Accardi L., Freudenberg W., Obya M. (Eds.) Quantum Bio-informatics IV: From Quantum Information to Bio-informatics
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450
Acknowledgments
Work
supported
by
FCT
and
POCl
2010
(project
MAT /58321/2004)
with
participation
of
FEDER,
and
by
the
Austrian
Science
Fund
(FWF)
under
project
P21450.
References
1.
A.-L.
Barabasi
and
Z.
N. Oltvai, Nature Reviews Genetics 5,
100(February
2004).
2.
M.
E.
J.
Newman,
Proceedings
of
the National
Academy
of
Sciences
103,
8577 (2006).
3. M. E.
J.
Newman
and
M. Girvan, Physical Review E (Statistical, Nonlinear,
and
Soft
Matter
Physics)
69,
p. 026113 (2004).
4.
F.
Chung
and
L. Lu,
Annals
of
Combinatorics 6, 125 (2002).
5.
S.
Fortunato,
Physics Reports
486,
75(February
2010).
6.
M. E.
J.
Newman,
Physical Review E (Statistical, Nonlinear, and
Soft
Matter
Physics)
69,
p. 066133 (2004).
7.
A.
Clauset,
M. E. J.
Newman
and
C. Moore, Physical Review E (Statistical,
Nonlinear, and
Soft
Matter
Physics)
70,
p. 066111 (2004).
8. M.
J.
Barber,
Physical Review E (Statistical, Nonlinear, and
Soft
Matter
Physics)
76,
p. 066102 (2007).
9.
S.
Fortunato
and
M.
Barthelemy,
PNAS
104,
36 (2007).
10.
U.
Brandes,
D. Delling, M.
Gaertler,
R. Gorke, M. Hoefer,
Z.
Nikoloski
and
D.
Wagner,
Knowledge and Data Engineering,
IEEE
Transactions on
20,
172(December 2007).
11. B. H.
Good,
Y.
A.
de Montjoye
and
A.
Clauset,
Physical Review E
81,
046106+(Apr
2010).
12. L.
Danon,
A. Diaz-Guilera,
J.
Duch
and
A.
Arenas,
J.
Stat. Mech.
,p.
P09008
(2005).
13.
J. P. Bagrow,
E.
M. BoHt, J. D. Skufca
and
D.
ben
Avraham,
EPL
(Euro-
physics Letters)
81,
68004+(March
2008).
Quantum
Bio-Informatics
IV
eds. L.
Accardi,
W.
Freudenberg
and
M.
Ohya
© 2011
World
Scientific
Publishing
Co.
(pp.
451-460)
STUDY OF
HIV-l
EVOLUTION BY CODING THORY AND
ENTROPIC
CHAOS DEGREE
KEIKO SATO
Department
of
Information Science, Tokyo University
of
Science,
Noda, 278-8510, Japan
We studied the evolution
of
HI V-I (Human Immuno-deficiency Virus Type
I)
by means
of
coding theory and information dynamics. More precisely,
(I)
we applied various
artificial codes to look for the similarity between these codes and the code
of
HI V-I; (2)
the entropic chaos degree was used
to
describe the evolution
of
HIV -1.
1. Introduction
Information
of
life
is
stored
as
a sequence
of
nucleotides, and the sequence
composed
of
four bases seems to be a sort
of
code [1, 2]. Therefore, we can
consider that the DNA or gene in each organism
is
a code showing its inherent
structure. Thus, we ask what kind
of
structure each code has. More precisely, we
ask what roles the code structure has for the emergence
of
life and how it
is
concerned with the changes
of
the living body. On such questions, by using the
sequences encoded by various artificial codes in information transmission, we
explored the code structure
of
the sequences
of
HIV
-1
envelope
V3
region
obtained longitudinally for each patient and then studied the evolutionary
changes during the course
of
HIV
-1
infection from the viewpoint
of
the code
structure.
We next examined the evolutionary changes in the
V3
region for each
patient with HIV-linfection by using the entropic chaos degree introduced on
information dynamics [3, 4], and then studied whether these evolutionary
changes are related with the disease progression
of
HIV
-1
infection.
It
is
very
interesting
to
find the' dynamics causing the changes
of
viruses from one type to
others.
If
one can get a rule to describe the dynamics such
as
Schrodinger
equation
of
motion and traces the changes in viruses, it will be a great step to
study viral genomes, even more the diseases caused by viruses. However it is
also very difficult
to
find such an equation
of
motion and more difficult
to
solve
the equation because the micro-dynamics
of
the virus change will be one
of
the
multi-body problems with very complicated interactions. In any case, we have
to
look for some rule representing the dynamics even in not complete micro-level,
that is, in a certain macro level. Most
of
biologists discuss the changes
of
viruses
by looking at each site
of
genome or counting the substitution rate
of
the sites.
451
452
We use the probability theory and the chaos degree
to
study such changes,
which provide us with a bit more rules than merely counting.
2. Methods
2.1.
The
Code Structure
of
HIV-I
First, we select artificial correcting codes satistying the hypothesis below, and
then we investigate which artificial code characterizes the nucleotide sequence
ofHIV-l
envelope
V3
region. Our hypothesis
is
the following.
Each codon determines an amino acid and the third nucleotide
of
the codon
will not have much influence on the amino acid, so that the third nucleotide
is
supposed to
playa
role
of
a check symbol in an error-correcting code. That is,
an error-correcting code that a gene has is considered
to
be a code which has the
code length that does not destroy the codon unit and changes the third nucleotide.
Under this hypothesis, we consider how the code structure
of
a gene
is
analyzed.
Since the Galois Field
GF(4)
consists
of
four elements,
0,
1,
a and a
2
,
the
four bases can be expressed
as
A~0,T~I,C~a,G~a2
.
We rewrite an important part
of
the sequence in a gene by that
of
these four
elements, and we make the error-correcting code by using an artificial code. The
total length
of
such a code
is
multiples
of
3 and the length
of
the information
symbols
is
multiples
of
2.
Artificial error-correcting codes used for our study are BCH codes, self-
orthogonal codes and Iwadare-codes. We encode the
n nucleotide sequences
of
each patient at a specific point in time, and then get the encoded amino acid
sequences
X~
(j
= 1,2,· .. , n) by code C . Here, a degree to measure the
similarity between an artificial code
of
X~
and the code
of
amino acid
sequences
Xj
(j
= 1,2,.··, n) before coding
is
defined by
Dc(j)
= p(
Xj,Xn
(0
s;
p(Xj,Xn
S;
l),
which
is
called "Entropy Evolution Rate"
[5]
and it indicates the difference
between
Xj
and
X~
. When the code structure
of
Xj
gets closer to an
artificial code C obtained in usual coding theory, the value
of
Dc(j)
gets
closer to
O.
Moreover, to look for a common code
of
a group
of
several
V3
sequences, we use the degree Dc defined by
Ip(Xj,xn
D
_-,-j_~l
____
_
c
n
453
By calculating
Dc
for various C codes, we can find a common code structure
ofV3
sequences obtained at each point in time after HIV-1 infection.
If
Dc
for
a code
is
smaller than that
of
any other codes, then we can infer that the group
has the property that code
C owns.
To study the code structure
of
the sequences
of
HIV
-1
envelope
V3
region
during the course
of
HIV
-1
infection, we encoded, by means
of
various
artificial
C codes, the sequences obtained from patients infected with HIV
-1
at
several points in time after infection or seroconversion, and then calculated the
index
Dc
mentioned above.
2.2.
The
Evolutionary Changes
of
HIV-I
by Entropic Chaos Degree
Entropic Chaos Degree (ECD for short) has been used to characterize the
chaotic aspects
of
the dynamics leading sequence changes [6, 7]. We will briefly
explain the ECD.
The ECD for the amino acid sequence
xn
of
HIV
-1
envelope
V3
region
obtained at
n years post-HIV-1 seroconversion with the one
xn+l
obtained at
n + 1 years post-HIV
-1
seroconversion from a patient
is
given
as
follows [8,9].
Let take the amino acid sequences
of
two sequences
xn
and
xn+l
. We
like
to
find a rule
xn
changes
to
xn+l
if
xn
is
supposed to be ahead
of
xn+l
.
As explained above, it
is
almost impossible to write down the equation
of
motion in the very micro level, say the level
of
quantum mechanics. However it
is
true that there exists some micro-dynamics causing the change from
xn
to
xn+l
even
if
we can not knows the exact form
of
the dynamics. We denote this
dynamics by
A:;cro
and its extension to a macro scale properly considered by
A'.
Even when the micro-dynamics
A:;cro
is
hidden, we can somehow find
the macro dynamics
A'
. The macro dynamics we consider here
is
one in usual
discrete probability theory, in which we can apply the usual Shannon's
information theory.
So the macro-dynamics
A'
is
nothing but a channel from
the probability space
of
xn
to the probability space.
After two amino acid sequences are aligned, they are denoted by the same
symbols
xn
and
xn+l
. The complete event system
of
xn
is
determined by
the occurrence probability
P;
of
each amino acid a;
(i
= 1,2,···,20) and
Po
of
the gap
*;
20
(Xn,P)=(
*
Po
where
LP;
=
1.
In the same way, the complete event system
xn+l
is
denoted
i::::O
by
454
...
~20).
...
P20
The compound event system
of
xn
and
xn+1
(Xnxxn+l,r)=(**
*a
l
roo
r
OI
where 'ij represents the joint probability for the event
a;
of
xn
and the event
20 20
a j
of
X n+ 1 satisfYing
2>ij
= Pi'
2>ij
=
Pj'
So
the dynamics A * describing
j =O ;=0
the change
of
sequence from
xn
to
xn
+1
is
given by a certain mapping called
a channel sending the probability distribution
p
to
p
==
A * P .
It
is
difficult
to
know the details
of
this dynamics
in
the course
of
sequence changes. The ECD
can be used to measure the complexity without knowing the exact dynamics [7],
which
is
one
of
the aspects due
to
the sequence change.
The ECD for the amino acid sequences
is
given by the following formula;
ECD(xn,x
n
+
I
)==
L>;S(A*8J,
{
I
(i
=
j)
where
SO
is
the Shannon entropy and P = ~ pA,
8;
(j)
= 0
(i"*
J)'
Note
that the
ECD(X
n
,xn+l)
is
written as
ECD(p,A*)
to
indicate p and A*
explicitly.
This chaos degree
is
originally considered for how much chaos
is
produced
by the dynamics
A * [10, 11]. Therefore it
is
considered as:
(1) A * produces a chaos
iff
ECD > 0
(2) A* does not produce a chaos iffECD =
O.
Moreover, the chaos degree
ECD(X",X
n
+
l
)
provides a certain difference
between
xn
and
xn+1
through a change from
xn
to
xn+1
, so that the chaos
degree characterizes the dynamics changing
xn
to
xn+1
.
We calculated the ECD
of
the dynamics leading sequence changes
of
the
V3
region which were obtained from patients infected with HIV
-1
at several
points in time after infection or seroconversion.
2.3. Longitudinal Sequence Data
We obtained envelope
V3
sequence data from several longitudinal studies
of
HIV
-1
infection [12-16]. Some patients had progressed
to
AIDS and died
of
AIDS-related complications, while others have been asymptomatic during the
period
of
follow-up. This paper has the results for four patients
as
representatives
of
many patients.
455
3. Results
and
Discussion
The value
of
Dc
by various C codes and that
of
the ECD for the
V3
sequences
are shown in Figure 1 for Patient A and Patient B who have been asymptomatic
during the follow-up period, and those values are shown in Figure 2 for Patient
C and Patient D who had progressed to AIDS and died
of
AIDS-related
complications during the follow-up period.
Patient
A
0.2
G-
9
~
I"l
-8
0.15
Dc
0.1
0.05
0
0
13
22 33 46
59
M>nths Post Seroconversion
035
0.3
0.25
0.2
0
()
0.15
UJ
0.1
0.05
0
(0,
13) (13, 22) (22, 33)
(33,46)
(46, 59)
456
Patient B
Dc
0.2
0.15
0.1
0.05
0
0.3
0.25
0.2
0°15
()
w
01
0.05
°
---
-
3
(3,
29)
..n
..tl
-
~
---
29
42
58
70
100
",ntlls
Post
Seroconversion
(29, 42) (42, 58) (58, 70)
(70, 100)
Figure
1.
The value
of
Dc
and the value
of
the ECD for theV3 sequences obtained at each point
in time from two patients (Patient A and Patient
8)
who have been asymptomatic during the follow-
up period.
Codes
Error·correcting Capacity GB'lerator Polynomial
-+-(15, lO)-cyclic code
---
(21, 14)-cyclic code
...... Self-orthogmal code (Constraint Leng
th
69) 2 (randcrn erroc)
--
(15, lO)-cyclic code
.....
(21, 14)-cyclic code
.........
Self-orthogcnal code (Constraint Length 120) 3 (random error)
-&-
(15, lO)-cyclic code
tj+a~t4+t'+at+ci
t
1
+
(it
G
+al'
+t4
+cit
3
+ci
d
J
+D
J1
+rJ+d+l
.
d
8
+ D
U
+
LfD
+ n
4
+1
t!J+
t
4
+t
J
+1
t
'
+ar+arl+at
J
+at+l
[j9
+
nY,
+[jB+
n21+
Dl4
+1
•
d8+DD+D2S+D21
+£1+1
"+1
457
The values
of
Dc for the
V3
region obtained from asymptomatic patients
were low for artificial codes represented by dark blue, pink and light blue at all
points in time. In addition, all codes had a constant value
of
Dc'
Although
Patient B
is
a long term non-progressor, CD4+ T-Iymphocyte counts gradually
decrease from around
100 months post-HIV-I seroconversion. We observed that
the code structure
of
V3
region showed a change previous to the decrease
of
CD4+ T-Iymphocyte counts.
The ECD
of
patients who have not diagnosed AIDS during follow-up like
Patient A has maintained stable and low value. For Patient B, the value
of
the
ECD for the
V3
sequences obtained at 70 months and 100 months post-HIV-I
seroconversion was larger compared with that
of
other points in time.
Patient C
Dc
o
()
0.2
0.15
0.1
0.05
0.35
0.3
0.25
W 0.2
0.15
0.1
0.05
o
G-eeeeee~eee~
9 16
19
21 25 28 34 40 42 43 49
56
62 68
81
Mmths
Post
Seroconversion
(3.
9)
(9,21)
(21,34)
(34,
49) (49,68) (68,81)
458
Patient D
0_2
0.15
Dc
0.1
0_05
3
14
24 34 45
51
61 66 68
17
80 87 94 98
105
Mmths
Post
Seroconversion
0.35
0.
_3
80.·25
w
0..2
015
0.1
0
0.5
0.
(3,14)
(14
,24
) (24, 34) (34,45) (45,61)
(61,77)
<77,87) (8
7,
98)
(98,105)
Figure 2, The value
of
Dc and the value
of
the
ECD
for theV3 sequences obtained at each point
in time from two patients (Patient C and
Patient D)
who
had progressed to AIDS and died
of
AIDS-
related complications during the follow-up period, Color scheme
of
various codes is the same as that
used in Figure
1.
The
points in time after AIDS diagnosis (CD4+ T -lymphocyte counts
of
less than
200 cells/J-lL) are represented by red color. Patient C and Patient D died at 97 and 109 monthes post-
HIV-l
seroconversion, respectively,
Codes
Error
-cocrecting
Capacity Generatcr Polynomial
--+- (15,10)-cyciIc co
de
~
(21, 14)-cyclic code
-4-
Self-crthogonal code (Constraint Length 69) 2 (rand
cm
efTC!')
-to-
(15, lO)-cyclic code
.....-
(21,
14
)-cyclic code
__
Self-orthogonal code (Constraint Length 120) 3 (random error)
---<>--
(15,lO)-cyciIc code
t
5
+
a?t
4
+t
2
+at+ci
[7
+ah
6
+a
t
j
+t4
+a~t3
+0:"
Jj" +D21+D9+
D
"+1.
d
S
+
Di'j
+ D
TJ
+
D4
+1
[5+
t
4+
t
2+
1
t
1
+a:f
+at
3
+ at"
+at+1
g9
+
n)6
+ [?3 + n-:ll+
[j4
+1.
d
9
+ n
D
+
D-:l
S
+ n
27
+ rJ
+1
t
5
+1
459
The code structure
of
the V3 region obtained from patients who had
progressed to AIDS and died
of
AIDS-related complications during the follow-
up period changed throughout the entire course
of
HIV
-1
infection. The values
of
Dc
for Patient C and Patient D were low for artificial codes represented by
dark blue, pink and light blue during the early stage
of
HI V-I infection
.However, as
HIV-l
disease progressed, the V3 region was completely
different from the structure
of
such codes. In contrast, the V3 region was close
to the structure
of
the code represented by orange or pale purple during the late
stage
of
HIV
-1
infection. The code structure
of
V3 region changed previous to
the decrease
ofCD4+
T-Iymphocyte counts.
We
consider that the changes
of
the
code structure are related with the stage
of
HIV
-1
disease.
The values
of
the ECD for asymptomatic patients with HIV
-1
infection
were comparatively low. The value
of
the ECD gradually increases from
asymptomatic HIV
-1
infection to around the time
of
AIDS diagnosis, and then it
continues to decrease up to death for duration
of
AIDS. Therefore we can infer
patient's stage
of
disease progression from the variation pattern
of
the ECD.
References
1.
M. Ohya and
K.
Sato, USE OF INFORMATION THEORY TO STUDY
GENOME SEQUENCES, Rep. Math.
Phys., vol.46, 419-428 (2000).
2. M. Ohya and
K.
Sato, The Code Structure
of
HlV-l,
Open Sys. &
Information Dyn., vo1.14, 295-306 (2007).
3. R.
S.
Ingarden, A. Kossakowski and M. Ohya, Information Dynamics and
Open Systems, Kluwer Academic Publishers (1997).
4.
I.
V. Volovich and M. Ohya, "Mathmatical Foundation
of
Quantum
Information and Computation with applications to nano- and bio-sciences,
to
be
published from Springer-Verlag.
5.
M. Ohya, Information theoretical treatment
of
genes, IEICE E72,
No.5,
556-560 (1989).
6.
M. Ohya, Information Dynamics and Its Applications to Optical
Communication
Processes, Lect. Note Phys., vo1.378, 81-92 (1991).
7.
M. Ohya, Complexities and their application to characterization
of
chaos,
Int.
1.
Theor. Pliys., vol. 37, 495-505 (1998).
8.
K. Sato,
T.
Tanabe and M. Ohya, How to Classify Influenza A Viruses and
Understand their
Severity," Open Sys. & Information Dyn.,
VoU7,
297-310
(2010).
9.
K. Sato and M. Ohya, Analysis
of
the disease course
of
HIV
-1
by entropic
chaos degree, Amino Acids, vol.20, 155-162
(2001).
10.
M. Ohya, Adaptive Dynamics and its Applications to Chaos and
NPC
Problem, QP-PQ: Quantum Probability and White Noise Analysis,
Quantum Bio-Informatics, vol. 21, 181-216
(2007).
11.
K.
Inoue, M. Ohya and K. Sato, Application
of
chaos degree to some
dynamical systems, Chaos, Solitons and Fractals, vol. 11, 1377-1385
(2000).
12. R. Shankarappa,
1.
B. Margolick,
S.
J.
Gange, A. G. Rodrigo, D. Upchurch,
H. Farzadegan,
P. Gupta,
C.
R. Rinaldo, G. H. Learn, X. He, X. L. Huang