Computational Methods in Systems Biology

Detecting Attractors in Biological Models with Uncertain Parameters 47

first

iter.

second

iter.

(incorrect)

second

iter.

(correct)

V̂ F̂=B̂=B̂′ V̂\B̂′

s t

u v

s t

u v

s t

u v

t

u v

t

u v

t

u v

t

u v

t

u v

t

u v

Fig. 2.Illustration of Algorithm 1. (Color Figure Online)

Otherwise, we remove the parameter valuations fromV̂and repeat the process.
As for the counter, instead of a single number, we use a mappingP→Nthat
assigns to each parameter valuation the number of tSCCs in its induced graph.
The actual implementation of the counter depends on the (symbolic) represen-
tation of the parameter valuations and is discussed in Sect.4.1.
Note that the algorithm as presented in Algorithm 1 solves both the tSCCs
Detecting Problem and its counting version. If only thecountingversion is con-
sidered, we simply remove lines 17 and 18. Furthermore, if we are only interested
in thethresholdversion of the problem, we may stop considering all parameter
valuationspfor whichcountphas already reached the threshold. Moreover, in
theexistential thresholdversion, we stop the whole algorithm once any parame-
ter valuation has reached the threshold.

3 Applications

We apply the method to several models used in systems biology. Since most of the
existing and widely used models are represented by means of ordinary differential
equations (ODEs), we employ the piece-wise multi-affine approximation [ 16 ]and
rectangular abstraction procedures [ 2 , 11 ] to obtain a discrete representation of
the systems dynamics in the form of a finite parametrised graph.