Database of Dynamic Signatures Generated 303constant are semi-algebraic sets, and that the decomposition of parameter space
into regions of constancy is explicitly computable. This decomposition of high
dimensional parameter space is codified via aParameter GraphPG. The nodes
ofPGcorrespond to regions of parameter space with identicalF, and edges
correspond to co-dimension 1 boundaries between the regions which capture the
geometry of the decomposition.
Storing the entire collection of multivalued maps over all parameter ranges
is prohibitive. We use the concept of aMorse graph to extract the essential
recurrent dynamics information from the combinatorial mapF:X⇒X.Weuse
linear time algorithms to identify maximal sets of nodes inXmutually related
by directed paths from one node to another. These maximal sets of nodes are
calledMorse setsofF and are identified by nodes in the Morse graph. The
directed edges in the Morse graph indicate the reachability from one Morse set
to another via paths inF. Thus, minimal nodes in the Morse graph represent
stable or attracting dynamics (see Fig. 1 (c), (e), (g), (i), (k)).
The parameter graph along with the associated Morse graphs provides an
extremely condensed representation of the global dynamics over all of parameter
space. We augment the Morse graph with labels on the Morse sets that describe
the recurrent dynamics they represent. In our graphical output, described in
detail in the Supplement, we denote by FP a Morse set that corresponds to a
stable equilibrium (Fixed Point). FP OFF means that every protein concentra-
tion is below its lowest threshold, and FP ON means that every variable is above
its lowest threshold. We use FC (Full Cycle) to denote a Morse set in which each
coordinate crosses at least one threshold (this must happen an even number of
times).
The output of the DSGRN software is a SQL database, which allows the
user to query for particular types of dynamics and/or for the dynamics in dif-
ferent regions of parameter space. Although the DSGRN database is computed
using switching systems ( 2 ), the Morse graph structures are valid for smooth
systems taking the form ( 1 ) under the assumption thatfiandΛiare sufficiently
close. Furthermore, explicit expressions for what sufficiently close means can be
obtained [ 12 ]. Thus the DSGRN framework allows us to make mathematically
rigorous statements about the global dynamics of regulatory networks even if
the nonlinearities are not explicitly known.
3 Examples
(1) Characterizing the dynamics of a network over global parameter
space.To illustrate the range of dynamics being detected by DSGRN, we show
five STGs and the associated Morse graphs from the two-dimensional network
showninFig. 1 (a). These STGs and associated Morse graphs arise from five of
the 120 regions in parameter space for this system.
The network graph in Fig. 1 (a) implies thatX has two thresholds andY
has one, decomposing phase space into six domains. We label each domain by a
pair of integers denoting the locations ofXandYcompared to their thresholds.