In the early 1970s, Rene' Thomas, Leon Glass, and Stuart Kauffman began
their pioneering efforts in exploring the possibility of modelling what
was then called "Genetic Control Circuits" (Thomas) and "Biochemical
Control Networks" (Glass and Kauffman) by using concepts and ideas from
mathematical logic. Combining these ideas with earlier ideas from Monod
and others on allostery and cooperativity, leading to a sigmoidal rate
dependence of key metabolites, they proposed discrete, Boolean models
of gene regulation (Thomas) and ordinary differential equation models
with switch-like interaction terms, i.e. step functions or sigmoid
functions (Glass and Kauffman).

From these early attempts, phenomenologic frameworks for the modelling
of Gene Regulatory Networks (GRNs) have been developed, based on a few
fundamental premises: (i) genes are controlled by transcription factors
(TFs) which combine into logical input functions which can be described
by Boolean logic; (ii) the effect of a transcription factor on the
transcription rate of a gene (the response function) can be described
by a sigmoidal function of its concentration with a pronounced
threshold behaviour or by a Heaviside step function (binary response);
(iii) this can be modelled in a discrete way in which transcription
factors are either absent of present, and proteins are either
transcribed or not, or in a continuous way by means of ordinary
differential equations; (iv) proteins act as transcription factors, so
that networks become closed by feedback loops; (v) posttranscriptional,
translational and posttranslational regulation, transport processes,
metabolic processes etc. can be phenomenologically encompassed by the
sigmoidal or binary response functions.

I will show how these basic premises naturally lead to GRNs with steep
sigmoid or binary interactions, how such networks can be analysed
mathematically, and present some generic results. If time allows, I
will also discuss to which extent recent experimental results confirm
these basic premises and justify the GRN model framework.