The Shea-Ackers formalism

The Shea-Ackers formalism is a mathematical model that describes the activity of a regulatory sequence as a function of the concentrations of activating and inhibiting factors²⁰. Assuming that the regulated gene is active when the binding sequence is occupied by an activator (∈ act), and inactive when it is unoccupied or in the presence of an inhibitor (∈ inh), the formalism borrows techniques from statistical mechanics to compute the average activity at the site assuming thermodynamic equilibrium:

where kT,X are dissociation constants describing the interaction strength between factor T and the regulatory sequence of X, and hX,T is a non-linearity parameter representing cooperativity between multiple factors of the same type.

We use this equation to build a system of ordinary differential equations that describes the time evolution of concentrations of the relevant regulators X. Assume that the expression rate of X is proportional to the activity of the regulatory site (defined by Eq. ¹) as the average fraction of time the site is occupied by an activator). Assume also that each gene product undergoes exponential decay, that is, its concentration decreases at a rate proportional to itself. Then the change in gene product concentration over time can be described as

where αX and δX are scaling coefficients for the rate of transcription and decay, respectively (decay half-life is ln2/δX).

This approach requires that Eq. (²), for each node X in the network, depends only on other variables that are explicitly modelled in the GRN. In practice, most genes are influenced by other genes that are not present in the network because they are not considered relevant in the context of the GRN model. The influence of these interactions on the master equation of the quantitative model (Eq. ²) can be greatly simplified with a change of variables.

Considering Eq. (¹), each constitutive inhibitor and activator T will be represented by a term [T]/kT,XhT,X in the rate of expression αXSAX. Assuming that the concentration T is constant in the relevant conditions for the model, this term can be simplified to the constant value βT,X=[T]/kT,XhT,X. Furthermore, the combined influence of all constitutive activators is also constant βact,X=∑T∈actβT,X, and for the constitutive inhibitors we have that their sum is equal to the constant βinh,X=∑T∈inhβT,X. This leads to the equation

where act′ and inh′ are the sets of non-constitutive activators and inhibitors. By dividing the numerator and denominator by 1+βinh,X and defining βX=βact,X/1+βinh,X, and kT,X′=kT,X1+βinh,XhT,X, the parameter set of the equation can be further reduced:

For nodes which are not activated by non-constitutive factors, the background parameter βX is also redundant. By dividing the numerator and denominator by 1+βX and defining αX′=αXβX/1+βX and kT,X″=kT,X′1+βXhT,X we obtain the expression

Henceforth we will not write the prime ′ on these transformed variables.

The Shea–Ackers formalism is derived with the assumption that all activations and inhibitions are physically realised by binding competition on a gene's regulatory sequence. The rate equations above assume that there is transcription at a constant rate whenever an activator is bound, thus ignoring differences in the rate at which transcription factors recruit the transcription machinery. Overlooking the complexity of different regulatory mechanisms allows for a mathematical approximation that can describe a wide range of dynamic behaviours that are realistic for different gene interaction mechanisms while using only a small number of parameters.