The Hill function is frequently used to infer the degree of cooperativity of the chemical reaction in which ligand molecules bind to a protein [1]. Often, the binding of a ligand increases the association rate for the binding of the next ligand. Such reactions are said to be (positively) cooperative. There are examples of cooperative reactions in cell biology. The classical example is the binding of oxygen molecules by hemoglobin [1]. Other perhaps less well-known examples would be parts of the Notch signaling and 30 S ribosome assembly processes [2], as well as the assembly of cholesterol-sphingomyelin complexes [3]. Also, the noise characteristics of various ligand binding reactions were studied theoretically in [4] and some of the experimental systems could be classified as cooperative reactions. A cooperative reaction builds a final complex successively. If strong cooperativity is present, the dynamics of the system can be studied using Hill's model, at least to a first approximation [5].

Hill's model is a grossly simplified version of reality. The model is constructed by assuming that binding and unbinding of ligands occur in

*one* step as

${C}_{0}+hA\leftarrow {C}_{h}$

(1)

where *C*_{0} denotes a protein that binds ligands *A*, and *C*_{
h
} is the ligand-protein complex. The Hill coefficient *h* describes the number of binding sites on the protein. Both the forward and the back reactions are allowed.

Strictly speaking, the Hill coefficient in Hill's model (1) is a stoichiometry coefficient and should be an integer number larger than zero. However, in the calculations that follow, *h* will be allowed non-integer values. Thus in the context of this work the Hill model should be understood more from a model average perspective, where the Hill coefficient is an effective parameter.

An important quantity related to Hill's model is the fraction of the proteins that are bound

$\phi \equiv \frac{{c}_{h}}{{c}_{0}+{c}_{h}}$

(2)

In particular, the dependence of

*φ* on the amount of unbound ligand in the system

*a* is of considerable interest, and is referred to as a dose-response curve. A function frequently used to fit a dose-response curve is the expression derived by Hill, the so-called Hill function, given by

${\phi}_{H}(a)=\frac{\frac{{a}^{h}}{K}}{1+\frac{{a}^{h}}{K}}$

(3)

where *c*_{0}, *c*_{
h
}, *a* are used to denote the amounts of unbound proteins, bound proteins, and free ligands, respectively. Please note that the Hill function is only parameterized by *K* and *h*. When fitting experimental data to extract *K* and *h*, it is useful to allow *h* to be a real number. Also, the Hill function is used frequently in theoretical studies to model cooperativity effects.

In general, *c*_{0}, *c*_{
h
} and *a* can denote average particle numbers, particle concentrations or partial pressures. It really depends on the types of experiments one wishes to describe. The dissociation constant is essentially controlled by the ratio of the forward and the backward reaction rates.

The original Hill's model is unrealistic since a truly multiparticle reaction with a high Hill's coefficient would be a very unlikely reaction event. The probability that all required ligand molecules meet at the right place, at the right time, is very small. The model was already criticised by Hill himself [6, 7]. Subsequently, more realistic models were suggested in a series of studies: Adair [8]; Monod, Wyman, Changeux [9]; and Koshland, Nemethy, Filmer [10]. The difference between the models was critically investigated on the mean field level in [5], which confirmed Hill's original claim that the Hill equation can be used in a case of strong cooperativity when intermediate states are short-lived. For a reaction set that appears strongly cooperative as in (1), the Hill coefficient provides a rough measure of the cooperativity degree of the reaction.

Despite the problems discussed above, the use of Hill's model has some merits [1], and the Hill equation is used frequently in many fields as discussed in review article [11]. Accordingly, in this work, Hill's model will be taken as a basic standard for describing multiparticle (cooperative) reactions. The validity of the model has been extensively investigated previously. The conditions for safe usage of Hill's model can be easily verified.

From now on, it will be assumed that the Hill model under investigation is a valid alias for a more complicated multiparticle-like reaction scheme. The focus will be on investigating the correctness of the resulting Hill's function *φ*_{
H
}(*a*) in a low particle number limit. The ultimate goal of this study is to investigate in what ways the effects of the noise related to the low copy numbers affect the form of the dose-response curve predicted by Hill. Please note that such a goal enforces consideration of a closed system. For an open system, where injection and the decay of particles are allowed, one cannot use the Hill function at all.