Abstract
It is easy to assume that a potent drug is simply one that acts at a low concentration. This paper counts that the span from first effect to full effect is set by a different quantity from the effective concentration. No new theorem or law is claimed.
First, the span is $81^{1/n}$-fold: solving for $10$ % and $90$ % effect by bisection gives $6561$-fold at $n=0.5$, $81$ at $1$, $9$ at $2$ and $3$ at $4$, matching $81^{1/n}$ within a relative difference of $10^{-9}$ (Section 2).
Second, and this is the core. Moving the effective concentration does not move the span: with $n=2$, setting the effective concentration to $0.01$, $1$ and $100$, ten-thousand-fold apart, gives a span of $9.000000$ each time (Section 2).
Third, two binding sites without mutual help still give a coefficient of $1$: with independent sites, $n=1.000000$ and the span is $81$-fold, the same as a single site (Section 3).
Fourth, stronger help narrows the span, but not beyond the number of sites: with a help factor $c$ of $10$, $n=\mathbf{1.519494}$; at $100$, $1.818182$; at $10000$, $1.980198$; the span narrows only to $9.243221$-fold, never below $9$ (Section 3).
Fifth, numerical slopes match the formula: slopes computed numerically at half binding matched $2/(1+1/\sqrt{c})$ within $10^{-6}$ (Section 3).
Sixth, the separator is whether one asks about position, the concentration at which it acts, or spread, the range over which it goes from starting to full effect: position is set by the effective concentration and spread by cooperativity, and the two move independently (Section 4).
Keywords: Hill coefficient, dose-response, cooperativity, Adair equation, effective concentration.
Scope (what is not claimed)
No new theorem or law is claimed: the Hill equation, the $81^{1/n}$-fold span from $10$ % to $90$ %, the Adair equation for two binding sites, and that its coefficient does not exceed the number of sites, are all known. No real values are claimed: concentrations and coefficients are model values matching no drug and no receptor. No advice is given. What amount should be used is not stated. Mechanisms are not entered: models in which a change of shape produces the mutual help, and binding rates, are not treated. Relation to earlier papers: Paper 679 showed that a buffer does not hold the pH but only the neighbourhood of its pKa. This paper stands beside it. There the span of an equation for one proton binding at one site was seen in pH. Here the same form of equation has its coefficient varied, counted through the mutual help of two binding sites. What is added is solving the span by bisection for four values of $n$ and matching the formula, checking that moving the effective concentration ten-thousand-fold leaves the span unchanged, measuring slope and span for each help factor with two sites, and placing the separator on position versus spread.
1. Notation and the Question
Setting: the effect $R$ at concentration $C$ is given by the effective concentration $\mathrm{EC}_{50}$ (where the effect is half) and a coefficient $n$. Alongside it is the Adair equation for two binding sites, where $c$ is how much binding at one site makes binding at the other easier.
$$R=\frac{C^{n}}{\mathrm{EC}_{50}^{\,n}+C^{n}},\qquad \frac{C_{90}}{C_{10}}=81^{1/n},\qquad \theta=\frac{x+c\,x^{2}}{1+2x+c\,x^{2}}$$$x$ is concentration multiplied by the affinity of one site, and $\theta$ the fraction bound. The coefficient was computed numerically as the slope of $\log(\theta/(1-\theta))$ against $\log x$ at half binding.
Question: what sets the span of concentration from first effect to full effect?
2. Coefficient and Span
Computation 2.1: the concentrations for $10$ % and $90$ % effect were solved by bisection ($\mathrm{EC}_{50}=1$).
| $n$ | concentration at $10$ % | at $90$ % | span (fold) | span (decades) |
|---|---|---|---|---|
| $0.5$ | $0.012346$ | $81.000000$ | $6561.000000$ | $3.816970$ |
| $1$ | $0.111111$ | $9.000000$ | $81.000000$ | $1.908485$ |
| $2$ | $0.333333$ | $3.000000$ | $9.000000$ | $0.954243$ |
| $4$ | $0.577350$ | $1.732051$ | $3.000000$ | $0.477121$ |
Observation 2.2 this is the core. Moving the effective concentration does not move the span. With $n=2$ and effective concentrations $0.01$, $1$ and $100$, the span was $9.000000$-fold each time: the effective concentration only shifts the curve left or right; it does not change how steeply it rises. What narrows the span is $n$, and the bisection spans matched $81^{1/n}$ within a relative difference of $10^{-9}$ for all four $n$.
3. Two Binding Sites
Computation 3.1: varying the help factor $c$ between two sites, the coefficient at half binding and the span from $10$ % to $90$ % were computed.
| $c$ | numerical coefficient | formula $2/(1+1/\sqrt{c})$ | span (fold) |
|---|---|---|---|
| $1$ | $1.000000$ | $1.000000$ | $81.000000$ |
| $10$ | $1.519494$ | $1.519494$ | $20.436504$ |
| $100$ | $1.818182$ | $1.818182$ | $11.741239$ |
| $10000$ | $1.980198$ | $1.980198$ | $9.243221$ |
Observation 3.2 the number of sites only caps the coefficient; mutual help sets it. With independent sites ($c=1$) the coefficient is $1.000000$ and the span $81$-fold, the same as one site: stronger help raises the coefficient to $1.519494$, $1.818182$ and $1.980198$ but never beyond $2$, and the span narrows only to $9.243221$-fold, not below the $9$ of $n=2$. Numerical coefficients matched the formula within $10^{-6}$.
4. The Separator
Observation 4.1 the separator is whether one asks about position, the concentration at which it acts, or spread, the range over which it goes from starting to full effect. Position is set by the effective concentration, which shifts the curve: spread is set by cooperativity, which steepens its rise. The two move independently, so acting at a low concentration and changing effect sharply with a small change in concentration are different properties.
Observation 4.2 placed beside Paper 679. 679 showed that a buffer holds only the neighbourhood of its pKa. There the span of an equation for one proton at one site was seen in pH. That equation has coefficient $1$, taking $1.908485$ decades from $10$ % to $90$ %. Here the coefficient of the same form was varied, counting that mutual help is what narrows the span.
Observation 4.3 what is not said. Real drug or receptor values, amounts to use, and mechanisms of help through shape change are not treated. What is shown is that the span of concentration is set not by the effective concentration but by cooperativity.
5. Census and Closing
- span $81^{1/n}$-fold (Section 1): the Hill equation.
- $81$-fold at $n=1$ (Computation 2.1): $1.908485$ decades.
- $9$ at $n=2$, $3$ at $n=4$ (Computation 2.1): steeper.
- $9.000000$ despite a ten-thousand-fold shift (Observation 2.2): the core.
- bisection matches the formula (Observation 2.2): within $10^{-9}$.
- coefficient $1$ with two independent sites (Computation 3.1): span $81$.
- $\mathbf{1.519494}$ at $c=10$ (Computation 3.1): help raises it.
- $1.980198$ at $c=10000$ (Computation 3.1): not beyond $2$.
- span down to $9.243221$ (Observation 3.2): not below $9$.
- position, or spread (Observation 4.1): the separator; Paper 679.
Closing: The span of concentration from first effect to full effect is set not by the effective concentration but by the degree of cooperativity. The span from $10$ % to $90$ % is $81^{1/n}$-fold, and with $n=2$ it stays $9.000000$-fold even when the effective concentration moves ten-thousand-fold, because the effective concentration only shifts the curve and does not change how steeply it rises. Even with two binding sites, without mutual help the coefficient is $1$ and the span $81$-fold, and however strong the help, the coefficient reaches only $1.980198$ and the span does not narrow below $9.243221$-fold: the number of sites only caps it; mutual help sets it. Placed among the earlier papers: beside the span of single-site binding that Paper 679 saw in pH, we placed what narrows the span in the same form of equation. To be honest: concentrations and coefficients are model values, and neither the mechanism of help nor binding rates are included.
References
- Brunton, L. L., Hilal-Dandan, R., & Knollmann, B. C. (Eds.). (2018). Goodman & Gilman’s The Pharmacological Basis of Therapeutics (13th ed.). McGraw-Hill.
- Hill, A. V. (1910). The Possible Effects of the Aggregation of the Molecules of Haemoglobin on Its Dissociation Curves. The Journal of Physiology.
- Yamagishi, Y. (2026). A Buffer Does Not Hold the pH; It Holds Only the Neighbourhood of Its pKa [Paper 679]. YOI: https://www.yamagishiyuuki.com/yoi/679; DOI: 10.5281/zenodo.23014789
Reference
Yamagishi, Y. (2026). The Concentration Span from First to Full Effect Is Set Not by the Effective Concentration but by Cooperativity [Paper 878]. YOI: https://www.yamagishiyuuki.com/yoi/878
ORCID: 0009-0006-8587-3450
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The record page for this paper (in Japanese, with the PDFs in each language and the date proof): YOI 878. This paper is a preprint and has not been peer reviewed.