23.2 Network Modelling
315
lated protein, measured using microarrays (Sect. 18.1). To a first approximation, it
is useful to represent expression as “1” and the absence of expression as “0”. Alter-
natively, since many proteins are nearly always expressed to some extent, increased
transcription–translation (“upregulation”) can be represented as “1”, and decreased
transcription–translation (“downregulation”) as “0”. The system can then be analysed
as a Boolean network.
Genes observed to be close to each other in expression space are likely to be
controlled by the same activator. Each gene can have its own promoter sequence;
co-expression is then achieved by the transcription factor binding to a multiplicity
of sites. Indeed, given that several factors may have to bind simultaneously to the
TFBS region in order to modulate expression, control appears to be most commonly
of the “many to many” variety, as anticipated many years ago by Sewall Wright.
Since genes code for proteins, which, in turn, control the expression of other genes,
the network is potentially extremely interconnected and heterarchical.
Each gene will have its experimentally determined expression profile, and once
these data are available, the genes can be clustered (Sect. 13.2.1) or arranged into a
hierarchy (Sect. 17.7). The principal task, however, is to deduce the state structure
from such data.
It is a very useful simplification to consider the model networks to be Boolean (i.e.,
genes are switched either on or off). To give a flavour of the approach, consider an
imaginary mini-network in which gene A activates the expression of B, B activates A
and C, and C inhibits A. 11 This is just an abbreviated way of saying that the translated
transcript of A binds to the promoter sequence of B and activates transcription of
B, and so on. Hence, A, B, and C form a network, which can be represented by a
diagram of immediate effects (cf. Fig. 3.1) or as a Boolean weight matrix:
StartLayout 1st Row 1st Column Blank 2nd Column normal upper A 3rd Column normal upper B 4th Column normal upper C 2nd Row 1st Column normal upper A 2nd Column 0 3rd Column 1 4th Column negative 1 3rd Row 1st Column normal upper B 2nd Column 1 3rd Column 0 4th Column 0 4th Row 1st Column normal upper C 2nd Column 0 3rd Column 1 4th Column 0 EndLayout period
A B C
A 0 1 −1
B 1 0
0
C 0 1
0
.
(23.1)
Reading from top to bottom gives the cybernetic formalization; reading horizontally
gives the Boolean rules: A equals= B NOT C, B equals= A, C equals= B. Matrix (23.1) can be
transformed to produce a stochastic matrix (a probabilistic Boolean network) and the
evolution of transcription given by a Markov chain. Different external circumstances
engendering different metabolic pathways can be represented by hidden Markov
models (Sect. 17.5.2). Noise can be added in the form of a random fluctuation term.
Alternatively, the system can be modelled as a neural net in which the evolution of
the expression levela Subscript iai (i.e., the number of copies produced) of theiith protein in time
tauτ is
tau StartFraction d a Subscript i Baseline Over d t EndFraction equals script upper F Subscript i Baseline left parenthesis sigma summation Underscript j Endscripts w Subscript i j Baseline a Subscript j Baseline minus x Subscript i Baseline right parenthesis minus a Subscript i Baseline commaτ dai
dt = Fi
(Σ
j
wi ja j −xi
)
−ai ,
(23.2)
11 After Vohradský (2001).