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Table 3 Conditions for alloster effects on initial efficacy and maximal efficacy in ATSM

From: Allosteric transition: a comparison of two models

Assumptions for product Reduced equation Lower level product assumptions Reduced equation at lower level product assumptions IntEff / MaxEff, their dependence on product of
parameters ·[M]
possible allo-agonism of IntEff for [orthoster] → 0: L/[L+((1+A m· M)/(1+b·A m· M))]
b·A m· M >> 1 L·b·A m·M = X A m·M >> 1 L·b /(L·b +1) L·b vs 1
X/(X+1+A m· M)
A m·M = 1 L·b /(L·b +2) L·b vs 2
A m·M << 1 L·b /(L·b +1) L·b vs 1
b·A m· M = 1 L·2/(L·2+1+A m· M) A m· M >> 1 L·2/(L·2+A m· M) L·2 vs A m· M
A m· M = 1 L/(L+1) L vs 1
A m· M << 1 L·2/(L·2+1) L·2 vs 1
b·A m· M << 1 L/(L+1+A m·M) A m· M >> 1 L/(L+A m· M) L vs A m· M
A m· M = 1 L/(L+2) L vs 2
A m· M << 1 L/(L+1) L vs 1
possible allo-synergy of MaxEff for [orthoster] →∞ : L·a/[L·a+((1+c·A m·M)/(1+b·c·d·A m·M))]
b·c·d·A m·M >> 1 with L·a·b·c·d·A m·M =Y c·A m·M >> 1 L·a·b·d/(L·a·b·d+1) L·a·b·d vs 1
Y/(Y+1+c·A m·M) c·A m·M = 1 L·a·b·d/(L·a·b·d+2) L·a·b·d vs 2
b·d >> 1
c·A m·M << 1 Y/(Y+1) Y vs 1
b·d >>> 1
b·c·d·A m·M = 1 with L·a·2 = Z c·A m·M >> 1 Z/(Z+c·A m·M) Z vs c·A m·M
Z/(Z+1+c·A m·M) c·A m·M = 1 L·a/(L·a+1) L·a vs 1
b·d = 1
c·A m·M << 1 L·a·2/(L·a·2+1) L·a·2 vs 1
b·c·d·A m·M << 1 L·a/(L·a+1+c·A m·M) c·A m·M >> 1 L·a/(L·a+c·A m·M) L·a vs c·A m·M
c·A m·M = 1 L·a/(L·a+2) L·a vs 2
b·d >> 1
c·A m·M << 1 L·a·/(L·a+1) L·a vs 1
  1. Initial and maximal response for ATSM with orthoster concentration as independent variable with an interfering alloster. M or [M] stands for alloster concentration. Conditions are listed with decreasing number of parameters from column 1 to 5 for products of M and parameters that affect the initial efficacy, IntEff, at very low concentrations of orthoster, S, and the final maximal efficacy, MaxEff, at very high concentrations of S.
  2. All conclusions for IntEff and MaxEff of ATSM are similar for the EXOM with the following exceptions: for EXOM 1) parameter L is replaced with 1in all statements for ATSM and 2) parameter b disappears out of all MaxEff statements as listed for ATSM.
  3. Below are further details about effects of parameters and alloster concentration on IntEff and MaxEff for ATSM and EXOM.
  4. Initial efficacy. IntEff for ATSM or spontaneous activity:
  5. For b = 1, IntEff = L/(L+1) and independent of the value of A m·M.
  6. For b > 1, IntEff > L/(L+1). With increasing values of A m·M above 1 the IntEff increases towards a ceiling value of L·b/(L·b+1), equal allo-agonism. For decreasing values of A m·M below 1, the IntEff goes towards L/(L+1).
  7. For b < 1, IntEff < L/(L+1). With increasing values of A m·M above 1 the IntEff reduces towards a ceiling value of L·b/(L·b+1). For decreasing values of A m·M below 1, the IntEff increases towards L/(L+1).
  8. Allo-agonism above spontaneous activity in ATSM, L/(L+1), is given by L·b/[L·b+1+1/(A m·M)], when both b·A m·M >> 1 and also parameter b > 1. The ceiling value of this allo-agonism is L·b/(L·b+#), where # is a value between 1 or 2, depending on the value of A m·M.
  9. IntEff for EXOM:
  10. Allo-agonism in EXOM is always given by b/[b+1+1/(A m·M)], and going towards zero for b → 0, independent of the value for b·A m·M, and with a ceiling level of b/[b+¤], where ¤ is a value between 1 or 2, depending on the value of A m·M. Examples of ceiling effects and their absence in ATSM and EXOM are shown in Figure 5. For 1/(A m·M) >> b+1 in EXOM, IntEff goes towards 0 if b < 1, while for 1/(A m·M) << b+1, IntEff approaches b/(b+1) as its ceiling level.
  11. Maximal efficacy. MaxEff for ATSM:
  12. When b·c·d·A m·M >> 1 and as long as c·A m·M ≥ 1, ATSM-MaxEff is always dependent on the product b·d and independent of the value of c·A m·M.
  13. For b·d = 1, MaxEff = L·a/(L·a+1), independent of c·A m·M.
  14. For b·d > 1, MaxEff > L·a/(L·a+1), = synergy. With increasing values of c·A m·M above 1, the MaxEff increases towards a ceiling value of 100%, i.e., above L·a/(L·a+1) if L·a >> 1. For decreasing values of c·A m·M below 1, the MaxEff goes towards L·a/(L·a+1).
  15. For b < 1, MaxEff < L·a/(L·a+1). With increasing values of c·A m·M above 1 the MaxEff reduces towards a ceiling value of L·a/(L·a+1) . For reducing values of c·A m·M below 1, the MaxEff increases towards L·a/(L·a+1).
  16. More details on dependence of MaxEff-ATSM on parameter combination are listed in the table.
  17. As mentioned above, for b·c·d·A m·M >> 1, and c·A m·M ≥ 1, MaxEff is always independent of the value of c·A m·M.
  18. MaxEff for EXOM:
  19. MaxEffs for EXOM are as well as for ATSM dependent on c·d·A m·M. Further, for c·A m·M >> 1 when d >> 1, EXOM-MaxEff goes to 100%, while for c·A m·M >> 1 but with c·d·A m·M << 1, it is determined by a/(a+c·A m·M). When c·A m·M ≤ 1 and c·d·A m·M >> 1, EXOM-MaxEff goes to 1, while for d << 1, it goes to zero.