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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 constantsCritical value of shape parameterShape 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=( qr ) 1/2 q+r  ac = 1/2 a=Δ11/21+Δ1 a=(1+Γ2)1/2 2 
3    
a=( qr+qs+rs ) 1/2 q+r+s  ac = (1/3)1/2 a=(Δ1+Δ2+Δ1Δ2)1/21+Δ1+Δ2 a=[(1+Γ2)+2(1+Γ2)1/2Δ2]1/2 2+(1+Γ2)1/2Δ2 
b=( qrs ) 1/3 q+r+s  bc = 1/3 b=(Δ1Δ2)1/31+Δ1+Δ2 b=(1+Γ2)1/2Δ21/3 2+(1+Γ2)1/2Δ2 
4    
a=( qr+qs+qt+rs+rt+st ) 1/2 q+r+s+t  ac = (61/2)/4 a=(Δ1+Δ2+Δ3+Δ1Δ2+Δ1Δ3+Δ2Δ3)1/21+Δ1+Δ2+Δ3 a=[(1+Γ2)(1+Δ2Δ3)+2(1+Γ2)1/2(Δ2Δ3)]1/22+(1+Γ2)1/2(Δ2+Δ3) 
b=( qrs+qrt+qst+rst ) 1/3 q+r+s+t  bc = (41/3)/4 b=(Δ1Δ2+Δ1Δ3+Δ2Δ3+Δ1Δ2Δ3)1/31+Δ1+Δ2+Δ3 b=[(1+Γ2)3/2(Δ2+Δ3)+2(1+Γ2)(Δ2Δ3)]1/32+(1+Γ2)1/2(Δ2+Δ3) 
c=( qrst ) 1/4 q+r+s+t  cc = 1/4 c=(Δ1Δ2Δ3)1/41+Δ1+Δ2+Δ3 c=(1+Γ2)1/2 (Δ2Δ3)1/42+(1+Γ2)1/2(Δ2+Δ3) 
n and shape parameters as function of PF constantsCritical value of shape parameterShape 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=( qr ) 1/2 q+r  ac = 1/2 a=Δ11/21+Δ1 a=(1+Γ2)1/2 2 
3    
a=( qr+qs+rs ) 1/2 q+r+s  ac = (1/3)1/2 a=(Δ1+Δ2+Δ1Δ2)1/21+Δ1+Δ2 a=[(1+Γ2)+2(1+Γ2)1/2Δ2]1/2 2+(1+Γ2)1/2Δ2 
b=( qrs ) 1/3 q+r+s  bc = 1/3 b=(Δ1Δ2)1/31+Δ1+Δ2 b=(1+Γ2)1/2Δ21/3 2+(1+Γ2)1/2Δ2 
4    
a=( qr+qs+qt+rs+rt+st ) 1/2 q+r+s+t  ac = (61/2)/4 a=(Δ1+Δ2+Δ3+Δ1Δ2+Δ1Δ3+Δ2Δ3)1/21+Δ1+Δ2+Δ3 a=[(1+Γ2)(1+Δ2Δ3)+2(1+Γ2)1/2(Δ2Δ3)]1/22+(1+Γ2)1/2(Δ2+Δ3) 
b=( qrs+qrt+qst+rst ) 1/3 q+r+s+t  bc = (41/3)/4 b=(Δ1Δ2+Δ1Δ3+Δ2Δ3+Δ1Δ2Δ3)1/31+Δ1+Δ2+Δ3 b=[(1+Γ2)3/2(Δ2+Δ3)+2(1+Γ2)(Δ2Δ3)]1/32+(1+Γ2)1/2(Δ2+Δ3) 
c=( qrst ) 1/4 q+r+s+t  cc = 1/4 c=(Δ1Δ2Δ3)1/41+Δ1+Δ2+Δ3 c=(1+Γ2)1/2 (Δ2Δ3)1/42+(1+Γ2)1/2(Δ2+Δ3) 

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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