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An algebraic-numeric algorithm for the model selection in kinetic networks
Hiroshi Yoshida (Faculty of Math., Kyushu Univ.)Koji Nakagawa (CBRC, AIST)Hirokazu Anai (Fujitsu Lab. LTD./CREST JST.)Katsuhisa Horimoto (CBRC, AIST)
CASC2007, Bonn
Contents
Introduction of kinetic networkModel and Method(1) Laplace transformation of model
formulae and observed data(2) Matching (3) Model consistency estimationResultSummaryAnnouncement of AB2008
Background
Metacore: database of networks describing interactions between genes or proteins D-sep test: can deal with only directed acyclic graph (DAG)
Cannot be dealt with
Aim: selection for the most/more consistent model with the observed data
Which is Consistent?
A CB
A B
C
Model I
Model II
(A)
(B)
More…
To select the model most/more consistent with the given sampling dataWe have performed model selection over Laplace domain using algebraic equations
Model (Example)
)()(
)()()(
)()()(
)()(
tAkAGtGdtd
tAkAGtNkNAtAdtd
tNkNAtSLAkSNtNdtd
tSLAkSNtSLAdtd
=
−=
−=
−=
SLA N A GkSN kNA kAG
Assuming a linear relation between the variables, the kinetics of the abovenetwork can be describedas the system ofdeferential
equations on the left.
More …
SLA N A GkSN kNA kAG
SLA N G AkSN kNA kGA
SLA N A GkSN kNA kAG
kSA
SLA N A GkSN kNA kAG
kSSLoop
Laplace domain
System of differential equations
Observed data
Laplace trans.
k1 fd(s)s+k2
sum of exponentialsLaplace trans.
MatchingStrategy
1 21 2
a as m s m
+ ++ +
Solution:
Method
① The kinetics for describing biological phenomena are expressed by a system of differential equations
② The observed data are numerically fitted as a sum of exponentials
③ Both the system of differential equations and the sum of exponentials are transformed into the corresponding system of algebraic equations by Laplace transformation
④ The two systems of algebraic equations are
compared according to measure
Consistency estimation (1)
Change the system of differential equations into algebraic equations by Laplace transformation
Ex.) The system of algebraic equations
Into polynomials over Laplace domain
)()(
)()()(
)()()(
)()(
tAkAGtGdtd
tAkAGtNkNAtAdtd
tNkNAtSLAkSNtNdtd
tSLAkSNtSLAdtd
=
−=
−=
−=
)]([)0()]([)]([)]([)0()]([
)]([)]([)0()]([)]([)0()]([
tALkAGGtGLstALkAGtNLkNAAtALs
tNLkNAtSLALkSNNtNLstSLALkSNSLAtSLALs
=−−=−−=−
−=−
The solution over Laplace domain is:
Consistency estimation (2)
The observed data are fitted as a sum of exponentials:
k is the number of exponentials which can theoretically be determinedInto functions in s over Laplace domain:
∑=
−k
iii t
1)exp( αβ
∑= +
k
i i
i
s1 αβ
Change the observed data into algebraic equations by Laplace transformation
Previous study: Identifiability problem
Cobelli, C., Foster, D. and Toolo, G.: Tracer Kinetics in Biomedical research: From data to model, KluwerAcademic/Plenum Publishers, 2000.
2dy k ydt
= − ⋅ ( 2 3)dy k k ydt
= − +
k2 =m, uniquely determinedglobally identifiable
k2+k3=m …. unidentifiable
On the assumption of error-free data, … unrealistic situation
Both data are fitted as: a exp(-m t)
Our model: we handle noisy data
We take the position that it is sufficient to fit a noisy observed data as a sum of exponentials:
1 2 31 2 3m t m t m ta e a e a e− − −+ + +
Example: the observed data intoLaplace domain
Consistency estimation procedure (3)
Comparison of coefficient List in s
Derive the coefficient List by comparing the algebraic equations of the model and the observed data over Laplace domain:
Ex.) Coefficient List
Without any error, these polynomials would be zero, but in the case of real noisy observed data, unfortunately not zero… => Least squares method (LSM)
Consistency measure:
The smallest sum-square value of the elements in Coefficient Listunder
k1 > 0, k2 > 0, ….., kn> 0 (1)Or
k1 >=0, k2 >=0, ….., kn >=0 (2)If e.g. k1 = 0 … subnetwork
SLA N A Gk1 k2 k3
Coefficient List:
The consistency measure can be calculated as the smallest values of f(k)among various ks : => Least squares method:
∑=
−=n
iii rklkf
1
2))(()(
nn rklrkl == )(,,)( 11
Concrete procedure
Measures (1) and (2)
Under ki > 0 … Measure (1)
Under ki >= 0 … Measure (2)
001
=∂∂
∧∧=∂∂
nkf
kf
001
=∂∂
∧∧=∂∂
nkf
kf
Including subnetworks
Algorithm to compute the measure (2)(Recursive procedure)
MinimizePositive.. Compute the measure (1)
MinimizePositive(f(k1, …., 0, …. kn))
Result
We used the following five models
Data generation for simulationWe have generated the time series of data for the consistentmolecules for the simulation study, before the model consistencyestimation.
The given and estimated parametersare as follows: S LA;1, 1 (given) and 1.00 (estimated); S LA;1, 10 and 10.0; N;1, 1/10 and 0.100;N;2, 1 and 1.00; N;1, 163/9 and 18.1; N;2, 100/9 and 11.1; A;1, 1/10 and 0.100; A;2, 1/2 and0.500; A;3, 1 and 1.00; A;1, 163/36 and 4.53; A;2, 15=4 and 3:75; A;3, 20/9 and 2.22; G;1,1/10 and 0.100; G;2, 1/2 and 0.500; G;3, 1 and 1.00; G;1, 815=36 and 22:6; G;2, 15/4 and3.75; G;3, 10=9 and 1:11; G;4, 21 and 21.0. Each figure corresponds to the four variables(molecules) in the model: (a) S LA, (b) N, (c) A, (d) G.
Table of Consistency measure
Measure (1) Measure (2).. Including subnetworks
SLA N A GkSN kNA kAG
SLA N A GkSN kNA kAG
kSSLoop
When kSS is small, Model E is almost equal to model A
Model A
Model E
The verification of our method , Measure 1
SLA N A GkSN kNA kAG
SLA N A GkSN kNA kAG
kSA
kNG
Model C
When kSA and kNG are exactly zero, Model C is equivalent to model A
Model A
The verification of our method , Measure 2
Query A
Summary
We have proposed a method to select a model which is more/most consistent with the time series of observed data. We have verified our method, using generated data, handling a cyclic relationship hitherto unavailable in previous methods.
Future works … scalability
Focusing on a local network within a large-scale network.Easy elimination of the unnecessary variables in virtue of algebraic equations
Eliminating C(t),
k1 kd
C(t)
C1(t)
C2(t)k2
2 1 1 2 1 2( )
0
( ) ( ) (dd
tk tk C t k C t k k e C dτ τ τ− −= − )∫
2 2 1 1 2( ) [ ( )]( ) [ ( )]( ) 0dsk k k L C t s sk L C t s− + + =
International Conference on Algebraic Biology
2005(1st)
2007 (2nd)
from Universal Academy Press
LNCS 4545 from Springer
November 28-30FUJITSU SOLUTION SQURETokyo, Japan
July 2-4RISC, Johannes Kepler UniversityLinz, Austria
organized by B. Buchberger, H. Hong, K. Horimoto
organized by H. Anai, H. Hong, K. Horimoto
2009
(4th)
2008(3rd)
July 31-August 2RISC, Johannes KeplerUniversityLinz, Austria
organized by B. Buchberger,K. Horimoto, R.
Laubenbacher, B. Mishra
SAMSI, NC, USAorganized by R. Laubenbacher
http://www.risc.uni-linz.ac.at/about/conferences/ab2008/
Paper deadline is Jan.-14, 2008