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23

Regulatory Networks

A Boolean network is a mathematical model used to represent the interactions

between components in a system; it is a directed graph in which each node repre-

sents a component, and the edges between nodes represent the interactions between

components. The states of the nodes in the network can be either true or false. Logic

rules governing the evolution of the network’s state structure are generally simple

(Chap. 12). Somewhat more sophisticated are Bayesian networks, a type of proba-

bilistic graphical model that uses Bayesian inference for probability computations

(Chap. 9). The networks represent a set of variables and their conditional dependen-

cies via a directed acyclic graph (DAG). Each node in the graph represents a variable,

and each directed edge represents a dependency between two variables. The nodes

in the graph can be used to compute probabilities based on the values of other vari-

ables in the network. The other great task involving networks is the inference of their

architecture from experiment data. Here it should be borne in mind that a network

encodes only pairwise correlations of node state variables, and ignores higher-order

correlations. For the latter, tensor analysis techniques can be brought to bear. ³ It is

clear that the living cell (and a fortiori the multicellular organism) comprises a great

variety of different components that must somehow be integrated into a functional

whole. The framework of this integration is directive correlation (Fig. 3.2) and it may

be considered as a problem of regulation.

The problem of defining a system by delineating its boundary has already been

raised (Sect. 12.1.4). In some cases, it might be meaningful to include multiple

organisms within the system being regulated, as in, for example, plant–microbe

interactions. ⁴

To recapitulate, regulation was introduced in Chap. 3 as a means of ensuring that

the system’s output remained within its essential variables while its environment was

undergoing change—in other words, as one of the mechanisms of adaptation (which

is itself a special case of directive correlation). We are perhaps most familiar with

regulation whereby the volition of the regulator is transformed into direct action—

such as pressing the accelerator pedal of a motor car. In a steam locomotive, the lever

with equivalent function is actually called the regulator. Stationary steam engines

providing mechanical power to a factory or mine are typically required to run at a

constant speed and are equipped with a “governor” (a device mounted on a spindle

turned by the engine that increases its radius with increasing angular velocity of the

spindle, due to centrifugal force and, via a system of cranks and levers, directly closes

a valve shutting off steam to the driving cylinders) that automatically regulates the

speed (this is another example of the “regulation by error” described in Sect. 3.2).

Some degree of quantification of a regulatory network can be gained by looking

at how the network elements interact with each other, and how the elements can be

tuned to optimize a desired outcome. The trade-offs between them can be quantified

by looking at the cost of changing the network structure, in terms of the amount

of energy it takes to maintain the network. Trade-offs between different forms of

3 Yahyanejad et al. (2019).

4 Baker et al. (1997). This work, incidentally, also demonstrates how a rational understanding of a

regulatory network can lead to practical guidance for designing crop protection strategies.