Kinetic and computational analysis of the reversible inhibition of porcine pancreatic elastase: a structural and mechanistic approach

Abstract

Structural and mechanistic insights were revealed for the reversible inhibition of Porcine Pancreatic Elastase (PPE); the kinetics of uninhibited and inhibited hydrolysis of substrate Suc-AAA-pNA was analyzed thoroughly. Additionally, the interactions between PPE and its inhibitor were studied by computational techniques. The uninhibited hydrolysis of Suc-AAA-pNA by PPE proceeds through a virtual transition state, involving an inferior physical and another dominating chemical step, where two stabilized reactant states precede the predominant acyl-enzyme. Different kinds of bonding with the PPE-backbone residues, including those of the catalytic triad, were found during the MD simulation of 5 ns, as key interactions favoring a higher stabilization of the best ranked complex PPE-CF₃C(O)-KA-NHPh-p-CF₃. The proton inventories of the inhibited hydrolysis of Suc-AAA-pNA by PPE, were ruled out the existence of any virtual transition state and thus they argue for a different mode of catalysis involving a structurally disturbed PPE molecule. Thereafter, a novel inhibition mechanism was suggested.

Keywords:

• Inhibition mechanism
• molecular docking
• molecular dynamics
• proton inventories

Introduction

Elastases are proteolytic enzymes, which mainly hydrolyze proteins of the connective tissue in lungs, arteries, skin and ligaments1,2; they predominate in the pathogenesis of emphysema in the absence of suitable inhibitors3. The molecule of Porcine Pancreatic Elastase (PPE) comprises a polypeptide chain of 240 residues4, whereas the occupancy of its S₅ and S₄ subsites, as well as the occurrence of proline and valine at P₂ and P₁, respectively, in its substrates is important for effective catalysis. Analysis of co-crystals of 1:1 complex of PPE and the reversible inhibitor CF₃C(O)-LA-NHPh-p-CF₃ (K_i = 23.6 × 10^− 9 M), at 1.8 Å effective resolution, showed the inhibitor molecule to be tightly bound on the PPE, with its trifluoromethylamide group to be bonded to the PPE active site, whereas both of its two fluorinated groups exhibited different specificities5.

In this work, the mechanism of reversible inhibition of PPE is investigated based on a series of suitable and useful probes6. Subsequently, three specific reversible inhibitors of PPE were synthesized, of the general form CF₃C(O)-XA-NHPh-p-CF₃, where X = {D,K,V}, and they were employed through appropriate methodologies which include (a) pH, absolute temperature profiles and proton inventories of the corresponding parameters, as well as the calculation of important rate and thermodynamic constants, (b) computational analyses (molecular modeling techniques i.e. docking and molecular dynamic simulations), and (c) inhibition kinetics. Throughout this work, the Suc-AAA-pNA has been used as substrate. Furthermore, the combined results of this work support a novel mechanism of the reversible inhibition of PPE, which could be useful in design future protection against the pathogenesis of emphysema, other diseases and related applications.

Materials and methods

The substrate Suc-AAA-pNA, the PPE, the deuterium oxide (D₂O) and all other chemicals were purchased from Sigma-Aldrich (St. Louis, MO). The trifluoroacetyl peptide-p-(trifluoromethyl)anilide inhibitors, 1-CF₃C(O)-KA-NHPh-p-CF₃, 2-CF₃C(O)-LA-NHPh-p-CF₃5,6 3-CF₃C(O)-VA-NHPh-p-CF₃ and 4-CF₃C(O)-DA-NHPh-p-CF₃, were synthesized in our Laboratory and their purity was checked, as it has been described previously6–8. The competitive nature of the inhibition was checked in all cases9. All kinetic measurements were performed spectrophotometrically by using the enzyme PPE ([E]₀=30 nM) and the peptide Suc-AAA-pNA as substrate ([S] = {70–4200 μM), in 0.1 M Tris HCl buffer solutions, pH = 8.0, at 25 °C, and in eight-fold experiments; the was monitored at 410 nm. In this way, appropriate probes, were applied herein.

Profiles of the Michaelis–Menten parameters k_cat/K_m, k_cat, K_m and 1/K_m were constructed versus both the pH-value at 25 °C and the absolute temperature at pH 8.00; the thermal stability of PPE had been examined and confirmed by previous experimentation10,11. Subsequently, all rate constants and their corresponding thermodynamic activation parameters ΔH^‡, ΔS^‡ and ΔG^‡ were estimated by fitting the experimental data with variants of the Arrhenius equation, and/or through a linear form of Eyring equation T × [ln(k/T)] = T × [ln(k_B/ħ) + ΔS^‡/R]−ΔH^‡/R, respectively12, the corresponding energy diagram of enzymatic hydrolysis of the used substrate Suc-AAA-pNA was drawn accordingly12,13. Furthermore the proton inventories of k_cat/K_m were performed in buffers of variable D₂O molar fraction n, at both in absence as well as in presence of inhibitors, at pD 8.00.

Computational analyses were applied in this study, on Linux Platform, by employing the Schrödinger Software suite, ver. 2014-4 LLC (New York, NY). The X-ray crystal structure of PPE (PDBID: 7EST) was obtained from the Protein Data Bank, and it was further prepared by the Protein Preparation Wizard module. The Molecular structures of the trifluoroanilides inhibitors were built and prepared using the Build and the LigPrep module, respectively. Different conformations were generated with the Advanced Search of ConfGen module, allowing amide bond trans-conformations only, by selecting water as solvent and the truncated Newton (TCNG) method for minimization (100 iterations). A grid box of 30 × 30 × 30 ų with a default inner box (10 × 10 × 10 ų) was centered on the inhibitor in the crystal structure (coordinates X = 14.1462, Y = 48.1472 and Z= −1.1562). All the output conformers were docked using the extra-precision (XP) mode of Glide docking module, and the five top poses (based on Glide XP score sorting) of each PPE-inhibitor complex were selected.

The obtained best docked conformations per inhibitor from the Glide XP were further rescored using the Prime-MM/GBSA method (Prime, version 3.1). Energy minimization was performed using the local optimization feature in Prime, whereas the contributing ΔG energies of each complex were calculated using the OPLS-2005 with VSGB solvation model, allowing a flexible configuration of the residues of PPE backbone within 5 Å around the inhibitor14,15. Subsequently, molecular dynamics simulations were carried out on the energy minimized PPE-inhibitor complexes, by the Desmond module of Schrödinger suite. The system was solvated using the TIP3P water model in an orthorhombic box with buffer space of 10 Å apart from the edges of protein, which was neutralized by adding 6 Cl⁻ counter ions, and the ionic strength was corrected by NaCl to 0.1 M. The system was subjected to energy minimization (at least 10 steps) until a gradient threshold i.e. 104.67 kJ mol⁻¹Å⁻¹ was obtained; then, the system was relaxed using the Desmond’s default relaxation protocol before simulation. To get structural insights of interactions, MD simulations were performed for 5 ns, using NPT ensemble as micro-canonical system with 1 bar pressure and at constant temperature of 300 K, using a time step of 2 fs. The trajectories were saved at intervals of 5 ps for analyses.

Thereafter, inhibition kinetics were performed by the use of all three reversible inhibitors in 0.1 M Tris HCl buffer solutions, at pH 8.0 and 25 °C, for five different concentrations per inhibitor ([I] = {30,60,90 180 240} expressed in nM) and for comparison purposes; in each inhibition kinetic run and before the addition of the appropriate substrate solution in the corresponding reaction mixture, the enzyme was incubated for 15 min with inhibitor. The well-known Morrison Equation (1)(1) , best fitted the experimental data and the estimated K_i values were collected and compared with previous referred ones6,16,17. Additional kinetic measurements were performed for all three employed reversible inhibitors, at 0.1 M suitable buffer solutions, at different pH-value runs and 25 °C. In each run, a K_i value was estimated as it is described above, using the substrate Suc-AAA-pNA; the experimental data of versus the pH-values were best fitted by the Equation (2)(2) and two pK_a-values were estimated one for each of the acidic and the alkaline limb of the corresponding pH-profile18,19. (1) (2)

For all three employed reversible inhibitors, the corresponding k_cat/K_m values, as well as their equilibrium dissociation constants K_i were calculated by using five different concentrations per inhibitor, at different values of the D₂O molar fraction n of the buffer solution, and the experimental data were best fitted by the Morrison equation; then, the estimated k_cat/K_m and K_i values, per inhibitor, were collected and best fitted by various forms of the Gross–Butler–Kresge equation, whereas all the values of the resulting fractionation factors, and other useful parameters, were estimated and evaluated accordingly11,18–26

Results and discussion

Hydrolysis of Suc-AAA-pNA by PPE in the absence of inhibitors

Dependencies of Michaelis–Menten parameters versus pH and absolute temperature

The existence of stabilized reactant states which are governed by either the k_cat and/or 1/K_m was validated from their pH-profiles which were found wider than that of k_cat/K_m; although all the three pH-profiles exhibited two pK_a values, one for each limb (acid and alkaline), which are similar in the cases of k_cat and 1/K_m parameters, however the pK_a values differ appreciably in the case of k_cat/K_m (Figure 1a). Therefore, the predominant enzyme species is the acyl-enzyme (E_acyl) i.e. to whom the estimated pK_a values are related, as it was found that k₂ ≫ k₃ (Scheme 1 and Figure 1a). The abovementioned reactant states follow the ionization of free enzyme (E) and substrate (S) toward the products, of the enzyme-substrate complex (ES), and of ES toward the products, for the parameters k_cat/K_m, 1/K_m and k_cat, respectively26. Consequently, it is likely that more intermediate enzyme species may occur according to Scheme 1 10,11,18.

Figure 1. (a) Merged graphs of the dependencies of Michaelis–Menten parameters, k_cat/K_m, k_cat and 1/K_m, versus the pH-value of the hydrolysis of Suc-AAA-pNA by PPE, and (b) energy diagram of hydrolysis of Suc-AAA-pNA by PPE (the value of Arrhenius pre-exponential factor was assumed as 6 × 10¹² s⁻¹).

Scheme 1. Accepted reaction scheme for the hydrolysis of substrate Suc-AAA-pNA, by PPE.

The profiles of the abovementioned Michaelis–Menten parameters versus the absolute temperature at pH 8.00 were carried out and the rate constants k₁, k₋₁, k₂ and k₃, were estimated, as well as the energy diagram (Figure 1b) has been drawn accordingly (Scheme 1)12,13; then, novel interesting points were shown i.e. (a) the reaction Scheme 1 was reconfirmed through a different probe, and (b) there is a full agreement with the estimated values of ΔH^‡, ΔS^‡ and ΔG^‡ in the cases of k_cat/K_m and k_cat, by pointing out the entropy effects, which are associated with the formation of ES and E_acyl, respectively (Δ S_kcat/K_m}^ddag−Δ S_kcat^ddag= 49.839 kJ mol⁻¹ K⁻¹). The profile of absolute temperature, which correspond to k_cat, was found as slightly shifted to higher temperatures verifying that processes associated with removal of substrate from the enzyme prevail at higher temperatures, where k_cat/K_m≠k₁, and thus k_cat/K_m=k₁k₂/(k_-1 + k₂), are valid20.

Proton inventories

The proton inventories for k_cat/K_m were found as bowed upward exhibiting a medium solvent isotope effect, and the following parameter values were estimated. (k_cat/K_m)₀=2525.99 ± 15.71 M⁻¹ s⁻¹, C_Ph = 0.30 ± 0.01, C_Ch= 0.70 ± 0.02, φ^T,Ch = 0.37 ± 0.01 and S.I.E. = 2.64, while from preliminary fitting were calculated the values of Z₁=2.00 and μ = 2.00. The values found for C_Ph, C_Ch, φ^T,Ch and μ argue for the existence of a virtual transition state during the reaction governed by the k_cat/K_m, comprising a process through two steps an inferior physical (C_Ph = 0.30) and a dominating chemical one (C_Ch = 0.70), which both of them contribute to the formation of a non-covalent complex (ES^‡) and of the subsequent product P₁; two following steps are the formation of acyl-enzyme (E_acyl) and of product P₂. These findings, in accordance to those from the pH-profiles of Michaelis–Menten parameters, confirm Scheme 1, and support the reported results. It has been considered that conformational changes of complex ES are associated directly with efficient interactions of substrate and the active site of PPE in the transition state toward acylation. Moreover, two protons (μ = 2) from different protonic sites are transferred in the transition state of the chemical step, whose fractionation factor φ^T,Ch value was estimated to be 0.37, and along with the observed solvent isotope effect (S.I.E.) could be assigned more likely to transition-state hydrogen bridges during solvation catalysis; these hydrogen bridges are mainly observed in proton transfer among oxygen and nitrogen atoms, which are usually located on all species i.e. on the enzyme, substrate and/or on inhibitor’s molecules as well18,21–24.

Analysis of the interactions between PPE and inhibitors

This work reported for the first time the investigation of binding modes of the used trifluoroacetyl peptide-p-(trifluoromethyl)anilide inhibitors5,6 with molecular docking. The best poses of inhibitors were identified based on the binding affinity, the Glide XP energy and the number of hydrogen bonds, which are formed with key-residues Table 1. In order to acquire more knowledge on the binding mode of these inhibitors, the five top ranked poses of each one of them have been further rescored using the MM-GBSA approach Table 2, implying similar binding modes with their trifluoroacetyl group (N-terminus) to occupy the S₁ pocket5. Analogous binding mode was observed for the trifluoroacetyl group of inhibitor-4 whose the remainder molecule adopts a dissimilar orientation versus those of the rest of inhibitors, more likely due to the negative charge of aspartic residue (Figure 2(a–d)). The inhibitors bind onto PPE through the formation of hydrogen bonds and π–π interactions, mainly, with four key-residues i.e. S²¹⁴, F²¹⁵, V²¹⁶ and R^217A. The C-terminus of inhibitor-1 has the proper geometry of sandwich-type, and it forms a stronger π–π stacking interaction with F²¹⁵ (−3.31 kJ mol⁻¹); this latter interaction contributes to a higher stabilization of that inhibitor into the active site of PPE, versus those of the remainder inhibitors. A different hydrogen bonding network is adopted in the binding of inhibitor-4, which includes the residues V²¹⁶, Q¹⁹², S¹⁹⁵, G¹⁹³ and T⁴¹.

Figure 2. Extra precision Glide docking showing the binding mode of inhibitors onto PPE: (a) Overlay of docked pose of inhibitor 2 (light grey/cyan in color) with its crystal structure conformation (grey/brown in color) confirming the validation of the docking protocol, (b) inhibitor-1 (grey/yellow in color) forming five H-Bonds (D¹⁰²/G¹⁹³/S²¹⁴/2 × V²¹⁶), one π–π stacking interaction (F²¹⁵), and one cation-π interaction (R^217A), (c) inhibitor-3 (dark grey/dark green in color) forming three H-Bonds (S²¹⁴/2 × V²¹⁶), and (d) inhibitor-4 (grey/magenta in color) forming five H-Bonds (T⁴¹/Q¹⁹²/G¹⁹³/S¹⁹⁵/V²¹⁶); in all four panels white/yellow in color dashed lines represent H-Bonds, grey/green in color dashed lines represent π–π stacking interactions and light grey/cyan in color dashed lines represent cation-π interactions.

Table 1. Extra precision Glide docking results and interacting residues in the PPE.

	Inhibitor	Glide XP docking score	Glide XP energy (kJ)	Hydrogen bonding residues	HB*
1		−7.16	−209.97	V^216, S^214	4
2		−6.95	−192.30	V^216, S^214	3
3		−6.55	−186.94	V^216, S^214	3
4		−6.44	−181.08	Q1^92,S1^95, V^216,G1^93, T^41	5

*Number of hydrogen bonds formed.

Table 2. Binding energies* calculated using MM-GBSA approach.

	Inhibitor	ΔG Binding energy	ΔG Coulomb energy	ΔG Hydrogen bonding energy	ΔG Lipophilic energy	ΔG van der Waals energy	ΔG π-π stacking energy	ΔG solvation energy GB
1		−393.22	−117.19	−5.57	−137.49	–199.63	–3.10	69.92
2		−380.20	−108.27	−3.10	−133.85	–202.43	–1.84	69.29
3		−344.66	−89.05	−3.14	−117.65	–186.90	–1.47	53.55
4		−346.63	−239.07	−7.58	−106.72	–186.56	0.00	193.30

*Expressed in kJ mol ⁻¹.

The estimated ΔG_bind values of PPE complexes Table 2, range from −344.66 to −397.41 kJ mol⁻¹; accordingly, and among the energy components it is remarkable that the strong Coulomb energy term of inhibitor-4 is counterbalanced by the unfavorable ΔG solvation GB-energy, which opposes the binding course. Inhibitor-1 shows more favorable binding energy terms as compared to the neutral inhibitors, ranked as second and third, although all these three adopt similar binding modes.

MD simulations of the top scored complex of PPE-inhibitor-1

Molecular dynamics (MD) simulations were performed for 5 ns and useful insights were obtained, herein, on how this inhibitor settles in the binding site of PPE. The simulations reached equilibrium within the first ns whereas the RMSD value of the PPE backbone with respect to its initial structure and the RMSD value of inhibitor were stabilized at distances less than 2 Å, indicating a steady conformation of the enzyme (Figure 3a) and that the inhibitor-1 has not diffused away from its initial binding site during the simulation. In Figure 3(b) are depicted the Root-Mean-Square Fluctuations of inhibitor-1 with respect to PPE; two regions of elevated fluctuations are observed corresponding to the two − CF₃ groups of inhibitor (N-terminal, and C-terminal), which indicate that these regions more likely do not contribute in the binding of the inhibitor5. The inhibitor interacts with PPE by forming hydrogen bonds, hydrophobic interactions (cation-π and other nonspecific interactions) and water bridges (Figure 4a,b).

Figure 3. (a) The RMSD values of both the PPE-backbone atoms and of the heavy atoms of inhibitor-1, (b) The plot of Root-Mean-Square Fluctuations of inhibitor-1 with respect to PPE.

Figure 4. Interactions of PPE and inhibitor-1 during the MD simulation: (a) types and percentages of PPE-inhibitor interactions of key-residues, (b) PPE-inhibitor detailed interactions occurring for > 5.0% of the simulation time.

Among the four predicted hydrogen bonds for the complex PPE-Inhibitor-1_, two of them with V²¹⁶ were maintained during the MD simulation; instead, a third hydrogen bond with S²¹⁴ was preserved at ∼80% of the MD trajectory, whereas the D¹⁰² and G¹⁹³ (of PPE backbone) were involved in new hydrogen bonding interactions with inhibitor-1.

In addition, water molecules were observed to form transient contacts with the inhibitor and to promote the formation of hydrogen bridges with PPE through the residues R^217A and V⁹⁹. Moreover, the C-terminus (–C₆H₄–CF₃ group) of inhibitor-1 interacts with F²¹⁵ by means of π–π interactions (15%) as well as with R^217A by cation-π interactions (63%). After the first ns of the MD simulation, the Solvent Accessible Surface Area (SASA) was reduced to 100 Ų (∼60%) indicating that the surface area of ihhibitor-1, which is accessible by the water molecules, was minimized i.e. the inhibitor binds more tightly onto PPE.

Inhibition kinetics

Equilibrium inhibition constants and dependencies of 1/K_i versus pH

The inhibition kinetics re-ranked the employed reversible inhibitors similarly as in the previous sections. Therefore, the S₂-P₂ interactions of PPE favor positively charged groups (K-residue), whereas they do not favor negatively charged groups (D-residue), and maintain a medium level of favor for short aliphatic groups (V-residue). In the 1/K_i pH-profiles (Figure 5), we observed that the maxima of plots for both inhibitors-3 and 4 are shifted to more acidic values, exhibiting two almost identical pK_a values one for each limb; the two pK_a values of 1/K_i pH-profile of inhibitor-1, differ substantially from those of the rest two inhibitors. Furthermore, the resulting graphs exhibit narrower curvatures as going from inhibitor-1 to inhibitor-3, and to inhibitor-4; this successive narrowness, in addition to the shift of pH maximum to more acidic values and the differences in the estimated pK_a values (equilibrium of free enzyme and inhibitor to enzyme-inhibitor complex) implies a progressive destabilization of the corresponding PPE-inhibitor complex (EI) for the inhibitors-3 and 418,20,27.

Figure 5. Merged graphs of the 1/K_I dependencies for all three used reversible inhibitors versus the pH-value of the reaction medium, where pK_a1=6.87 ± 0.10 and pK_a2=9.847 ± 0.12 for inhibitor-1, mean values of pK_a1=7.59 ± 0.08 and pK_a2=8.54 ± 0.21 for inhibitors-3 and 4, respectively.

Proton inventories

The proton inventories of K_i and 1/K_i, as well as of k_cat/K_m in the presence of all three reversible inhibitors, were found as nonlinear and bowed downward more likely due to multiple hydrogen bonds formed in the corresponding transition states; moreover, they exhibited high solvent isotope effects (inverse in the case of 1/K_i). All the values of fractionation factors and solvent isotope effects (S.I.E.), as well as of other significant factors were estimated by best fitting of the experimental data to suitable forms of the Gross–Butler–Kresge equation Table 3. In order to ensure the mechanistic interpretation of the aforementioned proton inventories, the corresponding experimental data were fitted also by various forms of equations as k_n=k₀(1-n + nφ^T₁)….(1-n + nφ^T₄) and/or k_n=k₀/[(1-n + nφ^G₁)….(1-n + nφ^G₄)], and equal φ^T and/or φ^G values were estimated in all cases (for the k_cat/K_m parameter and in the presence of inhibitors)25. From Table 3, it is easily deduced that for all three used inhibitors the estimated φ^T values (proton inventories of k_cat/K_m) follow an exponential decay course (y = y₀+A*e^-x*λ), i.e. they change proportionally to the variation of the concentration of inhibitor; the estimated values of the exponents μ of equation k_n=k₀ (1-n + nφ^T)^μ, as well as the corresponding estimated S.I.E., were best fitted by the well-known Hill equation y = ax^h/(b^h+x^h) including its logistic form28. These values of the Hill coefficients for the exponents μ, and for S.I.E. were estimated in the order as 0.71, 0.41, 0.11 and 0.46, 0.81, 0.90, for the inhibitor-1, inhibitor-3 and inhibitor-4, respectively, showing increased negative cooperativity for the exponents μ, and decreased negative cooperativity for the S.I.E. These results are novel and provide additional evidence on the evolution of interactions among PPE and its ligands (competitive inhibitors and substrate) up to the equilibrium. The negative cooperativity is increased for the μ exponents, in consistency to the ranking of the used inhibitors, by based on various findings (PPE-inhibitors interactions, MD simulations, K_i values and 1/K_i pH-profiles); the lesser negative cooperativity is attributed to the best inhibitor, and vice versa, which binds onto PPE by delivering a conformation of high affinity with both its ligands (inhibitor and substrate) toward the equilibrium. On the other hand, the observed decreased negative cooperativity in the case of S.I.E. is in a straightforward analogy to the decreased efficiency of the PPE-inhibitor binding modes (in the abovementioned order), which it favors a low-affinity conformation of PPE with both its ligands29. Besides, these phenomena in cases of monomeric enzymes have kinetic origins, they are referred to the extent of cooperativity which affects the equilibrium and the catalytic process, and they account for structural changes in the enzyme molecule. Eventually, the aforementioned φ^T values, along with the estimated values of exponents μ and those of the S.I.E. should be assigned mostly to transition-state hydrogen and/or water bridges for proton transfer without tunneling or rarely to hydrogen bonds which are stronger in the transition state25,30. The proton inventories of K_i and 1/K_i, and per inhibitor, they exhibited similar φ^T and/or φ^G values, which could be assigned as before equivalently to those for the k_cat/K_m parameter; these latter proton inventories are decreased linearly from inhibitor-4 up to inhibitor-1, whereas they were estimated almost equal values for all the exponents μ and/or ν (of either k_n=k₀(1–n + nφ^T)^μ and/or k_n=k₀/[(1–n + nφ^G)^ν] equation). The values of the S.I.E., per inhibitor, were estimated almost equal (i.e. S.I.E. for K_i equal to 1/(S.I.E) of 1/K_i), and are increased linearly in the order of the mentioned ranking of inhibitors Table 3. More findings, which further support these results are included in the Supplemental information.

Table 3. Values of fractionation factors, S.I.E. and other factors from proton inventories of k_cat/K_m, K_i and 1/K_i parameters in the inhibited hydrolysis of Suc-AAA-pNA by PPE; the experimental data were best fitted by suitable forms of the Gross–Butler–Kresge equation.

Parameter	φ^T	φ^G	Exponent μ	S.I.E.	Equation	[I] nM Inhibitor
kcat/Km	0.54		2.03	3.38	kn = k0 (1-n + nφ^T)^μ	30
	0.52		2.22	4.16		60
	0.50		2.41	4.87		90
	0.47		2.50	6.91		180
	0.46		2.54	7.54		240
Ki	0.54		2.23	3.91	Kn = K0 (1-n + nφ^T)^μ
1/Ki		0.55	2.37	1/4.05	kn = k0/[(1-n + nφ^G)^ν]		4
kcat/Km	0.55		2.31	3.95	kn = k0 (1-n + nφ^T)^μ	30
	0.43		2.61	8.24		60
	0.40		2.72	11.34		90
	0.36		2.96	17.12		180
	0.38		3.09	20.38		240
Ki	0.48		2.27	5.17	Kn = K0 (1-n + nφ^T)^μ
1/Ki		0.49	2.42	1/5.38	kn = k0/[(1-n + nφ^G)^ν]		3
kcat/Km	0.61		3.06	4.51	kn = k0 (1-n + nφ^T)^μ	30
	0.53		3.19	7.63		60
	0.51		3.28	8.76		90
	0.44		3.37	14.31		180
	0.41		3.41	15.15		240
Ki	0.43		2.22	5.98	Kn = K0 (1-n + nφ^T)^μ
1/Ki		0.45	2.40	1/6.42	kn = k0/[(1-n + nφ^G)^ν]		1

Conclusions

The objective of this work ensues inductively from the aforementioned results, whose the most decisive points are: In the absence of competitive inhibitors, the PPE initiates the catalytic hydrolysis of Suc-AAA-pNA by means of a virtual transition state, which is due to conformational changes toward the complex ES, and comprises two steps an inferior physical and another dominating chemical one. Two stabilized reactant states, which are governed by the parameters 1/K_m and k_cat, respectively, precede the predominant acyl enzyme species. The best ranked inhibitor-1, was used as the model to illustrate key interactions which favor a higher stabilization of the PPE-inhibitor complex; the latter has been verified also by the wider 1/K_i pH-profile arguing for the importance of the S₂–P₂ interactions that prefer positively charged groups.

Additionally, the RMSD values of both, PPE and inhibitor-1 are in full agreement, indicating that the inhibitor maintains its initial binding mode during the 5 ns of MD simulation. Consequently, the following PPE-inhibitor interactions were found to occur for more than 5.0% of the simulation time (Figure 4a,b): (a) seven hydrogen bonds (with V⁹⁹, D¹⁰², 2× G¹⁹², S²¹⁴ and 2× V²¹⁶), (b) eight water bridges (with T⁹⁶, 2× V⁹⁹, 2× G¹⁹², 2× S²¹⁴ and R^217A) and (c) two cation-π interactions (with H⁵⁷ and R^217A).

The exhibited exponentially decaying φ^T values of k_cat/K_m proton inventories, in the presence of inhibitors, could be due to that the formed water bridges (i.e. all possible water/deuterium oxide species) and/or hydrogen bonds, which are stronger in the transition state, they change gradually to hydrogen bonds between the catalytic residues H⁵⁷ and D¹⁰², and without tunneling in all cases, depending on the concentration of the competitive inhibitors. The estimated negative cooperativities of the exponents μ and/or of S.I.E. were interpreted as indicative for enzyme conformations of high affinity in the former case, and/or of decreased efficiencies of the PPE-inhibitor binding modes, in the latter case. Another important and decisive point should be taken into account. The reversible competitive inhibitors in this work, and before the addition of substrate, were incubated with PPE, and bound on it, while the system reached the equilibrium and a network of interaction was established (Figure 4b); after that, the addition of substrate in the reaction mixture and its consequent hydrolysis proceeds by means of setting up a new equilibrium where the formation of both complexes ES and ES^‡ is achieved by a structurally disturbed and more rigid PPE molecule, and thus, no further experimental evidence was given for either a virtual transition state or a physical step.

In the reaction mechanism of Scheme 2, and in relation to Scheme 1, the virtual transition state is illustrated and includes the two individual transition states [ES]₂ and [ES]^‡ plus the tetrahedral intermediate ES₂, i.e. the equivalent of the TS₂ of Scheme 1; therein, the physical step is symbolized by the so-called nitrogen inversion and/or rotation about the ex-scissile amide bond who loses its double bond character in the tetrahedral intermediates ES₁ and ES₂ before its breaking down7,31.

Scheme 2. The mechanism of acylation of PPE (formation of the E_acyl species), during the hydrolysis of substrate Suc-AAA-pNA, which is achieved through a charge relay system.

Next, and by taking into account all the above referred conclusions, a potential mechanism of the reversible inhibition of PPE should be represented by Scheme 3(a), whose the conceptual short version is illustrated in Scheme 3(b).

Scheme 3. (a) Illustration of a potential mechanism of reversible inhibition of PPE (inhibitor: CF₃C(O)-KA-NHPh-p-CF₃; substrate: Suc-AAA-pNA); the mechanism is achieved without the development of a charge relay system. (b) The brief version of the inhibition mechanism of Scheme 3a.

In fact, inhibition starts by the reaction of free PPE and inhibitor toward the formation of both a complex E*I, and of a modified PPE molecule to a more rigid conformation due to the binding of inhibitor on it, where more likely, a virtual transition state is not observed when the modified PPE molecule hydrolyzes its substrate in the presence of inhibitor. Therefore, they are easily deduced, the almost irreversible character of the reaction of free PPE and inhibitor toward the E*I complex, and that in the reaction of the modified PPE molecule with substrate toward acylation either the ring-flip hypothesis or the nitrogen inversion and/or rotation are not in function31,32; then, it follows the formation of the corresponding acyl-enzyme and its consequent transformation to free enzyme and products.

Declaration of interest

The authors report no declarations of interest.

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