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Hill in Hell 10CHAPTER Hill in Hell Chapter 10 begins with an account of how and why the 10.1.1. Start of De-ri-va-tion of Load Hill equation was introduced. The purpose of ‘the Hill’ From around 1850 on, somewhat different descriptions was if not to explain, then at least to describe the of various dose-response relations were formulated by aberrant behavior of dose-response relations and ease chemists/physicists (Wilhelmy 1850; Biot 1860; van’t calculations. In Sub-chapter 10.2, I illustrate and Hoff 1877; Guldberg & Waage 1967), physiologists discuss a frequently observed failure of linearity for working on for instance hemoglobin-oxygen binding experimental data in the so-called Hill plot Section (Hufner 1890; Bohr 1904a), and biochemists working 10.2.6. In Sub-chapter 10.3, there is an analysis and with enzymes (Brown 1902; Brown & Glendinning 1902; critique of what has been termed the ‘logistic’ Hill Henri 1903). equation. The ‘logistic’ Hill is rendered perspective Biot, a French physicist (1776Á1862), first realized against the original logistic equation. An explanation is that chemical reactions in solution follow a load-type given of how the ‘logistic’ Hill equation is a mere semi- rectangular hyperbolic function (Fig. 10.2) (Biot 1860, log presentation-possibility of data when it is inserted pp. 237Á243 and Plate II). He probably reached his pre- into the Hill scheme. The chapter concludes with a list Hill/pre-Langmuir load-equation on reading Wilhelmy’s of facts about the use of the Hill formulation (Sub- quantitative insight on chemical reactions (1850). As chapter 10.4). Finally, the reader is taken on a brief early as 1884, Hu¨fner also described binding of carbon- tour through the history and use of the term logistic in monoxide to hemoglobin by a load-type equation data analyses, summarized in a Sub-chapter, Appendix (Hufner 1884) (Fig. 10.3). 10.A, including statistical terms such as ‘logistic regres- After 1900, the dose-response relation in the form of sion’ and ‘polytomous logistic regression’. the load started to reappear in works by Ostwald (1902), Detailed descriptions of statistical theories, methods, Henri (1903), Hill (1909), and Micahelis-Menten and transformations can be found in various texts (see (1913). Meanwhile, at equilibrium, the hyperbolic load Sub-chapter 10.A). However, my purpose here is to reaction theory is usually referred to as the ‘isotherms of illustrate in a superior way the terms logistic and Langmuir’. This is due to Langmuir’s explicit theore- ‘logistic’, and thus maybe even resolve some of the tical formulation of the load and his confirmation of obfuscation about their use. the load by its experimental corroboration (Langmuir This chapter presents non-mechanistic models. 1918). Mechanistic models are detailed in Chapter 7. Here I shall refer to the ‘hyperbolic’ interaction both at equilibrium and at steady-state as the law of adsorption-desorption (load).1 The law is based on the 10.1. Deviations from the Law-of- fact that binding sites are limited (see Sections 1.1.2 adsorption-and-desorption and 1.1.8). Dose-response relations that follow the simple law-of- adsorption-desorption (load) give a plot that is a fraction of a rectangular hyperbolic curve (Fig. 1.3). In 1 Since Henri (1903) and Micahelis-Menten (1913) formulated steady- a semi-log plot, this fraction of the hyperbolic curve is an state expressions of the load, while Biot (1860), Hu¨fner (1884, 1890), Hill (1909), and Langmuir (1918) formulated equilibrium equations for S-shaped and symmetric curve. An example is the dotted the load, we may divide the law-of-adsorption-desorption into two line curves in Fig. 10.1AB (see also Fig. 8.1AB). types: an s-load (HMM-theory) and an e-load (BHHL-theory). # 2008 N Bindslev. This book and all matter and items published therein are distributed under the terms of the Creative Commons Attribution-Noncommercial 3.0 257 Unported License (http://creativecommons.org/licenses/by-nc/3.0/), permiting all non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited. DOI: 10.3402/bindslev.2008.14 258 Part III: Test of Tools for Data Analysis A B 100 100 load / lin-lin load / semi-log 80 80 60 60 40 40 20 20 fractional response (%) fractional response (%) 0 0 0 2 4 6 8 10 -3 -2 -1 0 1 2 3 [agonist] (rel-Ks) log [agonist] (rel-Ks) CD 100 100 Hill / lin-lin Hill / semi-log 80 80 60 60 40 40 20 20 fractional response (%) fractional response (%) 0 0 0 2 4 6 8 10 -3 -2 -1 0 1 2 3 [agonist] (rel-Ks) log [agonist] (rel-Ks) Figure 10.1. The load and modified Hill equations in linear and semi-log plots. (A) LinearÁlinear plot of load. (B) Semi-log (‘logistic’) plot of load. (C) LinearÁlinear plot of Hill. (D) Semi-log (‘logistic’) plot of Hill. The five Hill plots in (C) and (D) are generated by the modified Hill equation with Ymax 100, Ks 1 and nH changing in five steps with the values 0.3333 (*), 0.6667, 1.0, 1.3333, and 1.6667 (Á××Á). Hill coefficient1.0 generates the plots in panel A and B. 10.1.2. Start of De-vi-a-tions from Load. K × xn y ; (10:1) Formulation of Hill’s Equation 1 K × xn At the turn of the last century, it had already been in which y is the observed saturation (fully-oxygen- observed that an experimental dose-response relation saturated hemoglobin aggregates) as a fraction of all deviated from the simple hyperbolic rectangular dose- hemoglobin conformations, bound or unbound, ‘x’ is response relation. Synagics for the hemoglobin-O2 the free ligand concentration (oxygen tension), and K binding as a function of O2 tension did not follow the in this equation has no physical meaning, but relates to simple load (Fig. 10.4) (Bohr 1904a,b; Bohr et al. 1904). its origin, an equilibrium association constant. The power Instead, the Hb-O2 saturation against O2 tension gave a n is known as Hill’s coefficient. When Hill’s coefficient is sigmoidal dose-response relation, equal to positive co- 1, Hill’s mathematical expression is identical to the operativity. A theory for this behavior that is now Langmuirian mechanistic expression for adsorption of accepted was formulated in a preliminary equation by gases, the load, Eq. 1.11. Thus, Hill just elevated the Adair (1925).2 In 1910, Hill had also formulated an concentration (oxygen tension) to an arbitrary power n Adair-type equation (see Eq. 10.6), but due to the (or nH for ‘the power of Hill’). tedious work of calculating all the constants, Hill In the linear scaled Cartesian coordinate system, advocated for a simplified version, namely Eq. 10.1 dose-responses that follow the Hill equation with a Hill given here and henceforth referred to as the Hill slope"1 (Fig. 10.1C) give plots that deviate from a equation (Hill 1910). His equation reads: hyperbolic curve (dotted curves in Figs. 10.1AC), while these plots in a semi-log diagram are symmetric and S-shaped curves just as the simple load (Figs. 10.1BD, also see Figs. 8.1 and 8.8). As mentioned, employing the Hill equation was only 2 Edsall gives a fine account of this story (Edsall 1980), see also Klotz meant as a quick way to estimate the co-operativity in (2004). experiments without any mechanistic interpretation, as Chapter 10: Hill in Hell 259 Figure 10.2. The origin of load curves. From Biot (1860, Plance II). its inventor also attested to (Hill 1910). He writes ‘My plot is a plot of the logit formulation (see Section object was rather to see whether an equation of this type 10.2.2). can satisfy all the observations, than to base any direct physical meaning on n and K’. Notice in this context, that there are three Hill- 10.1.3. Hill’s Aggregation Theory for Hemoglobin concepts: (1) the Hill equation (Eq. 10.1), (2) the Hill Complexation Crumbles coefficient, -factor, -power, or -slope, nH, and (3) the For complexation of a multi-unit hemoglobin, Hill Hill plot (see Section 8.2.5 and Fig. 8.8CF). The Hill postulated a so-called ‘aggregation theory’ (Hill 1910). 60 ) 3 40 20 CO-Hb complex (x10 experimental data calculated data 0 0 20 40 60 80 100 3 vc = "active mass of CO" (x 10 ) Figure 10.3. The origin of experimental data plots combined with the load theory. Carbon-monoxide binding to hemoglobin. Á3 Á3 Maximal binding76.993.410 ,EC50 199110 . After Hu¨fner (1884). 260 Part III: Test of Tools for Data Analysis 100 80 60 40 -saturation (%) 2 O 20 CO2 pressure = 40 mm Hg 0 0 20 40 60 80 100 120 140 O2 pressure (mm Hg) Figure 10.4. The origin of load-deviant behavior. Demonstration of a genuine sigmoidal dose-response relationship. Oxygen binding to hemoglobin in linearÁlinear dose-response plot. The relationship is determined at 40 mmHg CO2 with a maximal O2-saturation at 10492%, EC50 3191 mmHg O2 and a Hill coefficient nH equal to 2.1790.14. Redrawn from Bohr et al. (1904). In this theory, hemoglobin units, each with one atom of Physiology’. Herein Hill commented: ‘There can, how- n iron, can aggregate when O2 is bound. Thus, for all ever, be little doubt now that my equation yKx /(1 hemoglobin units, the actual binding site in a single and Kxn), based on the aggregation theory, is wrong, or at free hemoglobin unit associates or dissociates one least a serious over-simplification’.
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