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 (1776 1862), 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. 237 243 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. 10.1A B (see also Fig. 8.1A B). 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) Linear linear plot of load. (B) Semi-log (‘logistic’) plot of load. (C) Linear linear 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 ( ×× ). Hill coefficient 1.0 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. 10.1A C), while these plots in a semi-log diagram are symmetric and S-shaped curves just as the simple load (Figs. 10.1B D, 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. 8.8C F). 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. 3 3 Maximal binding 76.993.4 10 ,EC50 1991 10 . 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 linear linear 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 y Kx /(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’. He continues: ‘The molecule of O2 with identical association or dissociation theory that the earth is a sphere is also wrong, but for constants regardless of the status of neighboring bind- practical purposes it is often convenient’. ing sites in hemoglobin aggregates. In the physical The story continues. Hill noted with surprise that his model of Hill’s aggregation there is an average number equation was still in use and the Hill coefficient of O2-complexed hemoglobin molecules equal to nH, maintained as a quick-quantifier of load-deviation. He which is therefore also the average number of bound O2 writes ‘The equation originally deduced in 1910 from molecules per hemoglobin aggregate. Hill’s equation the aggregation theory had been laid decently to rest in thus describes a coincidence or simultaneity model, the 1920s; its body lay mouldering in the grave, but based on the aggregation of only liganded units. Hill apparently its soul goes marching on’ (Hill 1965, pp. indeed realized that his simple formula was a mathema- 105 106). Currently, it still marches on. tical expression without any physical reality, a short-cut We may conclude that the aggregation theory for hemo- to analyze his aggregation theory with oxygen bound to globin is falsified and the physical meaning of Hill’s hemoglobin complexes (Hill 1910). equation is inappropriate for almost any adsorption- In the mid-1920s, based on experiments, Adair (1925) desorption system due to the coincidence requirement and Svedberg and Fa˚hreus (1926) argued that hemo- and its lack of site interaction. Meanwhile, applying Hill’s globin as such consisted of four subunits with four equation and using its coefficient as a quantifier for separate binding sites for oxygen, a fact that, as deviation from simple concentration-binding or dose- indicated, was generally accepted and soon proven function schemes is convenient. correct. Therefore, Hill’s aggregation theory crumbled. Together, the models by Adair (1925) and Pauling 10.1.4. All-or-none is Often Forgotten (1935), the X-ray work by Perutz and colleagues (1942, 1960; Muirhead & Perutz 1963), and the determination We may suppose that the aggregation theory for of four association constants for oxygen binding to hemoglobin is more likely than the coincidence hemoglobin by Roughton et al. (1955), finally con- resonance-theory to bring four hydrogen atoms together vinced Hill to give up his aggregation theory (Hill 1965, at once to form helium and followed by three helium p. 105). For hemoglobin, four units are bound together atoms to combine into carbon in stardust (Burbridge allowing interaction between their binding sites. et al. 1957; Davies 2006, pp. 151 158). Nonetheless, In 1965, 55 years after suggesting the quick test in even if the coincidence formulated by Hill’s equation is the form of Hill’s equation, Hill catalogued his life a remote possibility found in nature, his theory still as a scientist in a book entitled ‘Trails and Trials in seems to require continued luck. Or does it? Consider Chapter 10: Hill in Hell 261
n n this. Di-merization and multi-merization for functional- K H × [S] H 1 y n : (10:2) ity of only liganded receptors are ceaseless processes for 1 KnH × [S] H 1 1 n the tyrosine kinase receptor superfamily, with the (K × [S]) H insulin receptors, IRs, as a prototype example (de Myets 2004; Stoker 2005; Blanquart et al. 2006; Bublil & In this expression, y is the fractional response or Yarden 2007), and could certainly be described by a fractional binding, K is now an equilibrium association Hillesque aggregation theory in which coincidence is constant, itself raised to the power of a Hill coefficient, inherent. The resonance required for carbon-formation nH assuming all single-step association constants to in stars is here replaced by stability of the insulin-IR be identical, and [S] is the free ligand concentration. complex. On the other hand, subunits of G protein- Eq. 10.2 is a slightly modified form of the original Hill coupled receptors (GPCRs) seem to be aggregated equation (see Hill 1910, 1913). I will refer to Eq. 10.2 as before ligand binding (Ma et al. 2007). the modified Hill equation. In the meantime, the use of Hill’s formulation may be justified for a fast and easy documentation of load- deviation as long as: (1) it is realized that the synagic 10.2.2. The Logit or the Hill Plot model behind Hill’s equation is irrelevant for a host of If you subtract the nominator from the denominator on binding and functional dose-response systems with both sides of the equal sign in Eq. 10.2, i.e., invoking intermediate interaction; and (2) the conclusions about regula detri rule 1b (Section 1.1.7 and Box 1.1) you the Hill equation at the end of this chapter are taken have: into account. y n n Numerous researchers have exposed their data to the [S] H ×K H : (10:3) ‘Hill’-analysis and used the obtained Hill coefficient as a 1 y simple handle to express their data’s deviation from the As no intermediate steps of occupation are allowed in load. Many scientists have probably more or less Eq. 10.3, the fraction on the left of the equation recognized the limitations of Hill’s equation, but most explicitly expresses that the fraction of fully bound likely again, have quickly suppressed and forgotten all receptive sites, y, over fraction of free receptive sites, 1 y, about its tacit assumptions. is equal to the product of ligand concentration and association constant between ligand and receptor, as equated on the right side in Eq. 10.3; and both raised to 10.1.5. The Fickle Use of Hill’s Equation the power of nH. Thus, the ratio of bound over free receptive sites is proportional to the normalized con- What are the conclusions about the use of Hill’s 3 equation today? Hill’s argument for introducing the centration [S]×K raised to a power. On a log log form, Eq. 10.3 is a linear function: Hill equation, as justified due to the cumbersome work in 1910 of calculating dose-responses for systems with y log n log[S] n logK; (10:4) many constants, is of course no longer valid (see Sub- 1 y H H chapter 8.3) (Weiss 1997). The argument for continuing to use the Hill equation as a tool to analyze one’s with a slope of nH and a ‘cut-off’ at the ordinate of nH × experimental data is based on the fact that very often log K. there is too little information on studied systems to The expression ‘log [y/(1 y)]’ is a well-known term, justify a detailed description of the system constants. called the ‘logit’. Compare this with the logit derived in Here Hill’s coefficient is a convenient measure. The Section 8.2.4 (Eqs. 8.9 and 8.13) and Section 8.2.5 (Eqs. schism between convenience and realism has been the 8.17 and 8.21). Plots of Eq. 10.4 are referred to as ‘logit story for Hill’s equation ever since its emergence. plots’ or in synagics also as ‘Hill plots’ (Fig. 8.8) (Hill Therefore, the real problem with a choice of the Hill 1913; Brown & Hill 1923; Adair 1925). See Section 8.2.5 equation is once it works as a silencer of possible and Appendix 10.A for the naming and benefits of the explorative investigations and hence prevents a pene- logit formulation. trating analysis. As pointed out by Cornish-Bowden and Koshland (1975), the ‘logit’ plot in pharmacology is a plot identical to the Hill plot, and formally equivalent to 10.2. Deviations from the Linear Hill Plot Nernst’s plot of redox electrochemistry (Reed & Berk- son 1929, their example 5; Malmstrom 1974). The logit 10.2.1. The Modified Hill Equation plot is also equivalent to the gating transitions of voltage
3 The following equation is nearly identical to Eq. 10.1, [S]×K [S]/Ks is a normalized concentration, where Ks is a with slightly different symbols: dissociation constant. 262 Part III: Test of Tools for Data Analysis dependent ion channels (Yifrach 2004). See Section order to analyze such a system seriously for its equili- 10.3.9 for more on the relationship between the Nernst brium reactions, you may consider formulas derived and and the Hill equations. described for random or ordered sequential binding and In former times, linear transformation of theoretical function in Chapter 6 (Table 6.1 and 6.2) (Tuk & van expressions was the simplest mean of analyzing load- Ostenbruggen 1996; Weiss 1997), and for two-state deviant dose-response data. Levitzki (1978) has given a models presented in Chapter 7. detailed account of the development and usefulness of Hill’s analysis, arguing for the continued use of the Hill plot. For more on the use of the Hill slope as a measure 10.2.4. Hill’s Formulation for Multi-sited of co-operativity, see Hill (1985, pp. 64 66) and Giraldo Hemoglobin and co-workers (Giraldo et al. 2002; Giraldo 2003). In the world of formulae, the term ‘logit’ is sometimes Hill (1910) and Adair (1925) derived a general formula mixed up with the term ‘logistic’. However, the ‘logit’ is for multi-sited receptive units; both different from the Hill equation and with possible site-interaction at bind- not to be confounded with the concepts ‘logistic’, ing. Their type of formula was later expanded according ‘logistic equation’, or ‘logistic function’ (Tables 10.3 to Pascal’s triangle as summarized in Table 6.1, and and 10.4 and Appendix 10.A) (Berkson 1944; Prentice described for the random reaction scheme as the 1976; Collins et al. 1992). Evidence that ‘logit’ is easily ‘reformulated occupancy model’ (Tuk & vanOosten- mixed with ‘logistic’ can be seen in an article on ‘Generalized linear models’ in Wikipedia (updated 2 bruggen 1996), and by Weiss (1997) for both random March 2007), in which a link function as the logit or (independent) and ordered (sequential) reaction ‘logit model’ for logistic regression is mistakenly quoted schemes (see Chapter 6, Figs. 6.13 6.15, for details). Hill (1910) gave the following general formulation: as a logistic function. X r Factually, as formulated in Eq. 10.13, the original n K × x y a × r ; (10:6) logistic equation is the one for growth as a function of r r r 1 Kr × x time and reproduced here from Section 10.3.2: where y is the total fractional occupancy, factor ar is the 1 y : (10:5) maximal fraction of receptive units with a number of ‘r’ 1 exp( k × (t t0)) ligands bound, and Kr is the association constant for Here ‘exp’ means exponentiation of the parenthesis ligand binding with r 1 ligands already bound. The with the base for the natural logarithm, e 2.71828. implications of equating K with subscript ‘r’ allow for co-operativity, as the association constants may vary The logistic equation will be discussed further in Sub- independently. That is, alternating site-interaction chapter 10.3 and Appendix 10.A, as well as the term upon binding. ‘logistic’ which is sometimes just a cover-up for a simple It is interesting to note that even with ten binding sites semi-log re-scaling. in the random binding regime with full occupancy but without site interaction, the fit-determined Hill coeffi- 10.2.3. Non-linear Hill Plots cient never rises above 2.1 (Fig. 6.11) (Weiss 1997). One assumption about linearity of the logit (Hill) plot is that all sites have the same affinity for the tested ligands, 10.2.5. Site-interaction in One and ×××× i.e., association constants K1 K2 K3 Kn. Systems Two-state Models with different association constants between binding sites, which is the most likely, will have curvilinear logit Numerous researchers have equated explicit expressions (Hill) plots when experimental dose-response data are for such multi-sited receptive units with site-interaction drawn according to the logit equation (Eq. 10.4) (see in order to describe observed co-operativity, some in Figs. 8.7C F and 8.8C F for linear examples). At the one-state models (Hill 1910; Adair 1925; Pauling 1935; extreme ends of such a non-linear experimental curve Allen et al. 1950; Cornish-Bowden & Koshland 1975; based on the logit, you may estimate the association Segel 1975/93; Levitzki 1978; Dixon & Webb 1979, constant for the site with the highest affinity, Kn, for pp 79 138; Wyman & Gill 1990; Wells 1992; Tuk & small values of [S], while for very high values of [S], the vanOostenbruggen 1996; Weiss 1997) and others in two- affinity may be estimated for the binding site with lowest state models (Koshland et al. 1966; Eisenthal 1975), affinity, K1 (Cornish-Bowden & Koshland 1975). while at the same time a whole tradition was built Therefore, when the logit plot of an observed dose- around a two-state reaction scheme without site-interac- response is not a straight line, the actual reaction tion upon binding (Monod et al. 1965; Karlin 1967; schemes behind the relation do not follow a simple Janin 1970; Perutz 1970; Thron 1973; Colquhoun 1973; theory excluding interactions between binding sites. In Perutz et al. 1998). Chapter 10: Hill in Hell 263
For two-states models, Perutz and co-workers paved and Koshland point out ‘It is therefore rather puzzling the path taken by others (Monod et al. 1963, 1965; that straight Hill plots have so often been published in Harber & Koshland 1967; Perutz 1990; Perutz et al. which the Hill coefficients are appreciably different from 1998) by presenting the quaternary structural changes unity’ (Cornish-Bowden & Koshland 1975). Of course, an of oxy/deoxy-hemoglobin consisting of four hemoglo- escape explanation is that a narrow range of concentra- bin subunits (Perutz 1942, 1960; Muirhead & Perutz tions may yield a linear logit plot and an assumption of 1963, Perutz et al. 1964). intermediate steps in the process could explain the observed non-linearity (Levitzki 1978). However, accord- 10.2.6. Adair’s Equation Predicts Non-linear ing to a linearized Hill plot, the random or ordered Hill Plots sequential schemes with intermediate binding steps involved before full occupancy of function (Table 6.2), Both Hill’s multi-sited Eq. 10.6 and Adair’s equation does not define straight lines even if all the association with individual association constants predict non-linear (or dissociation) constants are equal (see Fig. 6.15B). In Hill plots. Historically, the Hill plot was used by Adair addition, as already stated, to find identical association (1925) to demonstrate that Hill’s aggregation theory constants as the binding process progresses is expected with no site interaction was too simple. The aggregation to be a seldom event. Therefore, the multitude of theory should show straight Hill plots, but Adair’s published linear Hill plots remains puzzling. experimental data gave a non-linear Hill (logit) plot (Fig. 10.5) (Adair 1925, Fig. 2). Based on the most recent theories for multi-sited receptive units, it is interesting that each association 10.3. The Original Logistic Equation and constant, as binding increases or decreases, has different the Original Hill Equation values and therefore must be considered separately in an expression for the magnitude of a response as indicated How are equations born? For all applications of data by Krs in Eq. 10.6. When Krs are not identical, the Hill analyses it is an art to find the formulation that best plot (the logit plot) can never be a straight line, except describes and most faithfully interprets the results of for a Hill coefficient 1. In addition, as Cornish-Bowden an experiment. However, due to lack of sufficient
Figure 10.5. Deviation by hemoglobin saturation with molecular oxygen from a linear logit, Hill, plot. The curve fitted to the data points is a plot of Adair’s equation with adjusted association constants. Taken from Adair (1925, Fig. 2). 264 Part III: Test of Tools for Data Analysis information, a possible mechanistic interpretation ob- 1971; DeLean et al. 1978) and depicted in Figs 8.7C F tained by implementing a realistic equation is often and 8.8C F (see also Sub-chapter 10.A). ignored in favor of a simple formulation that gives a quick pseudo-qualitative description of one’s data. In the 10.3.1. A Logistic Equation, What is That? biological sciences, a prototype example of this is the Hill equation from 1910 and its more recent transformation The classical logistic equation (Verhulst 1845; Pearl & into a so-called ‘logistic’ equation (Jenkinson et al. 1995; Reed 1920) is a formulation dealing with the description Jenkinson 2003; Neubig et al. 2003). of population growth and auto-catalytic chemical pro- In order to evaluate theories for dose-responses that cesses as functions of time (Verhulst 1845; Ostwald 1883; are applied to experimental data theories expressed as Robertson 1908; Pearl & Reed 1920; Reed & Berkson equations there may be an advantage in transforming 1929; Smith 1968). Development of this formulation is either the independent or the dependent variable, i.e., described in Section 10.3.2, and its history is recounted by the x-axis or the y-axis, or both axes simultaneously. This Cramer 2003b ((http://www.cambridge.org/resources/ is also called meta-metric re-scaling. Examples of such 0521815886/1208_default.pdf). transformations are (1) a normal distribution re-scaling Meanwhile, Berkson who was familiar with the logistic of the response axis as suggested by Gaddum (1933; growth curve transformed the classical dose-response 1953) equal to a probit scaling; that is, a Gaussian relation (load) for drug-dose versus death-rate data into cumulative distribution of the dependent variable (see an equation which he termed ‘logistic function’ due to Appendix 10.A) (Finney 1971). (2) A semi-logarithmic the resemblance of his semi-log transformed load equa- re-scaling of the dose axis as suggested by Berkson tion with the logistic growth curve (Berkson 1944). In the (1944) and Klotz (1982) equal to the semi-log plots in Berksonian transformation of the load, the independent Fig. 10.1B D and Fig. 8.1B. (3) A logarithmic re-scaling variable concentration S is expressed as the logarithm of of both axes in a logit transformation as described early normalized [S]. Therefore, the Berkson ‘logistic func- on by several authors (Adair 1925; Berkson 1944; Finney tion’ means semi-log plotting of load (Fig. 10.6).
Figure 10.6. The logistic equation as analyzed by Berkson for dose-death relations and termed a ‘logistic function’. Taken from Berkson (1944, Chart 1) with permission from the Journal of the American Statistical Association. Chapter 10: Hill in Hell 265
10
8
6
population growth (arbit) 4
2
0 5 10 15 20 time (arbit) Figure 10.7. Malthusian growth curve. Malthus’ geometrical (exponential) relationship according to the equation in Table 10.2. By some termed Gompertz curve. The rate parameter r, equal to the Malthusian parameter, is varied in five steps from 1×0.05 (*)to5×0.05 ( ×× ), increasing by 0.05 between steps. See Malthus (1798).
In fact, Berkson’s intention with the transformation 10.3.2. When Math meets Nature’s was to show that the ‘logistic function’ of the load and its Self-limiting Systems logit4 formulation yielded better statistics for the para- meters than the probit analysis suggested by Gaddum Setting up a mathematical formulation to describe processes in Nature and human society is an ever- (1933). Since Berkson’s semi-log transformation, the ‘logistic’ developing issue (Nowak 2006; Weisstein 2007 http:// scienceworld.wolfram.com). As an example, we can take re-scaling of load has also been adopted for Hill-type different types of formulations for growth. As such, formulations (van Rossum 1966) and characterized as a formulations for population growth are often referred ‘logistic equation’ (De Lean et al. 1978; Jenkinson et al. to as logistic equations. Nearly 200 years ago, Malthus 1995; Neubig et al. 2003; Motulsky & Christopoulos described population growth by a simple exponential 2004) (see also Sub-chapter 10.A). function (Fig. 10.7) (Malthus 1798, revised in 1803, Today, the purpose is to plot and analyze the introduction by MP Fogarty). Since the Malthusian hyperbola of the load or sigmoidality of dose-responses approach resulted in an unlikely unlimited growth, by the Hill equations in semi-log displays (Figs. 10.1B Gompertz (1825) invented and presented a popula- and 8.1D), for better statistical evaluation of the func- tion-mortality (growth) theory formulated as: tion parameters (Finney 1971; Motulsky & Christopou- los 2004). N Nmax ×exp( exp( (t t0)=b)): (10:7) The next sections will reveal that when we perform a so-called logarithmic transformation of the modified Here N is the ‘response’ equal to the actual number of Hill equation (Eq. 10.2), to a ‘logistic-Hill’ equation, it is deaths and Nmax is the total possible deaths, t0 the start a mere focus on logarithmic re-scaling of the indepen- of time, and b a time constant. The Gompertz equation dent variable for semi-log plotting and parameter may also be used to describe population growth evaluation, rather than the formulation of a new (Fig. 10.8). This growth is normalized to vary between equation for synagics. Furthermore, it is not in any way zero and one; one representing the maximum possible congruent with the original logistic equation dealt with in population. Gompertz’s theory has a rather steep analysis of growth. To illustrate, I will begin with the increase in population at the beginning (Fig. 10.8), a logistic equation for growth. phenomenon not characteristic of real population Note: in case there is no interest in logistic growth growth. Therefore, as we shall see, Verhulst introduced curves, you should go directly to Section 10.3.3 dealing yet another growth theory. with the ‘logistic’ Hill equation. Populations have maxima that depend on several conditions one being available volume or surface. As 4 The term ‘logit’ was coined by Berkson (1944) to differentiate it real populations reach a maximum and do not expand from the ‘probit’. rapidly with time at the beginning, which is the case for 266 Part III: Test of Tools for Data Analysis
100
80
60
40
fractional number (% of max) 20
0 -10 -5 0 5 10 time (arbit) Figure 10.8. Gompertzian growth curve. Plots of the growth law also known as Gompertz function (Table 10.2). Here the Malthusian parameter r is varied in five steps from 1×0.2 (*)to5×0.2 ( ×× ), increasing by 0.2 between steps. After Gompertz (1825).
the Gompertz equation, the model by Gompertz was d(ar) R ×ar×(1 ar)(10:9) soon revised by Verhulst employing a differential equat- dt max ion (Verhulst 1838, 1845, 1847) and here expressed in homology with a first order difference equation: (Verhulst 1938; Wilhelmy 1850; Ostwald 1883, 1902; Robertson 1908a) and limited by a maximal value of Nt 1 C ×Nt ×(1 Nt); (10:8) events, Rmax. Here ar is the actual reaction, homologous where Nt is the running number and C is a system to the response in the form of N in Eq. 10.7 and Nt 1 in constant (Table 10.2) (Peitigen et al. 1992, pp. 63 134; Eq. 10.8. Eq. 10.9 is a differential version of Eq. 10.8. We Kingsland 1995; Camazine et al. 2001, pp. 42 44; have now moved from integer math to thermodynamic Turchin 2003). Verhulst coined Eq. 10.8 a ‘logistic’ ensemble math. equation (Verhulst 1845). The logistic equation was Inserting TR for Rmax in Eq. 10.9 and integrating independently rediscovered in 1920 by Pearl and Reed yields: (1920). Almost simultaneously, Pearl recognized Ver- hulst’ earlier discovery of the difference equation (Pearl ar ln k×(t t0); (10:10) 1922). In an exchange of ideas with Pearl, Yule TR ar reintroduced the term logistic for the Verhulst/Pearl & Reed equation (Yule 1925; Kingsland 1995). Since then, in which k is a rate constant, see for instance Robertson formulations on growth have been known as logistic (1908a,b), who used this modified logistic equation to equations. describe growth in individuals with time. Accordingly, the transformed Verhulst formula (Eq. Note that the logistic equation in Eq. 10.10 has a 10.8) is a new logistic equation, generally referred to ‘logit-like’ expression to the left of the equal sign, but a as the logistic equation (Weisstein 2007; http://math non-logarithmic expression to the right of the equal world.wolfram.com). Eq. 10.8 may be compared with a sign for its independent variable time. Compare Eq. first order differential equation used by Ostwald (1883) 10.10 with Eq. 10.4 for the Hill (logit) plot in to describe auto-catalytic progression with time. Thus, Section 10.2.2. The Hill plot equation with its logit an auto-catalytic5 and mono-molecular reaction is expression is logarithmic on both sides. On the other formulated by: hand, as we shall see, the ‘logistic Hill’ equation (Eq. 10.15) has only a logarithmic re-scaled independent 5 There is modern use of autocatalytic processes in networks, see, e.g., variable. Thus, Eq. 10.15 merely has a semi-log adjusted King (1981), Lee et al. (1997), and Goldstein (2006). concentration. Chapter 10: Hill in Hell 267
Before proceeding, we must first reformulate Eq. maximal population found by data obtained for the 10.10 to its non-logit distribution form with a term as USA until 1910 (Fig. 10.10). Location at the time axis of ar/TR. The anti-log of Eq. 10.10 gives us: the inflection point for the fitted growth curve turned out to be the year 1914; four years ahead of the latest ar exp(k×(t t0)): (10:11) obtained population record (Fig. 10.10). TR ar The general form of Eq. 10.13 is given in Eq. 10.14: Then by using regula detri rule 1a from Chapter 1 (Box 1 1.1), we get: y ; (10:14) 1 ea b×x ar exp(k × (t t )) 0 ; (10:12) TR 1 exp(k × (t t0)) in which y is the level of sigmoidal increase as a function of any independent variable x, while a and b are system which, by dividing terms at the right with exp(k× constants. When the independent variable x is time, b is (t t0)), we can rewrite to: a rate constant. Using Fig. 10.10 as an example, growth ar 1 of the American population before 1920 had time : (10:13) × TR 1 exp( k × (t t )) constant 1/b equal to 31.9 and its ‘half-time’ was 31.9 0 0.693 22.1 years. The ‘half-time’ in the logistic equa- Eq. 10.13 is also generally considered as the logistic tion is not the time it takes to double a population, equation (Fig. 10.9). Here time t is the independent therefore it is better to use a term as ‘standard interval’ variable. The level of growth varies as time goes by for 1/b (Yule 1925). Parameter b is also known as the between zero growth and the maximum growth ex- Malthusian parameter. The US population followed the pressed as unity or 100%. If time is only defined in the logistic equation nicely in the beginning of the 20th range 05timeB , then the rate constant k is related century (Yule 1925, Fig. 4) and until the 1940s, still with to the initial level of growth, where constant t0 is the an inflection point around year 1914 (Snedecor & time for maximal growth rate at the inflection of the Cochran 1967). However, after 1940, luckily or unluck- growth curve (Fig. 10.9). ily, the rate constant b for growth of the US population I have to admit that to equalize the plots in Fig. 10.9 did not stay constant; it increased. Furthermore, the US with the plot in Fig. 10.1D is immensely tempting as they population did not saturate at around 200 million seem identical. Nevertheless, recognize that the mean- individuals, as suggested in Fig. 10.10, but is now over ing behind the two graphs is quite different. 300 million and still growing. In addition, the logistic Eq. 10.13 was the form of equation used for the equation is no longer a good descriptor for the growth growth of the USA population with time (Pearl & Reed of inhabitants in the USA. The two major reasons are a 1920). Nearly 200 million individuals, equal to TR or post-war baby boom in the 1940s and an improved ‘the maximal carrying capacity’, was the estimated health care system (Snedecor & Cochran 1967) not
100
80
60
40
fractional number (% of max) 20
0 -10 -5 0 5 10 time (arbit) Figure 10.9. Verhulstian growth curve. Logistic equation for simulation of population growth. Plots of the logistic equation according to Verhulst (see Table 10.2). Plots are spaced by changing parameter in five steps from 1×0.2 (*)to5×0.2 ( ×× ), increasing by 0.2 between steps. After Verhulst (1838, 1845). 268 Part III: Test of Tools for Data Analysis
100x103
80x103
60x103
40x103
20x103 recorded population (in thousands)
0 1800 1820 1840 1860 1880 1900 time (year) Figure 10.10. Population growth in the USA from 1790 to 1910. Analysis using the logistic equation. Maximal US population 196910×106 individuals, ‘time constant’ 3291 years, year of inflection point 191493. Based on Pearl and Reed (1920). None of the logistic functions, including the sigmoid function for population growth match the actual development of population growth in the 20th century (Snedecor & Cochran 1967; Kingsland 1995; Turchin 2003). incorporated in the formulation in Eq. 10.14. In fact, the modified equation, i.e., the logistic equation conjured logistic equation is not a good descriptor for population into the ‘logistic’ version of the modified Hill equation. growth anywhere (Kingsland 1995; Turchin 2003), but Fig. 10.1D is a graph of Eq. 10.15 with ln[S] as the may still be used for predicting an increase in industry independent variable (in the figure it is log[S]) and nH products such as cellular phones (Cramer 2003a). varying in five steps (see figure legend). In this, the graph in Fig. 10.1D is also just a simple semi-log presentation of the modified Hill equation. 10.3.3. Transforming Logistic Equations to If we assign a sign in reverse, i.e., a minus sign, for the × × Hill’s Equations and Vice Versa exponent (nH ln[Ks] nH ln[S]) in Eq. 10.15, then the transformation of the ‘logistic’ equation yields the well- Using mathematical manipulations, we can transform known reverse modified Hill equation in Eq. 8.4: the above logistic equation with time as independent 1 1 variable (Eq. 10.14), into an expression that is identical y ; (10:16) (n ×ln[K ] n ×ln[S]) nH to the modified Hill equation (Eq. 10.2), where K is an 1 e H i H 1 (S=Ki) association constant equal to 1/Ks, while the ligand with a Hill slope factor nH equal to ni described in concentration is the independent variable. Thus, assum- Chapter 8, a dissociation constant for an inhibitor Ki ing a in Eq. 10.14 to be a new constant equal to nH × instead of one for an agonist, and examples of its curves ln[Ks], and b yet another novel constant equal to nH, depicted in Fig. 8.8D F. and, in addition, the independent variable x converted The initiation of this ‘logistic’ transformation of the from time t as a variable to a different variable, viz. the Hill equation in Eqs. 10.15 and 10.16 can be traced natural log of the ligand concentration, ln[S], we can back to a comparison between a general form of rewrite the original logistic Eq. 10.14 as: logistic equations and Hill’s equation (van Rossum 1966; De Lean et al. 1978). However, one can argue 1 1 y ; (10:15) that the logistic-Hill-calamity if-you-will started with (n ×ln[K ] n ×ln[S]) nH 1 e H S H 1 (Ks=S) Joseph Berkson switching use of the logistic equation for time-dependent autocatalytic processes to processes in which y is the fraction of bound receptive units or the dependent on a concentration-like variable for data of fraction of active effectors as a function of the ligand mortality rate (Reed & Berkson 1929; Berkson 1944). concentration S, Ks is an equilibrium dissociation con- The above example on how the original logistic stant, and nH is the Hill coefficient for a modified Hill equation (Eq. 10.13), where time is the natural indepen- equation, equal to our earlier parameter ns in Chapter 8. dent variable, can be transformed into its dose-response The right hand-side of Eq. 10.15 is identifiable with the counterpart (Eq. 10.15), is described in Jenkinson et al. modified Hill equation (Eq. 10.2), and hence, the left (1995) and Neubig et al. (2003), see also for instance hand side of this expression is a ‘logistic’ version of Hill’s Giraldo et al. (2002) and Giraldo (2003). Chapter 10: Hill in Hell 269
Observe though, that the significance of parameters dose-response relations the ‘logit’ actually is often a Ks and nH and of independent variable S in the Hill and better method than the semi-log (‘logistic’) transforma- reverse-Hill ‘logistic’ equations is totally different from tion for statistical evaluation of log(EC50). Therefore, the physical meaning of parameters a and b and of why not quote and use the ‘logit’ instead of the ‘logistic’ independent variable x in the original logistic equation semi-log re-scaling? See section 10.A.11.7 (compare Fig. 10.1D and Fig. 10.9). The new parameter Notice that S 0 is not defined in the ‘logistic’-Hill or ln[Ks] and the variable ln[S] have no physical meaning. in the semi-log plot of Hill (Table 10.1). Thus, the In addition, see comments by Jenkinson (2003, pp. ‘logistic’-Hill has a disadvantage compared with a plot of 16 17) on the difference between the ‘logistic’-Hill Hill’s equation in the linear linear form. For all the equation and the logistic equation for growth. listed equations in Table 10.1, the dependent variable ‘y’ varies between zero and unity. The range of indepen- dent variable x, on the other hand, alternates between 10.3.4. Practical Use of a ‘Logistic’ Hill Equation types of logistic equations (Table 10.1 and 10.2). For instance, there is a difference between the range of the The similarity between the original logistic equation and independent variable x for non-transformed and trans- the ‘logistic’ transformed Hill/reverse-Hill equation has formed forms of Hill equations (Table 10.1). The led users of the Hill equations into quoting the modified independent variable, concentration, is not defined for Hill equation in its ‘logistic’ form (Accamazzo et al. x 0 in the ‘logistic’ transformed versions, while in 2002; Hornigold et al. 2003; Motulsky & Christopoulos reality the physical process is defined for x 0; equal to 2004; May et al. 2007) or just describing the Hill the control response and at zero ligand concentration. equation as ‘logistic’ (De Lean et al. 1978; Barlow & Therefore, when fitting to a semi-log plot of synagic Blake 1989; Lazareno & Birdsall 1993; Kenakin 2004, pp. data, it is recommended to insert an extra data point at a 94 and 144; Ehlert 2005). low ligand concentration value and with zero response. The modified Hill Eq. 10.2 in its ‘logistic’ form is That is, include a fictive data point at zero response two formulated as:6 or three orders of magnitude below the concentration of the first data point with an actual ligand concentra- 1 tion eliciting a small response and instead of the data y ; (10:17) point at zero ligand concentration (see also Motulsky & n ×(log[K ] log[S]) 1 10 H S Christopoulos 2004, p. 264).
in which log[Ks] might also be log (EC50). 10.3.5. ‘Logistic-Hill’ is a Technical Term In my opinion, Eq. 10.17 does not make things any easier to grasp. Rather, it will confound new users into a It is true that a semi-log plot of data from synagic belief of some special significance. This is not the case. It experiments analyzed by non-linear fitting of the Hill appears merely as vestiges of ‘The emperor’s new theory yields better statistics than in its linear linear clothes’. We are dealing with a simple pleonasm, since form (Motulsky & Christopoulos 2004, pp. 256 265). the log[S] is equated as a power for a base of 10, meaning This is correct in principle for synagic data if carried out that S still operates as straight forward S in Eq. 10.17. according to the probability laws (Finney 1971, Chapters Of course, one can argue that there is logic in the 2 and 3), but not necessarily for other types of data. For ‘logistic’ form of Hill’s equation when plotting data in a instance, population growth data analyzed by the logistic semi-log plot, where it may be said that by converting the equation is best handled by a linear linear scaling of independent variable now expressed by log[S] is the data (see Fig. 10.9). same as formulating Eq. 10.17, which is true (Giraldo Taken as a whole, it is the semi-log, the logit, or the et al. 2002; Giraldo 2003). probit technique used on synagic data that yields the Meanwhile, we have obtained nothing by just quoting better statistics, rather than just quoting exotic expres- the ‘logistic’ transformation of the Hill equation com- sions for the Hill equation. It may be said that it all boils pared with the ordinary modified Hill equation, except down to a matter of taste. My preference is to give the for a focus on the semi-log re-scaling for improved Hill equation as the modified Hill equation (Eq. 10.2), presentation and statistical evaluation. If this is the 7 purpose, why not just say as much in a Methods sec- For nH 1, Hill and ‘logistic’-Hill are both sigmoidal curves in linear linear plots, but symmetrically S-shaped when represented as tion? And, further, remember that for hyperbolic curves in semi-log plots (Fig. 10.1). Also, see Chapter 8 on S-shape versus sigmoidality. The original logistic growth curve in Eq. 10.13 is a 6 A section in the appendix at the end of the chapter gives references sigmoidal and symmetric curve in the relative linear linear plot but an to concentration dependent ‘logistic’ equations in Sigma-Plot software S-shaped and non-symmetric curve in the semi-log plot (see Section (Section 10.A.13 and Table 10.3). 10.3.6, Fig. 10.9 and Table 10.1). 270 Part III: Test of Tools for Data Analysis
Table 10.1. Various types of ‘logistic’ equations (for comments see Section 10.3.5)
Dependent Independent Variables Variables Plot type Plots in figures Range of x
Logistic equations
1 The original / x Symmetric sigmoid 10.9 BxB 1 e(a b×x) log x Skewed 0BxB
1 Modified Hill / x Skewed sigmoid 10.1A; 8.8 A x * nH B B 1 (Ks=x) log x Symmetric S-shaped 8.8B 0BxB
1 Semi-log transformed Hill / x Skewed sigmoid 10.1A; 8.8A 0BxB 1 10nH×(logKs logx) log x Symmetric S-shaped 8.8B 0BxB
1 Semi-log transformed reverse Hill / x Skewed sigmoid 8.8D 0BxB 1 10 nH×(logKi logx) log x Symmetric S-shaped 8.8E 0BxB
1 Reverse Hill / x Skewed sigmoid 8.8D x * nH B B 1 (x=Ki) log x Symmetric S-shaped 8.8E 0BxB
Constants a and b exist for all values. *Only range 05xB is relevant. Ks only for 0BKsB . and plot data in so-called semi-log plots with appropriate instance the term ‘distribution equation’ is a functional statistical evaluation of log(EC50) in either ‘logistic’ or descriptor with more meaning, as it also discriminates logit transforms. The transforms may not be needed the Hill equation from the original logistic equation. In with use of modern statistical software. line with this, Jenkinson also has a warning against The term ‘logistic’ in connection with Hill’s equation mixing the logistic equation and the Hill ‘logistic’ is a technical term referring to semi-logging, while for equation (Jenkinson 2003, Eq. 1.2.44 and Appendix 1.2D, pp. 16 17). Table 10.2. Terminology for logistic growth examples of In summary, quoting a ‘logistic’ Hill equation simply formulae for population growth equal to logistic equations implies semi-log re-scaling for better presentation and Name of equation Equation statistics.
Malthus’ equation with Malthusian See Gompertz curve parameter r $ 10.3.6. The Richards Equation qx b×ec×t Gompertz curve /y a×b y a×e r×t Gompertz standard model also known /N(t) N0 ×e Giraldo has specifically analyzed equations for asymme- as the law of growth try, lacking in Hill’s function (Giraldo et al. 2002; r×t Gompertz modified model /N(t) C N ×e 0 Giraldo 2003). One such equation is the Richards a 8 Logistic curve or when normalized /y equation (Richards 1959), where the whole denomi- 1 b × qt called the logistic distribution 1 nator is raised to a power (Van der Graaf & Schoemaker Logistic equation also called sigmoid /y 1 (1=y 1) × e rt 1999). Thus, for instance: function or Verhulst’ model 0 1 Sigmoid function also called the sigmoid /y 1 e x 1 curve, logistic function or sigmoid y ; (10:18) nH nR equation# [1 (Ks=x) ] dN rN × (K N) Verhulst’ e´quation logistique / dt K
Differential version of the logistic model /Xn 1 r×Nn ×(1 Xn) and its discrete quadratic version also known as the logistic map; which leads 8 Reference to a Richards equation in this section is due to Giraldo into deterministic chaos et al. (2002). I must admit that I do not see the correspondence between Richards’ formulations (Richards 1959) and those by Giraldo $Parameter r is the maximum population growth rate, equal to parameter c in Gompertz’ curve equation, where parameters b and c are negative numbers. et al. (2002). Meanwhile, in this text I shall stick with the Giraldo- e base of the natural logarithm, t time, and x any independent variable. nomenclature. Note, the conventional ‘Richards equation’ is used in a #‘Sigmoid equation’ in SigmaPlot manual (2004). different field; namely hydrology (Scienceworld.wolfram.com). Chapter 10: Hill in Hell 271
100
80
60
40
fractional number (% of max) 20
0 -1 0 1 log [ligand or 10-time] (arbit)
Figure 10.11. A Richards’ function also called the generalized Hill equation. Here the Richards coefficient nR is varied in five steps from 1×0.2 (*)to5×0.2 ( ×× ), increasing by 0.2 between steps, while keeping the Hill’s coefficient 3. Terminology after Giraldo et al. (2002). where y is a response as a function of independent cal formulation (Richards 1959), its relevance as a variable x. System constant nR signifies a Richards mechanistic tool should be questioned. coefficient. Both Hill’s symmetric equation and Gom- Eq. 10.18 can be applied to a sequential process of pertz’s asymmetric equation (http://mathworld. nR steps, in which only the final step renders activity wolfram.com) are included in the Richards equation or a conformational switch with change in ligand
(Giraldo et al. 2002). The effect of varying the Richards binding, RSn ?R Sn; governed by an isomerization ns ns coefficient, nR, on sigmoidality is shown in Fig. 10.11, constant equal to L , where L is assumed small depicting the skewed sigmoidality of Richards’ equa- compared to the other forward parameters as associa- tion for changing values of Richards’ coefficient in tion constants denoted Kax 1/Kdx. Besides assuming ns semilog plotting. For nH 1, this type of equation is L BBKax, we can further simplify the system by also called a ‘generalized logistic curve’ (Table 10.3). assuming that all the forward equilibrium association As an example, the Richards equation with nH 1 has constants are identical. Compare with the Weiss model been used frequently to analyze ion-pump data (Garay described in Chapter 6. & Garrahan 1973; Eisener & Richards 1981). Mean- Giraldo and coworkers (2002) show that the exponent while, since the Richards equation is purely an empiri- under these assumptions is equal to the number of steps.9 The slope at the point of inflection for four synagic models has been derived as a general expression, equal to 4×{(dR/d log[S]) }/{ln10×R } and includes Table 10.3. A comparison of expressions for various equa- 50 max tions used with those in the SigmaPlot software, some the Hill and Richards equations (Berkson 1944; Hill incorporating the term ‘logistic’ 1985, pp. 64 66; Giraldo 2003). However, both the Hill equation and the Richard type of equations as stated † Present terminology SigmaPlot terminology* are empirical equations that may be used for semi- Hill equation (modified) Hill equation quantitative estimates of deviation from the simple load Reverse Hill equation Logistic equation or four parameter formulation, but as mechanistic models they are only logistic curve relevant for systems with full co-operativity or complete Logistic equation# Sigmoid equation# Gompertz curve Gompertz growth model simultaneity. Generalized logistic curve§ or Modified hyperbola III$ See Giraldo et al. (2002) for a more detailed account Richard’s equation of the Hill, the modified Hill, the Gompertz, and Load plus diffusion as used in Rectangular hyperbola II or one site Richards equations as well as their asymptotic location, Chapter 9 saturation non-specific midpoint, and inflection points. *SigmaPlot terms from SigmaPlot User’s guide (2004, Chapter 20). #See Table 10.2. §Cramer (2003a) used the terms ‘logistic curve’ and ‘logistic function’ inter- 9 changeably; as did Verhulst (1845). Compare this with the meaning of nH in Hill’s aggregation theory $ For nH 1. described in Section 10.1.2. 272 Part III: Test of Tools for Data Analysis
10.3.7. Use of Double Hill or Seriatim an auto-intervention or co-operative process, likewise we Hill Equations use reversed Hill-type equations when the action of an interventor also indicates an intervention process or a Hill first suggested use of summed Hill equations modulator a modulatory process that deviates from the (Eq. 10.6) (Hill 1910). Using the sum of two or more simple load (Lazareno & Birdsall 1993, 1995; Trankle Hill equations to describe dose-response data with bell- et al. 2005). shaped or reverse bell-shaped appearance is even For the effects of combining agonists and interventors further from the mechanism of a functional system, or modulator molecules there are suggestions in the although this type of analysis is rather popular (Pliska literature for the use of embedded Hill-Richards for- 1994; Taleb & Betz 1994; Rovati & Nicosia 1994; Vogel et mulations in order to characterize parameters (Lazar- al. 1995; Bronnikow et al. 1999; Accomozzo et al. 2002; eno & Birdsall 1995; Trankle et al. 2005). Meanwhile, if Tucek et al. 2002; Hornigold et al. 2003; May et al. probing for a physical interpretation of synagic data, 2007), requiring determination of between six and once again my preference would be to try the interven- seven system constants when two Hill equations are tion model or the ATSM by Hall, and possible exten- combined. There are other examples for the use of sions thereof with different binding constants due to site seriatim Hill equations. They appear as products or interactions (Chapters 2 and 7). sums of Hill equations in analysis of genetic regulatory networks (Meir et al. 2002) and in pharmaco-dynamic and pharmaco-kinetic analyses of drug drug interaction in signal transduction, in ligand binding studies (Wells 10.3.9. On the Similarity Between the Hill Equation 1992), and in clinical drug treatments (Scaramellini and the Modified Boltzmann Equation et al. 1997; Short et al. 2002; Jonker et al. 2003), as well Gibbs free energy (Fig. 1.5) may be represented by as for synergy described in Chapter 12 (Chou & Talalay electrical potentials, concentration gradients, tempera- 1981, 1984; Berenbaum 1989; Greco et al. 1995; Chou ture gradients, pressure and other forms of physical or 2006). mechanical forces. Recently, Michel (2007) derived equations for a Boltzmann’s equation for probability distribution at combined Hill scheme and Adair approach with varying equilibrium10 recast into a chemical form is the Nernst association constants due to binding interactions. equation, which we can write as: Bell-shaped synagics may also be analyzed by a series of simple load functions (Szabadi 1977; Ehlert 1988; Jarv RT [S]1 Eeq E1 E2 ×log ; (10:19) 1994; Williams et al. 2000; Cladman & Chidiac 2002; zF [S]2 Griffin et al. 2003), requiring determination of between in which E is equal to the equilibrium potential four and five parameters when operating with the sum eq measured for example in milivolt given as the difference of two loads. between the potential at side 1 and 2, E E . RT/zF is a Ultra-sensitivity and switch-like function, for instance 1 2 constant equal to 61.5 mV at 378C and [S] and [S] are in signaling cascades, may be obtained by several levels 1 2 the passively distributed concentrations of S at side 1 of enzyme-induction and described by embedded Hill and side 2 (Nernst 1889; Hille 2001, p. 13). equations (Huang & Ferrell 1996; Ferrell 1997). For voltage operated ion channels (VOCs), a mod- Combining the Hill equation as well as other models ified form of the Boltzmann equation dictates a ratio of with a term for feed-forward and feedback was intro- open (O), to the sum of both open and closed (C) duced by Hofmeyr and Cornish-Bowden (1997; Cornish- channels (Fig. 5.2B), that is: Bowden 2004, Sub-chapter 12.10; Hofmeyr et al. 2006). However, these models are non-mechanistic. O 1 ; (10:20) For a description of models for bell-shaped responses, O C 1 exp[(zF=RT) × (V1=2 E)] but with a mechanistic implication, it is recommended to try other models such as the allosteric two-state model where ‘exp’ is the natural base e and zF/RT a constant as (ATSM) and the homotropic two-state model (HOTSM) before. When the potential is equal to V1/2 then 50% of presented in Chapter 7, or the random and ordered the channels are open. V1/2 is similar to the so-called reaction schemes including their expanded versions in Boltzmann constant. Chapter 6 (Table 6.2). The relation in Eq. 10.20 is often plotted in a log- linear form (Hille 2001, p. 55).
10.3.8. Use of Hill-Richards Type Formulations 10 In fact, developing this distribution equation was done nearly simultaneously by Maxwell (for rate of molecules) and by Boltzmann Just as we use Hill-type equations in analysis when (for energy of adsorption). Thus, this distribution equation is also agonism deviates from the load and seemingly involves known as das Maxwell-Boltzmannsches verteilungsgesetz. Chapter 10: Hill in Hell 273
Eq. 10.20 clearly resembles the Verhulst growth (2) The Hill analysis is a practical and convenient expression, the logistic Eq. 10.13, and since from the measure of co-operativity when information Nernst relation in Eq. 10.19 we have E proportional to about a system is limited. ln([S]/Kd), it is tempting to replace E in Eq. 10.20 with (3) Obtaining a non-linear Hill plot, i.e., a non- ln([S]/Kd) and rewrite Eq. 10.20 such that the expres- linear logit plot, is one way to uncover devia- sion for the number of charges, z, moved in the tions of experimental data away from the voltage gating is replaced by the number of ions binding modified Hill equation. In case the Hill (logit) and equated with Hill’s coefficient nH. Therefore, we plot of data with n"1 deviate significantly from can write: linearity, try the Pascal triangle expansion for O ar 1 multi-sited receptor complexes where indivi- n ; (10:21) dual association constants are allowed to vary O C TR 1 (K =[S] H ) d (Adair 1925; Ricard & Cornish-Bowden 1987) (see also Tables 6.1 and 6.2, or try allosteric where ln(Kd) replaces the V1/2. A transformation of the modified Boltzmann equation to the Hill equation. models such as the HOTSM for single ligand In fact, this transformation has its own life in analysis applications, Chapter 7). of VOCs (Spivak 1995; Yifrach 2004; Kubokawa et al. (4) The Hill equation is a special distribution 2005; Chapman et al. 2006; Zhang et al. 2007), and will equation for synagics rather than a logistic not be discussed further here, except that the transfor- equation. The ‘logistic’ formulation of Hill’s mation does not bring the logistic equation with time as equation is a technical term for semi-log plot- an independent variable any closer to the Hill equation ting and for statistical evaluation of data. It is a for dose-response analysis in synagics. mere reference to semi-log analysis of synagic data, on the same line as a logit or a probit analysis. Semi-log presentation of dose-response 10.4. Conclusions results, synagic data, is recommended. (5) Analysis may be performed for instance in semi- 10.4.1. Details on the Use of Hill’s Equation log plots of synagic data, thus evaluating the log(EC50). Semi-log plot of experimental data We can summarize the advantages and problems of analyzed by non-linear fitting to synagic data using Hill’s equation and its derived versions in the yields better statistical evaluation for the system following seven paragraphs. constant EC50, i.e., log EC50, compared to linear linear analysis of data (Motulsky & (1) As a physical model, the modified Hill equation Christopoulos 2004, pp. 256 265). This is true (Eq. 10.2) describes a coincidence system. The in principle if carried out according to the coincidence may come about by a simultaneous statistical laws, even when the Hill coefficient is and saturating binding of ligands to a multi- 1. On the estimation of the median effective sited preexisting complex or in the instant dose, ED , and its logarithmic transformation, formation of a multi-subunit complex where 50 logED , see for instance Finney (1971, 1978, each subunit is already liganded. The latter 50 possibility may be a scenario for di-merization of Chapters 3 and 4; Berkson 1944). The probit growth factor receptors (GFRs) (Bublil & and especially the logit methods for parameter Yarden 2007), though unlikely for GPCRs (Ma evaluation should also be considered. In theory, et al. 2007). Thus, for some GFRs, liganding of the logit method is even better than the semi- single unit receptors might increase the like- log (‘logistic’) method in using the modified lihood of di-merization or multi-merization. Hill equation as a base for analysis of dose Hill’s aggregation theory revitalized. The Hill response data (Berkson 1944). Modern software equation may also be used for a multi-sited may overcome such difficulties. system with a final dominating step for binding (6) In publishing results based on an analysis using or function. Thus, the Hill coefficient can be Hill’s modified equation and combined with equal to the number of binding sites, when either semi-log, logit, or probit transformation, reaction models behave as an all-or-non associa- write the following: modified Hill’s equation tion/dissociation reaction scheme, e.g., when was used as theory, and the statistical evaluation the final binding step has a much higher affinity was performed by semi-log (logistic) transfor- than the preceding binding steps, as in hemo- mation, by logit (double log of relative bound globin-O2 binding (Table 5.3) (Roughton et al. over free receptors) transformation, or by 1955; Monod et al. 1965; Weiss, 1997). probit (normal equivalent deviate of dependent 274 Part III: Test of Tools for Data Analysis
variable versus log of independent variable) From a narrower perspective, statistics may be said transformation. to be an analysis of the validity of data sets, testing 7. The modified Boltzmann equation (Eq. 10.20) probabilities, and comparing models. Examples of resembles the logistic equation (Eq. 10.13), but hypotheses and methods for data evaluation are chi- the two are used in different applications with square-, t-orF-tests, least square and/or maximum different meanings. The electrical potential as a likelihood fitting, and data transformations (Quinn & force appears similar to time as a ‘force’ and may Keough 2002; Samuels & Witmer 2003; Sokal & Rohlf mathematically be transformed to concentration 1995, 2004), and frequentists or Bayesians interpreta- as a force. tion of probability (Cox 2001; Quinn & Keough 2002; Winkler 2003; Gelman et al. 2003; Lee 2004; Sivia & In summary: (1) quoting ‘logistic Hill’ ought to cover Skilling 2006). As an example, Bayesian interpretations the same as a semi-log analysis of data and based on the relying on inference and filtering is now also used for modified Hill equation, and (2) the real problem with a optimal exclusion of spam E-mails (Zorkadis et al. choice of the modified Hill equation for theory is when 2005). When the variance alters for data in the data it works as a silencer of explorative investigations, set with time or another independent variable, this is thereby preventing possible penetrating analyses. an entirely separate subject. Its analysis in economics won Robert Engle the Nobel prize in 2003, solving the autoregressive conditional heteroskedasticity (http:// 10.4.2. Is Hill in Hell? nobelprize.org/nobel_prizes/economics/laureates/2003/ A century celebration of Hill’s equation now, then, and engle-lecture.html). On more general aspects of data in 2010, alia equibus.11 analysis see Ford (2000).
When co-operative logistics has to come ‘true’, ill 10.A.2. Statistical Methods shivers us as a hellish flu. For whether you don’t or you do have a clue, what do Names for some of the statistical methods are listed here: we do with a Hill of two? linear regression, curvilinear regression, non-linear regression, We use him here, we use him there, for Bells the and logistic regression (Snedecor & Cochran 1967; Pampel scientist uses ’im everywhere. 2000; Hosmer & Lemeshow 2000; Cramer 2003a; O’Con- Is Hill in Heaven, is Hill in Hell as he’s hill-hidden nel 2005; Vittinghoff et al. 2005). Logistic regression is no one can tell. for discontinuous and often binary responses as ‘dead’ (to be cont’d....) or ‘alive’ dependent on an independent predictor variable as for instance ‘level of toxic drug’ (Hosmer & Lemeshow 2000; Vittinghoff et al. 2005). Other theories or techniques are for instance Multinominal Logistic 10.A. Appendix A: A Short Course on the Regression, multi-way frequency analysis (MFA), Multi- Term ‘Logistic’ and its Uses in Statistics variate analysis, and Repeated Measures Models (Hos- mer & Lemeshow 2000; Vittinghoff et al. 2005; Here the term ‘logistic’ is put in a slightly different Kleinbaum et al. 2007). Viewing many of the statistical perspective compared to the term ‘logistic’ discussed in methods under the same framework led researchers to the preceding Sub-chapters; although several of the the generalized linear modeling (GLM)12 (Dobson 2001; related concepts are the same and also repeated here. Quinn & Keough 2005), not to be detailed here. The more recent Bayesian method is yet another approach using inference and filtering in decision 10.A.1. Statistical Analysis (Quinn & Keough 2002; Winkler 2003; Marin & Robert Statistics is the science of analyzing, interpreting, and 2007; Bernado & Smith 2007; Lage et al. 2007). understanding data. There are various statistical meth- ods for analyzing data. Which method to choose will 10.A.3. Statistical Theory the Logistic Equation depend on the distribution of data and the theory selected for their analysis. Before an analysis, the For analysis of data sets, the selected theory is an selected theory may further be transformed such that a equation. An example is the logistic equation (Eq. plot of data according to the transformation is more 10.13). This equation with its curvilinear regression is equally distributed, ‘linearized’ and skewness of resi- used for analysis of population growth, although now duals thus reduced. Examples of this are given below. recognized as insufficient (Kingsland 1995; Turchin
11 alia equibus everything else equal. 12 GLM is different from the general liner models in the SAS package. Chapter 10: Hill in Hell 275
2003). Parameters in the logistic equation are the Table 10.4. Concept examples from fields of statistics, some ‘maximum carrying capacity of the population that is including the term logistic approached as times runs, equal 100%, the standard Equation terms Parallel terms interval 1/r or reciprocal Malthusian parameter (Table 10.2), and the location of the inflection point on the Logistic equation Growth curves Hill/Reverse Hill Logistic dose-response curve time axis. The description obtained by the logistic equation is mathematical not physical. Oppositely, the Transformations Logarithmic re-scaling including ‘Logistic’ equation adsorption-desorption process equated by the load semi-log formulation is a physical description. Logit (Mistakenly called logistic) Probit Power, including reciprocal Hanes, Lineweaver-Burk, Eadie- 10.A.4. ‘Logistic’ and Logistic Equations Scatchard Presently there are expanding uses of the word ‘logistic’. Analysis Logistic regression ‘Logit analysis’ Pertinent to our discussion, you can find formulations Logistic map listed as ‘logistic’ functions and ‘logistic’ or logistic Logistic distribution equations when the independent variable is raised to a power or the independent variable itself is included in an exponent. Other examples are terms as ‘logistic operating with the Hill modified equation. The situation is similar to the analysis of density spectra by the maps’, ‘logistic distribution’, ‘logistic regression’, and 2 ‘polytomous logistic regression’ (Table 10.5). An exam- Lorentzian probability function: y 1/(1 (x/xc x0) ) ple of the ‘logistic’ equation is the modified Hill with the independent variable x as a frequency, xc as a equation (Eqs. 10.15 10.17), also referred to as a ‘corner frequency’, and x0 a location parameter (Verv- een & DeFilice 1974). With the Lorentzian equation, it is ‘logistic function’ when nH 1 (Berkson 1944). We also have the Verhulst/Pearl-Reed logistic equation for natural to look at a semi-log distribution of data rather population growth (Eqs. 10.13 and 10.14). than a linear linear distribution (Fig. 10.12). Mean- Berkson used the logistic equation to analyze dose- while, no devil would dare to dream or to talk about the response data and called the equation ‘the logistic Lorentzian equation as a ‘logistic’ equation. Note function’, in which his coefficient a/b for the indepen- further that the load equation with nH equal 1, which was termed ‘logistic’ by Berkson is seldomly referred to dent variable he claimed to be to equal to L.D. 50 (EC50) as a ‘logistic’ equation (Ehlert 2008). In fact, it is a although it is equal to log L.D. 50 (log EC50) (Fig. 10.6) (Berkson 1944). hyperbolic function, see Fig. 1.3. Furthermore, when equations have their independent In conclusion, for a dose-response formulation, the and/or dependent variables re-scaled to a logarithmic term ‘logistic function’ as introduced by Berkson (1944) dimension, such functions are also referred to as ‘logistic’ and transferred to the Hill equation is a misnomer, a equations or functions. For the history of the develop- nuisance, and a bad habit. ment of the original logistic equation, see Kingsland Remarkably, the modified Boltzmann distribution (1995), Turchin (2003), and Cramer (2003b at http:// function (Eq. 10.20), which is a true logistic equation, www.cambridge.org/resources/0521815886/1208_ does not carry this designator. default.pdf). For statistical analysis, the logistic equation (Eq. 10.13) 10.A.6. Logistic Regression appears relevant for many applications with a limited ‘Logistic regression’ is also frequently used as synon- number for response, which on a relative scale will vary ymous with the logit or probit analysis, although most between 0 and 1. The term ‘logistic regression’, on the use the term ‘logistic regression’ specifically for the other hand, is a statistical analysis based mainly on analysis of single or multiple predictor(s) with discon- discontinuous binary variables (Hosmer & Lemeshow tinuous outcome in contrast to single or multi-way 2000; Quinn & Keough 2002; Samuels & Witmer 2003, analyses for continuous variables. pp. 582 585; Cramer 2003a; Vittinghoff et al. 2005). Dependent on the type of outcome, ‘logistic regres- Logistic regression is also referred to as ‘logit analysis’ sion’ appears in various connections (see Table 10.5). covered by logit models (Table 10.4) (Cramer 2003a). 10.A.7. Transformations 10.A.5. ‘Logistic’ Hill’s Equation The theory for data in the form of an equation may be As mentioned in Section 10.3.4, there is no need for an transformed by altering the expression of the indepen- explicit use of the expression ‘logistic equation’ when dent (predictor) and/or the dependent (outcome) 276 Part III: Test of Tools for Data Analysis
100
80
60
40
20 probability density function (% of maximum) 0 -2 -1 0 1 log [frequency] (arbit) 2 Figure 10.12. Plots of the Lorentzian relation given by y Ymax/(1 (x C) ). Constant C is varied in five steps: from 2(*)to2 ( ×× ), increasing by 1 between steps. For values of C 0 the Lorentzian has a peak value. See also Christensen and Bindslev (1982). variable (Sokal & Rohlf 1994, 2004; Quinn & Keough 10.A.8. Logarithmic Re-scaling 2002). Examples are: In statistical analysis, a logarithmic transformation of Linear linear (no transformation). either the dependent or the independent variable, or Semi-log (‘logistic’ function, logarithmic transforma- both at a time, often have advantages over an analysis based on a linear linear representation, especially when tion of the axis for the independent variable). variables are log-normal, i.e., when the relationship Log-linear (logarithmic transformation of the axis for between dependent and independent variables is rela- the dependent variable as a response). tive. As already mentioned, semi-log, log-linear, and Log log (logarithmic scale of both axes). logit analysis are examples of such ‘logistic’ transforma- Logit (equal to use of regula detri rule 1b in Box 1.1 tions (Hosmer & Lemeshow 2001). and a logarithmic transformation of both axes). Probit (normal distribution of the dependent vari- able and logarithmic transformation of both axes). 10.A.9. Semi-log Re-scaling Power (including the power 1 for reciprocal trans- formations as the Hanes, Lineweaver-Burk, Eadie- The load equation was designated as a ‘logistic function’ when its independent variable, concentration, was re- Scatchard, or Schild plot in Fig. 11.7). scaled by logarithmic transformation (Berkson 1944). Transformation of the modified Hill equation (Eq. 10.2) to a so-called ‘logistic’ function (van Rossum 1966; De Lean et al. 1978; Giraldo et al. 2002; Giraldo 2003) is Table 10.5. Regression models divided according to the the same as transforming the independent variable dose to experimental outcome* a log-scale, and therefore equal to a semi-log transfor- mation. Semi-log plotting is for better visualization Response and/or improved statistical analysis. Semi-log plots of classification Response type Regression model the modified Hill’s equation and its use in analysis is the Numerical same as ‘logistic-Hill analysis’. Continuous* Linear, curvi- and non-linear Count Poisson Time-to-event Proportional hazards 10.A.10. Log-linear Re-scaling Categorical Binary Logistic Log-linear transformation is a log(-istic) transformat- Ordinal Proportional odds ion of the dependent variable often used in statistical Nominal Polytomous logistic evaluation of numerical (Vittinghoff et al. 2005)/cate- *Modeled after Vittinghoff et al. (2005, pp. 318 319). gorical (Quinn & Keough 2002) variables with Poisson Chapter 10: Hill in Hell 277 distribution, typical for economic and social sciences, 10.A.13. SigmaPlot logistic equations but certainly also relevant in biological sciences (Finney 1971, 1978; Hille 2001, p. 55; Quinn & Keough 2002; A catalog of equations for the SigmaPlot software Vittinghoff et al. 2005). Log-linear models are some- product refers to an equation as a ‘non-linear logistic thing different (see for instance Quinn & Keough 2002, dose-response’ equation. This SigmaPlot Logistic equa- Sub-chapter 14.3). tion, listed under Regression Wizard-Equation Catalo- Log(-istic)-linear transformation is commonly used gue, Sigmoid Equations, is in fact the reverse Hill for dependent variables that are continuous, such as the equation (see Section 8.2.2 and Table 10.3). Thus, response in the washout or loading of drugs in some of the terminology for equations in the SigmaPlot compartmental analysis often following mono- or User’s guide is at variance with their use in this book. multi-exponential decay or rise (Riggs 1970; Jacquez 1972 or 1996 3rd edition of his book). In addition, REFERENCES dependent responses in time-dependent pharmaco- kinetic studies are suitable for log-transformation as Accomazzo MR, Cattaneo S, Nicosia S & Rovati GE. Bell-shaped curves for prostaglandin-induced modulation of adenylate cyclase: two well (Chou et al. 2006). mutually opposing effects. Eur J Pharmacol 454: 107 114, 2002. Adair GS. The hemoglobin system. VI. The oxygen dissociation curve of hemoglobin. J Biol Chem 63: 529 545, 1925. 10.A.11. Logit Re-scaling Allen DW, Guthe KF & Wyman J Jr. Further studies on the oxygen equilibrium of hemoglobin. J Biol Chem 187: 393 410, 1950. The term ‘logit’ does not always refer to the logit re- Arie¨ns EJ. Molecular Pharmacology. The Mode of Action of Biologically Active scaling. Logit transformation is taking the ratio of Compounds, Vol. I II, 1st ed. New York: Academic Press, 1964. positive responses over negative responses for depen- Barlow R & Blake JF. Hill coefficients and the logistic equation. Trends dent categorical variables, also designated ‘odds ratio’, Pharmacol Sci 11: 440 441, 1989. and where both sides of the formulations are logarith- Berenbaum MC. What is synergy? Pharmacol Rev 41: 93 141, 1989. Berkson J. Application of the logistic function to bio-assay. J Am Statist mically re-scaled. Or, the ratio of actual response or Assoc 39: 357 365, 1944. probability over remaining potential response or prob- Bernado J & Smith AFM. Bayesian Theory, 2nd ed., Chichester, Wiley & ability for dependent quantitative variables is formulated Sons, 2007. in an equation and thereafter both sides of the equation Biot JB. Introduction aux recherches de me´anique chemique, dans are re-scaled logarithmically, a ‘logit’ transformation. In lesquelles la lumie`re polarise´e est employe´e auxiliairement comme re´actif. Ann Chim Phys (3) 59: 206 345, 1860. effect, it is a logarithmic re-scaling of a transformed Blanquart C, Gonzalez-Yanes C & Issad T. Monitoring the activation dependent variable (Figs. 8.7C F and 8.8C F) (Liao state of insulin/insulin-like growth factor-1 hybrid receptors using 1994; Chou 2006). Meanwhile, observe that in a fine text bioluminescence resonance energy transfer. Mol Pharmacol 70: on ‘logit models’ in econometrics, a ‘logit analysis’ is the 1802 1811, 2006. same as logistic regression (Cramer 2003a). Beware, a Bohr C. Theoretische behandlung der quantitativen verha¨ltnisse bei der sauerstoff aufnahme des ha¨moglobins. Zentralblatt Physiol 17: logit analysis can involve a logit transformation as a link 682 688, 1904a. function in logistic regression! See Table 10.4. Bohr C. Die sauerstoff aufnahme des genuinen blutfarbstoffes und des aus dem blute dargestellten ha¨moglobins. Zentralblatt Physiol 17: 688 691, 1904b. 10.A.12. Probit Re-scaling and Integration Bohr C, Hasselbalch K & Krogh A. Ueber einen in biologischer beziehung wichtigen einfluss, den die kohlensa¨uerspannung des In probit analysis the dependent variable, response, is blutes auf dessen sauerstoffbindung u¨bt. Scand Arch Physiol 16: re-scaled according to the inverse standard normal 402 412, 1904. cumulative distribution function, also designated ‘the Bronnikov GE, Zhang SJ, Cannon B & Nedergaard J. A dual normal equivalent deviate’, and the independent vari- component analysis explains the distinctive kinetics of cAMP accumulation in brown adipocytes. J Biol Chem 274: 37770 37780, able can be re-scaled percentage-wise or logarithmically 1999. depending on the analyzed theory (Finney 1971). This is Brown AJ. Enzyme action. J Chem Soc 81: 373 388, 1902. an alternative way of logarithmic transformation com- Brown HT & Glendinning TA. The velocity of hydrolysis of starch by pared to the examples above and is often used in diastase, with some remarks on enzyme action. J Chem Soc 81: 388 individual effective dose (IED) experiments (Newman 400, 1902. Brown WEL & Hill AV. The oxygen/dissociation curve of blood, and its & McCloskey 2000; Pena-Llopis et al. 2003). thermodynamical basis. Proc Roy Soc B 94: 297 334, 1923. Methods involving integration of the area ‘under’ the Bublil EM & Yarden Y. The EGF receptor family: spearheading a dose-response curve are used for better statistics as in merger of signaling and therapeutics. Curr Opin Cell Biol 19: 124 the probit analysis (Gaddum 1933). Similar integration 134, 2007. approaches have been employed for better statistics in Burbridge EM, Burbridge GR, Fowler WA & Hoyle F. Synthesis of the elements in stars. Revs Modern Phys 29: 547 650, 1957. dose-response relations in noise analysis of the Lorent- Camazine S, Deneubourg J-L, Franks NR, Sneyd J, Theraulaz G & zian type (Christensen & Bindslev 1982; Lindemann Bonabeau E. Self-organization in Biological Systems, 1st ed. Prince- 1984). ton, NJ: Princeton University Press, 2001. 278 Part III: Test of Tools for Data Analysis
Changeux JP & Edelstein SJ. Allosteric mechanisms of signal transduc- Gaddum JH. Reports on biological standards III. Methods of biological tion. Science 308: 1424 1428, 2005. assay depending on a quantal response. Med Res Council Spec Report Chapman H, Piggot C, Andrews PW & Wann KT. Characterisation of No 183: 5 46, 1933. large-conductance calcium-activated potassium channels Gaddum JH. Bioassays and mathematics. Pharmacol Rev 5: 87 134, (BK(Ca)) in human NT2-N cells. Brain Res 1129: 15 25, 2007. 1953. Chou TC. Theoretical basis, experimental design, and computerized Garay RP & Garrahan PJ. The interaction of sodium and potassium simulation of synergism and antagonism in drug combination with the sodium pump in red cells. J Physiol 231: 297 325, 1973. studies. Pharmacol Rev 58: 621 681, 2006. Gelman A, Carlin JB, Stern HS & Rubin DB. Bayesian Data Analysis. Chou TC & Talalay P. A simple generalized equation for the analysis of Boca Raton, Chapman Hall/CRC, 2003. multiple inhibitions of Michaelis-Menten kinetic systems. J Biol Giraldo J. Empirical models and Hill coefficients. Trend Pharmacol Sci Chem 252: 6438 6442, 1977. 24: 63 65, 2003. Chou TC & Talalay P. Generalized equations for the analysis of Giraldo J, Vivas NM, Vila E & Badia A. Assessing the (a)symmetry of inhibitions of Michaelis-Menten and higher-order kinetic systems concentration-effect curves: empirical versus mechanistic models. with two or more mutually exclusive and nonexclusive inhibitors. Pharmacol Ther 95: 21 45, 2002. Eur J Biochem 115: 207 216, 1981. Goldstein RA. Emergent robustness in competition between autocata- Christensen O & Bindslev N. Fluctuation analysis of short-circuit lytic chemical networks. Orig Life Evol Biosph 36: 381 389, 2006. current in a warm-blooded sodium-retaining epithelium: site Gompertz B. On the nature of the function expressive of the law of current, density, and interaction with triamterene. J Membr Biol human mortality, and on a new mode of determining the value of 65: 19 30, 1982. life contingencies. Philos Trans Roy Soc (Lond) 36: 513 585, 1825. Cladman W & Chidiac P. Characterization and comparison of RGS2 Grafen A & Hails R. Modern Statistics for the Life Sciences, 1st ed. Oxford: and RGS4 as GTPase-activating proteins for m2 muscarinic Oxford University Press, 2002. receptor-stimulated G(i). Mol Pharmacol 62: 654 659, 2002. Greco WR, Bravo G & Parsons JC. The search for synergy: a critical Collins J, Mancilulli M, Hohlfeld R, Finch D, Sandri G & Shtatland E. A review from a response surface perspective. Pharmacol Rev 47: random number generator based on the logit transform of the 331 385, 1995. logistic variable. Comput Phys 6: 630 632, 1992. Griffin MT, Hsu JC, Shehnaz D & Ehlert FJ. Comparison of the Colquhoun D. Lectures on Biostatistics, 1st ed. Oxford: Oxford University pharmacological antagonism of M2 and M3 muscarinic receptors Press, 1971. expressed in isolation and in combination. Biochem Pharmacol 65: Colquhoun D. The relation between classical and cooperative models 1227 1241, 2003. for drug action. In: Drug Receptors. A Symposium, 1st ed., edited by Guldberg CM & Waage P. E´tudes sur les Affinite´s Chimiques. Programme de Rang HP. London: MacMillan Press, 1973. l’Universite´ pour le 1er Semestre 1867. Christiania: Brøgger & Cornish-Bowden A & Koshland DE Jr. Diagnostic uses of the Hill Christie, 1867. (Logit and Nernst) plots. J Mol Biol 95: 201 212, 1975. Hatzimanikatis V, Emmerling M, Sauer U & Bailey JE. Application of Cox RT. Algebra of Probable Inference. Baltimore, MA: Johns Hopkins mathematical tools for metabolic design of microbial ethanol University Press, 2001. production. Biotechnol Bioeng 58: 154 161, 1998. Cramer JS. Logit Models. From Economics and Other Fields, 1st ed. Heijnen JJ. Approximative kinetic formats used in metabolic network Cambridge: Cambridge University Press, 2003a. modeling. Biotechnol Bioeng 91: 534 545, 2005. Cramer JS. The Origins and Development of the Logit Model. Cambridge: Heinrich R & Rapoport SM. The utility of mathematical models for the Cambridge University Press, 2003b. understanding of metabolic systems. Biochem Soc Trans 11: 31 35, Davies P. The Goldilocks Enigma. Why is the Universe Just Right for Life? 1983. London: Penguin/Allen Lane, 2006. Henri V. Lois Ge´ne´rales de l’Action des Diastases, 1st ed. Paris: A. De Lean A, Munson PJ & Rodbard D. Simultaneous analysis of families Hermann, 1903. of sigmoidal curves: application to bioassay, radioligand assay, Hill AV. The mode of action of nicotine and curari, determined by the and physiological dose-response curves. Am J Physiol 235: E97 form of the concentration curve and the method of temperature E102, 1978. coefficients. J Physiol (Lond.) 39: 361 373, 1909. De Meyts P. Insulin and its receptor: structure, function and evolution. Hill AV. The possible effects of aggregation of the molecules of Bioessays 26: 1351 1362, 2004. haemoglobin on its dissociation curve. J Physiol 40: iv vii, 1910. Dixon M and Webb EC. Enzymes. London: Longman Group Ltd, 1979. Hill AV. The combinations of haemoglobin with oxygen and carbon Dobson AJ. Introduction to Generalized Linear Models, 2nd ed. London: monoxide. Biochem J 7: 471 480, 1913. Chapman and Hall/CRC, 2001. Hill AV. Trails and Trials in Physiology, 1st ed. London: Edward Arnold Edsall JT. Hemoglobin and the origins of the concept of allosterism. Ltd., 1965. Fed Proc 39: 226 235, 1980. Hill TL. Cooperativity Theory in Biochemistry. Steady-state and Equilibrium Ehlert FJ. Analysis of allosterism in functional assays. J Pharmacol Exp Systems. Berlin: Springer-Verlag, 1985. Ther 3: 740 754, 2005. Hille B. Ionic Channels of Excitable Membranes, 3rd ed. Sunderland: Ehlert FJ. On the analysis of ligand-directed signaling at G protein- Sinauer Associates, 2001. coupled receptors. Naunyn Schmiedebergs Arch Pharmacol Feb 6; Hofmeyr JH & Cornish-Bowden A. The reversible Hill equation: how to [Epub ahead of print]: 2008. incorporate cooperative enzymes into metabolic models. Comput Eisner DA & Richards DE. The interaction of potassium ions and ATP Appl Biosci 13: 377 385, 1997. on the sodium pump of resealed red cell ghosts. J Physiol 319: Hofmeyr JH, Rohwer JM & Snoep JL. Conditions for effective allosteric 403 418, 1981. feedforward and feedback in metabolic pathways. Syst Biol Ferrell JE. How responses get more switch-like as you move down a (Stevenage) 153: 327 331, 2006. protein cascade. Trend Biochem Sci 22: 288 289, 1997. Hornigold DC, Mistry R, Raymond PD, Blank JL & Challiss RA. Finney DJ. Probit Analysis, 3rd ed. Cambridge: Cambridge University Evidence for cross-talk between M2 and M3 muscarinic acetylcho- Press, 1971. line receptors in the regulation of second messenger and Ford ED. Scientific Method for Ecological Research (paperback ed.). extracellular signal-regulated kinase signalling pathways in Chi- Cambridge: Cambridge University Press, 2004. nese hamster ovary cells. Br J Pharmacol 138: 1340 1350, 2003. Chapter 10: Hill in Hell 279
Hosmer DW & Lemeshow S. Applied Logistic Regression. Hoboken, NJ: Lazareno S & Birdsall NJ. Detection, quantitation, and verification of John Wiley & Sons, 2000. allosteric interactions of agents with labeled and unlabeled Huang CY & Ferrell JE Jr. Ultrasensitivity in the mitogen-activated ligands at G protein-coupled receptors: interactions of strychnine protein kinase cascade. Proc Natl Acad Sci USA 93: 10078 10083, and acetylcholine at muscarinic receptors. Mol Pharmacol 48: 362 1996. 378, 1995. Hufner G. Ueber die verteilung des blutfarbstoffs zwischen kohle- Lee DH, Severin K & Ghadiri MR. Autocatalytic networks: the noxyd und sauerstoff; ein beitrag zur lehre von der chemischen transition from molecular self-replication to molecular ecosys- massenwirkung. J Pract Chem 30: 68 84, 1884. tems. Curr Opin Chem Biol 1: 491 496, 1997. Hufner G. Ueber das gesetz der dissociation des oxyhaemoglobins und Lee PM. Bayesian Statistics: An Introduction. London, Hodder Arnold u¨ber einige daran sich knu¨pfende wichtige fragen aus der Publications, 2004. biologie. Arch Anat Physiol Physiol Abt 1 27, 1890. Levitzki A. Cooperativity in Multisubunit Proteins The Basic Concepts, 1st Jacquez JA. Compartmental Analysis in Biology and Medicine. Kinetics of ed. Berlin: Springer-Verlag, 1978. Distribution of Tracer-labeled Materials, 1st ed. Amsterdam: Elsevier, Lindemann B. Analysis of additively contaminated Lorentzians by 1972. integration. Biophys J 46: 409 411, 1984. Janin J. The study of allosteric proteins. Prog Biophys Mol Biol 27: 77 Ma AW, Redka DS, Pisterzi LF, Angers S & Wells JW. Recovery of 120, 1973. Oligomers and Cooperativity When monomers of the M(2) Jardel Scientific. Table Curve 3D. Automated Surface Fitting Software, 1st muscarinic cholinergic receptor are reconstituted into phospho- ed. San Rafael, CA: Jardel Scientific, 1993. lipid vesicles. Biochemistry 46: 7907 7927, 2007. Jarv J, Toomela T & Karelson E. Dual effect of carbachol on the Malmstrøm BG. Cytochrome c oxidase: some current biochemical and muscarinic receptor. Biochem Mol Biol Int 30: 649 654, 1993. biophysical problems. Q Rev Biophys 6: 389 431, 1974. Jenkinson DH. Classical approaches to the study of drug-receptor Malthus TR. An Essay on the Principle of Population, 2nd ed. London: St interactions. In: Textbook of Receptor Pharmacology, 2nd ed., edited Paul’s Church Yard, 1798, 1803. Reprinted by Dent & Sons, by Foreman JC & Johansen T. Boca Raton, FL: CRC Press, 2003. London, 1967. Jenkinson DH, Barnard EA, Hoyer D, Humphrey PP, Leff P & Shankley Marin JM & Robert CP. Bayesian Core: A Practical Approach to Computa- NP. International Union of Pharmacology Committee on Recep- tional Bayesian Statistics. New York: Springer, 2007. tor Nomenclature and Drug Classification. IX. Recommendations May LT, Avlani VA, Langmead CJ, Herdon HJ, Wood MJ, Sexton PM, on terms and symbols in quantitative pharmacology. Pharmacol et al. Structure-function studies of allosteric agonism at M2 Rev 47: 255 266, 1995. muscarinic acetylcholine receptors. Mol Pharmacol 72: 463 476, Jonker DM, Vermeij DA, Edelbroek PM, Voskuyl RA, Piotrovsky VK & 2007. Danhof M. Pharmacodynamic analysis of the interaction between May RM. Simple mathematical models with very complicated dynamics. tiagabine and midazolam with an allosteric model that incorpo- Nature 261: 459 467, 1976. rates signal transduction. Epilepsia 44: 329 338, 2003. McFadden DL. http://nobelprize.org/economics/laureates/2000/ Kacser H & Burns JA. The control of flux. Biochem Soc Trans 23: 341 mcfadden-lecture.pdf, 2003. 366, 1995. McFadden DL. Economic choices. In: From Nobel Lectures, Economics Karlin A. On the application of ‘a plausible model’ of allosteric 1996 2000, Editor, edited by Persson T. Singapore: World proteins to the receptor for acetylcholine. J Theor Biol 16: 306 Scientific Publishing Co., 2003. 320, 1967. Meir E, Munro EM, Odell GM & Von Dassow G. Ingeneue: a versatile King GA. Growth of a hypercycle and comparison with conventional tool for reconstituting genetic networks, with examples from the autocatalysis. Biosystems 13: 225 234, 1981. segment polarity network. J Exp Zool 294: 216 251, 2002. Kingsland SE. Modeling Nature. Episodes in the History of Population Michaelis L & Menten ML. Die kinetik der invertinwirkung. Biochem Ecology, 2nd ed. Chicago, IL: The University of Chicago Press, Zeitsch 49: 333 369, 1913. 1995. Michel D. Cooperative equilibrium curves generated by ordered ligand Kleinbaum DG & Klein M. Logistic Regression, 2nd ed. New York: binding to multi-site molecules. Biophys Chem 129: 284 288, 2007. Springer, 2005. Monod J, Changeux JP & Jacob F. Allosteric proteins and cellular Kleinbaum DG, Kupper LL, Nizam A & Muller KE. Applied Regression control systems. J Mol Biol 6: 306 329, 1963. Analysis and Multivariable Methods, 4th ed. Pacific Grove, CA: Monod J, Wyman J & Changeux J-P. On the nature of allosteric Duxbury Press, 2007. transitions: a plausible model. J Mol Biol 12: 88 118, 1965. Klotz IM. Numbers of receptor sites from Scatchard graphs: facts and Motulsky H & Christopoulos A. Fitting Models to Biological Data Using fantasies. Science 217: 1247 1249, 1982. Linear and Nonlinear Regression: A Practical Guide to Curve Fitting, 1st Klotz IM. Ligand-receptor complexes: origin and development of the ed. Oxford: Oxford University Press, 2004. concept. J Biol Chem 279: 1 12, 2004. Muirhead H & Perutz M. Structure of haemoglobin. A three-dimen- Kubokawa M, Sohma Y, Hirano J, Nakamura K & Kubota T. sional Fourier synthesis of reduced human haemoglobin at 55 a Intracellular Mg2 influences both open and closed times of a resolution. Nature 199: 633 639, 1963. native Ca(2 )-activated BK channel in cultured human renal Nernst W. Die elektromotorische wirksamkeit der ionen. Z Phys Chem 4: proximal tubule cells. J Membr Biol 207: 69 89, 2005. 129 181, 1889. Lage K, Karlberg EO, Storling ZM, Olason PI, Pedersen AG, Rigina O, Neubig RR, Spedding M, Kenakin T & Christopoulos A. International Hinsby AM, Tumer Z, Pociot F, Tommerup N, Moreau Y and Union of Pharmacology Committee on Receptor Nomenclature Brunak S. A human phenome-interactome network of protein and Drug Classification. XXXVIII. Update on terms and symbols complexes implicated in genetic disorders. Nat Biotechnol 25: in quantitative pharmacology. Pharmacol Rev 55: 597 606, 2003. 309 316, 2007. Newman MC & McCloskey JT. The individual tolerance concept is not Langmuir I. The adsorption of gases on plane surfaces of glass, mica the sole explanation for the probit dose-effect model. Environ and platinum. J Am Chem Soc 40: 1361 1403, 1918. Toxicol Chem 19: 520 526, 2000. Lazareno S & Birdsall NJ. Estimation of competitive antagonist affinity Nowak MA. Evolutionary Dynamics. Exploring the Equations of Life, 1st ed. from functional inhibition curves using the Gaddum, Schild and Cambridge, MA: The Belknap Press of Harvard University Press, Cheng-Prusoff equations. Br J Pharmacol 109: 1110 1119, 1993. 2006. 280 Part III: Test of Tools for Data Analysis
O’Connel AA. Logistic Regression Models for Ordinal Response Variables. Sak K, Barnard EA & Jarv J. Dual effect of nucleotides on P2Y Thousand Oaks, CA: Sage Publications Ltd., 2006. receptors. IUBMB Life 50: 99 103, 2000. Ostwald W. Studien zur chemischen dynamik. Zweite Abhandlung. Die Samuels ML & Witmer JA. Statistics for the Life Sciences, 3rd ed. Upper einwirkung der sa¨uren auf methylacetat. J Pract Chem 28 (136): Saddle River, NJ: Prentice Hall, 2002. 449 495, 1883. Scaramellini C, Bennett G & Leff P. Analysis of agonist agonist Ostwald W. Lehrbuch der Allgemeinen Chemie. Zweiten Bandes Zweiter Teil: interactions: the crucial influence of curve shape. J Pharmacol Verwandtschaftslehere Erster Teil, Vol. 2. Leipzig: Verlag von W. Toxicol Methods 37: 167 178, 1997. Engelmann, 1902. Segel IH. Enzyme kinetics. Behavior and analysis of rapid equilibrium and Pampel FC. Logistic Regression: A Primer. Sage Quantitative Applications in steady-state enzyme systems. New York: Wiley & Sons (reissued 1993), the Social Sciences, Vol. Series No. 132, 1st ed. Thousand Oaks, CA: 1975. Sage Publications, 2000. Short TG, Ho TY, Minto CF, Schnider TW & Shafer SL. Efficient Pauling L. The oxygen equilibrium of hemoglobin and its structural trial design for eliciting a pharmacokinetic-pharmacodynamic interpretation. Proc Nat Acad Sci USA 21: 186 191, 1935. model-based response surface describing the interaction between Pearl R & Reed LJ. On the rate of growth of the population of the two intravenous anesthetic drugs. Anesthesiology 96: 400 408, 2002. United States since 1790 and its mathematical representation. † Proc Natl Acad Sci USA 6: 275 288, 1920. SigmaPlot 8. SigmaPlot 9.0 User’s Manual, 1st ed. Point Richmond: Pearl R. The Biology of Death, 1st ed. Philadelphia, PA: Lippincott Co., Systat Software, Inc., 2004. 1922. Sivia D & Skilling J. Data Analysis: A Bayesian Tutorial. Oxford: Oxford Peitgen H-O, Ju¨rgens H & Saupe D. Classical Fractals and Self-similarity, University Press, 2006. Chaos and Fractals: New Frontiers of Science, 1st ed. Berlin: Springer- Smith JM. Mathematical Ideas in Biology, 1st ed. Cambridge: Cambridge Verlag, 1992. University Press, 1968. Pena-Llopis S, Ferrando MD & Pena JB. Fish tolerance to organopho- Snedecor GW & Cochran WG. Statistical Methods, 6th ed. Ames, IA: The sphate-induced oxidative stress is dependent on the glutathione Iowa State University Press, 1967. metabolism and enhanced by N-acetylcysteine. Aquat Toxicol 65: Sokal RR & Rohlf FJ. Biometry. The Principles and Practice of Statistics in Biological Research, 3rd ed. New York: WH Freeman, 1994. 337 360, 2003. Sorribas A, Hernandez-Bermejo B, Vilaprinyo E & Alves R. Coopera- Perutz M. X-ray analysis of hæmoglobin. Nature 149: 491 494, 1942. tivity and saturation in biochemical networks: a saturable form- Perutz M. Structure of hemoglobin. Brookhaven Symp Biol 13: 165 183, alism using Taylor series approximations. Biotechnol Bioeng 97: 1960. 259 277, 2007. Perutz M. Stereochemistry of cooperative effects in haemoglobin. Spivak CE. Correlation among Hill parameters reflect models of Nature 228: 726 739, 1970. activating ligand-gated ion channels. Trend Pharmacol Sci 16: 39 Perutz M. Mechanisms of Cooperativity and Allosteric Regulation in Proteins, 42, 1995. 1st ed. Cambridge: Cambridge University Press, 1990. Stoker AW. Protein tyrosine phosphatases and signalling. J Endocrinol Perutz M, Bolton W, Diamond R, Muirhead H & Watson HC. Structure 185: 19 33, 2005. of haemoglobin. An X-ray examination of reduced horse hae- Svedberg T & Fahraeus R. A new method for the determination of the moglobin. Nature 203: 687 690, 1964. molecular weights of the proteins. J Am Chem Soc 48: 430 438, Perutz M, Wilkinson AJ, Paoli M & Dodson GG. The stereochemical 1926. mechanism of the cooperative effects in hemoglobin revisited. Szabadi E. A model of two functionally antagonistic receptor popula- Annu Rev Biophys Biomol Struct 27: 1 34, 1998. tions activated by the same agonist. J Theor Biol 69: 101 112, 1977. Pliska V. Models to explain dose-response relationships that exhibit a Taleb O & Betz H. Expression of the human glycine receptor alpha 1 downturn phase. Trends Pharmacol Sci 15: 178 181, 1994. subunit in Xenopus oocytes: apparent affinities of agonists increase Prentice RL. A generalization of the probit and logit methods for dose at high receptor density. EMBO J 13: 1318 1324, 1994. response curves. Biometrics 32: 761 768, 1976. Thron CD. On the analysis of pharmacological experiments in terms of Quinn GP & Keough MJ. Experimental Design and Data Analysis for an allosteric receptor model. Mol Pharmacol 9: 1 9, 1973. Biologists, 1st ed. Cambridge: Cambridge University Press, 2002. Trankle C, Dittmann A, Schulz U, Weyand O, Buller S, Johren K, Ragozzino D, Woodward RM, Murata Y, Eusebi F, Overman LE & Heller E, Birdsall NJ, Holzgrabe U, Ellis J, Holtje HD & Mohr K. Miledi R. Design and in vitro pharmacology of a selective gamma- Atypical muscarinic allosteric modulation: cooperativity between aminobutyric acidC receptor antagonist. Mol Pharmacol 50: 1024 modulators and their atypical binding topology in muscarinic M2 1030, 1996. and M2/M5 chimeric receptors. Mol Pharmacol 68: 1597 1610, Reed LJ & Berkson J. The application of the logistic function to 2005. experimental data. J Phys Chem 33: 760 779, 1929. Tucek S, Michal P & Vlachova V. Modelling the consequences of Richards FJ. A flexible growth function for empirical use. J Exp Bot 10: receptor-G-protein promiscuity. Trends Pharmacol Sci 23: 171 176, 290 300, 1959. 2002. Robertson TB. On the normal rate of growth of an individual and its Tuk B & van Oostenbruggen MF. Solving inconsistencies in the biochemical significance. Arch Entwickmech Organ 25: 581 614, analysis of receptor-ligand interactions. Trend Pharmacol Sci 17: 1908a. 403 409, 1996. Robertson TB. Further remarks on the normal rate of growth of an Turchin P. Complex Population Dynamics: A Theoretical/Empirical Synthesis. individual and its biochemical significance. Arch Entwickmech Princeton, NJ: Princeton University Press, 2003. Organ 26: 108 118, 1908b. Van’t Hoff JH. Die grenzebene, ein beitrag zur kenntniss der Roughton FJW, Otis AB & Lyster RLJ. The determination of the esterbildung. Berichte D Chem Gesellshaft 10: 669 678, 1887. individual equilibrium constants of the four intermediate reac- Van der Graaf PH & Schoemaker RC. Analysis of asymmetry of agonist tions between oxygen and sheep haemoglobin. Proc Roy Soc B 144: concentration-effect curves. J Pharmacol Toxicol Methods 41: 107 29 54, 1955. 115, 1999. Rovati GE & Nicosia S. Lower efficacy: interaction with an inhibitory van Rossum JM. Receptor theory in enzymology. In: Molecular receptor or partial agonism? Trends Pharmacol Sci 15: 140 144, Pharmacology. The Mode of Action of Biologically Active Compounds, 1994. 1st ed., edited by Arie¨ns EJ. New York: Academic Press, 1964. Chapter 10: Hill in Hell 281 van Rossum JM. Limitations of molecular pharmacology. Some Wells JW. Analysis and interpretation of binding at equilibrium. In: implications of the basic assumptions underlying calculations Receptor ligand Interactions. A Practical Approach, 1st ed., edited by on drug receptor interactions and the significance of biological Hulme EC. Oxford: IRL Press at Oxford University Press, 1992. drug parameters. Adv Drug Res 3: 189 234, 1966. Williams BR, Barber R & Clark RB. Kinetic analysis of agonist-induced Verhulst P-F. Notice sur la loi que la population suit dans son down-regulation of the beta(2)-adrenergic receptor in BEAS-2B accroissement. Corrsp Math Phys 10: 113 116, 1838. cells reveals high- and low-affinity components. Mol Pharmacol 58: Verhulst P-F. Recherches mathe´matiques sur la loi d’accroissement de 421 430, 2000. la population. Bruxelles, Acad Sci Me´m 18: 1 41, 1845. Winkler RL. An Introduction to Bayesian Inference and Decision, 2nd ed. Verhulst P-F. Deuxie`me me´moire sur la loi d’accroissement de la Sugar Land, TX: Probabilistic Publishing; 2003. population. Me´m l’Academie Roy Sci, Lett & Baux-Arts Belgique 20: Witte RS & Witte JS. Statistics, 7th ed. Hoboken, NJ: John Wiley & Sons, 1 32, 1847. Inc., 2004. Verveen AA & DeFelice LJ. Membrane noise. In: Progress in Biophysics Wyman J & Gill SJ. Binding and Linkage. Functional Chemistry of Biological and Membrane Biology, Vol. 28, edited by Butler JAV & Noble D. Macromolecules, 1st ed. New York: Freeman, 1990. Oxford: Pergamon, 1974. Yifrach O. Hill coefficient for estimating the magnitude of coopera- Visser D, Schmid JW, Mauch K, Reuss M & Heijnen JJ. Optimal re- tivity in gating transitions of voltage-dependent ion channels. design of primary metabolism in Escherichia coli using linlog Biophys J 87: 822 830, 2004. kinetics. Metab Eng 6: 378 390, 2004. Yule GU. The growth of population and the factors that control it. J Roy Vittinghoff E, Shiboski SC, Glidden DV & McCulloch CE. Regression Statist Soc 88: 1 57, 1925. Methods in Biostatistics: Linear, Logistic, Survival, and Repeated Zhang Y, Niu X, Brelidze TI & Magleby KL. Ring of negative charge in Measures Models. New York: Springer Verlag, 2005. Vogel WK, Mosser VA, Bulseco DA & Schimerlik MI. Porcine m2 BK channels facilitate s block by intracellular Mg2 and muscarinic acetylcholine receptor-effector coupling in Chinese polyamines through electrostatics. J Gen Physiol 128: 185 202, hamster ovary cells. J Biol Chem 270: 15485 15493, 1995. 2006. Voit EO. Computational Analysis of Biochemical Systems: A Practical Guide Zorkadis V, Karras DA & Panayotou M. Efficient information for Biochemists and Molecular Biologists, 1st ed. Cambridge: Cam- theoretic strategies for classifier combination, feature extraction bridge University Press, 2000. and performance evaluation in improving false positives and Weiss JN. The Hill equation revisited: uses and misuses. FASEB J 11: false negatives for spam e-mail filtering. Neural Netw 18: 799 835 841, 1997. 807, 2005. 282 Part III: Test of Tools for Data Analysis