Commentary for S-system

A table of contents

1. Objectives .......................................................................................................................................1
2. Introduction of S-system.................................................................................................................1
2.1 S-system conversion .................................................................................................................1
2.2 Sensitivity analysis....................................................................................................................2
2.2.1. Logarithmic Gain of a metabolite .....................................................................................2
2.2.2. Logarithmic gain of a flux ................................................................................................2
2.2.3. Sensitivity with respect to a rate constant .........................................................................3
2.2.4. Sensitivity of dependent variables with respect to a kinetic order ....................................3
2.2.5. Sensitivity of a flux to a rate constant...............................................................................42.2.6.

Sensitivity of a flux with respect to a kinetic order........................................................4

2.3 Stability analysis .......................................................................................................................43 Process flow .....................................................................................................................................5i1. Objectives
Sensitivity analysis shows how a biochemical system responds to perturbations or uncertainties. Among manymethods for system analysis, the sensitivity and stability analysis is useful for characterizing the robustness of themathematical at a steady-state level, and allows one to determine which parameters have the most effect on themodel, or which factors cause to the system to oscillate. S-system is employed to analyze the various sensitivities
and stability of the system at its steady state, because the use of S-system solves them in symbolic form. In thismodule, ordinary differential equations such as TT, CMA, GMA, and MM are converted into S-system to analyze
the sensitivities and stability.
TT: ordinary transcription and translation equations
CMA : conventional mass action
GMA: general mass action
MM : simplified Michaelis-Menten equations

2. Introduction of S-system
We show how S-system is derived from ordinary differential equations, and how the stability and sensitivity aredefined at the steady state.
2.1 S-system conversion
Ordinary differential equations are divided into the positive terms and negative terms:

dX i= Vi + − Vi − ,dt(i = 1,....,n,....n+m) (1),where Vi+ is the sum of positive terms, and Vi− is the sum of negative terms. Generally, S-system is given by:
n+mn+mdX i
hg=αi ∏ X j ij −βi ∏ X j ij
dtj =1j =1(2),where X1,…, Xn are the dependent variables, whose values vary with time, and Xn+1,…,Xn+m are the independentvariables, whose values are fixed as constants. When the concentrations of the dependent variables are given at thesteady state, the coefficients (gij, hij, ai, bi) of S-system are solved from Eqs. (1, 2) in symbolic form:gij =∂Vi + X j
∂X j Vi +, hij =∂Vi − X j
∂X j Vi −V+V−αi = n + m i, βi = n + m igh∏ X j ij
∏ X j ijj =1(3)(4)j =1where gij and hij are the kinetic order and αi and βi are the rate constants. At the steady state:

1dX i= 0,dt
we can take logarithms on both sides of Eq. (2), the equation reduces to a linear equation:n+mn+mj =1

j =1

lnαi + ∑ g ij ln X j = lnβi + ∑ hij ln X j,orn∑ ( g ij − hij ) ln X j = lnj =1

Substituting ln X j = y j ,lnn+mβi− ∑ ( g ij − hij ) ln X jαij = n +1(5).

βi= bi . The equations (5) is changed to the vector-matrix expression:
αirrr(G D − H D )y D = b − (G I − H I )y I

(6).

We substitute the matrixes as follows,

G D − HD = AD ,G I − HI = AIto obtain the equation:rrr
AD y D = b − AI y I(7)where AD is the coefficient matrix of the kinetic order regarding dependent variables, AI is the coefficient matrix ofrr

the kinetic order regarding independent variables, y D is the solution vector of dependent variables, y I is thervector of independent variables, and b is the vector regarding the rate constants. Consequently, the vector of thedependent variables is symbolically solved as:rrr
y D = AD−1b − AD−1 AI y I(8).

2.2 Sensitivity analysis

We define five types of gains and sensitivities. The detailed derivation of them are described elsewhere. Here,
we explain the mathematical definition of them.
2.2.1. Logarithmic Gain of a metabolite
In order to answer the question of how a relative change in an independent variable affect the steady-staterrconcentration of a metabolite, the logarithmic gains of a metabolite L( X D , X I ) is defined in terms of thecoefficient matrixes A −D1 and A I as:r r
L( X D , X I ) = − A −D1 A I(9).

2.2.2. Logarithmic gain of a flux
In order to answer the question of how a relative change in a flux affect the steady-state concentration of a2r

r

−1metabolite, the logarithmic gains of a flux L(VD , X I ) is defined in terms of the coefficient matrices A Dand A I as:r rr r
L(V + , X I ) = G I + G D L( X D , X I )(10).

2.2.3. Sensitivity with respect to a rate constantIn order to answer the question of how a relative change in a rate constant affect the steady state concentration of ametabolite, we solve the sensitivity with respect to rate constants, which is defined as:∂ ln X i∂ ln α j(11),∂ ln X i∂ ln β j(12).

S( X i , α j ) = −andS( X i , β j ) = −These sensitivities are expressed using vectors and matrixes asr r
S ( X D ,α) = − AD−1(13),r r
S ( X D ,β) = AD−1(14).

2.2.4. Sensitivity of dependent variables with respect to a kinetic order
In order to answer the question of how a relative change in a kinetic order affect the steady-state concentration of ametabolite, we define the sensitivity with respect to the kinetic order as follows,

S( X i , g ij ) = −∂ ln X i∂ ln g ij(15),

S( X i , h ij ) = −∂ ln X i∂ ln h ij(16).

The partial derivatives of Eq. (8) regarding gij are provided by:orr
∂AD r
∂y D
= 0,y D + AD
∂g ij
∂g ij

(17)

r
∂y D
∂A r
= − AD−1 D y D .
∂g ij
∂g ij

(18)

Therefore,rr
∂A r
∂y D
S ( X D , g ij ) = g ij
= −g ij AD−1 D y D
∂g ij
∂g ij

(19)

3r
r
∂A r
∂y D
= −h ij AD−1 D y D
S ( X D , h ij ) = h ij
∂h ij
∂h ij

(20).

2.2.5. Sensitivity of a flux to a rate constantIn order to answer the question of how a relative change in a rate constant affect the steady-state concentration of aflux, we define the sensitivity with respect to a rate constant.
We take logarithms in the first term of the right hand side of S-system:
n+mln Vl = lnαl + ∑ g lk ln X k(21)k =1Here, we differentiate to both side regarding lnαi to obtain:n

S (Vl ,αi ) =δil + ∑ g lk S ( X k ,αi )

(22)

k =1

In analogy, partial derivatives of Eq.(21) with respect to lnβi gives:nS (Vl ,βi ) = ∑ g lk S ( X k ,βi )

(23).

k =1

2.2.6. Sensitivity of a flux with respect to a kinetic order
In order to answer the question of how a relative change in a kinetic order affect the steady-state concentration of aflux, we define the sensitivity with respect to a kinetic order.
The partial derivatives of Eq. (21) with respect of ln gij is provided as:nS (Vl , g ij ) =δil g ij ln X j + ∑ g lk S ( X k , g ij )

(24).

k =1

In analogy, the partial derivatives of Eq. (21) with respect of ln hij is provided as:nS (Vl , h ij ) =δil hij ln X j + ∑ hlk S ( X k , hij )

(25).

k =1

2.3 Stability analysis
Defining the equation:
δXk = Xk － XkS (XkS steady state),
we apply the first order approximation to Eq. (2) to obtain:n+mg ikαi ∏ X j ijn

δX& i = ∑k =1

j =1

Xkn+ mhikβi ∏ X j ijng

δX k − ∑X k = X kS

k =1

j =1XkhδX k (26),

X k = X kSorng ikViS+h V−
δX k − ∑ ik iSδX kk =1 X kS
k =1 X kSnδX& i = ∑(27),As ViS+ = ViS− , we define:4

V iS+
V iS−=X kS
X kS

f ik =
to deduce to:nδX& i = ∑ ( gik − hik ) f ikδX k(28).

k =1Stability at the steady state is investigated by solving the eigenvalues of the coefficient matrix consisting of:( gik − hik ) f ik .

3 Process flow
Fig. 1 shows the whole processes of the simulator, where the process flow of S-system mainly consists of threeparts:
1) creation of the S-system coefficient file, which consists of the kinetic orders and rate constants that are solved insymbolic form at the steady state,2) creation of the S-system parameter file (the coefficients and steady-state concentrations) necessary forsensitivity/stability analysis, which is made using the S-system coefficient function and the steady state
concentrations.
3) simulation of S-system.
The checkdae module carries out the process of 1), following that the user function for ordinary differentialequations (TT, CMA, GMA, MM) have been established. However, the checkdae module does not convert thedifferential and algebraic equations (DAEs), because they cannot be converted into S-system directly. The program
module carries out the analysis of the sensitivities and stability of S-system by linking the S-system coefficient
function to the steady-state concentrations. The concentrations at the steady state must be solved before thisanalysis.

5parsedae

DAEs(checkdae file)

Editionprocess

Edited DAEs
Filecheckdae

User Fucntion

Parameters

Sensitivity/stability analysis
S-system paremters

SolverDynaics

S-system coefficient
function

Steady state
anaylisSensitivity/stabil
ity results

S-systemparameters

S-systemuser function

solverS-systemdynamics/steady state
analysisFig.1 Process flow for S-system analysis

6