Stability of HIV-1 integrase–ligand complexes: the role of coordinating bonds

Abstract

It is known that the HIV-1 integrase (IN) strand transfer inhibitors include the chelating fragments forming the coordinating bonds with two Mg²⁺ ions placed in the IN active site. The subject of the article is the role of these coordination bonds on stability of ligand–IN complexes. For this purpose, a set of ligand–IN complexes was investigated theoretically and experimentally. The theoretical model is based on the quantum-chemistry calculations of coordinating bonds geometry and energy. Solvent effects were taking into account using the implicit water model and the two-stage calculation scheme developed previously. For the experimental part of our study a set of the ligands was synthesized, and their IC₅₀ values of IN inhibiting have been measured. It is shown that the main contribution to ligand–IN complexes stability is caused by the substitution of water molecules by the ligand in the first coordination sphere of two Mg²⁺ ions, and the change in the polarization energy of the bulk water. It is shown, that acid–base equilibrium and tautomeric forms of the ligands should be taken into account to improve the prediction ability of the theoretical estimations. All these factors are controlled by the chelating fragments of the ligands. It is demonstrated that our theoretical approach based on the consideration of the coordinating bonds allows to separate active ligands (inhibitors) from inactive ones.

Appendices

Appendix A

Let’s consider the Henderson–Hasselbalch equation [17, 18] for the equilibrium of the charged (A⁻) and neutral (AH) states of the molecule:

$$ {\text{pH}} = {\text{p}}K_{\text{a}} + \log \frac{{\left[ {A^{ - } } \right]}}{{\left[ {AH} \right]}}, $$

(6)

The ΔG _exp is defined as:

$$ \Updelta G_{\exp } = - {\text{RT}}\,\ln \frac{{[{\text{AP}}]}}{{[{\text{A}}][{\text{P}}]}}, $$

(7)

where [A] = [AH] + [A⁻], [AP] = [AHP] + [A⁻P] (P is the protein). Because only ionized product form the chelate complex with protein, [AHP] ≪ [A⁻P]. So the can rewrite (7) as

$$ \Updelta G_{\exp } = - {\text{RT}}\,\ln \frac{{\left[ {{\text{A}}^{ - } {\text{P}}} \right]}}{{\left( {\left[ {\text{AH}} \right] + \left[ {{\text{A}}^{ - } } \right]} \right)\left[ {\text{P}} \right]}}. $$

(8)

On the other hand,

$$ \Updelta G_{\text{calc}} = - {\text{RT}}\,\ln \frac{{\left[ {{\text{A}}^{ - } {\text{P}}} \right]}}{{\left[ {{\text{A}}^{ - } } \right]\left[ {\text{P}} \right]}}. $$

(9)

From (6):

$$ \left[ {\text{AH}} \right] = \left[ {{\text{A}}^{ - } ]} \right]10^{{({\text{p}}K_{\text{a}} - {\text{pH}})}} , $$

(10)

so

$$ \Updelta G_{\exp } = - {\text{RT}}\,\ln \frac{{\left[ {{\text{A}}^{ - } {\text{P}}} \right]}}{{\left( {1 + 10^{{({\text{p}}K_{\text{a}} - {\text{pH)}}}} } \right)\left[ {{\text{A}}^{ - } } \right]\left[ {\text{P}} \right]}} = \Updelta G_{\text{calc}} + {\text{RT}}\,\ln \left( {1 + 10^{{\left( {{\text{p}}K_{\text{a}} - {\text{pH}}} \right)}} } \right) $$

(11)

Appendix B

The free energy of the formation of the IN–ligand complexes ΔG _b is expressed in terms of the molar concentration [I], [L], [LI] of the IN, ligand, and complexes IN–ligand, respectively:

$$ \Updelta G_{\text{b}} = - {\text{RT}}\,\ln \frac{[LI]}{[I][L]} = - {\text{RT}}\,\ln K_{\text{d}} , $$

(12)

where $K_{\text{d}} = \frac{[{\text LI}]}{[{\text I}][{\text L}]}$ is the equilibrium constant.

Rewrite (12) as

$$ \Updelta G_{\text{b}} = - {\text{RT}}\,\ln \frac{[{\text LI}]}{{\left( {[{\text I}]^{0} - [{\text LI}]} \right)\left( {\left[ {L\left( {\text IC}{_{50} } \right)} \right] - \left[ {\text LI} \right]} \right)}}, $$

(13)

where [I]⁰ is the total concentration of the IN which is fixed under the experiments with different inhibitors, [L(IC₅₀)] is the total concentration of the inhibitor providing the double decreasing of the IN activity.

It can be shown that the [LI] remains constant for different inhibitors if the total concentration of the substrate [S]⁰ is fixed in the experiments for the [L(IC₅₀)] determination. Let us consider the total free energy F of the system containing N _I, N _S, N _L, N _SI, N _LI molecules of the IN, substrate, ligand in unbound state and complexes IN–substrate and IN–ligand, respectively:

$$ F = - {\text{RT}}\,\ln \frac{Z}{{N_{\text{I}} !N_{\text{S}} !N_{\text{L}} !N_{\text{SI}} !N_{\text{LI}} !}}, $$

(14)

where the total partition function Z in the dilute solution limit can be expressed as:

$$ Z = Z_{\text{I}}^{{N_{\text{I}} }} Z_{\text{S}}^{{N_{\text{S}} }} Z_{\text{L}}^{{N_{\text{L}} }} Z_{\text{SI}}^{{N_{\text{SI}} }} Z_{\text{LI}}^{{N_{\text{LI}} }} Z_{0} , $$

(15)

where Z _I is the individual partition function of the IN, other terms in (15) are defined on the similar way, Z ₀ is the part of the Z not dependent on N _I, N _S, N _L, N _SI, N _LI. Since

$$ N_{\text{I}} = N_{\text{I}}^{0} - N_{\text{SI}} - N_{\text{LI}} ;\;N_{\text{S}} = N_{\text{S}}^{0} - N_{{{\text{S}}I}} ;\quad N_{\text{L}} = N_{\text{L}} (IC_{50} ) - N_{\text{LI}} $$

(16)

only two independent variables are in (14): N _LI and N _SI. The equilibrium condition is:

$$ \frac{\partial F}{{\partial N_{\text{SI}} }} = \frac{\partial F}{{\partial N_{\text{LI}} }} = 0 $$

(17)

Using (15–18) and Stirling’s approximation $\ln \,N! \cong N(\ln \,N - 1)$ for large N, we obtain:

$$ - {\text{RT}}\,\ln Z_{\text{SI}} + {\text{RT}}\,\ln Z_{\text{I}} + {\text{RT}}\,\ln Z_{\text{S}} + {\text{RT}}\,\ln N_{\text{SI}} - {\text{RT}}\,\ln N_{\text{I}} - {\text{RT}}\,\ln N_{\text{S}} = 0, $$

(18)

and

$$ - {\text{RT}}\,\ln \frac{{Z_{\text{SI}} VN_{\text{A}} }}{{Z_{\text{S}} Z_{\text{I}} }} = - {\text{RT}}\,\ln \frac{{[{\text{SI}}]}}{{[{\text{S}}][{\text{I}}]}}, $$

(19)

where V, N _A is the total volume of the system and Avogadro number, respectively. Using (16) rewrite (19) as

$$ - {\text{RT}}\,\ln \frac{{Z_{\text{SI}} VN_{\text{A}} }}{{Z_{\text{S}} Z_{\text{I}} }} = - {\text{RT}}\,\ln \frac{{[{\text{SI}}]}}{{\left( {[{\text{S}}]^{0} - [{\text{SI}}]} \right)\left( {[{\text{I}}]^{0} - [{\text{LI}}] - [{\text{SI}}]} \right)}} $$

(20)

On the similar way, we can obtain for the system without ligands with same numbers of the molecules of the IN and substrate:

$$ - {\text{RT}}\,\ln \frac{{Z_{\text{SI}} VN_{\text{A}} }}{{Z_{\text{S}} Z_{\text{I}} }} = - {\text{RT}}\,\ln \frac{{[{\text{SI}}]_{\text{pure}} }}{{\left( {[{\text{S}}]^{0} - [{\text{SI}}]_{\text{pure}} } \right)\left( {[{\text{I}}]^{0} - [{\text{SI}}]_{\text{pure}} } \right)}} $$

(21)

Since, the left parts of (20) and (21) are equal, we obtain:

$$ \frac{{[{\text{SI}}]}}{{\left( {[{\text{S}}]^{0} - [{\text{SI}}]} \right)\left( {[{\text{I}}]^{0} - [{\text{LI}}] - [{\text{SI}}]} \right)}} = \frac{{[{\text{SI}}]_{\text{pure}} }}{{\left( {[{\text{S}}]^{0} - [{\text{SI}}]_{\text{pure}} } \right)\left( {[{\text{I}}]^{0} - [{\text{SI}}]_{\text{pure}} } \right)}} $$

(22)

Since the [L(IC₅₀)] provides the double decreasing of IN activity, $[{\text{SI}}] = \frac{{[{\text{SI}}]_{\text{pure}} }}{2}$ and the [LI] in (22) depends only on [S]⁰, [I]⁰, [SI].

So using (13) we can write for two different ligands L₁, L₂:

$$ \Updelta G_{\text{b1}} - \Updelta G_{\text{b2}} = - {\text{RT}}\,\ln \frac{{\left[ {L_{2} \left( {{\text{IC}}_{ 5 0} } \right)} \right] - \left[ {\text LI} \right]}}{{\left[ {L_{1} \left( {{\text{IC}}_{ 5 0} } \right)} \right] - \left[ {\text LI} \right]}} $$

(23)

In (23), we take into account that values of [LI], [I]⁰ do not depend on ligand. Under the condition [L₂(IC₅₀)], [L₁(IC₅₀)] ≫ [LI] (23) can be rewritten as:

$$ \Updelta G_{\text{b1}} - \Updelta G_{\text{b2}} \cong - {\text{RT}}\,\ln \frac{{\left[ {L_{2} \left( {{\text{IC}}_{ 5 0} } \right)} \right]}}{{\left[ {L_{1} \left( {{\text{IC}}_{ 5 0} } \right)} \right]}} $$

(24)

The expression (24) is used in this work for the comparison of the experimental and calculated relative stability of the IN–ligand complexes.