Database of Dynamic Signatures Generated 301dynamics across hundreds of networks with data allowing rigorous exploration
of the space of networks. To illustrate this aspect, in Sect. 3 we examine 4994
networks that are a perturbations of a transcriptional network underlying cell
cycle progression in yeast. We evaluate each network by the prevalence of stable
oscillatory behavior in the parameter space and doing so we find those networks
that most robustly exhibit oscillatory behavior.
2 Database for Dynamics
The current state of modeling gene network dynamics is characterized by a
trade-off between the model’s ability to quantitatively match the experimental
data, and the need for a large number of kinetic parameters to parameterize the
model [ 1 – 3 ]. A popular modeling approach uses Boolean networks, where each
protein, ligand or mRNA is assumed to have two states (ON and OFF), and
the discrete time evolution of the states is based on logic-like update functions
[ 4 – 6 ]. The highly constrained character of the states and the update rules allows
relatively easy parameterization of the model from data, but it limits the power
of generalization and typically results in a poor quantitative match with data.
In contrast, properly parameterized ordinary differential equation models can
provide a good quantitative match and are easily generalized [ 7 , 8 ], but we lack
first principle methods to select proper nonlinearities, and the parameters are
usually poorly constrained, or unknown.
Our approach is derived from Conley theory [ 9 – 11 ] and the computations
we perform allow us to identify trapping regions. As a consequence we are able
to combinatorialize the approximation of the dynamics and the parameter rep-
resentation of the system, while retaining the capabilities of providing rigorous
descriptions of the dynamics without explicit knowledge of the nonlinearities [ 12 ].
Furthermore, we obtain computational efficiencies similar to those of Boolean
nets while preserving the quantitative richness of ODEs.
It should be noted that there are a variety of techniques that have been
developed and implemented that are similar in spirit, but vary in focus and
detail. A complete review is beyond the scope of this note, but we remark on
the following examples. The focus of [ 13 ] is on minimizing models that exhibit
a particular transition system over a finite set of states. The goals of [ 14 ]a
similar in spirit, but make use of piecewise linear nonlinearities. As a consequence
the decomposition of parameter space is more involved and dealt with via a
hierarchical decomposition [ 15 ]. Perhaps closest to our approach in the context
of this work is [ 16 ] in which the dynamics and parameterization makes use of [ 17 ].
With regard to [ 17 ] our approach allows for broader state space decompositions
and less restrictions on parameters. However, [ 16 ] focuses on detailed matching
of dynamics to time series and thereby providing more powerful tools for model
and parameter rejection.