Commentary for the sensitivity and stability analysis
by S-system

A table of contents
1 Objectives ........................................................................................................................................12 Function ...........................................................................................................................................1
2.1 Creation of S-system parameter file..........................................................................................1
2.2 Sensitivity analysis....................................................................................................................1
2.3 Stability analysis .......................................................................................................................1
3 Input/Output Specification...............................................................................................................1
3.1 Input specification.....................................................................................................................1
3.1.1 S-system parameter file......................................................................................................1
3.2 Output specification ..................................................................................................................2
3.2.1 S-system sensitivity and stability result file.......................................................................2

1 Objectives
Sensitivity analysis shows how a biochemical system responds to perturbations or uncertainties. S-system is able tosolve the various sensitivities and stability of the system at the steady state in symbolic form. The checkdae module
converts ordinary differential equations (TT, CMA, GMA, and MM) into S-system with the coefficient function for
calculating the kinetic order and kinetic rate constants at the steady state. The concentrations at the steady statemust be solved before this analysis. The checkdae module carries out the sensitivity and stability analysis ofS-system.
2 Function
See the commentary of S-system.
2.1 Creation of S-system parameter file
See the commentary of S-system. Following calculating the coefficients of S-system, the S-system parameter file iscreated. The S-system parameters provide the values of the coefficients at the steady state for the S-system userfunction.
2.2 Sensitivity analysis
See the commentary of S-system. The coefficient matrix of S-system is created to solve the various kinds of thesensitivities.
2.3 Stability analysis
See the commentary of S-system. After the coefficient matrix of the system is created, the eigenvalues of the matrix
are calculated by the Hessenberg QR method.
3 Input/Output Specification
3.1 Input specification
The input files necessary for calculating the sensitivity and stability are the S-system user function that has beenobtained at the steady state, and the S-system parameter file. Their specifications are described in detail elsewhere.
3.1.1 S-system parameter file
The parameter file necessary for simulating S-system is output as follows,##### Model ################
N_VAR;
3; # num of variables(all)
N_ALGEBR;
0; # num of variables(all)#Y_START ; index; initial_value; tag #commentY_START ;1; 6.7000e-02; X1#Y_START ;2; 4.6501e-01; X2#Y_START ;3; 1.5000e-01; X3##PARAM;name;index;val;val_start;num_survey;D/R/S;change-val;GA_min;GA_max;tag #comment
PARAM;constantPlayer;
1; 1.0000e+01;;0;D;;;; X4#1

PARAM
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;constantPlayer;
2; 5.0000e+00;;0;D;;;; X5;constantPlayer;
3; 3.0000e+00;;0;D;;;; X6;constantPlayer;
4; 4.0000e+01;;0;D;;;; X7;constantPlayer;
5; 1.3600e+02;;0;D;;;; X8;constantPlayer;
6; 2.8600e+00;;0;D;;;; X9;constantPlayer;
7; 4.0000e+00;;0;D;;;; X10;kplus;
1; 7.7884e-02;;0;D;;;; alpha_1;kplus;
2; 5.8501e-01;;0;D;;;; alpha_2;kplus;
3; 7.9346e-04;;0;D;;;; alpha_3;kminus;
1; 1.0627e+00;;0;D;;;; beta_1;kminus;
2; 7.9346e-04;;0;D;;;; beta_2;kminus;
3; 1.0588e+00;;0;D;;;; beta_3;g;1; 6.6000e-01;;0;D;;;; gi_1_1;g;2; 1.0000e+00;;0;D;;;; gi_1_3;g;3; 9.5000e-01;;0;D;;;; gd_2_1---omission--;h;5; -3.0600e+00;;0;D;;;; hd_2_3;h;6; 1.0000e+00;;0;D;;;; hi_2_5;h;7; 3.0000e-01;;0;D;;;; hd_3_3;h;8; 1.0000e+00;;0;D;;;; hi_3_6

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#T_EVENT ; name; index; time; value; #comment######## Don't edit the following lines. ######PARAM_NAME ;constantPlayer #PARAM_NAME ;kplus#PARAM_NAME ;kminus#PARAM_NAME ;g#PARAM_NAME ;h#3.2 Output specification
3.2.1 S-system sensitivity and stability result file
Following the analytical solution, the various sensitivities and stability are provided as follows.
##### Fri Mar 28 16:14:40 2003 ###### Logarithmic Gains and Sensitivities.
# The following data is the things when calculating STD.
##### Fri Mar 28 16:14:39 2003 #####
Solver No.
:
11
Param Survey:0
NR_TRIAL:20
NR_TOL_F: 1.000E-15
NR_TOL_X: 1.000E-15---omission--h[ 8]= 1.0000e+00#hi_3_6# Below this line is as a result of sensitivity analysis etc.
$ Fluxes $index,
y value ,1,6.700e-02,2,4.650e-01,3,1.500e-01,

Flux
1.068e+00, #X1
1.714e+00, #X2
1.714e+00, #X3

2

$ Eigenvalues $No. ,Real ,
1, -2.508e+01,
2, -1.037e+00,
3, -5.281e+01,

Imaginary
0.000e+00
0.000e+00
0.000e+00$ Logarythmic Gains of Metabolites $param/var,y[ 1],
y[ 2],
y[ 3]constantPlayer[ 1], 8.283e-01, 1.029e+00, 1.216e+00
constantPlayer[ 2], 3.100e-01, 8.038e-01, 9.497e-01
constantPlayer[ 3], 1.255e+00, 1.560e+00, 1.843e+00
constantPlayer[ 4], -6.545e-01, -2.298e-03, -2.716e-03
constantPlayer[ 5], -8.648e-02, -2.243e-01, 3.264e-02
constantPlayer[ 6], -8.821e-01, -2.288e+00, -3.000e+00
constantPlayer[ 7], 3.681e-01, 9.545e-01, 1.128e+00$ Sensitivities of Metabolites with respect to rate constants $param/var,y[ 1],y[ 2],y[ 3]alpha[ 1], 1.255e+00, 1.560e+00, 1.843e+00
alpha[ 2], 9.686e-01, 2.512e+00, 2.968e+00
alpha[ 3], 8.821e-01, 2.288e+00, 3.000e+00$ Sensitivities of Metabolites with respect to kinetic orders $param/var,y[ 1],
y[ 2],y[ 3]g[ 1], 1.907e+00, 2.370e+00, 2.800e+00
g[ 2], 1.379e+00, 1.713e+00, 2.024e+00- - - - - omission - - - - h[ 7], 5.020e-01, 1.302e+00, 1.708e+00
h[ 8], -9.269e-01, -2.404e+00, -3.153e+00$ Logarythmic Gains of Fluxes $param/flux,v[ 1],v[ 2],v[ 3]constantPlayer[ 1], 6.600e-01, 3.649e-01, 3.649e-01
constantPlayer[ 2], 0.000e+00, 2.849e-01, 2.849e-01
constantPlayer[ 3], 1.000e+00, 5.528e-01, 5.528e-01
constantPlayer[ 4], 0.000e+00, -8.147e-04, -8.147e-04
constantPlayer[ 5], 0.000e+00, 9.791e-03, 9.791e-03
constantPlayer[ 6], 0.000e+00, 9.987e-02, 9.987e-02
constantPlayer[ 7], 0.000e+00, 3.383e-01, 3.383e-01$ Sensitivities of Fluxes with respect to rate constants $param/var,
v[ 1],
v[ 2],v[ 3]alpha[ 1], 1.000e+00, 5.528e-01, 5.528e-01
alpha[ 2], 0.000e+00, 8.903e-01, 8.903e-01
alpha[ 3], 0.000e+00, -9.987e-02, 9.001e-01$ Sensitivities of Fluxes with respect to kinetic orders $param/var,
v[ 1],
v[ 2],
v[ 3]g[ 1], 1.520e+00, 8.401e-01, 8.401e-01
g[ 2], 1.099e+00, 6.073e-01, 6.073e-01
g[ 3], 0.000e+00, -2.286e+00, -2.286e+00
g[ 4], 0.000e+00, 2.795e-01, 2.795e-01- - - - - omission - - - - h[ 6], -3.170e-16, 5.387e-01, -4.374e+00
h[ 7], -1.545e-17, -5.684e-02, -5.684e-02
h[ 8], 2.851e-17, 1.049e-01, 1.049e-01

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