23.4 Inference of Regulatory Networks

317

binding to the promoter sequence and hence initiates transcription of the lactose-

metabolizing genes. Note that a certain basal level of production of the lactose-

metabolizing proteins is necessary.

A relatively simple genetic toggle switch has also been analysed cybergeneti-

cally. ¹⁴ This is a paradigm for the bistable systems that are ubiquitous in cellu-

lar decision-making, which underlies differentiation pathways and so forth. Inter-

estingly, periodic external perturbations can maintain an “undecided” state; such

dynamic stabilization may be ubiquitous in biology. The phenomena uncovered in

the toggle switch form part of a wider phenomenology that can be modelled with

simple first-order nonlinear differential–delay equations of the kind

StartFraction d x Over d t EndFraction equals lamda minus gamma x^dx

dt ^{= λ −γx}

(23.3)

where xx is a variable of interest, tt is time and lamdaλ and gammaγ are positive constants giv-

ing, respectively, xx’s production and decay rates. Applied to physiology, lamdaλ and gammaγ

may not actually be constant but depend on x Subscript tauxτ, the value of xx at some earlier time

t minus taut −τ. Solutions to the equation then show a richly diverse behaviour, encompassing

aperiodic (“chaotic”) solutions. ¹⁵

The lactose inducible operon is an example in which the regulatory protein acts

as a negative (repressor) element in the control of gene expression; in other exam-

ples (e.g., the arabinose inducible operon) the protein acts as a positive (activator)

element. Savageau (1974) noticed that the repressor type is correlated with the fre-

quent presence of the system’s substrate, and the activator type is correlated with

infrequent presence of the substrate in the natural environment. This arrangement

serves to minimize the resources required to achieve regulation.

Problem. Construct a Boolean model of the lac operon. Hint: Start with a very simple

model and progressively add features. Can the effects of noise and delays in signal

transmission be incorporated?

Problem. Explore the behaviour of solutions to Eq. (23.3) where lamdaλ is a nonlinear

function of xx. ¹⁶

23.4 Inference of Regulatory Networks

Given the experimental microarray data consisting ofgg gene transcripts measured at

tt successive epochs, one seeks to find how expression is controlled by a relatively

14 Lugagne et al. (2017).

15 Mackey and Glass (1977).

16 The paper by Mackey and Glass (loc. cit.) can be consulted for physiological context. For a more

general exposition of how feedback delay can lead to chaos, see Pippard (1985).