Qiang Li

Estimation of evolutionary distance
based on K-string composition
Qiang Li
Department of Combinatorics and Geometry
CAS-MPG Partner Institute for Computational Biology
Difficulties in phylogenomics
• Species trees versus
gene trees
• Too few common
genes
“The tree of one
percent”
• Model selection
A science, or an art?
• Computational
complexity
• …
A tree based on 31 genes (Ciccarelli ’06)
Phylogenetics above sequence-level
• Rare genomic change
• Gene content & gene order
• K-string composition
“In summary, the methods for inferring trees based on
whole-genome features are at an early stage of their
development, which might be comparable to that of
sequence-based methods in the early 1970s. In particular,
they generally lack a global probabilistic modeling.”
(Philippe 2005)
CVTree: Compositional vector
• “Random background”: (K – 2)-order Markov chain
f (α1 α K −1 ) f (α 2 α K )
f ( α 1 α K ) =
f (α 2 α K −1 )
0
• “Subtraction”
 f (i ) − f 0 (i )
,

0
f (i )
ai = 
0,

f 0 (i ) ≠ 0
f 0 (i ) = 0
• Distance (dissimilarity)
1 − cos(a, b)
d (a, b)
=
∈ [0,1]
2
i ∈ ΣK
f versus f0
Courtesy of Alexander Grossmann
CVTree
•
•
•
•
simple
efficient
parameter free
truly whole
genomic
…
• mysterious
(Qi et al ’04)
CVTree of chloroplasts
CVTree of coronavirus
CVTree of dsDNA viruses
CVTree of fungi
Mycge
Mycpn
Mycpe
Urepa
Mycp
Borb u
Trep u
Chl a
Ch mu
Ch ltr
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Ch hlpnC
A lp
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R rt5
hi W
m
e
pe
Aer ae
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t
l
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l
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Th lsp
Ha tma
e
M
e
um su
r
B ru
B hilo r
R auc
C oliC
Ec oliE
Ec oliO
Ec liK
Eco l
Shif
Salti
Salty
YerpeC
YerpeK
Haein
Pasmu
Sheon
Vibch
Vibvu
Neim
Neim eM
Pse eZ
Ps ae
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X nc
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ca i
p
S
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a u
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m
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c
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csu
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m
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e
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e
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p
n
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a
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br
ig n
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M
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et c
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ra
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cu
Plafa
Worm
Yeast
Schpo
Arath
Fusnu
Thete
Clope
Cloab
StrpnT
R
Strpn u
Strm S
y
Strp yM
p
Str pyG
Str agN
Str agV a
r
St acl p
L ae
St auW
a
St
0.05
The tree plotted to scale
“Concentrating on the
topology of the trees in
the first place, we did
not scale the branch
lengths on the tree.
However, these lengths
should reflect evolution
rates in terms of Kstring composition
changes. The
calibration of branch
lengths is further
complicated by the
overlapping nature of
the K-strings when K ≥
2.”
• “However, the methods proposed so far are
extremely crude. Oligonucleotide or oligopeptide
frequencies are transformed into distances
without any underlying model of evolution. It is
nevertheless remarkable that something
considered as a bias in standard sequence-based
methods (Lockhart et al. 1994) contains a
phylogenetic signal, but it is not yet clear whether
accurate methods can be developed to extract it.”
(Philippe et al. 2005)
• “Why, for example, does the K-string complement
of a proteome yield a tree that is similar to
sequence-based trees?”
(Snel et al. 2005)
A heuristic probabilistic model
• Transition of frequency distribution:
Assume that the evolution process is stationary, and
the average mutation rate per site is 1 – α. Let
L = |S| - K + 1, then the conditional distribution
(
( t +1)
f S
)
(i ) S (i=
) n= binomial(n, α )
(t )
K

K E S(i ) 
*binomial  L, (1 − α )

L 

• Time evolution of expectation
(
)
n =+
E S (i ) S (i ) =
E S(i ) α
( t ')
(t )
K ( t ' −t )
(n − E S(i ))
Calibration of CVTree
• Generalized CV
w(S,
=
i ) (S(i ) − S (i ))c(i )
0
where S0(i) is an unbiased estimator of S(i),
and c(i) is arbitrary “normalization” factor.
• Angle between vectors vs. sequence identity
E cos( w(S), w(S')) = q K
• Calibration
pˆ (S,S') = 1 − cos1/ K ( w(S), w(S'))
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The calibrated tree
Using pre-/absence of words
• Any sequence S as above can be represented
also by just the set X(S) of its K-strings. In a
“gedanken alignment”, ignoring repeats and
collisions, we may put
1  X (S)  X (S') X (S)  X (S')
(K )
+
Q (S,S') : 
X (S)
X (S')
2
and
q
(K )
(S,S') := Q
(K )
1/ K
(S,S')



K = 9 Tree
M
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et
pe
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ae
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l
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rfu
y
P
Su
o
rh b
Py yra cfu
P Ar
K = 14
Using pre-/absence of words
• Any sequence S as above can be represented
also by just the set X(S) of its K-strings. In a
“gedanken alignment”, ignoring repeats and
collisions, we may put
1  X (S)  X (S') X (S)  X (S')
(K )
+
Q (S,S') : 
X (S)
X (S')
2
and
or
q ( K ) (S,S') := Q ( K ) (S,S ')1/ K
:= Q ( K +1) (S,S') Q ( K ) (S,S')



Making up alignments
• Pattern:
1: key, must match
0: space, to watch
• “Align” pairwise:
For keys occur in both
sequences, concatenate
letters in spaces.
• Sequence-based distance
111110
X: VKAAWG,HAGEYG…
Y: VKAAWS,HAGEYG…
GG…
SG…
pˆ = nd / n
Y eas
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D
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et
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Synpc
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R
Bru me hilo
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a
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p
pe
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Pl
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o
ev
Th eac sp
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m
et
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011111110 tree
The missing words approach
• Probabilities of double- vs. single-missing
1/ K
p0 = u
2−q K
ln p0 

⇒ qˆ =  2 −

u
ln


• Estimation of probabilities
pˆ 0 =
n0
Σ
K
n0 + nB n0 + nA
·
uˆ =
K
K
Σ
Σ
E nc
M
e
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uc
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pyG
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gV
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a
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so
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n
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lm
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r n te
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lo
C
0.05
Biflo
Coref
Corgl
Mycle
Myctu
Myct C
uH
Dei Strco
r
An a
a
S
s
Th ynpc p
e
C h el
lte
Le
A p
Ag grt5 in
rt5
W
pe
A er
ae
P yr
lto
Su so
l
S u e vo
T h eac h
t
Th Met
ka
et
M
MW9 Tree
ilo r
Rh auc n
C icc
R icpr
R cai
Bu cap
Bu gbr
Wi liC
Eco liE
Eco O
Ecoli
EcoliK
Shifl
Salti
Salty
YerpeC
YerpeK
Haein
Pasmu
Sheo
Vibc n
Vib h
Ne vu
Ne imeM
Ra imeZ
Ps lso
P ea
Xa sep e
Xa na u
nc x
a
S
St trpy
St rpy S
St rpyG M
r
Str agN
ag
La V
Sta cla
S ta e p
au
Staa W
Staa uN
uM
Lismo
Lisin
Oceih
Bachd
Bacsu
Bacan
Thema
Aquae
lfa je
Xy m
Ca elpj y
H lp
He asp
An npc
Sy eel
Th te
Chl bu
Bor a
Trep
Lepin
M
M e ta
et c
M ka
Me etth
Py tja
Py rfu
Py rho
ra
Arc b
Enc fu
cu
Plafa
Worm
Yeast
Schpo
Arath
Deira
Strco
MyctuH
MyctuC
Mycle
l
Corg f
Core o
Bifl u
n
Fus ete
Th pe
Clo oab
Cl pnT
r R
St rpn u
St trm
S
Mycge
Mycpn
Mycpe
Urepa
Mycp
Chlm u
Chlt u
Ch r
Ch lpnA
Ch lpnC
Ag lpnJ
A rt5
Rh grt5
B im W
Br rum e
us e
u
pe
Aer lto
Su
lso
Su ae
r
Py evo
Th eac
Th lsp
Ha tma
e
M
A tree based on
“Jensen-Shannon”
distance JSh(S, S’),
the “symmetrized”
Kullback-Leibler
distance between the
“induced K-string
probabilities”
ilo r
Rh auc n
C icc
R icpr
R cai
B u ca p
Bu gbr
Wi liC
Eco liE
Eco O
Ecoli
EcoliK
Shifl
Salti
Salty
YerpeC
YerpeK
Haein
Pasmu
Sheo
Vibc n
Vib h
Ne vu
Ne imeM
Ra imeZ
Ps lso
P ea
Xa sep e
Xa na u
nc x
a
S
St trpy
St rpy S
St rpyG M
r
Str agN
ag
La V
Sta cla
S ta e p
au
Staa W
Staa uN
uM
Lismo
Lisin
Oceih
Bachd
Bacsu
Bacan
Thema
Aquae
lfa je
Xy m
Ca elpj y
H lp
He asp
An npc
Sy eel
Th te
Chl bu
Bor a
Trep
Lepin
M
M e ta
e
M tka c
e
Me tth
Py tja
Py rfu
Py rho
ra
Arc b
Enc fu
cu
Plafa
Worm
Yeast
Schpo
Arath
Deira
Strco
MyctuH
MyctuC
Mycle
l
Corg f
e
Cor o
Bifl u
n
Fus ete
Th pe
Clo oab
Cl pnT
r R
St rpn u
St trm
S
0.1
Mycge
Mycpn
Mycpe
Urepa
Mycp
Chlm u
Chl u
Chl tr
Ch pnA
Ch lpnC
Ag lpnJ
A rt5
Rh grt5
B im W
Br rum e
us e
u
pe
Aer lto
Su
lso
Su ae
r
Py evo
Th eac
Th lsp
Ha tma
e
M
The tree drawn to scale
0.05
N
au uW
a
St taa ep
S ta cla
S La agV
StrtragN
S rpyG
St rpyM
St rpyS
St mu
S tr n R
Strp T
Strpn
Cloab
Clope
Thete
Fusnu
Mycge
Mycp
Mycp n
Urep e
Agr Mycpa
u
A t5
Rhgrt5W
B i
Br rumme
R us e
Ca hil u
uc o
r
cn
ic r
R icp
R
M
et
k
M a
e
Th Ar tja
Th evo cfu
ea
Me c
Me tma
H tac
Plaf alsp
Wor a
m
Yeas
Schpot
Enccu
Arath
Thema
Aquae
Chlte
Theel
Synppc
AnasJ
n
Chlp nC
p
Chl lpnA r
Ch Chlt u
lm
Ch epina
L ep u
Tr orb py
B el
H
Deira
Biflo
Coref
Co
Mycrlegl
My
MycctuC
tu
Ba Strco H
c
B a
B acs n
Ocachd u
L
Li isineih
St sm
aa o
uM
1/ K
H
Ca el
m pj
Xa Xyl je
Xa ncafa
Ps nax
Pseepu
Ra ae
Neimlso
Neim eZ
e
VibvuM
Vibch
Sheon
Pasmu
Haein
YerpeK
YerpeCy
Salt i
Salt fl
ShiK
li
Eco liO
o
Ec oliE
Ec oliC
Ec igbr p
W uca ai
B uc
B
1 − (1 − JSh(S,S') )
pe
Aer lto
Su
l so
Su ae rfu
r
Py Py rho b
Py yra tth
P Me
The tree using the calibrated
distance
Comparison with “standard” taxonomy
12
10
# diff
8
6
JSh
4
cali.
2
0
8
9
10
11
12
Word length
13
14
Outlook
• More realistic evolutionary dynamics
• Taking account of the variance of the
frequency distribution
– Better construction of composition vector
– Optimal value of K
Acknowledgement
• Bailin Hao
• Andreas Dress
• Zhao Xu