Research Article On the Existence of Wavelet Symmetries in Archaea DNA

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HindawiPublishingCorporation
ComputationalandMathematicalMethodsinMedicine
Volume2012,ArticleID673934,21pages
doi:10.1155/2012/673934
Research Article
On the Existence of Wavelet Symmetries in Archaea DNA
CarloCattani
DepartmentofMathematics,UniversityofSalerno,ViaPonteDonMelillo,84084Fisciano,Italy
CorrespondenceshouldbeaddressedtoCarloCattani,[email protected]
Received13September2011;Revised27October2011;Accepted29October2011
AcademicEditor:Sheng-yongChen
Copyright©2012CarloCattani.ThisisanopenaccessarticledistributedundertheCreativeCommonsAttributionLicense,which
permitsunrestricteduse,distribution,andreproductioninanymedium,providedtheoriginalworkisproperlycited.
ThispaperdealswiththecomplexunitrootsrepresentationofarcheaDNAsequencesandtheanalysisofsymmetriesinthewavelet
coefficientsofthedigitalizedsequence.Itisshownthatevenforextremophilearchaea,thedistributionofnucleotideshastofulfill
some (mathematical) constraints in such a way that the wavelet coefficients are symmetrically distributed, with respect to the
nucleotidesdistribution.
1.Introduction theearthunderextremeconditionsoflife,theirDNAhasto
fulfillthesameconstraintsofthemoreevolvedDNAs.
In some recent papers the existence of symmetries in nu- Inordertoachievethisgoalsomefundamentalstepshave
cleotide distribution has been studied for several living or- tobetakenintoconsiderationanddiscussed.
ganisms [1–6] including mammals, fungi [1–4], and vir-
(1)SinceDNAisasequenceofsymbols,amapofthese
uses [5, 6]. Thus showing that any (investigated) DNA
symbols into numbers has to be defined. In the fol-
sequence, when converted into a digital sequence, features
lowingwewillconsiderthecomplexunitrootsmap,
some fractal shape of its DNA walk and an apparently
which has the advantage of being unitary and dis-
random-like distribution. However, when the short wavelet
tributedalongtheunitcircle.
transformmapsthedigitalsequenceintothespaceofwavelet
coefficients,andthesecoefficientsareclusteredthentheyare (2)Theindicatormatrixisdefinedonthetheindicator
locatedalongsomesymmetricalshapes. map. This matrix is important in order to draw the
One of the main tasks of this paper is to show that al- dotplotoftheDNAsequenceandfromthisplotwe
thoughthedistributionofnucleotide,inanyDNAsequence, can see that apparently nucleotides seem to be ran-
can be considered as randomly given, when we compare a domlydistributed.However,wewillshowbywavelet
random sequence (and the corresponding random walk) analysis that they look randomly distributed, while
with a DNA sequence (and walk) it can be seen that there theyarenot.
existssomedistinctions.Sothatthenucleotidesdistribution (3)TheUlamspiraladaptedtoDNAsequencesisdefined
seems to side with a random distribution with some con- inordertosingleoutsomegeometricalpatterns.
straints.Theseconstraints(rules)aresingledoutinthefol-
(4)Random walks on DNA, or short DNA walks, show
lowing,byshowingtheexistenceofhiddengeometrywhich
thattherandomwalkslooklikefractals.
underliesthestructureofaDNAsequence.
(5)The analysis of clusters of wavelet coefficients show
In other words, nucleotides are distributed along any
thatDNAwalkshavetofulfillsomegeometricalcon-
DNA sequence at first apparently randomly but at second
straints.
analysis according to some (statistical) mathematical con-
straintswhichdoesnotallowagivennucleotidetobearbi- InallDNAsequences,analyzedsofar,fordifferentkinds
trarilyfollowedbyanyotherremainingnucleotides. of living organisms, this geometrical symmetry [1–6] has
It is interesting to notice that even in the primitives or- beendetected.Inthefollowingthisanalysisisextendedalso
ganisms which billions of years ago have been colonizing to archaea, since they might be considered at the early
2 ComputationalandMathematicalMethodsinMedicine
a1 a2
50 50
1 50 1 50
h3 b1
50 50
1 50 1 50
Figure1:Indicatormatrixfor:(a1)pseudorandom70-lengthsequence;(a2)pseudo-periodic70-lengthsequencewithperiodπ=35;(b1)
70-lengthDNAsequenceofMycoplasmaKS1bacter;(h3)70-lengthDNAsequenceofAcidilobusArchaea.
stage of life and their DNA is compared with more evolved that DNA has to fulfill not only some chemical steady state
microorganismsasbacteria. given by the chemical ligands but also some symmetrical
Itwillbeshownthat,inspiteofthemanysimilaritieswith distribution of nucleotide along the sequence. In other
random sequences, only the wavelet analysis makes it pos- words,basepairshavetobeplacedexactlyinsomepositions.
sibletosingleoutsomedistinctions.Inparticular,thewavel- According to previous results, it will be shown that as
et coefficients of all (analyzed) organisms tend to fulfill a anyotherlivingorganismsalsotheseelementaryorganisms
minimum principle for the energy of the signal. Also the have DNA walks with fractal shape and wavelet coefficients
archaea which often live in extreme environments have to boundedonashort-rangewavelettransform.Inotherwords,
fulfillthesamegeometricalruleofanyotherlivingorganism. alsoanaerobicorganismwhichshouldbeunderstoodasthe
TheanalysisofDNAbywavelets[7–9],asseenin[8–12], mostelementaryatthefirststepoflifehavethesamesymme-
helps to single out local behavior and singularities [7, 13] tries on wavelet coefficients as for more evolved organism,
or to express the scale invariance of coefficients [14]. Also so that life has to fulfill some constrained distribution of
multifractal nature of the time series [15–17] can be easily nucleotides in order to give rise to some organism even at
detectedbywaveletanalysis. themostelementarystep.
Some previous paper have studied various sequences of In particular, in Section2, some remarks about the an-
DNAsuchasleukemiatetvariants,influenzavirusessuchas alyseddataaregiven.Section3dealswithsomeelementary
the A (H1N1) variant, mammalian, and a fungus (see [1– plots which can easily visualize the distribution of nu-
3, 14]) provided by the National Center for Biotechnology cleotides.TheUlamspiralplotisalsoproposedforthefirst
Information [18–21]. In all these papers it was observed timeanditisobservedadifferentdistributionofweak/strong
ComputationalandMathematicalMethodsinMedicine 3
b1 b2 b3
100 100 100
50 50 50
1 50 100 1 50 100 1 50 100
h1 h2 h3
100 100 100
50 50 50
1 50 100 1 50 100 1 50 100
Figure 2: Indicator matrix for the first 100 amino acids of (h1) Aeropyrum pernix K1, (h2) Acidianus hospitalis W1, (h3) Acidilobus
saccharovorans345-15(b1)MycoplasmaputrefaciensKS1,(b2)Mortierellaverticillata,and(b3)Blattabacteriumsp.
T T A A G short wavelet trasform is given in order to single out some
symmetriesatthelowerorderoftransform.
A A G G G 2.MaterialsandMethods
In the following we will take into consideration some ge-
T A A T A A nome,completesequencesofDNA,concerningthefollowing
archaea:
h1:AeropyrumpernixK1,completegenome.DNA,circu-
T G A T A A
lar,1669696bp,[18–21],accessionBA000002.3.Lin-
eage:Archaea;Crenarchaeota;Thermoprotei;Desul-
furococcales;Desulfurococcaceae;Aeropyrum;Aero-
A A G A A A
pyrumpernix;AeropyrumpernixK1.
This organism, which was the first strictly aerobic
Figure3:Distributionofnucleotidesonarectangularspiral. hyperthermophilicarchaeonsequenced,wasisolated
fromsulfuricgasesinKodakara-JimaIsland,Japanin
1993.
hydrogen bonds. Section4 provides some definitions about h2:AcidianushospitalisW1,completegenome.DNA,cir-
parameters of complexity. We will notice that all these cular,2137654bp,[18–21],accessionCP002535.Lin-
parameters give rise to the same classification of organism. eage:Archaea;Crenarchaeota;Thermoprotei;Sulfol-
Section4 proposes a complex numerical representation of obales;Sulfolobaceae;Acidianus;Acidianushospitalis;
DNAchainsandrandomwalks,whileinfinalSection6the AcidianushospitalisW1.
4 ComputationalandMathematicalMethodsinMedicine
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G T
Figure4:Spiraldistributionofthefirst3752nucleotidesfortherandomsequence.
h3:Acidilobussaccharovorans345-15.completegenome. lineages; Mucoromycotina; Mortierellales; Mortier-
DNA, circular, 2137654bp, [18–21], accession ellaceae;Mortierella;Mortierellaverticillata.
CP001742.1.
b3:Blattabacterium sp. (Periplaneta Americana) str.
Lineage: Archaea; Crenarchaeota; Thermoprotei;
BPLAN, complete genome. DNA, circular, length
Acidilobales; Acidilobaceae; Acidilobus; Acidilobus
636994nt, [18–21], accession NC 013418. Lineage:
saccharovorans;Acidilobussaccharovorans345-15.
Bacteria; Bacteroidetes/Chlorobi group; Bacteroide-
Anaerobicbacteriafoundinhotsprings.
tes; Flavobacteria; Flavobacteriales; Blattabacteriace-
ae;Blattabacterium;Blattabacteriumsp.(Periplaneta
tobecomparedwiththefollowing(aerobic/anaerobic)bac-
Americana);Blattabacteriumsp.(PeriplanetaAmeri-
teria/fungi:
cana)str.BPLAN.
b1:MycoplasmaputrefaciensKS1chromosome,complete
MoreoverwewillcompareDNAsequenceswithartificial
genome. DNA, circular, length 832603bp, [18–21],
sequencesofnucleotidesrandomlytaken(seeSection4).
accessionNC015946,.Lineage:Bacteria;Tenericutes;
Mollicutes; Mycoplasmatales; Mycoplasmataceae;
Mycoplasma; Mycoplasma putrefaciens; Mycoplasma 2.1. Archaea. Archaea are a group of elementary single-cell
putrefaciensKS1. microorganisms,havingnocellnucleusoranyothermem-
brane-boundorganelleswithintheircells.Theyaresimilarto
b2:Mortierella verticillata mitochondrion, complete ge- bacteria,sincetheyhavethesamesizeandshape(apartfew
nome. dsDNA, circular, length 58745bp, [18–21], exceptions)andthegenerallysimilarcellstructure.However,
accession NC 006838. Lineage: Eukaryota; Opistho- the evolutionary history of archaea and their biochemistry
konta; Fungi; Fungi incertae sedis; Basal fungal hassignificantdifferenceswithregardtootherformsoflife.
ComputationalandMathematicalMethodsinMedicine 5
A C
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Figure5:Spiraldistributionofthefirst3752nucleotidesforMycoplasmaputrefaciensKS1.
Thereforetheyareconsideredasmembersofaphylogenetic the ancestral organism were evolving by random permuta-
groupdistinctfrombacteriaandeukaryota. tionsfromaprimitiveassemblyofnucleotides. Sothatthe
Archaea during their evolution have been spreading all evolutioncanbeseenasatendencytoasteadystatefarfrom
over the Earth in almost all habitats [22, 23] existing in the randomness. Therefore, the bacteria’s DNA (and other
a broad range of habitats, being one of the major con- eukaryotes’ [1–6]), as a result of the evolution, shows the
tribution(20%)toearth’sbiomass.Themostpeculiarfeature existenceofsomehiddenstability.
of archaea is that they can live in some environments with
extreme life conditions (thus being considered as extremo- 3.CorrelationPlots
philes[22,24]).Indeed, some archaeasurvive to high tem-
peratures,over100◦C,whileotherscanliveinverycoldhabi- Inthissectionwewillconsidersomeelementaryplotsfrom
tats or highly saline, acidic, or alkaline water. Nevertheless whereitispossibletovisualizeautocorrelation,distribution
somearchaeaarelivinginmildconditions. lawofnucleotidesandtomeasuresomefundamentalparam-
It has been also recognized that the archaea may be the etersbyusingfrequencycount.
most ancient organisms on the Earth, so that archaea, and Let
eu-karyotes are probably diverged early from an ancestral
Ad=ef {A,C,G,T} (1)
colonyoforganisms.
Wewillsee,inthefollowing,thatarchaeaDNAitlooks be the finite set (alphabet) of nucleotides (nucleic acids):
veryclosetorandomsequencessothatwecanassumethat adenine (A), cytosine (C), guanine (G), thymine (T), and
6 ComputationalandMathematicalMethodsinMedicine
A C
G T
Figure6:Spiraldistributionofthefirst3752nucleotidesforMortierellaverticillata.
x∈Aanymemberofthealphabet.Nucleicacidsarefurther Ingeneralwecandefinean(cid:3)-lengthalphabetasfollows:
groupedaccordingtotheirligandpropertiesas letthe(cid:3)-lengthDNAwordbedefinedbythe(cid:3)-combination
of the 4 nucleotides (1). For each fixed length (cid:3) there are
(a)purine{A,G},pyrimidine{C,T}, 4(cid:3) words, however not all of them can be considered, from
(b)amino{A,C},keto{G,T}, biologicalpointofview,asindependentinstances(see,e.g.,
(c)weakhydrogenbonds{A,T},stronghydrogenbond Table1),forthiswedefinethe(cid:3)-lengthalphabetasthesetof
{G,C}. (cid:3)-lengthindependentwords:
(cid:2) (cid:3)
ADNAsequenceisthefinitesymbolicsequence A d=ef a ,a ,...,a , M d=ef |A |≤4(cid:3) (5)
(cid:3) 1 2 M(cid:3) (cid:3) (cid:3)
S =N×A (2) with|···|cardinalityofthesetand
(cid:4) (cid:5) (cid:6) (cid:7)
sothat (cid:3)d=ef length a , j =1,...,M . (6)
j (cid:3)
S d=ef {xh}h=1,...,N, N <∞ (3) For instance with (cid:3) = 1, the alphabet is A = A =
1
{A,C,G,T},with(cid:3)=3thealphabetisgivenbythe20amino
with
acids
x d=ef (h,x)=x(h), (h=1,2,...,N; x∈A) (4)
h A ={M,E,Q,D,R,T,N,H,V,G,L,S,P,F,I,C,A,K,Y,W}
3
(7)
beingthenucleotidexatthepositionh.
ComputationalandMathematicalMethodsinMedicine 7
A C
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Figure7:Spiraldistributionofthefirst3752nucleotidesforBlattabacteriumsp.
each amino acid being represented by a 3-length word of suchthat
Table1. ⎧
{A,LCe,tGS,NT}baenadnAN(cid:3)-tlhenegcthhosoerndearlepdhasbeqetu,eanDceNoAfsneuqculeenotciedoesf u(xh,xk)d=ef ⎨⎩10 iiffxxhh==/ xxkk,, (xh∈S, xk ∈S), (12)
wordsisthefinitesymbolicsequence
D (S )=N×A (8) with
(cid:3) N (cid:3)
u(x ,x )=u(x ,x ), u(x ,x )=1 (13)
sothat h k k h h h
D(cid:3)(SN)d=ef {xh}h=1,...,N, (x∈A(cid:3); N <∞) (9) and,whereforshort,wehaveassumed
with S d=ef D (S ). (14)
1 N
xhd=ef (h,x), (h=1,2,...,N; x∈A(cid:3)) (10) Accordingto(12),theindicatorofanN-lengthsequencecan
beeasilyrepresentedbytheN×Nsparsesymmetricmatrixof
beingthewordxatthepositionh. binaryvalues{0,1}whichresultsfromtheindicatormatrix
(seealso[3–5])
3.1. Indicator Matrix. The 2D indicator function, based on
the1Ddefinitiongivenin[25],isthemap u d=ef u(x ,x ), (x ∈S, x ∈S; h,k=1,...,N),
hk h k h k
u:S×S −→{0,1} (11) (15)
8 ComputationalandMathematicalMethodsinMedicine
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Figure8:Spiraldistributionofthefirst3752nucleotidesforAeropyrumpernixK1.
being,explicitly This squared matrix can be plotted in 2 dimensions by
puttingablackdotwhereu =1andwhitespotwhenu =
hk hk
... ... ... ... ... ... ... ... ... ... ... ... 0wh(Ficighuirsea1s)ptehcuiaslgciavsiengofritsheetoretchuerrtewnoce-dpilmote[n2s6io].naldotplot,
G 0 1 0 0 0 0 0 0 0 1 ··· Asimplegeneralizationofthismatrixcanbeconsidered
forthealphabetsA ,asfollows.Bychoosingthe3alphabet
C 0 0 0 1 0 0 0 0 1 0 ··· (cid:3)
ofaminoacids,the2Dindicatorfunctionisthemap
A 1 0 0 0 1 0 1 1 0 0 ···
u:D (S )×D (S )−→{0,1} (17)
A 1 0 0 0 1 0 1 1 0 0 ··· 3 N 3 N
T 0 0 1 0 0 1 0 0 0 0 ··· suchthat
(16)
A 0 0 0 0 1 0 0 1 0 0 ··· ⎧
⎨1 ifx =x ,
C 0 0 0 1 0 0 0 0 1 0 ··· u(xh,xk)d=ef ⎩ h k (xh∈D3(SN), xk ∈D3(SN)),
0 ifxh=/ xk,
T 0 0 1 0 0 1 0 0 0 0 ···
(18)
G 0 1 0 0 0 0 0 0 0 1 ···
with
A 1 0 0 0 1 0 0 1 0 0 ···
u A G T C A T A A C G ··· u(x ,x )=u(x ,x ), u(x ,x )=1. (19)
hk h k k h h h
ComputationalandMathematicalMethodsinMedicine 9
Table1:Correspondencecodonstoaminoacids.
Aminoacid Codon
1 M Methionine ATG
2 E Glutamicacid GAA,GAG
3 Q Glutamine CAA,CAG
4 D Asparticacid GAT,GAC
5 R Arginine CGT,CGC,CGA,CGG,AGA,AGG
6 T Threonine ACT,ACC,ACA,ACG
7 N Asparagine AAT,AAC
8 H Histidine CAT,CAC
9 V Valine GTT,GTC,GTA,GTG
10 G Glycine GGT,GGC,GGA,GGG
11 L Leucine TTA,TTG,CTT,CTC,CTA,CTG
12 S Serine TCT,TCC,TCA,TCG,AGT,AGC
13 P Proline CCT,CCC,CCA,CCG
14 F Phenylalanine TTT,TTC
15 I Isoleucine ATT,ATC,ATA
16 C Cysteine TGT,TGC
17 A Alanine GCT,GCC,GCA,GCG
18 K Lysine AAA,AAG
19 Y Thyroxine TAT,TAC
20 W Tryptophan TGG
Stop TAA,TAG,TGA
According to (12), the indicator, on the 3-alphabet of 3.2. Test Sequences. In the following, in order to single out
amino acids of an N-length sequence can be easily repre- the main features of biological sequences, we will compare
sentedbytheN×Nsparsesymmetricmatrixofbinaryvalues theDNAsequencewithsometestsequences.
{0,1}:
(1)Pseudorandom N-length sequence of nucleotides is
uhk d=ef u(xh,xk), the sequence {Ri}(cid:3)i=1,...,N where ri is a symbol ran-
(20) domly chosen in the alphabet A , like for example,
(cid:3)
(xh∈D3(SN), xk ∈D3(SN); h,k=1,...,N), ((cid:3)=1):
being,explicitly {A,C,A,G,T,A,T,G,G,A,T,T,A,C,C,G,...}. (22)
... ... ... ... ... ... ... ... ... ... ... ... (2)Pseudoperiodic N-sequence of nucleotides with
M 1 0 0 0 0 0 0 0 0 1 ··· period π is the direct sum of a given π-length
pseudorandomsequence,suchthatN =kπ, (k∈N)
Q 0 0 0 0 0 0 0 0 1 0 ··· andR =R ,forexample,
i i+π
R 0 0 1 1 0 0 0 1 0 0 ···
{A,C,A,G,A,C,A,G,A,C,A,G,A,C,A,G,...},
T 0 0 0 0 0 1 1 0 0 0 ··· (23)
(π =4).
T 0 0 0 0 0 1 1 0 0 0 ···
(21)
E 0 0 0 0 1 0 0 1 0 0 ··· Whenπ =1wehaveapseudorandomsequence.
R 0 0 1 1 0 0 0 0 0 0 ··· If we plot the indicator matrix of some bacteria and
R 0 0 1 1 0 0 0 1 0 0 ··· compareitwithapseudorandomandperiodicsequence,we
canseethat(Figure1)
K 0 1 0 0 0 0 0 0 0 0 ···
(1)themaindiagonalisasymmetryaxisfortheplot;
M 1 0 0 0 0 0 0 0 0 1 ···
(2)therearesomemotifswhicharerepeatedatdifferent
u M K R R E T T R Q M ···
hk scaleslikeinafractal;
Withthegraphicalrepresentationofthismatrixwecanalso (3)periodicity is detected by parallel lines to the main
showthecorrelationofaminoacids. diagonal(Figure1(a2));
10 ComputationalandMathematicalMethodsinMedicine
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Figure9:Spiraldistributionofthefirst3752nucleotidesforAcidianushospitalisW1.
(4)emptyspacesaremoredistributedthanfilledspaces, as a consequence of the evolution, nucleotides move from
in the sense that the matrix u is a sparse matrix a disordered aggregation toward a more organized stru–
hk
(havingmore0’sthan1’s); cture,shownbythegrowingislandsinthedotplot.Thebio–
logical evolution is such that the challenge for the self-
(5)itseemsthattherearesomesquare-likeislandswhere
organization might follow from random permutations of
blackspotsaremoreconcentrated;theseislandsshow
a primitive disordered sequence so that the organization,
the persistence of a nucleotide (Figures 1(a2) and
that is, the complexity, is only the result of many arbitrary
1(b1));
permutationsofrandomness.Duringthechallengeforcom-
(6)thedotplotofarchaeaisverysimilartothedotplot
plexity, DNA sequence becomes “less random” and it loses
ofarandomsequence(Figures1(a1)and1(h3)).
somekindofenergy.
ItcanbenoticedthatDNAsequencesofalivingorgan- Fromthegraphicalrepresentationoftheindicatormatrix
ismresemble(Figure1)randomsequences,withsomeshort forbacteriaandaminoacidswecanseeamoresparsematrix,
range influence, built on the same alphabet. This has been butwithsometypicalplots(Figure2).
takenasanaxiomofnucleotidesdistribution, sothatDNA
sequencesareoftenconsideredasMarkovchain[27].How- 3.3.SpiralPlot. Inthissectionweconsidera2Ddistribution
ever, there are some hidden rules in combining the nucle- of nucleotides, following the idea given by Ulam for the
otidesandtheseruleslead,duringtheevolution,toasteady distribution of primes, along an Ulam-like spiral [28]. In
distribution.Infact,themoreprimitivethesequenceis,the ordertofindsomepatternsintheirdistribution,nucleotides
morerandomlydistributedthenucleotidesare.Itseemsthat arearrangedalongarectangularspiral.Thisisequivalentto