Computational Methods in Systems Biology

246 R. Schwieger and H. Siebert

{0001}

{0100}

{1101, 1111,

1010, 0011,

0101, 1100,

0110, 0000,

0010, 1001,

0111, 1000}

{1110}

{1011}

0000

0001

0010

1000

1010

0011

0100

0101

0110

1100

1110

0111

1001

1101

1011

1111

Fig. 4.This example illustrates how statements aboutMB(Σ) can be derived from
Σusing its strongly connected components (see Example 7 ). Left: Strongly connected
components ofGBooleanQDE. Right: The graphGBooleanQDE. Self-loops are not shown.

SinceGQDE(Σ) summarizes the behavior of the corresponding ODE systems,
one could argue that the more realistic trajectories of the Boolean system are to
be found inGasync(f)

/

φfrather than the ASTG.

The nodes of the graphGasync(f)

/

φfcan be interpreted as the slope signs of
the component trajectories in the regulatory networks directly without using the
ASTG at all. This addresses one of the difficulties often occurring when informing
a Boolean model with experimental data, where it is often not at all obvious what
discretization thresholds and basal values to assign when processing quantitative
data. Here, considering just changes in the measurements, e.g., when exploiting
time series data, allows the direct comparison with the ESTG resp.Gasync(f)

/

φf.

6 Conclusion

Motivated by the theory of QDEs, we presented the notion of an ensemble state
transition graphGBooleanQDE (Σ) for a family of monotonic Boolean functions with
coinciding global interaction graphs without negative loops. Every asynchro-
nous state transitions graph for a functionfin the ensemble can be mapped by
a graph homomorphism via their natural quotient graphGasync(f)

/

finto the
graphGBooleanQDE (Σ). Consequently, analysis of the ensemble graph yields informa-
tion valid across all ASTGs in the ensemble, in particular, on trap sets, no-return
sets and reachability properties in general. This may be exploitable when looking
for control strategies for model ensembles, e.g., for identifying knock-out candi-
dates in the interaction graph that assure certain reachability properties in the
ASTGs of the corresponding functions as has been done for differently defined
ensembles [ 5 ].