Phylogenetic Inference via Categorical Spectra
Categorical Time-series of Protein Amino-acid Sequences
Sumaira Qureshi1
Roger Brown2
Phylomania: Hobart Tasmania. October 2009
Sumaira Qureshi, Roger Brown
Phylogenetic Inference via Categorical Spectra
Sumaira Qureshi
Insert portrait here.
Sumaira Qureshi, Roger Brown
Phylogenetic Inference via Categorical Spectra
Stimulus
Ph.D. Thesis topic for Sumaira (Part-II)
Submitted for examination: July-2009
Examiners reports received a few days ago.
Outcome not known at this time.
Sumaira Qureshi, Roger Brown
Phylogenetic Inference via Categorical Spectra
Categorical Time-series
What is a Categorical Time-series ?
Consider the sequence: ABFDEHBFDEHBCDKAJGDE
Assign categories:
α = {A, H } , β = {B , J } , γ = {C , F , G } , δ = {D } , ε = {E , K }
List by categories:
αβ γδ ε αβ γδ ε αβ γδ ε αβ γδ ε
Fourier techniques are useful for establishing categories for
sequences with inherent periodicities.
Sumaira Qureshi, Roger Brown
Phylogenetic Inference via Categorical Spectra
Categorical Time-series
What is a Categorical Time-series ?
Consider the sequence: ABFDEHBFDEHBCDKAJGDE
Assign categories:
α = {A, H } , β = {B , J } , γ = {C , F , G } , δ = {D } , ε = {E , K }
List by categories:
αβ γδ ε αβ γδ ε αβ γδ ε αβ γδ ε
Fourier techniques are useful for establishing categories for
sequences with inherent periodicities.
Sumaira Qureshi, Roger Brown
Phylogenetic Inference via Categorical Spectra
Categorical Time-series
What is a Categorical Time-series ?
Consider the sequence: ABFDEHBFDEHBCDKAJGDE
Assign categories:
α = {A, H } , β = {B , J } , γ = {C , F , G } , δ = {D } , ε = {E , K }
List by categories:
αβ γδ ε αβ γδ ε αβ γδ ε αβ γδ ε
Fourier techniques are useful for establishing categories for
sequences with inherent periodicities.
Sumaira Qureshi, Roger Brown
Phylogenetic Inference via Categorical Spectra
Categorical Time-series
What is a Categorical Time-series ?
Consider the sequence: ABFDEHBFDEHBCDKAJGDE
Assign categories:
α = {A, H } , β = {B , J } , γ = {C , F , G } , δ = {D } , ε = {E , K }
List by categories:
αβ γδ ε αβ γδ ε αβ γδ ε αβ γδ ε
Fourier techniques are useful for establishing categories for
sequences with inherent periodicities.
Sumaira Qureshi, Roger Brown
Phylogenetic Inference via Categorical Spectra
Categorical Time-series
What is a Categorical Time-series ?
Consider the sequence: ABFDEHBFDEHBCDKAJGDE
Assign categories:
α = {A, H } , β = {B , J } , γ = {C , F , G } , δ = {D } , ε = {E , K }
List by categories:
αβ γδ ε αβ γδ ε αβ γδ ε αβ γδ ε
Fourier techniques are useful for establishing categories for
sequences with inherent periodicities.
Sumaira Qureshi, Roger Brown
Phylogenetic Inference via Categorical Spectra
Previous Applications of Fourier Analysis to Protein
Sequences
Far too many to mention them all, but two notable examples are:
The Atchley group: Prof. William R. Atchley, North Carolina
State University
The Resonance Recognition Method (RRM), under continuing
development by Irena Cosic et al.
Almost all previous studies have two features in common:
1
2
Each amino-acid is assigned a numerical value derived from a
table physio-chemical properties, such as electropositivity,
hydrophobicity etc.
A frequency analysis method that focuses on the local
instantaneous frequency. (e.g. Burg and Lewis correlation
matrices)
Sumaira Qureshi, Roger Brown
Phylogenetic Inference via Categorical Spectra
Previous Applications of Fourier Analysis to Protein
Sequences
Far too many to mention them all, but two notable examples are:
The Atchley group: Prof. William R. Atchley, North Carolina
State University
The Resonance Recognition Method (RRM), under continuing
development by Irena Cosic et al.
Almost all previous studies have two features in common:
1
2
Each amino-acid is assigned a numerical value derived from a
table physio-chemical properties, such as electropositivity,
hydrophobicity etc.
A frequency analysis method that focuses on the local
instantaneous frequency. (e.g. Burg and Lewis correlation
matrices)
Sumaira Qureshi, Roger Brown
Phylogenetic Inference via Categorical Spectra
Previous Applications of Fourier Analysis to Protein
Sequences
Far too many to mention them all, but two notable examples are:
The Atchley group: Prof. William R. Atchley, North Carolina
State University
The Resonance Recognition Method (RRM), under continuing
development by Irena Cosic et al.
Almost all previous studies have two features in common:
1
2
Each amino-acid is assigned a numerical value derived from a
table physio-chemical properties, such as electropositivity,
hydrophobicity etc.
A frequency analysis method that focuses on the local
instantaneous frequency. (e.g. Burg and Lewis correlation
matrices)
Sumaira Qureshi, Roger Brown
Phylogenetic Inference via Categorical Spectra
Features of this approach
1
2
Numerical values are assigned to each amino-acid type in a
manner to optimize some chosen metric of the sequence. No
consideration is given to any of the physical or chemical
properties of any particular residue type. Furthermore , the
assigned numerical values are not constant but depend on
spatial frequency. In other words a dierent spectrum of values
is assigned to each amino-acid for each protein sequence.
Frequency analysis techniques adopt a global rather than local
perspective. i.e. Long-range spatial frequency rather than the
instantaneous frequency.
Sumaira Qureshi, Roger Brown
Phylogenetic Inference via Categorical Spectra
What is instantaneous frequency ?
sin (α + δ ) + sin (α − δ ) = 2sin (α) cos (δ )
Sumaira Qureshi, Roger Brown
Phylogenetic Inference via Categorical Spectra
Illustration of acoustic beats sin (α + δ ) + sin (α − δ ) = 2sin (α) cos (δ )
Sumaira Qureshi, Roger Brown
Phylogenetic Inference via Categorical Spectra
The Canadian, Delhousie Group
Examining Protein Structure and Similarities by Spectral
Analysis Technique (2006)
Krista Collins, Hong Gu, Chris Field
Spectral analysis for categorical time series: Scaling and the
spectral envelope (1993)
Stoer, D S. and Tyler, D E. and McDougall, A J.
(Biometrika)
Sumaira Qureshi, Roger Brown
Phylogenetic Inference via Categorical Spectra
Variations on the Stoer method
Scalar algorithm
E
(ω) = min
φ , |β i
1
∑ sin (ω
a
N
t
{t }
+ φ ) − βτ(t )
2
Complex algorithm
E
(ω) = min
|β i
1
∑
a
N
{t }
Sumaira Qureshi, Roger Brown
e
iωt − β ∗
τ(t )
e
−i ω t
− βτ(t )
Phylogenetic Inference via Categorical Spectra
Sperm-whale Myoglobin Protein Sequence
>P02185|MYG_PHYCA Myoglobin - Physeter catodon (Sperm whale)
(Physeter macrocephalus).
VLSEGEWQLVLHVWAKVEADVAGHGQDILIRLFKSHPETLEKFDRFKHLKTE
AEMKASEDLKKHGVTVLTALGAILKKKGHHEAELKPLAQSHATKHKIPIKYL
EFISEAIIHVLHSRHPGDFGADAQGAMNKALELFRKDIAAKYKELGYQG
Sumaira Qureshi, Roger Brown
Phylogenetic Inference via Categorical Spectra
Scalar Spectral Envelope
Least-squares algorithm
Spectral envelope
0.40
0.42
0.44
0.46
0.48
2.65
P02185|MYG_PHYCA Myoglobin
Physeter catodon (Sperm whale)
2.70
2.75
2.80
Spatial frequency
Sumaira Qureshi, Roger Brown
2.85
2.90
-1
(x10 )
Phylogenetic Inference via Categorical Spectra
α−helix
Structure
Sumaira Qureshi, Roger Brown
Phylogenetic Inference via Categorical Spectra
Spatial wavelength - Spatial frequency
1/3.6
=
Sumaira Qureshi, Roger Brown
0.277777 . . .
Phylogenetic Inference via Categorical Spectra
Myoglobin Structure
Sumaira Qureshi, Roger Brown
Phylogenetic Inference via Categorical Spectra
Phylogenetic inference from Categorical Spectra
Scalar Spectral Envelope
Scalar Spectral Envelope
0.36
0.40
Spectral envelope
Spectral envelope
0.38
0.42
0.44
0.40
0.42
0.44
0.46
0.46
0.48
P02033: Red Colobus
Hemoglobin Beta chain
0.26
Q7M3C2: Bamboo Lemur
Hemoglobin Beta chain
0.28
Spatial frequency
Scalar Spectral Envelope
0.30
0.26
0.40
0.28
Spatial frequency
Scalar Spectral Envelope
0.30
0.38
Spectral envelope
Spectral envelope
0.40
0.42
0.44
0.42
0.44
0.46
0.46
P19885: Western black-and-white Colobus
Hemoglobin Beta chain
0.26
0.28
0.48
0.30
Sumaira
SpatialQureshi,
frequency Roger Brown
P02053: Brown Lemur
Hemoglobin Beta chain
0.26
0.28
0.30
Phylogenetic Inference
viafrequency
Categorical Spectra
Spatial
Rotor part of the ATP-synthase molecule
Sumaira Qureshi, Roger Brown
Phylogenetic Inference via Categorical Spectra
An ATP-synthase helix pair
Sumaira Qureshi, Roger Brown
Phylogenetic Inference via Categorical Spectra
ATP-Synthase gapped sequences
LIPLLR TQFFIVMGLV
DAIPMIAVGL GLYVMFAVA
LIPLLR TQFFIVMGLV - DAIPMIAVGL GLYVMFAVA
LIPLLR TQFFIVMGLV DAIPMIAVGL GLYVMFAVA
LIPLLR TQFFIVMGLV DAIPMIAVGL GLYVMFAVA
Table: Amino-acid sequences of ATP-synthase outer helices.
Sumaira Qureshi, Roger Brown
Phylogenetic Inference via Categorical Spectra
Complex Categorical spectra
Complex Spectral Envelope
Complex Spectral Envelope
0.5
0.55
Spectral Envelope
Spectral Envelope
0.60
0.65
0.70
0.6
0.7
0.75
1C17h2
0.1
1C17h2
0.2
0.3
Spatial frequency
Complex Spectral Envelope
0.4
0.1
0.2
0.3
Spatial frequency
Complex Spectral Envelope
0.4
0.5
Spectral Envelope
Spectral Envelope
0.5
0.6
0.6
0.7
0.7
1C17h2
0.1
1C17h2
0.2
0.3
0.4
Sumaira
SpatialQureshi,
frequency Roger Brown
0.1
0.2
0.3
0.4
Phylogenetic Inference
viafrequency
Categorical Spectra
Spatial
Fractional spacing
Complex Spectral Envelope
Spectral Envelope
0.5
0.6
0.7
Outer helix ATP-synthase
Punctuated sequence
0.15
0.20
Spatial frequency
Sumaira Qureshi, Roger Brown
0.25
Phylogenetic Inference via Categorical Spectra
Punctuated Sperm-whale myoglobin sequence
> P 02185|MYG − HYCA Myoglobin
VL
< 2.51235 > SEGEWQLVLHVWAKVEA
< 3.48089 > DVAGHGQDILIRLFKS
< 0.96800 > HPETLEKFDRFKHLK
< 3.18381 > TEAEMKA
< 1.75174 > SEDLKKHGVTVLTALGAILKKKGHHEAE
< 1.32164 > LKPLAQSHATKHKI
< 2.41373 > PIKYLEFISEAIIHVLHSRHPGDF
< 0.94676 > GADAQGAMNKALELFRKDIAAKYKELGYQG
Sumaira Qureshi, Roger Brown
Phylogenetic Inference via Categorical Spectra
Raw and punctuated sperm-whale complex spectra
Complex Spectral Envelope
Complex Spectral Envelope
0.80
0.82
Spectral Envelope
Spectral Envelope
0.80
0.84
0.86
0.88
0.90
0.92
0.24
0.85
0.90
P02185: Sperm Whale
Raw myoglobin sequence
0.26
0.28
Spatial frequency
P02185: Sperm Whale
Punctuated sequence
0.30
Sumaira Qureshi, Roger Brown
0.32
0.24
0.26
0.28
Spatial frequency
0.30
Phylogenetic Inference via Categorical Spectra
0.32
Distance measure between two categorical spectra
If S is a residue sequence of length N , consider
(S ; t ) = βτ(t ) = hY (t ) | β (ω)i. X (S ; t ) is a complex (or real)
sequence associated with the categorical sequence S
X
Λ (S1 , S2 ) =
1
Z ω=∞
Na
ω=0
W
(ω) ∑ kX (S1 ; t ) − X (S2 ; t )k2 d ω
Sumaira Qureshi, Roger Brown
{t }
Phylogenetic Inference via Categorical Spectra
Challenging rodent data-set
Common name
Scientic name
Short name
Q36461
Platypus
Ornithorhynchus anatinus
Platypus
B2WVN5
Leopard
Panthera pardus
Leopard
O03207
White Rhinoceros
Ceratotherium simum
W Rhino
O47561
European Hare
Lepus europaeus
Hare
O63910
Dormouse
Myoxus glis
Dormouse
P00156
Human
Homo sapiens
Human
P00157
Cow
Bos taurus
Cow
P00158
Mouse
Mus musculus
Mouse
P00159
Rat
Rattus norvegicus
Rat
P24950
Fin Whale
Balaenoptera physalus
Fin Whale
P24959
Sheep
Ovis aries
Sheep
P24964
Pig
Sus scrofa
Pig
Rabbit
P34863
Rabbit
Oryctolagus cuniculus
P41285
Blue Whale
Balaenoptera musculus
Blue Whale
P41303
American Opossum
Didelphis marsupialis
A Opossum
Horse
Equus caballus
Horse
Cat
Inference via Categorical
Spectra
Felis Phylogenetic
silvestris catus
Cat
P48665
P48886
Sumaira Qureshi, Roger Brown
Molecular structure of Cytochrome-b
Sumaira Qureshi, Roger Brown
Phylogenetic Inference via Categorical Spectra
Comparison against established distance measures
Distance matrix correlation
1.0
0.8
0.8
0.6
0.6
ScoreDist
Spectral Complex
Distance matrix correlation
1.0
0.4
0.2
0.0
0.0
0.4
0.2
0.2
0.4
0.6
Jones Taylor Thornton
0.8
1.0
Sumaira Qureshi, Roger Brown
0.0
0.0
0.2
0.4
0.6
Jones Taylor Thornton
0.8
Phylogenetic Inference via Categorical Spectra
1.0
More comparisons
Distance matrix correlation
1.0
0.8
0.8
Dayhoff PAM
Categories
Distance matrix correlation
1.0
0.6
0.4
0.2
0.0
0.0
0.6
0.4
0.2
0.2
0.4
0.6
Jones Taylor Thornton
0.8
1.0
Sumaira Qureshi, Roger Brown
0.0
0.0
0.2
0.4
0.6
Jones Taylor Thornton
0.8
Phylogenetic Inference via Categorical Spectra
1.0
Goodness of t rankings
A remarkable result is that phylogenetic trees as determined by well
established software packages have considerably lower least-squares
residual values compared against other accepted distance measures.
With respect to a phylogenetic tree T appropriate to the data, if
dj ,k is the sum of lengths (weights) for the edges that comprise the
path between leaf j and leaf k and Dj ,k is the measures or
estimated distance between the same leaves then the weighted sum
of squared residual values is
S
(T ) =
∑
j ,k ε{L}
dj ,k
− Dj ,k
Dj ,k
Sumaira Qureshi, Roger Brown
2
;
{L} is the set of all leaves
Phylogenetic Inference via Categorical Spectra
Tree structure consistency
Distance measure\Tree algorithm
Spectral Complex
Spectral Scalar
ScoreDist
Jones Taylor Thornton
Heniko/Tillier PMB
Dayho PAM
Kimura
Categories
GNJ
2.70953
2.34364
3.52057
3.60840
3.61953
3.71021
3.85859
3.41611
Fitch-Margoliash
2.75540
2.34365
3.52057
3.60840
3.62030
3.71021
3.85859
3.41612
BIONJ
3.27449
3.21843
3.82674
4.00037
3.85857
3.99255
4.24072
3.68582
Table: Weighted least-squares t results returned from trees determined
by the distance and tree-reconstruction algorithms listed, applied to the
mitochondrial cytochrome-b sequences of the taxa from the previous
data-set.
Sumaira Qureshi, Roger Brown
Phylogenetic Inference via Categorical Spectra
Cytochrome-b Phylogenetic trees I
Fin Whale
Fin Whale
W Rhino
W Rhino
Horse
Blue Whale
Horse
Ind Rhino
Blue Whale
Donkey
Donkey
Hippo
Ind Rhino
Hippo
Cow
Leopard
Cow
Tiger
Hbr Seal
Leopard
Sheep
Cat
Pig
Hbr Seal
Tiger
Pig
Sheep
Cat
Rabbit
Dog
Dog
Gibbon
Hare
Human
Dormouse
Gibbon
Orangutan
Rabbit
Grey Marmot
Human
Gorilla
Hare
Squirrel
Dormouse
Orangutan
Grey Marmot
Guinea Pig
Gorilla
Guinea Pig
Squirrel
Platypus
Platypus
Rus Hamster
Rus Hamster
A Opossum
A Opossum
Mouse
Wallaroo
Rat
Mouse
Wombat
Cytochrome-b Categories BIONJ
Sumaira Qureshi, Roger Brown
Wallaroo
Wombat
Rat
Cytochrome-b ScoreDist BIONJ
Phylogenetic Inference via Categorical Spectra
Cytochrome-b Phylogenetic trees II
Fin Whale
W Rhino
Fin Whale
W Rhino
Horse
Horse
Ind Rhino
Blue Whale
Blue Whale
Ind Rhino
Donkey
Donkey
Hippo
Hippo
Pig
Cow
Leopard
Leopard
Hbr Seal
Pig
Hbr Seal
Cow
Sheep
Sheep
Tiger
Tiger
Cat
Dog
Cat
Dog
Dormouse
Rabbit
Dormouse
Grey Marmot
Hare
Grey Marmot
Rabbit
Squirrel
Squirrel
Gibbon
Hare
Orangutan
Gibbon
Guinea Pig
Guinea Pig
Orangutan
Human
Human
Gorilla
Gorilla
Platypus
Platypus
Rus Hamster
Rus Hamster
A Opossum
A Opossum
Mouse
Wallaroo
Wombat
Mouse
Rat
Cytochrome-b Spectral BIONJ
Sumaira Qureshi, Roger Brown
Wallaroo
Wombat
Rat
Cytochrome-b Spectral GNJ
Phylogenetic Inference via Categorical Spectra
Two strands of Human NADH-ubiquinone oxidoreductase
Sumaira Qureshi, Roger Brown
Phylogenetic Inference via Categorical Spectra
NU4M Phylogenetic Trees I
Chimp
Chimp
Rt Lemur
W Lemur
S Lemur
Bonobo
N Lemur
Human
Rt Lemur
Bonobo
W Lemur
S Lemur
Human
N Lemur
Gorilla
Gorilla
Echidna
Echidna
Orangutan
Platypus
Gibbon
Platypus
Orangutan
Gibbon
Wallaroo
Mouse
Rabbit
Wallaroo
A Opposum
Mouse
Rabbit
Rat
A Opposum
Rat
Dugong
Dugong
A Elephant
A Elephant
Armadillo
Armadillo
I Elephant
Blue Whale
Pygmy Whale
Blue Whale
I Elephant
Pygmy Whale
Pig
Pig
Hippo
Fin Whale
W Rhino
Fin Whale
W Rhino
Hippo
Sheep
Sheep
I Rhino
I Rhino
Cat
Donkey
Gray Wolf
Zebu
Horse
Dog
Cow
H Seal
Cat
Yak
Yak
Gray Seal
NU4M Categories BIONJ
Sumaira Qureshi, Roger Brown
Gray Wolf
Zebu
Donkey
Horse
Dog
Cow
H Seal
Gray Seal
NU4M JTT BIONJ
Phylogenetic Inference via Categorical Spectra
NU4M Phylogenetic Trees II
Chimp
Rt Lemur
Bonobo
Chimp
W Lemur
N Lemur
Human
Rt Lemur
Bonobo
S Lemur
W Lemur
S Lemur
N Lemur
Human
Echidna
Gorilla
Gorilla
Platypus
Orangutan
Orangutan
Wallaroo
Gibbon
Echidna
Platypus
Gibbon
Rat
Mouse
A Opposum
Wallaroo
Mouse
A Opposum
Rat
Rabbit
Dugong
Dugong
Rabbit
I Elephant
Blue Whale
Blue Whale
Pygmy Whale
W Rhino
Pig
Hippo
Hippo
Fin Whale
Sheep
W Rhino
Sheep
I Rhino
Yak
Pig
Cat
Yak
Gray Wolf
Cow
I Rhino
Cat
Donkey
Zebu
I Elephant
Armadillo
Pygmy Whale
Fin Whale
A Elephant
A Elephant
Armadillo
Dog
H Seal
Horse
Gray Wolf
Dog
Zebu
Gray Seal
NU4M JTT FastME
Sumaira Qureshi, Roger Brown
Cow
Donkey
Horse
H Seal Gray Seal
NU4M Spectral 0.08-0.46 BIONJ
Phylogenetic Inference via Categorical Spectra
Work in progress
(Second and third cytochrome-b
Argand plot of complex beta values
Sumaira Qureshi, Roger Brown
α−helices)
Argand plot of complex beta values
Phylogenetic Inference via Categorical Spectra