J. Biomedical Science and Engineering, 2013, 6, 213-222 JBiSE
http://dx.doi.org/10.4236/jbise.2013.62A026 Published Online February 2013 (http://www.scirp.org/journal/jbise/)
Robust estimation of stochastic gene-network systems
Chia-Hua Chuang, Chun-Liang Lin*
Department of Electrical Engineering, National Chung Hsing University, Taichung, Chinese Taipei
Email: *chunlin@dragon.nchu.edu.tw
Received 15 December 2012; revised 14 January 2013; accepted 21 January 2013
ABSTRACT
Gene networks in biological systems are highly com-
plicated because of their nonlinear and stochastic
features. Network dynamics typically involve cross-
talk mechanism and they may suffer from corruption
due to intrinsic and extrinsic stochastic molecular
noises. Filtering noises in gene networks using bio-
logical techniques accompanied with a systematic
strategy is thus an attractive topic. However, most
states of biological systems are not directly accessible.
In practice, these immeasurable states can only be
predicted based on the measurement output. In the
lab experiment, green fluorescent protein (GFP) is
commonly adopted as the reporter protein since it is
able to reflect intensity of the gene expression. On this
basis, this study considers a nonlinear stochastic
model to describe the stochastic gene networks and
shows that robust state estimation using Kalman fil-
tering techniques is possible. Stability of the robust
estimation scheme is analyzed based on the Ito’s
theorem and Lyapunov stability theory. Numerical
examples in silico are illustrated to confirm perfor-
mance of the proposed design.
Keywords: Biological System; Stochastic Model;
Stability; Estimation
1. INTRODUCTION
The gene network in biological systems plays an impor-
tant role in recent diagnoses of diseases such as cancer
and autoimmune diseases. Because this network is highly
complicated and extremely nonlinear, investigating re-
lated problems using a systematic strategy is highly de-
sirable.
Systems biology aims to understand the internal be-
haviors of biological systems from a system level view. It
is different from traditional biology, which focuses on
individual cellular components [1-3]. Researchers have
recently designed and constructed biological models us-
ing molecular biology techniques and engineering ap-
proaches. For example, microarray technology uses high-
throughput methods to measure a large amount of gene
expression states, and is a useful tool in biotechnology.
Measured data makes it possible to reconstruct the struc-
tures of gene networks, perform qualitative an d quantita-
tive analyses, systematically control biological states and
design desired biological process, and ultimately exam-
ine dynamic behavior using computational simulations.
Biological models describing the behavior of biologi-
cal systems can be classified into a logical model in the
discrete-time domain and a differential equation set in
the continuous-time domain [4-6]. Unlike the determi-
nistic case, the gene networks of real biological systems
are generally non-ideal and invariably noisy. These mo-
lecular noises generally involve the intrinsic noises re-
sulting from molecular birth and death, and extrinsic
noises caused by environmental influences such as
changes in temperature, PH, or nutrient levels and may
affect the quantitative and qualitative characteristics of
biological systems [7-9]. To ensure modeling accuracy,
the influence of noise contamination should not be ig-
nored. The parameters of gene network are estimated to
reconstruct its model form noisy measured data [10].
The robustness of biological systems is defined as the
capability of the system to resist noise corruption while
ensuring satisfactory performance or stability [11]. Dis-
ease and malfunction represent a decay in robustness and
the noise filtering ability of the corresponding biological
networks. Drug design is an effective way to improve the
robustness and filtering ability of biological networks to
resist fluctuation and noise, much like the robust control
design in engineering problems. Nonlinear feedback
control methods have also been used to regulate the
steady state of biological systems [12]. Other issues that
have directed greater attention to stochastic biological
systems include the development of control strategies
when ensuring robust stability and filtering ability. Chen
and Wu proposed a robust filtering circuit design based
on
H
-control theory by regulating kinetic parameters
[13].
Before performing any feedback control designs, all
biological state information should be available. How-
ever, most of the internal states of these systems can only
*Corresponding author.
OPEN ACCESS
C.-H. Chuang, C.-L. Lin / J. Biomedical Science and Engineering 6 (2013) 213-222
214
be observed partially. In this situation, a state estimator is
appropriate to reconstruct full states, especially in noisy
environments. The Kalman filter (KF) has been adopted
to estimate full states in engineering for decades [14,15].
Liang and Lam designed a linear state estimator to esti-
mate the concentrations of mRNA and protein for sto-
chastic gene regulatory networks by considering pa-
rameter uncertainties [16]. However, there are relatively
few applications of the extended KF (EKF) in state esti-
mation for nonlinear biochemical networks [17,18].
Moreover, a state estimator was implemented based on
the fluorescence probe, a dynamic state model of the
plant cell bioreactor and online GFP fluorescence meas-
urement [19].
Although a few papers discuss state estimation for
biological networks, most approaches are based on the
traditional Kalman filtering theory. This theory assumes
that noise covariances, including process noise and
measurement noise, are known a priori. The KF can
identify optimal state estimation against noise using
Gaussian distributions. However, noise distribution may
not be Gaussian in b iological systems; its autocorrelation
may not be known exactly, or may be difficult to model
precisely [7-9]. Chuang and Lin proposed a robust EKF
to handle gene network systems with uncertain process
noises [20].
This paper extends the design to a more general class
of perturbative gene networks with uncertain extrinsic
noise, process noise, and multiple in trinsic noise sources.
A state estimator for this class of gene networks is de-
signed based on a generalized robust EKF. This study
also presents quantitative error analysis for the robust
EKF based on Ito derivatives and Lyapunov stability
theory. After this analysis is completed, establishing the
convergence condition for estimation error, which is ex-
pressed in terms of the linearization error of the given
gene network and the amplification factors of intrinsic
noises, is then possible. Numerical experiments for an in
silico example verify the theoretical results obtained.
2. PRELIMINARIES
To clarify the notation in the derivations, let the vector
norm of , denoted by
n
x
x
, be defined as
T
x
xx. Some preliminary lemmas are introduced a
priori.
The following lemma provides the covariance propa-
gation equation for stochastic linear systems.
Lemma 1 [21]. For the following linear stochastic sys-
tem:
 
 
22
,
w
dx tAx tdtDx tdWt
EWt Wt






where
x
t is state,
E
denotes expectation, and
Wt is the zero mean Gaussian white noise. The co-
variance propa gati on for
x
t is governe d by

T2
w
T
tAXt XtADXtD
 
where the state covariance is
 

T
X
tExtxt.
The following lemma is known as the Ito Lemma for
the differential of the given stochastic process.
Lemma 2 [22]. For the following nonlinear time-
varying stochastic system:



,,dxtfxt tdthxttdWt
and the differential of a given stochastic process
,Vxtt is




 








 






 




T
2
2
22
2
T
2
,
,
,,,
,
1,,
2
,
1
2
,,,
T
Vxt t
dVx ttdt
t
Vxttfxttdth xtt dWt
xt
Vxtt
h xtthxtt
xt
Vxttdt
t
Vxtt
f
xt tdthxttdWtdt
xt t

















where
,
f
xt t and are time-varying
nonlinear function vectors, and is a Brownian
motion.

,hxt t
dW

t
2.1. Problem Description
To explain this problem, consider a stochastic nonlinear
synthetic gene network of a cascade loop of transcrip-
tional inhibitions built in E. coli with 4 genes
(, and ), 3 repressor proteins (TetR,
LacI, and CI), and the fluorescent protein EYFP as the
output (shown in Figure 1). The fluorescence of the sys-
tem resulting from EYFP is the only measured output,
and other gene products
, , tetR lacIcIeyfp
, , tetR lacIcI are not acces-
sible. By considering the disturbance effect, the dynamic
response of the system can be described by the following
equations [23]:
tetR lacI cI eyfp
TetR LacI CI EYFP
tetR
eyfp
lacl
cl
lacl
keyfp
k
cl
k
tetR
k
Figure 1. Example of a synthetic gene network. The dashed
line and the solid line represent, respectively, the repression
effect and the activation effect.
Copyright © 2013 SciRes. OPEN ACCESS
C.-H. Chuang, C.-L. Lin / J. Biomedical Science and Engineering 6 (2013) 213-222 215
 



 
 




 
 




 

,0 1
11
,0 2
22
,0 3
33
,
,
,
,
tetRtetRtetRtetRtetR cI
tetR tetRtetR
lacIlacIlacIlacIlacI tetR
lacI lacIlacI
cIcIcIcIcI lacI
cI cIcI
eyfp eyfp
xt ntrx
nt xtwt
xt ntrx
ntxt wt
xt ntrx
ntxtwt
xt







 

 

 





 
04
44
,
eyfpeyfpeyfp cI
eyfp eyfpeyfp
ntr x
ntxt wt




(1)
where the initial state is

T
00000
tetR lacIcIeyfp
xExxxx
,
the production rate parameters of the corresponding pro-
teins is i, , the decay rate pa-
rameters of the corresponding proteins is
,,,itetR lacIcIeypf
,,,,
iitetRlacI cI eypf
, represent the
intrinsic parameter fluctuations with uncertain magni-
tudes i and i
, 1,,4
i
ni
, ,0i and ,0i
are the basal pro-
duction and decay rates, and reflect the
effect of environmental noises. The stochastic molecular
noises in the host cells are assumed uncertain but
bounded. The Hill function for the repressors is
, 1,
i
wi,4

, ,,,
1
r
kn
r
rxktetRlacI cI eyfp
x
K




with r
being the maximal expression level of the
promoter and r
K
the repression coefficient. The EYFP
protein, a green fluorescent protein (GFP), is a useful
reporter protein consisting of several amino acid residues.
The EYFP protein exhibits bright green fluorescence
when exposed to blue light. Based on this feature, it is
possible to obtain information about the concentration
variations of other proteins and mRNAs by measuring
the fluorescent intensity generated by GFP. According to
the Beer-Lambert law [19], the measurement model can
be expressed as follows:



01e eyfp
lx
eyfp
ytF vt
  (2)
where

eyfp
y
t denotes the measurement output of GFP,
is the measurement noise, 0

vt
F
is the basal light
intensity, is path length of light, and absorption coef-
ficient l
with 0.05
eyfp
lx
.
This study discusses an approach for estimating the
gene concentration of a class of stochastic gene networks
in the form of (1)-(2) with multiple intrinsic noises when
their states are not directly accessible. In this situation,
estimating state information based on measurement out-
put is a key issue.
Mathematical models provide a platform for the sys-
tematic analysis of various gene networks. One type of
ordinary differential equation (ODEs) is the stoichiomet-
ric model, which is known for representing biochemical
reactions. Gene networks often suffer from intrinsic
noises resulting from molecular birth and death, but also
from extrinsic noises caused by environmental perturba-
tions. The dynamical variation of concentrations for bio-
logical systems shown in (1)-(2) can be applied to a more
general perturbative gene network using the following
nonlinear stochastic differential equation, which incor-
porates intrinsic and extrinsic noises:
 



 
1
M
ii
i
x
tfxt gxtntwt
 
(3)
where
M
represents the number of intrinsic noise
sources,
 



0
T
00
0, 00,
x
xExP Exxxx
0


and the measurement model is given by


yt hxtvt
(4)
where
n
x
tR denotes a concentration vector that
indicates the concentration of mRNA and protein.
ytr
denotes the measurement output. The terms
,f
, 1
i,2, ,
g
iM and are nonlinear
functions that respectively denote the interactions of
gene networks, coupling vectors of intrinsic noises, and
the function of sensors. The intrinsic noises

h
nt, 1,2,,i
i, the extrinsic noise , and the
measurement noise
M

wt
vt are uncorrelated and assumed
to be zero-mean Gaussian white noise processes:

0,
i
Ewt Evt Entt


M
(5)
and








T
2
11 1
2
00 0
00 0
00 0
00 0
MM
wt wtQ
vt vtR
Ent nt
nt nt

























 
(6)
The noise uncertainties satisfy
01 020
, ,
ii 3
QQ RRi

 
T
(7)
where T
0, 0QQ RR
 and are
positive definite matrices, and
20,
ii

0,
ii
00
,QR
are
their corresponding nominal parts, and 12
,
and
3,
ii
are positive constants.
Remark 1. Equations (3) and (4) can be rewritten as
the following Ito stochastic equations:
 



 
1
M
ii
i
dx tfx tdtgx tdNtdWt
 
Copyright © 2013 SciRes. OPEN ACCESS
C.-H. Chuang, C.-L. Lin / J. Biomedical Science and Engineering 6 (2013) 213-222
216
and
 


dz thx tdtdVt
where
 
,
y
tzt

,
i
Nt Wt and are stan-
dard Wiener processes or Brownian motions with

Vt
 
,
ii
dNtntdtdWtw tdt
and . This formulation is widely appli-
cable to general nonlinear gene networks with
 
v tdtdVt
M
in-
trinsic noise sources.
2.2. Estimator Design
Biological processes for gene networks include DNA to
mRNA transcription and mRNA to protein translation,
and generated protein regulates other genes. However,
the internal states of most biological systems are not di-
rectly accessible.
As described, gene networks in the real world are al-
ways noisy. The corresponding dynamic model is thus
stochastic. To tackle the situation, this study presents a
design approach for robust estimation with the estimator
given in the following form:
 





 

0
ˆˆˆˆ
, 0
ˆˆ
x
tf xtKtytytxx
yt hxt
 
(8)
where

ˆn
x
tR
r
R is the estimated state vector,

ˆ
yt is the estimated output, and

nr
K
tR
is
the estimator gain. The computational algorithm (8) is
implemented in a computer to conduct state integration
with

y
t

obtained from the GFP expression while
filtering measurement noises. Figure 2 shows the system
configuration. The sensor measures GFP fluorescence
intensity and conv erts it into electrical signals for further
processing on the computer. The green fluorescence in-
tensity per cell can be measured using a flow cytometer.
The computer computes an appropriate estimation gain
K
t using the measured data. The process of propagat-
ing the estimation error to further determine
K
t is
independent of the gene network. The following deriva-
Estimator
G ene networkSensor
 



 
1
M
ii
i
x
tfxtgxtntwt

x
t
 

yt hxt
 

ˆˆ ˆ,
ˆˆ
x
tfxtKtytyt
yt hxt
 
y
t
vt
interface
compute r
biolog ical b o dy electronic device

wt
Figure 2. System structure for realization of the state estimator.
tions are given to determine the estimation gain so that
the estimated states will track the noise-free states.
Let the estimation error state be

ˆ
x
txtxt
then
 

 




 
1
ˆˆ
M
ii
i
x
tfxt fxtKthxthxt
gxtntwtKtvt

 


(9)
The augmented system can then be constructed as


 

 
1
ˆ
,
M
iii
i
tAttLtAxtxt
BtBxt ntLtt




 

(10)
where

TT
TT TT
, ,txtxttwtvt




 
  



 

 

 

00
,,
00
0, ,
ˆ
,
ˆ
,ˆ
,
i
ii
ii
FG
At B
FKtH G
II
LtB xtG xt
IKt I
Fxt xt
Axt xtHxt xt







 

 







The partial derivative matrices evaluated at the esti-
mated state are given by

 



 



 
ˆˆ
ˆ
, ,
x
txt xtxt
i
i
xtxt
fxthxt
FH
xt xt
gxt
Gxt




and




 





ˆˆ
,,
ˆˆ
,
F
xt xtFxtFxt
H
xtxtH xtHxt


where

F
xtf xtFxt
and
H
xthxt Hxt
denote the linearization
errors. The errors are assumed to be bounded as follows:
 

 
 

 

 




 
T
T
1
T
T
2
T
T
ˆˆ
,,
ˆ
, ,
ˆˆ
,,
ˆ
, ,
, ,
ii
i
EFxtxtFxtxt
Extxtxt xt
EHxtxtHxtxt
Extxtxt xt
EGxt Gxt
Ex txtxti
















(11)
Copyright © 2013 SciRes. OPEN ACCESS
C.-H. Chuang, C.-L. Lin / J. Biomedical Science and Engineering 6 (2013) 213-222 217
where 12
,
, and i
are finite constants.
For the EKF design, consider the nominal case with
the linearization error ignored. In this case, the aug-
mented system becomes
 
1
M
ii
i
tAtt BtntLtt

 
(12)
First consider a case in which all noise covariances are
exactly measured so that 123
0,
ii

. By de-
fining the augmented covariance matrix


T
tEtt




and applying Lemma 1, the error propagation equation
with the Kalman gain1
22
 
T
K
ttH R
can be
determined by solving the stochastic Riccati equation:
T
T
 
 
 
T2T
1
TT
T2 T
1diag ,
i
M
ii i
i
M
ii
i
tAttABEntt tB
LEttL
A
ttABtBL QR





 
L
,
I
(13)
Next, consider the case with noise uncertainties pre-
sented in (7). Based on results obtained by [24], and tak-
ing the least favorable noise covariances of
0102
,QIR
 and 03
, 1,2,,
ii
iM
into
consideration, the robust Kalman gain
K
can be de-
termined by solving the following Riccati equation:
T



T
2T
03
1
M
iii i
i
tAttA
BtB LQL



(14)
where

0102
diag ,QQIRI

and
 
1
220 2
T
K
ttHRI
 (15)
Equation (15) indicates that

K
t is closely related
to the amount of measurement noise reflected by the
magnitude of 0 and the extent of the uncertain noise
covariance specified by
R
2
.
When
F
is time-invariant, it is easy to verify the
stability of the linearized system (12) with the Kalman
gain 0 2
1
T
22
K
HR I
 . Stability can be analyzed
by observing that the Riccati matrix equation (14) is re-
duced to the algebraic Riccati equation

2
TT
03
1
0M
iiii
i
T
A
ABB

 
LQL
and

T2T T
1
diag ,0
M
ii i
i
AABBL QRL
 
for all , and
,QR i
satisfying (7). Based on the result
given in [13], the above inequality represen ts a Lyapunov
inequality guaranteein g the stability of the system (12).
However, if one considers a system disturbed not only
by noises with uncertain covariances, but also by lin-
earization errors, the KF presented in this study is possi-
bly not robust. To make the estimation scheme robust
and stable, more constraints must be imposed on lineari-
zation errors and intrinsic noises. Thus, advanced analy-
sis of the augmented system (10) with linearization er-
rors is required.
3. STABILITY ANALYSIS
This section analyzes stability of the stochastic gene sys-
tem based on the Ito Lemma and Lyapunov stability the-
ory. Stability condition derivation is developed on the
basis of the following definition.
Definition 1 [15]. Given a stochastic process denoted
by
t
, assume that there is a stochastic process
t
V and real numbers min max
, , ,0VV

such
that
 


22
minmax , VtVtVt
 
t

and


Vt Vt


are fulfilled. In this case,

t
is exponentially
bounded in mean square with
determining its decay-
ing rate.
To proceed the stability analysis, we choose a Lya-
punov candidate function as [15]
 
,T
Vttttt
 

where
1
t
t
and is the solution of (14).
Taking the time derivative of by Lemma 2
and ignoring the higher order terms gives

t

,Vtt




  

 


 
 





 


 

 
T
1
2
T
2
2
1
2
TT
2
,
,
,ˆ
,
,
1
2
,
1
2
M
iii
i
M
ii i
i
ii
Vtt
EVt tEt
Vtt AttLtAxt xt
t
BtBxtntLt t
Vtt
nB tBxtt
Vtt
B
tBxttLLt
t









 






 


Then, using ( 5) and (6) yields
Copyright © 2013 SciRes. OPEN ACCESS
C.-H. Chuang, C.-L. Lin / J. Biomedical Science and Engineering 6 (2013) 213-222
218


 
 
 


 


 



T
TT
T
2TT
03
1
T
T
,
ˆ
2,
trace
M
iii i
i
ii
EVt tttt
tAtttAtt
EttLtAxt xt
tBtBt
EBxt tBxt
LtQL tt


 









 

 



Using

ttt
t
then


  
 
 


 


 



T
T
T
2TT
03
1
T
T
,
()
ˆ
2,
trace
T
M
iii i
i
ii
EVtt ttttt
tAtttAtt
EttLtAxt xt
tBtBt
EBxt tBxt
LtQL tt


 

  





 

 


i
(16)
Substituting (14) in to (16) gives




 
 
 


 


 

 

TT
2TT
03
1
TT
T
2TT
03
1
T
T
,
ˆ
2,
trace
M
iii i
i
M
iii i
i
ii
EVt tttAtttAt
BtBLtQLttt
tAtttAtt
EttLtAxt xt
tBtB t
EBxt tBxt
LtQL tt

 

 



 






 

 


or


 


 

 


 


 

 

2T
03
1
T
T
2TT
03
1
T
T
,
ˆ
2,
trace
M
Tiii
i
M
iii i
i
ii
EVt tttBtB
LtQL ttt
EttLtAxt xt
tBtBt
EBxt tBxt
LtQL tt
 
 

 







 


It is easy to see that
  
 

 

 
T
TT
1
T
1
T112
1
ˆ
2,
1
ˆˆ
,,
1max ,
T
EttLtAxt xt
ttLtLttt
EAxt xtAxt xt
ttLtLtt I





 

 


 


t

(17)
and




 









 
T
T
T
max
T
max
()
2
2
2
ii
ii
ii
i
EBxttB xt
EGxttG xt
tE GxtGxt
ttt





 

 

(18)
According to (17) and (18), and by using the
Lyapunov stability theory, it is possible to obtain


 


  





 

 
2
TT
03
1
T112
1
2T
max0 3
1
T
T
,
1max ,
2
trace
M
iii
i
M
ii iii
i
EVt tttBtB
LtQI LttI
tIBBt
CtQC tt
ttt
 


 

 


 


 

 
i
where

T
trace LtQL tt

(from the non-
singularity of the last term of (14) we know that
t
is singular and
t is bounded) and 0
provided
that


 





2T
03
1
T112
1
2T2
max0 3
1
min
1max ,
20
M
iiiii
i
M
ii iii
i
tBtB
LtQI LtI
tIBBt





 


,0t

(19)
Equation (19) was obtained by applying the Rayleigh
principle and algebraic manipulations, where
de-
notes the eigenvalue.
According to Definition 1, the stochastic gene network
becomes exponentially more stable with the exponen-
tially decaying rate
. This means that the estimation
error never diverges if noises do not force the gene net-
work to diverge when the stability condition is satisfied.
Copyright © 2013 SciRes. OPEN ACCESS
C.-H. Chuang, C.-L. Lin / J. Biomedical Science and Engineering 6 (2013) 213-222 219
4. DEMONSTRATIVE EXPERIMENTS
Consider the nonlinear stochastic gene network illus-
trated in Figure 1. The system model can be mathemati-
cally described by (1), and the values of the parameters
are taken from [23]:













,0
,0
,0
,0
,,150,2000,1.98 ,
,50,0.3,
,,587,2000,0.05,
,200,0.3,
,,210,2000,0.7 ,
,50,0.3,
,,3487,15000,0.57 ,
,
tetRtetR tetR
tetR tetR
lacIlacIlacI
lacI lacI
cIcIcI
cI cI
eyfpeyfp eyfp
eyfp








 
 
 



200,0.3
eyfp
(20)
The initial state is assumed to be
and the Hill function
takes the Hill coefficient

T
0200 40000 200 20000x2n
, the repression coeffi-
cient , and the maximal expression level of
the promoter
1000
r
K1
r
5
010F10l
. The extrinsic noise, intrinsic noise,
and measurement noise are zero-mean white Gaussian
noises with the standard deviations of , , and 1,
respectively. For the measurement model in (2), th e basal
light intensity , and .
2
0.5
6
2
0.1
Figure 3 shows the dynamic simulation of the noisy
states. For this synthetic gene network, the protein CI
inhibits gene and gene te, the protein TetR in-
hibits gene , and the protein LacI inhibits gene .
However, one only possesses the measurement output
response depicted in Figure 4.
eyfp
lacI tR cI
These linearized matrices can obtain the following
bounds on linearization errors using the remainder for-
mula of the Taylor approximation [25]:
010 2030 40 5060 708090 100
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5 x 104
time (sec)
c o ncen t ration
x
tetR
x
lacI
x
cI
x
eyfp
Figure 3. Dynamic simulation of the noisy states.
010 2030 4050 60 70 8090100
1800
2000
2200
2400
2600
2800
3000
3200
time (sec)
intensity
y
Figure 4. Dynamic simulation of the measurement output.
2
2
6
2
2
6
2
2
6
2
2
6
ˆ
1.98 00
ˆ
250 110
ˆ0.050 0
ˆ
250 110 ,
ˆ
00.7
ˆ
250 110
ˆ
3
0 00.57
ˆ
100 110
cI
cI
tetR
tetR
lacI
lacI
cI
cI
x
x
x
x
Fx
x
x
x
0












12
2
46
1
ˆ
0.3 00
ˆ
10110 ,
0000
0000
0000
cI
cI
x
x
G



2
2
26
000
ˆ0.3 0 0
ˆ,
2500 110
000
000
tetR
tetR
x
x
G
0
0
0



2
32
46
00 0
00 0
ˆ
00
,
ˆ
10110
00 0
lacI
lacI
x
Gx
0
0
.30
0



Copyright © 2013 SciRes. OPEN ACCESS
C.-H. Chuang, C.-L. Lin / J. Biomedical Science and Engineering 6 (2013) 213-222
220
4
2
2
6
00 00
00 00
00 00
ˆ
00 0.3
ˆ
2500 110
cI
cI
Gx
x













and
6ˆ
10
0000.1eeyfp
x
H



These linearized matrices reveal that the bounds on the
linearization errors are as follows:


T
T
ˆˆ
,,
ˆˆ
237 ,
EFxxFxx
Exx xx






 

T
T
ˆˆ
,,
ˆˆ
0.2642 ,
EHxxHxx
Exx xx










 
 
T
33
TT
11
0.0025 ,
EGxGx
EGxGx Exx




 

 

T
44
TT
22
0.04
EGxGx
EGxGx Exx




 

4
for all .
ˆ
, xx
Suppose that the nominal covariance matrices are
and
2
040
0.5, 1QIR 2
00.1,
ii
300 20000

01
, and the initial
state of the estimator is .
For the initial covariance of 8

T
0
ˆ10040000x00
I
 , Figure 5
shows the simulation results of the noise-free state re-
sponse and the case of state estimation using the pro-
posed method for the stochastic gene network without
noise uncertainties. The steady-state KF gain is


T
0.05360.1420.0929 1.1602K
It’s seen that the estimator tracked the noise-free state
well when there were intrinsic, extrinsic, and measure-
ment noises.
Next, consider the existence of uncertainty regarding
extrinsic noise, measurement noise, and intrinsic noise
with 12
0.05, 0.1
, and 30.01,
ii
. Figure 6
shows the results of dynamic simulation of the noise-free
state response and the case of state estimation using the
proposed method for the stochastic gene network with
noise uncertainties. For the case, the steady-state KF gain
is obtained as


T
0.05360.14220.0929 1.1616K 
010 2030 4050 60 708090 100
0
1000
2000
x
tetR
Nois e-f ree s tate
Esti m ation state
010 2030 4050 60 7080 90
6x 10
4
100
2
4
x
lacI
4000
2000
010 2030 4050 60 708090 100
0
x
cI
010 2030 4050 60 7080 90
4x 10
4
3
2100
x
ey fp
time (sec)
Figure 5. Dynamic simulation of the noise-free state response
and the cases of state estimation using the proposed method for
the stochastic gene network without noise uncertainties.
01020 30 4050 60 70 80 90100
0
1000
2000
x
tetR
Nois e-f ree stat e
Est i m ati on state
01020 30 4050 60 70 80 90
6x 10
4
100
2
4
x
lacI
4000
2000
01020 30 4050 60 70 80 90100
0
x
cI
01020 30 4050 60 70 80 90
4x 10
4
3
2100
x
ey fp
time (sec)
Figure 6. Dynamic simulation of the noise-free state response
and the case of state estimation using the proposed method for
r as
the stochastic gene network with noise uncertainties.
For comparison, we define the estimation erro
 


2
0ˆd
error
f
t
f
2
0d
f
t
f
x
txt t
xtt
where
f
x
t is the state of the noise-freesystem, and
f
t is thl time. Table 1 has compared the error per-
vi
Previous literature indicates that the state variables of
e fina
of thcentagee estimation error under the noise-free en-
ronment and the estimated states of the system with
and without noise uncertainties via the traditional EKF
design and our proposed method. Under the same initial
conditions and settings, the dynamic simulation of the
noise-free state response and the case of state estimation
using the traditio nal EKF for the stochastic g ene network
with noise uncertainties yield larger estimation error.
5. DISCUSSION
Copyright © 2013 SciRes. OPEN ACCESS
C.-H. Chuang, C.-L. Lin / J. Biomedical Science and Engineering 6 (2013) 213-222 221
Table 1. Comparison of the estimation error for the stochastic
gene network.
state
estimation error
tetR
x
lacl
x
cl
x
eyfp
x
system without n
under the proposed
oise uncertainties
EKF design 0.0918 0.1784 0.81450.0372
system with noise uncertainties
under the proposed EKF design 0.0918 0.1784 0.81430.0373
system with noise uncertainties by
the traditional EKF design 0.0981 0.1780 0.84660.0453
orks cannot be fully acquired [23]. For the steady ge
This paper proposes a robust estimation scheme to ac-
or a class of gene networks that
This research was sponsored by National Science Council, Taiwan,
-1664. doi:10.1126/science.1069492
experimental systems in the field of biological frame-
wne
concentration tracking problem, fluorescent proteins
(with red, green, and cyan color) can be used to observe
gene expressions. This experimental design makes it
possible to determine whether all state variables can ap-
proach desired states. However, this approach does not
solve the problem of noise corruption because the ob-
served state variables can still be noisy and seriously
deteriorate the accuracy of state information. Neverthe-
less, a robust state estimator based on the EKF is a useful
design for predicting network states when there are vari-
ous noise sources. The resulting state information can
next be used to analyze and track gene concentration.
6. CONCLUSION
quire state information f
are suffered from uncertain extrinsic and intrinsic noise
corruption. Quantitative performance and stability analy-
ses based on the Ito Theorem and Lyapunov stability
theory for state estimation are presented. In silico ex-
periments confirm the propo sed method for designing th e
estimator. Simulation results demonstrate the poten tial of
the presented design method in bridging engineering
approaches and specific biological pr oblems.
7. ACKNOWLEDGEMENTS
under the Grant NSC-98-2221-E-005-087-MY3.
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