_list. The indexingfrequency_matrix[’C’][i]

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andthevaluesareexactly asinthelastexample;theonlydiﬀerenceiswhether

frequency_matrix[’C’]isa listordictionary.

Ourfunctionworkingwithfrequency_matrixasadictofdictsiswrittenas

def freq_dict_of_dicts_v1(dna_list):

n = max([len(dna) for dna in dna_list])

frequency_matrix = {base: {index: 0 for index in range(n)}

for base in ’ACGT’}

for dna in dna_list:

for index, base in enumerate(dna):

frequency_matrix[base][index] += 1

return frequency_matrix

UsingDictionarieswithDefaultValues. Themanualinitializationofeach

subdictionary tozero,

frequency_matrix = {base: {index: 0 for index in range(n)}

for base in ’ACGT’}

canbesimpliﬁedby usingadictionarywithdefaultvaluesforanykey. Thecon-

structiondefaultdict(lambda: obj) makesadictionarywithobjasdefault

value. Thisconstructionsimpliﬁesthepreviousfunctionabit:

from collections import defaultdict

def freq_dict_of_dicts_v2(dna_list):

n = max([len(dna) for dna in dna_list])

frequency_matrix = {base: defaultdict(lambda: 0)

for base in ’ACGT’}

for dna in dna_list:

for index, base in enumerate(dna):

frequency_matrix[base][index] += 1

return frequency_matrix

Remark. Dictionary comprehensionswerenewinPython2.7and3.1,but

can be simulated in earlier versions by making (key, value) tuples via list

comprehensions. Adictionarycomprehension

d = {key: value for key in sequence}

isthenconstructedas

d = dict([(key, value) for key in sequence])

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Using Arrays andVectorization. . Thefrequency_matrixdictoflistsfor

caneasilybe changedtoadictof f numpyarrays: justreplacetheinitialization

[0]*n bynp.zeros(n, dtype=np.int). The indexingremainsthesame:

def freq_dict_of_arrays_v1(dna_list):

n = max([len(dna) for dna in dna_list])

frequency_matrix = {base: np.zeros(n, dtype=np.int)

for base in ’ACGT’}

for dna in dna_list:

for index, base in enumerate(dna):

frequency_matrix[base][index] += 1

return frequency_matrix

Havingfrequency_matrix[base]asanumpyarrayinsteadofalistdoesnot

give anyimmediate advantage, asthestorageandCPUtimeisaboutthe same.

Theloopoverthedna stringandthe associatedindexingiswhatconsumesall

the CPUtime. However, thenumpy arraysprovide a potential for increasing

eﬃciencythroughvectorization,i.e., replacingtheelement-wiseoperationson

dna andfrequency_matrix[base] byoperationsontheentirearraysatonce.

LetususetheinteractivePythonshelltoexplore thepossibilitiesofvector-

ization. Weﬁrstconvertthestringtoanumpy arrayofcharacters:

>>> dna = ’ACAT’

>>> dna = np.array(dna, dtype=’c’)

>>> dna

array([’A’, ’C’, ’A’, ’T’],

dtype=’|S1’)

Foragivenbase,sayA,wecaninone vectorizedoperationﬁndwhichlocations

indnathatcontainA:

>>> b = dna == ’A’

>>> b

array([ True, False, True, False], dtype=bool)

Byconvertingbtoanintegerarrayiwecanupdatethefrequencycountsforall

indicesbyaddingi tofrequency_matrix[’A’]:

>>> i = np.asarray(b, dtype=np.int)

>>> i

array([1, 0, 1, 0])

>>> frequency_matrix[’A’] = frequency_matrix[’A’] + i

Thisrecipecanberepeatedforallbases:

for dna in dna_list:

dna = np.array(dna, dtype=’c’)

for base in ’ACGT’:

b = dna == base

i = np.asarray(b, dtype=np.int)

frequency_matrix[base] = frequency_matrix[base] + i

It turns out that we do not need to convert the boolean array y b to an

integerarrayi,becausedoing arithmeticswithb directlyispossible: : False is

interpretedas0andTrueas1inarithmeticoperations. Wecanalsousethe+=

operatorto update allelementsoffrequency_matrix[base]directly,without

ﬁrstcomputingthe sumoftwoarraysfrequency_matrix[base] + i andthen

assigningthisresulttofrequency_matrix[base]. Collectingalltheseideasin

one functionyieldsthe code

def freq_dict_of_arrays_v2(dna_list):

n = max([len(dna) for dna in dna_list])

frequency_matrix = {base: np.zeros(n, dtype=np.int)

for base in ’ACGT’}

for dna in dna_list:

dna = np.array(dna, dtype=’c’)

for base in ’ACCT’:

frequency_matrix[base] += dna == base

return frequency_matrix

Thisvectorizedfunctionrunsalmost10timesasfastasthe(scalar)counterpart

freq_list_of_arrays_v1!

1.5 Analyzing the Frequency Matrix

Havingbuiltafrequencymatrix outof acollectionof DNA strings,itistimeto

useitforanalysis. TheshortDNA stringsthatafrequency matrixisbuiltout

of,istypicallyasetofsubstringsofalargerDNAsequence,whichsharessome

commonpurpose. Anexampleofthisistohaveasetofsubstringsthatservesas

akindofanchors/magnetsatwhichgivenmoleculesattachtoDNAandperform

biologicalfunctions(like turninggenesonoroﬀ). Withthefrequencymatrix

constructedfromalimitedsetofknownanchorlocations(substrings),we can

now scan forothersimilarsubstrings that have the potential toperformthe

samefunction. Thesimplestwaytodothisistoﬁrstdeterminethemosttypical

substringaccordingtothefrequency matrix,i.e.,the substringhavingthe most

frequentnucleotideateachposition. Thisisreferredtoastheconsensusstring

ofthe frequency matrix. We can thenlook for occurrencesof the consensus

substringin alargerDNAsequence, and considerthese occurrencesaslikely

candidatesforservingthesamefunction(e.g.,asanchorlocationsformolecules).

Forinstance, giventhree substringsACT, CCAandAGA, the frequency

matrixwouldbe(listoflists,withrowscorrespondingtoA,C,G,andT):

[[2, 0, 2]

[1, 2, 0]

[0, 1, 0]

[0, 0, 1]]

We see thatforposition0, which corresponds tothe left-mostcolumninthe

table, the symbolAhasthe highestfrequency(2). The maximumfrequencies

for the other positionsare seen to be C for position 1, and Afor position 2.

Theconsensusstringistherefore ACA.Notethattheconsensusstringdoesnot

needtobeequaltoanyofthe substringsthatformedthebasisof thefrequency

matrix (thisisindeedthecasefortheabove example).

List of Lists FrequencyMatrix. . Letfrequency_matrixbealistoflists.

Foreachpositioni werunthroughtherowsinthefrequencymatrixandkeep

trackofthemaximumfrequencyvalueandthe correspondingletter. Iftwo or

more lettershavethesame frequencyvalueweuseadashtoindicatethatthis

positionintheconsensusstringisundetermined.

The followingfunctioncomputesthe consensusstring:

def find_consensus_v1(frequency_matrix):

base2index = {’A’: 0, ’C’: 1, ’G’: 2, ’T’: 3}

consensus = ’’

dna_length = len(frequency_matrix[0])

for i in range(dna_length): # loop over positions in string

max_freq = -1

# holds the max freq. for this i

max_freq_base = None

# holds the corresponding base

for base in ’ATGC’:

if frequency_matrix[base2index[base]][i] > max_freq:

max_freq = frequency_matrix[base2index[base]][i]

max_freq_base = base

elif frequency_matrix[base2index[base]][i] == max_freq:

max_freq_base = ’-’ # more than one base as max

consensus += max_freq_base # add new base with max freq

return consensus

Since thiscode requiresfrequency_matrix to be a listoflists we should

inserta testandraiseanexceptionifthetypeiswrong:

def find_consensus_v1(frequency_matrix):

if isinstance(frequency_matrix, list) and \

isinstance(frequency_matrix[0], list):

pass # right type

else:

raise TypeError(’frequency_matrix must be list of lists’)

...

Dict of Dicts FrequencyMatrix. . Howmustthefind_consensus_v1func-

tionbealterediffrequency_matrixisadictofdicts?

1. Thebase2indexdictisnolongerneeded.

2. Accessofsublist,frequency_matrix[0],totestfortypeandlengthof

the strings,mustbereplacedby frequency_matrix[’A’].

Theupdatedfunctionbecomes

def find_consensus_v3(frequency_matrix):

if isinstance(frequency_matrix, dict) and \

isinstance(frequency_matrix[’A’], dict):

pass # right type

else:

raise TypeError(’frequency_matrix must be dict of dicts’)

consensus = ’’

dna_length = len(frequency_matrix[’A’])

for i in range(dna_length): # loop over positions in string

max_freq = -1

# holds the max freq. for this i

max_freq_base = None

# holds the corresponding base

for base in ’ACGT’:

if frequency_matrix[base][i] > max_freq:

max_freq = frequency_matrix[base][i]

max_freq_base = base

elif frequency_matrix[base][i] == max_freq:

max_freq_base = ’-’ # more than one base as max

consensus += max_freq_base # add new base with max freq

return consensus

Here isatest:

frequency_matrix = freq_dict_of_dicts_v1(dna_list)

pprint.pprint(frequency_matrix)

print find_consensus_v3(frequency_matrix)

withoutput

{’A’: {0: 0, 1: 0, 2: 0, 3: 2, 4: 0},

’C’: {0: 0, 1: 0, 2: 0, 3: 0, 4: 2},

’G’: {0: 3, 1: 3, 2: 0, 3: 1, 4: 1},

’T’: {0: 0, 1: 0, 2: 3, 3: 0, 4: 0}}

Consensus string: GGTAC

Letustryfind_consensus_v3withthedictofdefaultdictsasinput(freq_dicts_of_dicts_v2).

Thecoderunsﬁne,buttheoutputstringisjustG!Thereasonisthatdna_length

is1,andthereforethatthelengthoftheAdictinfrequency_matrixis1. Print-

ingoutfrequency_matrixyields

{’A’: defaultdict(X, {3: 2}),

’C’: defaultdict(X, {4: 2}),

’G’: defaultdict(X, {0: 3, 1: 3, 3: 1, 4: 1}),

’T’: defaultdict(X, {2: 3})}

whereourXisashortformfortextlike

‘<function <lambda> at 0xfaede8>‘

Weseethatthelengthofadefaultdictwillonlycountthenonzeroentries. Hence,

touse adefaultdictourfunctionmustgetthelengthoftheDNAstringtobuild

asanextraargument:

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