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Table 2 Literature related to the Hill equation

From: Input-output relations in biological systems: measurement, information and the Hill equation

The Hill equation or related expressions form a key part of analysis in many areas of chemistry, systems biology and pharmacology. Each subdiscipline has its own set of isolated conceptual perspectives and mutual citation islands. Here, I list a few publications scattered over that landscape. My haphazard sample provides a sense of the disconnected nature of the topic. I conclude from this sample that the general form of input-output relations is widely recognized as an important problem and that no unified conceptual approach exists.
Cornish-Bowden’s [7] text on enzyme kinetics frequently uses the Hill coefficient to summarize the relation between input concentrations and the rate of transformation to outputs. That text applies both of the two main approaches. First, the Hill equation simply provides a description of how changes in input affect output independently of the underlying mechanism. Second, numerous specific models attempt to relate particular mechanisms to observed Hill coefficients. Zhang et al. [2] provide an excellent, concise summary of specific biochemical mechanisms, including some suggested connections to complex cellular phenotypes.
Examples of the systems biology perspective include Kim & Ferrell [15], Ferrell [16], Cohen-Saidon et al. [17], Goentoro & Kirschner [18] and Goentoro et al. [19]. Alon’s [20] leading text in systems biology discusses the importance of the Hill equation pattern, but only considers the explicit classical chemical mechanism of multiple binding. Those studies share the view that specific input-output patterns require specific underlying mechanisms as explanations. In pharmacology, the Hill equation provides the main approach for describing dose-response patterns. Often, the Hill equation is used as a model to fit the data independently of mechanism. That descriptive approach probably follows from the fact that many complex and unknown factors influence the relation between dose and response. Alternatively, some analyses focus on the key aspect of receptor-ligand binding in the response to particular drugs. Reviews from this area include DeLean et al. [21], Weiss [22], Rang [23] and Bindslev [24]. Related approaches arise in the analysis of basic physiology [25].
Other approaches consider input-output responses in relation to aggregation of underlying heterogeneity, statistical mechanics or aspects of information. Examples include Hoffman & Goldberg [26], Getz & Lansky [27], Kolch et al. [28], Tkačik & Walczak [29] and Marzen et al. [30].
Departures from the mass-action assumption of Michaelis-Menten kinetics can explain the emergence of Hill equation input-output relations [31, 32]. Many studies analyze the kinetics of diffusion-limited departures from mass action without making an explicit connection to the Hill equation [3335]. Modeling approaches in other disciplines also consider the same problem of departures from spatial uniformity [3638]
Studies often use the Hill equation or similar assumptions to describe the shapes of input-output functions when building models of biochemical circuits [13, 39]. Those studies do not make any mechanistic assumptions about the underlying cause of the Hill equation pattern. Rather, in order to build a model circuit for regulatory control, one needs to make some assumption about input-output relations.