# On the Reducible Character of Haldane-Radić Enzyme Kinetics to Conventional and Logistic Michaelis-Menten Models

## Abstract

**:**

## 1. Introduction

^{199}→ Asp

^{199}mutation in the sequence Phe-Glu-Ser-Ala-Gly at the active center of the three-dimensional structure of Torpedo californica acetylcholinesterase (AChE; EC 3.1.1.7) surprisingly appears to affect similarly the binding of the peripheral and active center site ligand; as a consequence the allosteric coupling between the sites is diminished, and the substrate inhibition is no longer observed indicating the substrate inhibition constant (K

_{SS}) becomes infinitely large:

_{297}I, F

_{338}G, F

_{297}Y, or D

_{74}N of mouse AChE-BuChE enzyme chimeras [23]. Such allosteric studies introduced the conceptual need the substrate (S) may combine at two discrete sites of an enzyme forming two binary complexes ES and SE that both end up within the ternary complex SES whose hydrolysis efficiency relative to the Michaelis-Menten binary complex ES is now quantified by the catalytic parameter (b), see Scheme 2.

## 2. Background Theories

#### 2.1. Haldane-Radić Equation

#### 2.2. Logistic Enzyme Kinetics

**Figure 1.**Michaelis-Menten and logistic initial velocities plotted against initial substrate concentration for the E-S mono-substrate enzymic reaction. The dashed curve corresponds to the Michaelis-Menten equation (1) while the continuous thick curve represents its logistic generalization from (21): .

## 3. Results and Discussion

#### 3.1. Probabilistic form of the Haldane-Radić Equation

#### 3.2. Temporal Solution of Haldane-Radić Equation by W-Lambert Functional

#### 3.3. Temporal Solution of Haldane-Radić Equation by Analytic Logistic Transformation

_{0}] as previously performed for the instantaneous free substrate [S](t), see eqs. (46) and (48), viz.:

_{S }= 190 ± 30 µM; K

_{SS}= 8,700 ± 2,200 µM ; V

_{max}= 2.45 ± 0.15 ∆OD/min; b = 0.12 ± 0.03; instead, with the same parameters in logistic related derived velocity of Equation (61) the departure is recorded for initial substrate concentrations higher than 100 µM, see Figure 2 (b), targeting the Michaelis-Menten kinetics of Figure 1 in Figure 2(c).

- The hAChE-ATC kinetics (Figure 3) differs from hBChE-ATC kinetics (Figure 4) essentially only in the lowering the V
_{max}and increasing b parameters for the last case, in accordance with the prescription associated with activation mechanism; moreover, the W-Lambert and logistic curves depart clock-wise from experimental record and more quickly for logistic case; - The hBChE-ATC kinetics (Figure 4) differs from hBChE-BTC kinetics (Figure 5) essentially by further lowering the V
_{max}accompanied by decrease of K_{S}parameter for the BTC kinetics, while the W-Lambert and logistic computationally fitting curves show in Figure 5 a departure tendency in anti-clock-wise respecting the experimental evidence; here is also recorded the clear failure of the numerical W-Lambert progress curve to reach the initial substrate concentration, a matter fully satisfied by the logistic counterpart instead; - Comparison between hBChE-BTC kinetics (Figure 5) and BSCh-BTC hydrolysis (Figure 6) reveals that by maintaining the same kinetic parameters between these two cases, in the latter, the computational fitting with respect the experimental data oscillate from clock-wise to anti-clock-wise departure of the logistic model as the initial substrate concentration goes from lower (<100 µM) to higher (>100 µM) values, respectively; here, again, the initial time discrepancy between W-Lambert and logistic kinetics is obviously in the favor of the latter approach.

**Figure 2.**(a): The fitting curves for original Haldane-Radić velocity equation (14) corresponding with the Scheme 2 for large human acetylthocholine substrates’ concentration intervals; (b) & (c) the same fits with logistic based velocity equation (61) corresponding to the “reduction” of the Scheme 2 to the consecrated Michaelis-Menten mechanism of Scheme 1; the kinetic fitting parameters are K

_{S }= 190 ± 30 µM; K

_{SS}= 8,700 ± 2,200 µM ; V

_{max}= 2.45 ± 0.15 ∆OD(optical density)/min; b = 0.12 ± 0.03.

**Figure 3.**The W-Lambert and logistic progress curves as they fit with experimental data for the hAChE-ATC kinetics, according to the equations (55) and (57), through considering the kinetic parameters from (36) and (39) with the actual values K

_{S }= 160 µM; K

_{SS}= 8,700 µM; V

_{max}= 162.45 µM/min; b = 0.12, for various initial substrate concentrations.

**Figure 4.**The same type of plots as in Figure 3, here for hBChE-ATC kinetics and parameters K

_{S }= 160 µM; K

_{SS}= 8,700 µM; V

_{max}= 31.0 µM/min; b = 3.

**Figure 5.**The same type of plots as in Figure 3, here for hBChE-BTC kinetics and parameters K

_{S }= 7.5 µM; K

_{SS}=8,700 µM; V

_{max}= 7.2 µM/min; b = 3.

**Figure 6.**The W-Lambert and logistic progress curves as their fit with the experimental data for hydrolysis of various concentrations of butyrylthiocholine by a fixed concentration of butyrylcholinesterase according to the equations (55) and (57), through considering the kinetic parameters from (36) and (39) with the actual values K

_{S }= 7.5 µM; K

_{SS}= 8,700 µM; V

_{max}= 7.2 µM/min; b = 3, for various initial substrate concentrations.

- The Haldane-Radić kinetics may be quite well modeled by its Michaelis-Menten counterpart progress curves for substrate kinetics below 100 µM in all studied cases, being this condition susceptible to be a general fact that is independent of ideal approach for the Haldane-Radić kinetic parameters K
_{SS}and b as prescribed in Equation (7); - Haldane-Radić kinetics display full specificity in looping S-E mechanisms of inhibition/activation for higher concentration of the substrate, i.e. within the mili-molar range;

## 4. Conclusions

- Higher dissociation constant for inhibition/activation substrate site interaction to enzyme;
- Equal catalytic efficiency of inhibition/activation substrate-enzyme loop as provided by E-S hydrolysis;
- Lower substrate concentration, typically in the range up to the 100 µM.

## Acknowledgements

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**MDPI and ACS Style**

Putz, M.V.
On the Reducible Character of Haldane-Radić Enzyme Kinetics to Conventional and Logistic Michaelis-Menten Models. *Molecules* **2011**, *16*, 3128-3145.
https://doi.org/10.3390/molecules16043128

**AMA Style**

Putz MV.
On the Reducible Character of Haldane-Radić Enzyme Kinetics to Conventional and Logistic Michaelis-Menten Models. *Molecules*. 2011; 16(4):3128-3145.
https://doi.org/10.3390/molecules16043128

**Chicago/Turabian Style**

Putz, Mihai V.
2011. "On the Reducible Character of Haldane-Radić Enzyme Kinetics to Conventional and Logistic Michaelis-Menten Models" *Molecules* 16, no. 4: 3128-3145.
https://doi.org/10.3390/molecules16043128