Sloppy/Stiff Parameters Rankings in Sensitivity Analysis of Signaling

Pathways

Malgorzata Kardynska

, Jaroslaw Smieja

, Anna Naumowicz

, Patryk Janus

1,2

, Piotr Widlak

and Marek Kimmel

1,3

Institute of Automatic Control, Silesian University of Technology, Akademicka 16, 44-100 Gliwice, Poland

Maria Sklodowska-Curie Memorial Cancer Center and Institute of Oncology, 44–100 Gliwice, Poland

Department of Statistics, Rice University, Houston, TX 77005, U.S.A.

Keywords: Sensitivity Analysis, Sloppy Parameters, Signalling Pathways, NF-кB, HSF, HSP.

Abstract: Sensitivity analysis methods have been developed for over half a century. However, their application to

systems biology is a relatively new concept and has not been fully investigated. In this paperwe focus on

creating parameter rankings based on sloppy/stiff parameter sensitivity analysis, that can be used to find the

most important parameters and processes (that have the greatest impact on the system output) and

subsequently can be used to reduce the number of experiments needed to precisely estimate parameters

values or to indicate molecular targets for new drugs. In order to test the proposed procedure we performed

sensitivity analysis of the HSF/NF-кB pathway model - a model combining two signaling pathways

essential for cell survival.

1 INTRODUCTION

A biomathematical model is a description of a

biological system using mathematical language.

Such models are created to describe processes taking

place at different levels: from a single cell to the

entire population. In addition to many tiers of

biological system, there are also many methods that

can be used to describe them with mathematical

language. In this paper we focused on deterministic

models of so called signaling pathways, described

by ordinary differential equations. Such models are

powerful tools that allow us to develop and test

several hypotheses about complex biological

systems (Locke et al., 2005, Voit et al., 2006). In the

literature there is a growing number of high

dimensional models with a large number of

parameters. As an example, we used a model

combining two signaling pathways: HSF and NF-

кB. However, methods for measuring biochemical

parameters are limited and may introduces

substantial inaccuracies (Maerkl and Quake, 2007).

Therefore, each model should be checked with

respect to its sensitivity to parameter changes.

The sensitivity analysis is an important tool used

to determine how the change of parameters influence

the system behavior. It provides information about

the most important parameters that have the greatest

impact on the system output (and as a consequence

should be determined with the highest accuracy).

Moreover it gives us information about robustness of

the systems (Rand, 2008), which helps us validate

the model. Most of the pathways should be robust

with respect to changes in parameters in a relatively

wide range which may represent the differences

between individual cells, e.g. in the rate of

biochemical reactions (characterized by different

parameter values). Sensitivity analysis provides also

a valuable insight into the importance of particular

processes.

Sensitivity analysis methods are used to test

mathematical models for over half a century.

However, the methods used e.g. in automatic control

cannot always be directly used in systems biology,

and may lead to false conclusions. For this reason it

is necessary to develop methods which take into

account the specificity of biological systems and

experimental data. In this paper we propose a new

measure of parameter sensitivity. We use one of the

most common methods used currently in sensitivity

analysis of signaling pathways, known as

sloppy/stiff parameter sensitivity analysis

(Gutenkunst et al., 2007), however our work is

278

Kardynska, M., Smieja, J., Naumowicz, A., Janus, P., Widlak, P. and Kimmel, M.

Sloppy/Stiff Parameters Rankings in Sensitivity Analysis of Signaling Pathways.

DOI: 10.5220/0005820002780283

In Proceedings of the 9th International Joint Conference on Biomedical Engineering Systems and Technologies (BIOSTEC 2016) - Volume 3: BIOINFORMATICS, pages 278-283

ISBN: 978-989-758-170-0

focused on creating parameter rankings, that can be

subsequently used either to reduce the model

complexity (Kim et al., 2011) or indicate

prospective molecular targets for new drugs (Marin-

Sanguino et al., 2011). We compared our method of

creating parameter rankings with the ranking based

on the areas under curve of sensitivity function.

2 SLOPPY/STIFF PARAMETER

SENSITIVITY ANALYSIS

Let the model be described by the state equation:



,







,

,,,

(1)

where y

s,c

denoting number or concentration of

molecules of species s in condition c, u is an input

variable and θ are model parameters.

The change in model behavior as parameters θ

varied from their nominal values θ

can be quantified

by the average squared change in molecular species

time course (Gutenkunst et al., 2007):

=

2













,

∙

∙



,



,





,





∗

,



















(2)

where N

and N

are the total number of species and

conditions, respectively, T

is the sampling time and

is the maximum value of species s across the

conditions considered.

To analyze model sensitivity to parameter

variation we considered the Hessian matrix

corresponding to cost function C(θ). Since

biochemical parameters very often have different

units and widely varying scale to eliminate the

impact of relative changes in parameter values the

derivatives with respect to logθ are taken:



,

















, (3)

where j and k denotes j−th and k-th parameter,

respectively. The Hessian describes the quadratic

behavior of the cost function C near the point θ

. H

can be calculated as (Gutenkunst et al., 2007):



,























,

∙

∙



,



∗

,







,



∗

,













(4)

Based on Eq. (4) the sensitivity of the entire

model (for all species s across all considered

conditions c) to parameter variation can be

calculated. However, the sensitivity of individual

species or sensitivity of the model in specific

condition could be also examined by taking into

account only one species or condition.

The Hessian matrix is positive, definite and

symmetric, so it has real eigenvalues λ and

eigenvectors v. Analyzing H

corresponds to

approximating the surfaces illustrating deviations

from nominal system response. The surface is N

dimensional ellipsoids, where N

is the number of

parameters in the model. The principal axes of the

ellipsoids are the eigenvectors of H

, and the width

of the ellipsoids along each principal axis is

proportional to one over the square root of the

corresponding eigenvalue λ

(Gutenkunst et al.,

2007):











.

(5)

Figure 1: An example of ellipse illustrating deviations

from nominal system response for a simple model with

two parameters θ

and θ

. d

and d

denotes the width

the ellipse along each principal axis, corresponding to

eigenvalues λ

and λ

respectively, while v

and v

denotes

the eigenvectors defining the position of the ellipse.

The narrowest axes are called “stiff”, and the

broadest axes “sloppy”. The meaning of eigenvalues

and eigenvectors of H

is illustrated on a simple

example, where Hessian describes an ellipse in the

/θ

parameter space (Figure 1).

The relative widths d

and d

, shown in the

Figure 1, allow us to identify “sloppy” and “stiff”

principal axis of the ellipse. However, the degree to

which the principal axes of the ellipsoids are aligned

to the bare parameter axes is also important. It can

be estimated by comparing the ellipsoids

Sloppy/Stiff Parameters Rankings in Sensitivity Analysis of Signaling Pathways

279

intersections I

with each bare parameter axis i,

calculated as:









,



, (6)

and projections P

onto each bare parameter axis i,

calculated as:















,

.

(7)

If I

/ P

= 1, then one of the principal axes of the

ellipsoids lies along bare parameter direction i,

however in biological systems this occurs very

rarely. More often the ellipses are skewed from

single parameter directions (Gutenkunst et al.,

2007).

Although I

/ P

ratio provides some useful

information, it does not link the skewing rate with

the width of particular principal axes of the

ellipsoids, which is also very important because it

would help us to identify the most significant

parameters in the model. To relate these width

(corresponding to “sloppy” and “stiff” principal

axes) with specified parameter changes we propose

another index, which is used to create the parameters

ranking. The index is defined for the j-th parameter

as:





=



,









(8)

where the sum is calculated over all principal axes,

is the width of the ellipsoid along i-th principal

axis, and v

j,i

is the element of the i-th eigenvector

corresponding to the j-th parameter.

3 SENSITIVITY ANALYSIS OF

THE HSF/NF-КB PATHWAYS

MODEL

In order to test the applicability of the procedure

described above we performed sensitivity analysis of

the HSF/NF-кB pathway model, which described in

(Smieja et al., 2015). The model combines two

signaling pathways essential for cell survival.

NF-κB is a family of transcription factors that

regulate the transcription of hundreds of genes,

including genes that determine cell fate. It has been

proved that NF-κB can play an antiapoptotic role in

cancer cells, e.g. via activation of anti-apoptotic

genes (Cataldi et al., 2003). Upregulation of the NF-

κB pathway is frequently observed in cancer cells,

which contributes to their resistance to the

anticancer treatment (Hayden and Ghosh, 2012;

Perkins, 2012). Therefore inhibition of NF-κB

pathway may constitute one of the goals in

anticancer therapies. Experimental results show that

heat shock induces such inhibition in cancer cells

(Janus et al., 2011). However, the precise

mechanisms of interactions between HSF and NF-

κB pathways are not fully understood yet.

Development of a combined mathematical model of

these pathways and its subsequent computational

analysis should help to develop the most efficient

anticancer therapy protocols.

Figure 2: The eigenvalues (a) and I/P (b) spectrum of the

HSF/NF-кB pathways model.

So far, numerous models of NF-κB pathway

have been developed, whereas much fewer models

of HSF pathway have been published. The model

proposed in our work was based on the previously

published ones, which described either NF-кB

(Lipniacki et al., 2004) or HSF (Szymanska and

Zylicz, 2009) pathways separately. In order to

incorporate crosstalk between HSF and NF-κB

pathways, they had to be modified: nuclear and

cytoplasmic levels of proteins and complexes had to

be separated and constitutive and inducible HSPs

were described by separate variables. The

interactions between the HSF and NF-κB pathways

take into account creation HSP:IKK complexes,

temperature-dependent inactivation of proteins

located upstream of IKK activation and inhibition of

NF-κB import to the nucleus under heat shock

condition. The reactions taken into account are

summarized in the Table 1.

We checked the sensitivity of the system

BIOINFORMATICS 2016 - 7th International Conference on Bioinformatics Models, Methods and Algorithms

280

following the procedure described in the previous

chapter. We have chosen nuclear NF-кB as one of

the state variables to illustrate applicability of the

method. The eigenvalues and I/P spectrum is plotted

on Figure 2, while parameter ranking based on index

(Eq. (8)) is shown on Figure 3. To compare the

results with other commonly used procedure, a

parameter ranking based on area under curve of

sensitivity functions is show on Figure 4. Due to

high number of parameters the horizontal axes

contains only numbers corresponding to the

parameters listed in Table 2.

In both presented rankings the position of most

parameters is comparable. However there are some

significant differences, e.g. in parameters 37 and 45,

corresponding to the ratio of cytoplasmic to nuclear

volume (k

) and IkBa mRNA degradation rate (c

respectively. The parameter k

is indicated as the

most important by the sloppy/stiff method, while in

the ranking based on sensitivity function parameter

seems to be more important. To check which of

these two parameters has greater influence on the

system response, we performed three simulations: 1)

for nominal parameter values, 2) for parameter k

increased by 30% and 3) for parameter c

increased

by 30%. The results of these three simulations are

shown in Figure 5. By comparing these three time

courses, we can see that parameter k

significantly

increases the maximum concentration of free nuclear

NF-κB and in this term the ranking based on

sloppy/stiff method seems to be more reliable.

However, changing the parameter c

results in phase

shift in system response, what in biological systems

can also be very important.

Table 1: Reaction list for the HSF/NF-кB pathways model (Smieja et al., 2015).

NF-кB subsystem HSF subsystem

IKKn

⎯⎯→⎯

deg

IKKn

⎯⎯⎯⎯→⎯

nTRAF2,TNF,

IKKa

⎯⎯⎯⎯→⎯

nTNF,A20,

IKKi

IKKa

⎯→⎯

IKKi

IKKa

⎯⎯→⎯

deg

IKKi

⎯⎯→⎯

deg

NF-κB

nuc

⎯→⎯

NF-κB

nuc

+ A20

A20

⎯→⎯

A20

+ A20

A20

⎯→⎯

A20

⎯→⎯

IKKa + IκBα

⎯→⎯

IKKa:IκBα

⎯→⎯

IKKa

IκBα

⎯→⎯

IκBα

⎯→⎯

NF-κB + IκBα

⎯→⎯

NF-κB:IκBα

⎯→⎯

IKKa + NF-κB:IκBα

⎯→⎯

IKKa:NF-κB:IκBα

⎯→⎯

IKKa + NF-κB

NF-κB

⎯⎯→⎯

,k i

NF-κB

nuc

NF-κB

nuc

+ IκBα

nuc

⎯→⎯

NF-κB

nuc

:IκBα

nuc

NF-κB

nuc

:IκBα

nuc

⎯→⎯

NF-κB:IκBα

NF-κB

nuc

⎯→⎯

NF-κB

nuc

+ IκBα

IκBα

⎯→⎯

IκBα

+ IκBα

IκBα

⎯⎯→⎯

,k i

IκBα

nuc

IκBα

nuc

⎯→⎯

IκBα

Prot

⎯⎯→⎯

k T,

mfProt

mfProt + HSP

cons

⎯→⎯

HSP

cons

:mfProt

mfProt + HSP

ind

⎯→⎯

HSP

ind

:mfProt

HSP

cons

:mfProt

⎯⎯→⎯

−1

k a,

HSP

cons

+ Prot

HSP

ind

:mfProt

⎯⎯→⎯

−1

k a,

HSP

ind

+ Prot

HSP

cons

+ HSF

⎯→⎯

HSP

cons

:HSF

HSP

ind

+ HSF

⎯→⎯

HSP

ind

:HSF

HSP

ind

:HSF

⎯→⎯

-2

HSP

ind

+ HSF

HSP

cons

:HSF + mfProt

⎯→⎯

-3

HSP

cons

:mfProt + HSF

3HSF

⎯→⎯

HSF

HSF3 + HSP

ind

⎯→⎯

-4

HSP

ind

:HSF + 2 HSF

HSP

ind

⎯→⎯

HSPmRNA

⎯→⎯

HSF

⎯→⎯

HSF

+ mRNA

HSPmRNA

⎯→⎯

HSPmRNA + HSP

ind

HSF

cons

+ IKKa

⎯→⎯

HSP

cons

:IKK

HSF

ind

+ IKKa

⎯→⎯

HSP

ind

:IKK

HSP

cons

:IKK

⎯→⎯

-6

HSP

cons

+ IKKn

HSP

ind

:IKK

⎯→⎯

-6

HSP

ind

+ IKKn

TRAF

⎯⎯⎯→⎯

TRAF5

k T,

mf TRAF

mf TRAF2 + HSP

cons

⎯⎯→⎯

TRAF1

HSP

cons

:mf TRAF2

mf TRAF2 + HSP

ind

⎯⎯→⎯

TRAF1

HSP

ind

:mf TRAF2

HSP

cons

:mf TRAF2

⎯⎯⎯→⎯

− TRAF1

k a,

HSP

cons

+ TRAF2

HSP

ind

:mf TRAF2

⎯⎯⎯→⎯

− TRAF1

k a,

HSP

ind

+ TRAF2

HSF

3 cyt

⎯⎯→⎯

1iv

t,k

HSF

3 nuc

HSF

3 nuc

⎯→⎯

HSF

3 cyt

HSP

cons,cyt

⎯⎯→⎯

2iv

t,k

HSP

cons,nuc

HSP

cons,nuc

⎯→⎯

HSP

cons,cyt

HSP

ind,cyt

⎯⎯→⎯

3iv

t,k

HSP

ind,nuc

HSP

ind,nuc

⎯→⎯

HSP

ind,cyt

Sloppy/Stiff Parameters Rankings in Sensitivity Analysis of Signaling Pathways

281

Figure 3: Parameter ranking based on sloppy/stiff method.

Figure 4: Parameter ranking based on sensitivity functions.

Table 2: List of parameters in the HSF/NF-кB pathways

model (Smieja et al., 2015).

No. Name No. Name No. Name

-1

1TRAF

-1TRAF

-2

-3

-4

5TRAF

-6

prod

deg

It should be noted that contrary to standard

rankings based on sensitivity function, the proposed

ranking reflects the influence of parameter changes

on the system output, not only in the case when a

single parameter is varied but also when it changes

together with other ones. However, computational

complexity is the same as for calculating sensitivity

functions. The variance-based approaches (e.g.

Sobol, 2001), would require much more

computational power.

Figure 5: The comparison of three simulation runs:1) for

nominal parameter values (dotted line), 2) for parameter k

increased by 30% (gray line) and 3) for parameter c

increased by 30% (black line).

4 CONCLUSIONS

Parameters rankings are a useful tool that allows us

to indicate parameters that are most important for

the dynamics of a given pathway. In this paper we

presented a new method for creating the parameters

ranking based on the popular sloppy/stiff parameter

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282

sensitivity analysis. Taking into account the example

presented in this work we showed that the method

can provide valuable information about the most

important parameters that have the greatest impact

on the system output.

Moreover, the work shows that the parameters

rankings for the same model may vary depending on

the applied methodologies. Various parameters

rankings may be sensitive to various changes in

response (e.g. quantitative or qualitative changes).

For this reason the choice of sensitivity analysis

method must be adapted to the purpose of research

and the type of model we investigate. Furthermore it

is a good practice to examine the sensitivity of the

system using various methods and compare the

results.

ACKNOWLEDGEMENTS

The work has been supported by the NCN grants

DEC-2013/11/B/ST7/01713 (MKardynska, JS) and

DEC-2012/05/B/NZ2/01618 (AN, PJ, PW,

MKimmel).

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