On the Filtering of Molecular Noise in Genetic Regulatory Networks

Bor-Sen Chen and Yu-Chao WangNational Tsing Hua University

AbstractNoise has many important roles in biological functions of cells. At present, no good theory exists for identifying all possible mechanisms of genetic regulatory networks to attenuate or exploit the molecular noise. Based on stochastic dynamic regulation equation, the intrinsic fluctuation is modeled as state-dependent stochastic process and the robust stability of genetic regulatory network is discussed under intrinsic noise. Then the mechanisms of genetic regulatory network to attenuate or exploit extrinsic fluctuation are revealed from nonlinear stochastic filtering point of view. Furthermore, a simple measure of attenuation level or exploitation level of extrinsic noise by genetic regulatory networks is also introduced by nonlinear H∞ filtering technique via solving Hamilton-Jacobi inequality (HJI). In the linear stochastic regulatory network, the attenuation level of extrinsic noise can be calculated by solving a corresponding linear matrix inequality (LMI).

Introduction

• Noises have played many roles in biological functions of genetic regulatory networks, including noise-driven divergence of cell fates and population heterogeneity, noise-induced amplification of signals, generation of errors in DNA replication leading to mutation and evolution, and maintenance of the quantitative individual of cells (Rao et al., 2002).

• Other cellular processes influenced by noise include ion-channel gating (White et al., 2000), neural firing (Meir et al., 2002), developmental module (Von Dassow et al., 2000; Houchmandzadeh et al., 2002), cytoskeleton dynamics (Van Oudenaarden and Theriot, 1999) and motors (Simon et al., 1992).

• Even though the molecular fluctuations leading to phase variation seem random in the individual, regulatory factors tune the variation to ensure mean levels of heterogeneity for the population, i.e., the random noises showed to filtered or attenuated by the genetic regulatory networks (Rao et al., 2002).

• As molecular events in cells are subject to significant thermal fluctuations, gene expression is best viewed as a stochastic process. Many observations suggest that molecular events underlying cellular physiology are subject to fluctuations and have lead to the proposal of a stochastic model for gene expressions and biofunctions (Thattai and Van Oudenaarden, 2001 and 2002; Tegner et al., 2003).

• Several observations also provided explicit evidence that noise arises during translation (Ozbudak et al., 2002). Protein translation can amplify translational noise as a simple mRNA transcript can produce multiple copies of a protein (Kepler and Elston, 2001; Kierzek et al., 2001).

• Elowitz and colleagues (2000 and 2002) have found that extrinsic noise dominates at intermediate level, and both noises are absent at high levels. It is also found that the particular fate of each cell seems random, although biased by environmental and intercellular signals. Once a particular fate is chosen, the resulting process is tightly controlled (Msadek, 1999; Hallet, 2001; Rao et al., 2002). Obviously, the extrinsic noises are attenuated efficiently by the corresponding genetic regulatory networks.

• In terms of signal processing, these pathways function as low-pass filters to transduce low-frequency signal and to attenuate high-frequency noise (Arkin, 2000; Paulsson et al., 2000). Negative feedback schemes are the most common noise-attenuation regulatory mechanism. Intrinsic chemical damping, integral feedback and redundancy are also found to be the efficient noise filtering schemes in genetic regulatory systems (Yi et al., 2000; Tegner et al., 2003).

• Although it is straightforward to understand how a simple feedback loop shapes noise, it is far more difficult to understand the composite noise shaping behavior of multiple mechanisms interconnected in complex regulatory networks (Hasty et al., 2000 and 2002).

• Several examples have shown that noise attenuation arise from systematic properties of network rather than a single mechanism (Mehr and Seifert, 1998).

• It is clear that large, complex networks are able to function reliably despite inherent noise attributable to molecular fluctuations. Apparently, noise attenuation arises from complex network mechanisms involving multiple feedback loops (Barkai and Leibler, 1997; Rao et al., 2002).

• Although noise attenuation and exploitation examples are abound, how cells are able to manipulate biochemical noise remain unknown. By what mean do regulatory networks attenuate the noise? And how and why do networks exploit noise? As pointed out by Rao et al. (2002), these questions present one of the most challenging and fascinating problems for systems biologists, as they open questions in physiology, development and evolutionary biology.

• The answer likely resides in complex gene regulatory networks. Stochastic dynamic models are the ideal tools for such investigations, because they allow us to describe quantitatively the current states of network structure and component interactions to explore the network stochastic dynamics under intrinsic fluctuations and extrinsic noise.

• In this study, based on a stochastic dynamic model of genetic regulatory networks with intrinsic and extrinsic noises, the scheme for noise attenuation or amplification is revealed from Lyapunov (energy) function point of view and then a measure of noise attenuation level or amplification level of network is developed

• In this study, the intrinsic noises due to parametric fluctuations are modeled as state dependent noise, which will influence the stability of the gene regulatory network. The robust stability to tolerate these intrinsic parameter fluctuations by gene regulatory networks is discussed by the Lyapunov stability theory of nonlinear stochastic systems.

• We need to solve an HJI to measure the robustness of stability of nonlinear gene regulatory networks (Zhang and Chen, 2005). The ability to attenuate the extrinsic noises of nonlinear gene regulatory networks is measured based on nonlinear H∞ filtering theory (Zhang et al., 2005).

• For the convenience of computation, LMI techniques (Boyd et al., 1994), which can be efficiently solved by LMI Toolbox of Matlab, are used to calculate the robust stability with respect to parametric fluctuations and the attenuation level or exploitation level with respect to extrinsic noises in gene regulatory networks, respectively.

Method• Figure 1. The linear genetic regulatory network with intrinsic

fluctuation and extrinsic fluctuation .ijN )(ti

)(1 tx )(2 tx2121 NN

3131 NN

4141 NN 3232 NN

)(3 tx

)(4 tx

)(txi

ii NN 11

ii NN 22

ii NN 33

)(tx j

ijij NN

22 jj NN

)(txn

njnj NN

inin NN

44 jj NN

)(1 t

)(2 t

)(tj

)(4 t

)(tnnH

4H

1H

jH

2H

• (1)

where concentration vector and stoichiometric matrix N are given by

in which denotes the concentration of the ith gene and Nij denotes the interaction between gene j and gene i.

)(tx

)()( tNxdttdx

nnn

ij

n

n

NNN

NNN

tx

txtx

tx

1

1112

1

,

)(

)()(

)(

)(txi

• Suppose the linear genetic regulatory network suffers intrinsic molecular fluctuations mainly due to thermal fluctuation so that stoichiometric matrix N is perturbed as N+△N, where the components of the perturbation △N could be modeled by a stochastic Wiener process on a probability space , which is a mathematical description of the so-called Brownian motion (Chen et al., 1995).

• In intrinsic fluctuations, the perturbed dynamics of nominal genetic regulatory network in equation (1) could be modeled as the following linear stochastic equation

(2)

)(t),( p

)()()()( tdtMxdttNxtdx

• This is a standard linear stochastic dynamic equation with state dependent noise. denotes the term due to the intrinsic fluctuation , where the stochastic property of fluctuation △N is extracted out to the standard Wiener process (or Brownian motion) (which, roughly speaking, is integral of white noise) with correlation function

with unit covariance . For example, , where

denotes the deterministic part (amplitude) of fluctuation and n(t) is a white Gaussian noise with zero mean and unit variance to denote the stochastic part of fluctuation, i.e., the stochastic part of

)()( tdtMx

)(tNx

)(t )()()( 22 ttE

12

)()()()(

)()()(

1

111

1

111

tMntntNtN

tNtNtN

N

nnn

ij

n

nnn

ij

n

ij

fluctuation is absorbed to n(t) with (Chen and Zhang, 2004; Zhang et al., 2005; Zhang and Chen, 2005). If some components ij of N are free of intrinsic fluctuation, then the corresponding should be equal to zero. And the covariance of

.

dttntd )()(

ij

2))((Cov)(Cov)( ijijijij tnNtN

• In the conventional notation of engineering and system science, the stochastic dynamic equation could be represented by

(3)

where denote a normalized white Gaussian noise with unit covariance. Since is a stochastic process, in equation (2), or (3) is also a stochastic process. Actually, the stochastic dynamic equations of genetic regulatory networks are always nonlinear. In order to meet the nonlinear stochastic regulatory networks, equation (3) should be generalized as

(4) where denotes the nonlinear stoichiometric equ

ation of nonlinear genetic regulatory network and is due to nonlinear intrinsic fluctuation.

)()()()( tntMxtNxtxdtd

)()()()( tdxMdtxNtdx

dttdtn )()(

)(t

)(tx

)(xN

)()( tdxM

• For example, the standard chemical Langevin equation (Gillespie, 2000 and 2002) based on Fokker-Plank equation, which has been widely employed to model stochastic genetic regulatory network, is given by (Rao et al., 2002)

(5) where

Remark : At different operation points, the linearization of nonlinear stochastic regulatory network of equation (4) will be of the form in equation (3), i.e., at an operation point , and

)())(())(()( 21

tdtxNfdttxNftdx

))(())(())(())((,))(())(())(())(( 2

121

21

121

1 txftxftxftxftxftxftxftxf nini

oxx oxx

xxNN

)(

oxxxxMM

)(

• In this study, the white noise is normalized with unit variance, in which the covariance of stochastic noise is absorbed by M. Since the effect of intrinsic biochemical kinetic parametric fluctuation is state dependent and will influence the stability of genetic regulatory networks, it will be discussed at first. Let denote the Lyapunov (energy like) function of a stochastic genetic regulatory network, with , as . In the linear genetic regulatory network, the Lyapunov function is always chosen as for some symmetric positive definite matrix P. Then the stochastic genetic regulatory network is stable in probability if the expectation of the derivative of

is not positive (Chen et al., 1995), i.e., the total energy (squares of concentrations) of the genetic regulatory network could not increase again in probability.

(6)

)()( tdtdtn

0)( xV

0)0( V )(xV x

PxxxV T)(

)(xV

0)(

xVdtdE

• In the linear stochastic genetic regulatory network (3), the robust stability under intrinsic fluctuation is given as follows

Theorem 1: The linear perturbative regulatory network in equation (3) is stable in probability if the following Lyapunov-type equation

(7)has a symmetric positive definite solution .

• Remark 1: (i) In the intrinsic noise free case, i.e., the nominal case in equation (1), the stable condition is that the matrix inequality has a symmetric positive definite solution . Obviously, the symmetric positive definite solution in (7) is more strict because the eigenvalues of stoichiometric matrix N should be located at the more left hand side of complex domain with large negative real values to overcome the extra term due to intrinsic noise in equation (7)

0 PMMPNPN TT

0 PNPN T

0P

PMM T

0P

• Theorem 2: The nonlinear perturbative genetic regulatory network in equation (4) is stable in probability if the following Hamilton Jacobi inequality H-J-I has a positive solution with

(8)Then the nonlinear stochastic regulatory netwo

rk in equation (4) is locally stable in probability.

0)( xV 0)0( V

0)()()(21)()(

2

2

xM

xxVxMxN

xxV T

T

• Remark 2: (i) The inequality of equation (8) is an extension of inequality in equation (7) from linear stochastic case to nonlinear stochastic case. (ii) In the special nonlinear stochastic case of standard chemical Langevin equation, the robust stability under intrinsic fluctuation in (8) should be modified as the existence of a nonnegative solution to the following H-J inequality

(9) (iii) In general, it is not easy to solve nonlinear inequ

ality in equation (8) to see whether the stability of nonlinear stochastic genetic network is guaranteed under intrinsic fluctuation. A convenient method based on the so-called global linearization (Boyd et al., 1994) is proposed in the following.

)(xV

0)()())((21)()( 2

1

2

221

xNf

xxVNxfxNf

xxV TT

T

• Suppose the linearization of is defined as

and suppose is bounded by the

following polytope with n vertices (10)

where denotes the convex hull consisted by the vertices . Therefore, all linearized systems of

are bounded in the polytope consisted of the linearized vertices

Then the perturbative nonlinear regulatory network is robustly stable under intrinsic fluctuation if the following inequalities have a common positive solution

(11)

)(xNnnR

xMxN

x

2

)()(

n

n

MN

MN

MN

, ,Co2

2

1

1

Co

)()()()( tdxMdtxNtdx

LitdtxMdttxNtdx ii ,2,1 ),()()()(

0 TPP

LiPMMPNPN iTi

Tii ,2,1for 0

• In general, it is very easy to find common to solve the above LMIs by LMI Toolbox in Matlab if it exists. Therefore, it is more appealing to solve LMIs than to solve HJI in equation (10), but the result may be more conservative.

• Attenuation and Exploitation of Extrinsic Noise• After the robust stability of genetic regulatory network is guara

nteed under the intrinsic biochemical parametric fluctuation, the effect of the extrinsic fluctuation on the network will be discussed. If the linear regulatory network in equation (2) also suffers the extrinsic signals (or noises, see Figure 1) outside the network, then stochastic equation (2) is modified as follows

(12) where H is a coupling matrix which denotes the influence of ext

rinsic signal on the state . denotes the concentration of some genes we are interested. For example, we only want to discuss the effect of noises of and on gene i, i.e., , then we let , i.e., every element of C is zero except 1 in the ith element. If we want to discuss the effect of noises on the whole genetic regulatory network, we let , the identity matrix.

)(t

)()(

)()()()()(tCxtZ

tdtMxdttHtNxtdx

)(tx )(tZ

)(t )(t)(txi 0 010 0 0 C

IC

• Similarly, the nonlinear stochastic regulatory networks in equation (4) and (5) under extrinsic fluctuation should be modified, respectively, as

(13)and

(14)Then the attenuation and exploitation of extrinsic noise

of stochastic regulatory networks in equations (12)-(14) will be discussed in the following.

Let us denote L2-norm of as (15)

where E denotes the expectation.

)()(

)()()()()()(tCxtZ

tdxMdttxHxNtdx

)()(

)()()()()( 21

tCxtZtdxNfdttHxNftdx

)(tx

02)()( )( dttxtxEtx T

• Then the positive value in the following inequality is called the attenuation level from extrinsic noise

to in the perturbative genetic regulatory network (16)

If , we say the extrinsic noise is attenuated by the genetic regulatory network. If , we say the extrinsic noise is exploited by the genetic regulatory network. In this situation, is called attenuation level if or exploitation level if .If , we call it lossless. In the inequality (16), it is under the assumption of . If the initial condition is not zero, i.e., , then an extra term of initial condition should be added as follows (Chen and Zhang. 2004; Zhang et al., 2005)

(17) for some positive function of initial condition

)(t

)(tZ

222

2 )()(or )()(

ttZttZ

11

1 1 1

0)0( x0)0( x

22

)())0(()( txVEtZ

))0((xV )0(x

• Based on analysis above, some results about the attenuation or exploitation of extrinsic noise of perturbative genetic regulatory networks are discussed in the following

• Theorem 3: The attenuation level of linear perturbative genetic regulatory network in equation (12) is guaranteed if the following inequality has a positive definite solution

(18)By Shur complement (Boyd et al., 1994), inequality (18)

is equivalent to the following LMI (19)

0P

01 PPHHCCPMMPNPN TTTT

0)(

IPH

PHPMMCCPNPNT

TTT

• The optimal attenuation level of the linear stochastic genetic regulatory network (12) can be obtained by solving the following constrained optimization

(20)• The above LMI optimization can be solved easily by

decreasing until no positive definite solution in equation (19) can be found. Software package such as LMI Optimization Toolbox in Matlab have been developed for easily solving the above LMI optimization.

• Remark 3: (i) If in equation (20), the extrinsic noise is attenuated by the genetic regulatory network at . (ii) If , the extrinsic noise is exploited by the genetic regulatory network at . (iii) If you only want to check if the genetic regulatory network has a prescribed attenuation level , we only check if inequality (19) has a positive definite solution .

o

(19) and 0 ,0 subject tomin

Po

0P

1o)(t

)(tZ 1o )(t)(tZ

0P

• For nonlinear perturbative genetic regulatory network in equation (13), the attenuation or exploitation of extrinsic noise at is discussed in the following

• Theorem 4: The attenuation level of nonlinear genetic regulatory network (13) is guaranteed if the following Hamilton-Jacobi equation has a positive definite solution

(21)

• The optimal attenuation level of the nonlinear stochastic regulatory network can be obtained by solving the following constrained optimization

(22)

)(t )(tZ

0)()(41)()()(

21)()(

2

2

xxVHH

xxVxM

xxVxMxN

xxVCxCx T

TT

TTT

0)( xV

o

(21) and 0 ,0 subject to

min)(

PxVo

• Remark 4: (i) The optimal attenuation level of standard chemical Langevin equation (14) can be obtained by solving the following constrained optimization

(23) (ii) In general, there exists no systematic method to s

olve the constrained optimizations in (22) and (23). It should be solved case by case by decreasing until no positive solution in the Hamilton-Jacobi inequality in (21) or (23) can be found.

o

0)()(41)()())((

21)()(

,0)( ,0subject to

min

21

2

221

)(

xxVHH

xxVxNf

xxVNxfxNf

xxVCxCx

xV

TT

TTT

TT

xVo

)(xV

(iii) An approximation solution for (22) based on the “global linearization” techniques of (10) and (11) in Remark 2 (iii) is introduced in the following, i.e., if the nonlinear perturbative genetic regulatory network (13) could be bounded in the polytope consisted of L linearized vertices ,

then, after some arrangements, the optimal attenuation level in Theorem 4 can be obtained by solving the following constrained optimization problem

(24)

• This result is similar to the constrained optimization in (20) except a set of LMI constraints.

LitdtxMdttHtxNtdx iii ,2,1 ),()()()()(

o

LiIPH

PHCCPMMPNPN

P

Ti

iT

iTi

Tii

Po

,,2,1for ,0)(

,0 ,0subject to

min

Experimental Simulation

• Genetic regulatory network

X1 mRNAGene 1

X2 enzyme

X3 inducer

AA

substrate

+

X4 mRNAGene 4

X5 regulatorAA+

- -

+

+-

-

2510

2495

248

15

2374

23

126

1253

224

2132

212

15311

XkXkX

XkXXkX

XXkXkX

XkXkX

XkXXkX

5.0)0(5.0)0(5.0)0(5.0)0( 5.0)0(

5

4

3

2

1

XXXXX

• Nominal case

Network Robust Stability under Intrinsic Fluctuation • Suppose there are some perturbations making become , respectively, where

By remark 2(iii), we can linearize the system and solve the LMIs to s

ee if the perturbed system is stable or unstable. In the perturbative case, the solution P of the LMIs is

where eigenvalues of P = 0.074151, 0.15722, 0.20491, 0.22885, 0.50216

• The system is stable.

ikii kk ),(2 ),(1 21 tnktnk

),(4.1 ),(8.0 ),(3.0 ),(2 6543 tnktnktnktnk )(8.1),(2 ),(2.1 987 tnktnktnk

0.187590.0257020.065039-0.033226-0.033869-0.0257020.259560.00219010.026898-0.1377-0.065039-0.00219010.125030.01921-0.024633-0.033226-0.026898-0.01921-0.213150.0761870.033869-0.1377-0.024633-0.0761870.38196

P

• Suppose there are other perturbations making become , respectively, where

then the solution P of the LMIs is

where the eigenvalues of P = -7.6686×10-12, -4.6163×10-13, -2.268×10-14, 1.7721×10-12, and 6.5188×10-12 , which are not all positive • The system is unstable.

ik

ii kk ),(3 ),(5.1 21 tnktnk

),(3 ),(2 ),(2 ),(1 ),(3 76543 tnktnktnktnktnk )(1 ),(2 ),(5.2 1098 tnktnktnk

13-12-15-12-12-

12-13-12-13-12-

15-12-12-12-12-

12-13-12-12-13-

-12-12-12-13-12

105.8391101.1423109.5987101.2688-102.0621-101.1423103.2196-102.0094-107.0144101.0044-109.5987102.0094-101.95-104.7701103.0598101.2688-107.0144104.7701101.1992-102.9683-102.0621-101.0044-103.0598102.9683-103.0253

P

Attenuation and Exploitation of Extrinsic Noise

• Suppose the coupling matrix H is an identity matrix

• The attenuation or exploitation level can be evaluated by solving the LMIs of equation (24)

• of (C=[1 0 0 0 0]) is 0.5723 • of (C=[0 1 0 0 0]) is 0.5328 • of (C=[0 0 1 0 0]) is 0.5489 • of (C=[0 0 0 1 0]) is 0.6980 • of (C=[0 0 0 0 1]) is 0.6964

1x

2x

3x

4x

5x

• Noise attenuation of extrinsic noise (White Gaussian noise mean=2, variance=1) at x1, x2, x3,x4,x5

The noise is attenuated at x1, x2, x3,x4,x5

)(t

)(t