Table 1. Relationship between shape... | Rockefeller University Press

Table 1.

Relationship between shape parameters of n-site binding curves and one-site binding parameters derived from PF expansion

n and shape parameters as function of PF constants	Critical value of shape parameter	Shape parameters as function of spacing factors (q, r, s, t = real)	Shape parameters as function of spacing and slope factors (q, r = complex conjugates)
2
$a = \frac{{(q r)}^{1 / 2}}{q + r}$	a_c = 1/2	$a = \frac{{Δ_{1}}^{1 / 2}}{1 + Δ_{1}}$	$a = \frac{{(1 + Γ^{2})}^{1 / 2}}{2}$
3
$a = \frac{{(q r + q s + r s)}^{1 / 2}}{q + r + s}$	a_c = (1/3)^1/2	$a = \frac{{(Δ_{1} + Δ_{2} + Δ_{1} Δ_{2})}^{1 / 2}}{1 + Δ_{1} + Δ_{2}}$	$a = \frac{{[(1 + Γ^{2}) + 2 {(1 + Γ^{2})}^{1 / 2} Δ_{2}]}^{1 / 2}}{2 + {(1 + Γ^{2})}^{1 / 2} Δ_{2}}$
$b = \frac{{(q r s)}^{1 / 3}}{q + r + s}$	b_c = 1/3	$b = \frac{{(Δ_{1} Δ_{2})}^{1 / 3}}{1 + Δ_{1} + Δ_{2}}$	$b = \frac{{(1 + Γ^{2})}^{1 / 2} {Δ_{2}}^{1 / 3}}{2 + {(1 + Γ^{2})}^{1 / 2} Δ_{2}}$
4
$a = \frac{{(q r + q s + q t + r s + r t + s t)}^{1 / 2}}{q + r + s + t}$	a_c = (6^1/2)/4	$\begin{array}{l} a = \\ \frac{{(Δ_{1} + Δ_{2} + Δ_{3} + Δ_{1} Δ_{2} + Δ_{1} Δ_{3} + Δ_{2} Δ_{3})}^{1 / 2}}{1 + Δ_{1} + Δ_{2} + Δ_{3}} \end{array}$	$\begin{array}{l} a = \\ \frac{{[(1 + Γ^{2}) (1 + Δ_{2} Δ_{3}) + 2 {(1 + Γ^{2})}^{1 / 2} (Δ_{2} Δ_{3})]}^{1 / 2}}{2 + {(1 + Γ^{2})}^{1 / 2} (Δ_{2} + Δ_{3})} \end{array}$
$b = \frac{{(q r s + q r t + q s t + r s t)}^{1 / 3}}{q + r + s + t}$	b_c = (4^1/3)/4	$b = \frac{{(Δ_{1} Δ_{2} + Δ_{1} Δ_{3} + Δ_{2} Δ_{3} + Δ_{1} Δ_{2} Δ_{3})}^{1 / 3}}{1 + Δ_{1} + Δ_{2} + Δ_{3}}$	$\begin{array}{l} b = \\ \frac{{[{(1 + Γ^{2})}^{3 / 2} (Δ_{2} + Δ_{3}) + 2 (1 + Γ^{2}) (Δ_{2} Δ_{3})]}^{1 / 3}}{2 + {(1 + Γ^{2})}^{1 / 2} (Δ_{2} + Δ_{3})} \end{array}$
$c = \frac{{(q r s t)}^{1 / 4}}{q + r + s + t}$	c_c = 1/4	$c = \frac{{(Δ_{1} Δ_{2} Δ_{3})}^{1 / 4}}{1 + Δ_{1} + Δ_{2} + Δ_{3}}$	$c = \frac{{(1 + Γ^{2})}^{1 / 2} {(Δ_{2} Δ_{3})}^{1 / 4}}{2 + {(1 + Γ^{2})}^{1 / 2} (Δ_{2} + Δ_{3})}$

n and shape parameters as function of PF constants	Critical value of shape parameter	Shape parameters as function of spacing factors (q, r, s, t = real)	Shape parameters as function of spacing and slope factors (q, r = complex conjugates)
2
$a = \frac{{(q r)}^{1 / 2}}{q + r}$	a_c = 1/2	$a = \frac{{Δ_{1}}^{1 / 2}}{1 + Δ_{1}}$	$a = \frac{{(1 + Γ^{2})}^{1 / 2}}{2}$
3
$a = \frac{{(q r + q s + r s)}^{1 / 2}}{q + r + s}$	a_c = (1/3)^1/2	$a = \frac{{(Δ_{1} + Δ_{2} + Δ_{1} Δ_{2})}^{1 / 2}}{1 + Δ_{1} + Δ_{2}}$	$a = \frac{{[(1 + Γ^{2}) + 2 {(1 + Γ^{2})}^{1 / 2} Δ_{2}]}^{1 / 2}}{2 + {(1 + Γ^{2})}^{1 / 2} Δ_{2}}$
$b = \frac{{(q r s)}^{1 / 3}}{q + r + s}$	b_c = 1/3	$b = \frac{{(Δ_{1} Δ_{2})}^{1 / 3}}{1 + Δ_{1} + Δ_{2}}$	$b = \frac{{(1 + Γ^{2})}^{1 / 2} {Δ_{2}}^{1 / 3}}{2 + {(1 + Γ^{2})}^{1 / 2} Δ_{2}}$
4
$a = \frac{{(q r + q s + q t + r s + r t + s t)}^{1 / 2}}{q + r + s + t}$	a_c = (6^1/2)/4	$\begin{array}{l} a = \\ \frac{{(Δ_{1} + Δ_{2} + Δ_{3} + Δ_{1} Δ_{2} + Δ_{1} Δ_{3} + Δ_{2} Δ_{3})}^{1 / 2}}{1 + Δ_{1} + Δ_{2} + Δ_{3}} \end{array}$	$\begin{array}{l} a = \\ \frac{{[(1 + Γ^{2}) (1 + Δ_{2} Δ_{3}) + 2 {(1 + Γ^{2})}^{1 / 2} (Δ_{2} Δ_{3})]}^{1 / 2}}{2 + {(1 + Γ^{2})}^{1 / 2} (Δ_{2} + Δ_{3})} \end{array}$
$b = \frac{{(q r s + q r t + q s t + r s t)}^{1 / 3}}{q + r + s + t}$	b_c = (4^1/3)/4	$b = \frac{{(Δ_{1} Δ_{2} + Δ_{1} Δ_{3} + Δ_{2} Δ_{3} + Δ_{1} Δ_{2} Δ_{3})}^{1 / 3}}{1 + Δ_{1} + Δ_{2} + Δ_{3}}$	$\begin{array}{l} b = \\ \frac{{[{(1 + Γ^{2})}^{3 / 2} (Δ_{2} + Δ_{3}) + 2 (1 + Γ^{2}) (Δ_{2} Δ_{3})]}^{1 / 3}}{2 + {(1 + Γ^{2})}^{1 / 2} (Δ_{2} + Δ_{3})} \end{array}$
$c = \frac{{(q r s t)}^{1 / 4}}{q + r + s + t}$	c_c = 1/4	$c = \frac{{(Δ_{1} Δ_{2} Δ_{3})}^{1 / 4}}{1 + Δ_{1} + Δ_{2} + Δ_{3}}$	$c = \frac{{(1 + Γ^{2})}^{1 / 2} {(Δ_{2} Δ_{3})}^{1 / 4}}{2 + {(1 + Γ^{2})}^{1 / 2} (Δ_{2} + Δ_{3})}$

Symbol n represents number of binding sites. Factors a, b, and c are shape parameters of n-site total binding curves (see main text). Factors q, r, s, and t are association constants of one-site binding curve components obtained by PF decomposition of n-site binding curves. Critical values of shape parameters are values above which PF constants (q, r, s, etc.) cannot all have real values. Spacing parameters (Δ₁, Δ₂, Δ₃) are defined in Fig. 13. The parameter Γ is the ratio of the imaginary to the real parts of complex conjugate PF constants (assumed to be q and r in the last column).

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