Multivariate Time Series Analysis of Metabolic Network
Using the E-CELL Simulation SystemYusuke Saito

Kouichi Takahashi

13

t96406ys@sfc.keio.ac.jp

Masaru Tomita

12shafi@sfc.keio.ac.jp

13

mt@sfc.keio.ac.jp

1 Laboratory for Bioinformatics
2 Graduate School of Media and Governance
3 Department of Environmental Information
Keio University, 5322 Endo, Fujisawa 252-8520, Japan

1 Introduction
We applied multivariate time series analysis to metabolic models constructed using the E-CELL system. The E-CELL system [1] is a generic simulation environment for modeling cellular processes atlevel of chemical reaction. A state of the cell at a point in time can be represented with variables suchas concentration of molecular species and ux rates of reactions. The output of the system throughout
simulation is time series of those variables, and we employed the following two mathematical methods
in order to analyze the output of the E-CELL system.

2 Impulse response
Impulse is a sudden shock to the system and impulse response is the system's response to the impulse.

Network function H (s) of the system can be expressed as
H (s) =L[ImpulseResponse]

where L is Laplace transformation. Thus, system function H (!) can be also de ned as
H (! ) = F [ImpulseResponse] = A(! )e0j(!)
where F is Fourier transformation, A(!) is amplitude spectrum, and (!) is phase spectrum.
From the impulse response, information about (a) distances in time between reactions in a metabolic
pathway, (b) dependencies among reactions and (c) other characteristics of transfer function between
reactions, can be obtained.

3 Power contribution ratio
We applied the multivariate auto regressive (AR) model, which assumes that (a) the value of the series
at time t depends only on its previous values and random disturbance, and (b) the dependency canbe expressed as a linear combination of the values. The equation of the AR model is as follows:

X X ( ) ( 0 )+ ( )( )=kMxi t

aij m xj t

mei t

j =1 m=1

where xi (t) is the value of xi at the time t, k is the number of variables, M is the order of the model,a(m) is the AR coecients of the model, and ei (t) is the original innovation of xi .
252

An Example of Inpulse Response
Y x 103
ABCD

1.60
1.50
1.40
1.30
1.20
1.10
1.00
0.90
0.80
0.70
0.60
0.50
0.40
0.30
0.20
0.10
0.00
-0.10
-0.20
-0.30

X
60.00

70.00

80.00

90.00

100.00Figure 1: An Example of Impulse Response. For the sake of pilot study on impulse response analysis,
we have modeled a small pathway which have only three enzymes and one feedback loop. An impulse
is given to substance A. Fig. 1 shows transduction of the impulse to downstream substances (B, Cand D). The distance between peaks can be viewed as distance of these two substances in time. The
height of peak indicates the degree of correlation to substance A.
For oscillating time series, we analyze them in frequency domain. Speci cally, we use power
contribution ratio, which is an index of how uctuation of one variable depends on the other variables
at each frequency. The power contribution ratio rij (f ) is de ned as follows:rij (f ) =qij (f )pii (f )pii (f ) =X ()kqij fj =1

where pii (f ) is power spectrum of variable xi and qij (f ) is contribution ratio of variable xj 's innovation.

4 Summary
These multivariate time series analysis methods are supposed to be e ective in analyzing metabolic
network, particularly when feedback loops are present.
However, it is nearly impossible for laboratory experiments to obtain a large collection of sample
values enough to employ these analytical methods. Since computer simulation can produce ample
data sucient for time series analyses, we believe that the E-CELL system will be a promising meansfor investigation of dynamic behavior of metabolic network.

Acknowledgements
This work was supported in part by a Grant-in-Aid for Scienti c Research on Priority Areas `GenomeScience' from The Ministry of Education, Science, Sports and Culture in Japan. We would like tothank other members of the E-CELL project for their technical support in an early stage of this work.

References[1] Tomita, M., Hashimoto, K., Takahashi, K., Shimizu, T., Matsuzaki, Y., Miyoshi, F., Saito, K., Tanida,S., Yugi, K., Venter, J.C., and Hutchison, C., E-CELL: Software environment for whole cell simulation,

Bioinformatics, (to appear).

253