48 J. Barnat et al.
3.1 Discretisation of ODE Models
In this section, we briefly describe the format of the ODE models used and the
subsequent procedures of approximation and abstraction that allow us to apply
the method defined in Sect. 2.
Model.We considerP ⊆Rm≥ 0 as thecontinuous parameter valuation space
of dimensionm.Abiological modelMis given as a system of ODEs of the
form ̇x=f(x, μ) wherex=(x 1 ,...,xn)∈Rn≥ 0 is a vector of variables,μ=
(μ 1 ,...,μm) is a vector of parameters such thatμis evaluated inP,andf=
(f 1 ,...,fn) is a vector where each component is a function constructed as a
sum of reaction rates where every sum member is an affine or bi-linear function
ofx, or a sigmoidal function ofx. An important requirement is that eachfi
must be affine inμand there exist nok, lsuch thatk =landμk,μlboth
occur in some fi. Moreover, we assume that every variablexihas a bound
denoted byxmaxi. In consequence, we require for allp∈Pthat no trajectory
can exit the bounds. Formally,∀p∈P,∀i∈{ 1 , ..., n}:(xi=0⇒fi(x, p)>
0)∧(xi=maxi⇒fi(x, p)<0). Similarly to [ 2 ], we assumePincludesalmost
allparameter valuations excluding singular cases for which some trajectory can
slide along a threshold plane. In particular, any parameter valuationpfor which
some component off(x, p) can be zero on a boundary of some rectangle (as
defined below) is not allowed. In consequence, a fixed point can appear only in
a rectangle interior.
The restriction imposed onfcovers mass action kinetics with stoichiometric
coefficients not greater than one and any sigmoidal kinetics such as all significant
variants of enzyme or Hill kinetics. Parameters must be independent and cannot
appear in an exponent or a denominator of the kinetic function employed.
Approximation.To proceed with discretisation, the model ̇x=f(x, μ) has to
satisfy the criterion that everyfiis piecewise multi-affine (PMA) inx. To trans-
form the model into this form, we employ the approach defined in [ 16 ]. In par-
ticular, each sigmoidal function member infiis approximated with an optimal
sequence of piecewise affine ramp functions. In this procedure, a finite number
of thresholds is introduced for every component ofx. The crucial factor of the
approximation error is the number of piecewise affine segments. Though there is
not yet a method that would somehow propagate the information on approxima-
tion error into the trajectories of the resulting PMA model, it has been shown
on several case studies that the approximation does affect the system’s vector
field only negligibly [ 12 , 16 ].
Abstraction.We employ the rectangular abstraction [ 3 , 16 ]. We assume that we
are given a set of thresholds{θi 1 ,...,θini}for each variablexisatisfyingθi 1 <θ 2 i<
···<θini.Eachfiis assumed to be multi-affine on eachn-dimensional interval
[θ^1 j 1 ,θ^1 j 1 +1]×···×[θjnn,θjnn+1]. We call these intervals rectangles. Each rectangle
is uniquely identified via ann-tuple of numbers:R(j 1 ,...,jn)=[θj^11 ,θj^11 +1]×
··· ×[θnjn,θjnn+1], where the range of eachjiis{ 1 ,...,ni− 1 }. We also define
VR(j 1 ,...,jn) to be the set of all vertices ofR(j 1 ,...,jn).