SensSB is an easy to use Matlab® based Sensitivity Analysis software toolbox. This tool integrates a variety of local and global sensitivity methods (Saltelli et al, 2008) that can be applied to biological models described by ordinary differential equations (ODEs) or differential algebraic equations (DAEs). SensSB is also able to import models in the Systems Biology Mark-up Language (SBML) format.

SensSBSensSB..-- A software toolbox for sensitivity A software toolbox for sensitivity

analysis in systems biology modelsanalysis in systems biology modelsMaría Rodríguez-Fernández and Julio R. Banga*

(Bio)Process Engineering Group, IIM-CSIC, Vigo, Spain, e-mail: julio@iim.csic.es

References:S. Kucherenko, M. Rodriguez-Fernandez, C. Pantelides, N. Shah. Monte Carlo evaluation of Derivative based Global Sensitivity Indices, Reliab, Eng. Syst. Safety, 2008.A. Saltelli, M. Ratto, T. Andres, F. Campolongo, J. Cariboni, D. Gatelli, M. Saisana, S. Tarantola. Global Sensitivity Analysis: The Primer. Willey, 2008.

References:S. Kucherenko, M. Rodriguez-Fernandez, C. Pantelides, N. Shah. Monte Carlo evaluation of Derivative based Global Sensitivity Indices, Reliab, Eng. Syst. Safety, 2008.A. Saltelli, M. Ratto, T. Andres, F. Campolongo, J. Cariboni, D. Gatelli, M. Saisana, S. Tarantola. Global Sensitivity Analysis: The Primer. Willey, 2008.

Acknowledgements: The authors acknowledge financial support from project Kosmobac-SysMo and Spanish MICINN project MultiSysBio DPI2008-06880-C03-02.

Several examples regarding signal transduction pathways, gene regulation networks and metabolic pathways were implemented. The importance of using sensitivity analysis techniques for fixing unessential parameters is reflected on the improvement of the identifiability of the models and the decrease of the confidence intervals of the estimated parameters.This software tool is freely available for academic users:

httphttp://://www.iim.csic.eswww.iim.csic.es//~gingproc~gingproc//software.htmlsoftware.html

9 15181312198 201 145 166 10177 213 114 2 0

0.5

msq

r

9 15181312198 201 145 166 10177 213 114 2 0

0.5

mab

s

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00.5

mea

n

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max

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-1-0.5

0

min

1 2 3 4 5 6 7 8 9 1011121314151617181920211 2 3 4 5 6 7 8 9 101112131415161718192021

-1

-0.5

0

0.5

1

Fig 1.- Local sensitivities Fig 3.- Variance based global sensitivities Fig 4.- Parameters ranking Fig 5.- Correlation analysis

0 2 4 6 8 100

0.2

0.4

0.6

0.8

1

Time (h)

u1 (h

-1)

0 2 4 6 8 100

10

20

30

40

50

u2(g

/l)

Fig 6.- Optimal Experimental DesignFig 2.- Derivative based global sensitivities

ODEs DAEs

SensSB

5 10 15 20 25 30 350

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

0.009

0.01

parameter

SI

local

5 10 15 20 25 30 350

0.02

0.04

0.06

0.08

0.1

parameter

SI

DGSM

5 10 15 20 25 30 350

0.05

0.1

0.15

0.2

parameter

SI

Sobol

SOBOL’ INDICESConsider a model:

Y is the model output:

p is a vector of input variables:

pi is the i-th element of p varying in

),...,,( 21 kpppp =)( pfY =

10 ≤≤ ip

Model, f(p)Ω∈p Y

( ) ( ) ( )kki ij

jiij

k

iii pppfppfpffpfY ,...,,...,)( 21,...,2,1

10 ++++== ∑∑∑

>=

kljiijl

jiij

k

ii SSSS ,...,2,1

1...1 ++++= ∑∑∑

<<<=

first order ormain effect of pi second order index:

effect of pure interactionbetween pairs of variables

…higher-order indices...

High Dimensional Model Representation (HDMR):

Sobol’ Indexes:

LOCAL SENSITIVITIESPartial derivatives:

Numerical methods:

Finite-Difference Approximation

Direct Method: ODESSA

The Green Function Method

Efficient in computer time

Unwarranted when- the model input is uncertain

- the model is of unknown linearity

( * )ii

fS xx∂

=∂

DERIVATIVE BASED GLOBALSENSITIVITY MEASURES (DGSM)

Partial derivatives:

Average over :

Variance of :

Combining and :

Compute them numerically using Sobol’numbers for sampling the parameter space.

More efficient than Sobol’ indices in computational time

( *)ii

fE xx∂

=∂

*ni iH

M E dx= ∫

( )1/ 22* *

ni i iHE M dx⎡ ⎤Σ = −⎢ ⎥⎣ ⎦∫

( *)iE x nH

*iM

*iM *

ii∑2 2 2

ni i i iHG M E dx= Σ + = ∫

PARAMETER RANKINGSquare root of the sensitivities squared:

Absolute values mean:

Mean square:

Maximum sensitivity:

Minimum sensitivity:

PRACTICAL IDENTIFIABILITY

Expected value of J for (p+dp):

Fisher Information Matrix (FIM):

Covariance Matrix: (Godfrey & DiStefano II, 1985)

Correlation Matrix:

Analyse possible correlations among parameters

Check FIM-based criterions (practical identifiability):

Singular FIM: unidentifiable parameters

Large condition number of FIM: low identifiability

1−= FIMC

( ) ( )∑=

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

=N

iii

T

i tpzQt

pzFIM

1

⎪⎩

⎪⎨

⎧

==

≠=

jiR

jiCC

CR

ij

jjii

ijij

,1

,

( )[ ] ( ) ( ) ( )∑∑==

+⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

≅+N

iii

N

iii

T

iT QVtrpt

pzQt

pzpppJE

11

δδδ

OPTIMAL EXPERIMENTAL DESIGNFIM based criteria (traditional approach):

A criterion =

D criterion =

OED novel approach based on global SA:

( )[ ]FIMminmax λ

( )[ ]FIMdetmax

( )[ ]1min −FIMtrace

Main advantage: based on global SI allows to consider a range of

values for the parameters to be estimated

⇒

( )GSIMudetmaxr

( ) ( )[ ]∑=

=N

iii

Ti tQtQGSIM

1( )

( ) ( ) ( )( ) ( ) ( )

( ) ( ) ( )⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

=

ipsisis

ipii

ipii

i

tSItSItSI

tSItSItSItSItSItSI

tQ

,2,1,

,22,21,2

,12,11,1

L

MOMM

L

L

,

( )kijkij tS,

min min=δ

( )∑∑= =

=xN

i

N

kkij

x

msqrj tS

NN 1 1

211δ

( )∑∑= =

=xN

i

N

kkij

x

mabsj tS

NN 1 1

11δ

( )∑∑= =

=xN

i

N

kkij

x

meanj tS

NN 1 1

11δ

( )kijki

j tS,

max max=δ

θ2

θ1

A-optimality

E-optimality

D-optimality

( )( ) ⎥⎦

⎤⎢⎣

⎡FIMFIM

min

maxminλλ

Main drawback: based on local SI linear and local assumptions⇒

E criterion =

Modified-E criterion =