Affirmation of the Separate Rate-law Method for Determining vmax and KM of the Michaelis Reaction

The maximum-velocity, v max, and the Michaelis-constant, KM , are the essence of the Michaelis-Menten equation. The value of v max is determined when the reaction is in zeroth-order and KM is in first-order. With a slight change in the notion of v max concept, both were used as the basis for a new determination method called Separate Rate-law Method (SRlM). As yet, the Lineweaver-Burk and Eadie-methods were the most commonly used. Both combine zeroth-order and first-order data. The purpose of this study was to validate the quality of all methods. The research was carried out theoretically. All were tested by applying them to the same literature data, and the results were compared. The assessment is based on the v max and KM values obtained; doubling [E] will double v max but the same KM . The results show that the two present methods are inaccurate. Both give v max and KM values even if the reaction is always first-order. Based on the integral method, two variants of the new method, SRlMIM , and initial rate, SRlMIRM , were also introduced. All new methods give more accuracy, which results in the method accuracy of SRlM ≈ SRlMIM > SRlMIRM .

Using the Steady-State Approximation and after some steps, the rate law is written in Eq. (3).
Where v, k2, [E], [S], and KM are rate, enzyme rate constant, enzyme concentration, substrate concentration, and Michaelis constant. KM is a measure of the enzyme's affinity for its substrate, and it has value in the range of 10 -6 to 10 -2 M. In experiments, the initial concentration of the enzyme is usually made very small compared to substrate concentration. In data analysis, the initial rate method (IRM) is the most commonly used.  (4), in which the reaction is zeroth-order to [S] and first-order to [E].
In this condition, all enzymes react with the substrate, and the reaction is in the maximum velocity or v will approach vmax. Then, Eq. (4) becomes Eq. (5).
Three problems arise. First, there is a need for a new method that followed the basic assumptions of Michaelis reaction. The method must be based on Eq. (5) and Eq. (7) and can be used as a standard method. Second, during the reaction process, the enzyme/catalyst concentration will be constant, but the substrate decreases. Depending on the ratio of the initial concentration of substrate and enzyme, the reaction can be zeroth-order at the beginning reaction; however, it then becomes first order at near-end reaction or always being the first-order reaction. There is no check for the real rate law at these two conditions in all of the present equations for Michaelis-Menten or catalyzed reaction fitting-constants determination. When both data are combined, this can cause problems. For example, if the first happens, it cannot give the correct values for vmax and KM because the reaction changes from zeroth-order to first-order at the near-end. If the latter happens, it will contradict the essence of Eq. (5) and Eq. (7). Third, v max. will be achieved if all enzymes/catalyst reacts with the substrate. However, the Michaelis reaction is first-order to enzyme/catalyst concentration. Thus, doubling enzyme/catalyst concentration will double the value of vmax as well. That is to say, vmax is not a constant. In the present equations, vmax is used to determine KM without considering this fact. In addition, save for first-order reaction, IRM or the differential method, will give different rate constant values at a constant time of reaction.
The present methods for determining vmax and KM are based on Eq. (8). However, it is difficult to determine both fitting-constants in this equation because the plot of v against [S] is hyperbolic. To cope with, the linear form is usually used, such as that introduced by Lineweaver-Burk (L-B) [1] and by Eadie [2]-Hofstee (E-H) [3], Hanes [4], and Espenson. The most commonly used is L-B and E-H, as presented in Eq. (9-10), respectively.
The linear plot of (vo) against (vo/[S]o) gives vmax = intercept and KM = -slope. Both equations use and share the outset and near-end data. Based on the above theoretical description, this study aims to introduce new methods based on Eq. (5) and Eq. (7) to prove that the present methods for vmax and KM determination in Michaelis reactions are ambiguous and to introduce a better notion of vmax and the real constant in Michaelis reactions.

Reaction model
This study theoretically used the enzymatic reaction data taken from the hydrolysis of phenylacetate catalyzed by enzyme acetylcholinesterase [6] and the hydrolysis of methyl-hydro cinnamate catalyzed by enzyme chymotrypsin [7]. These two hydrolysis data are presented in Table 1 and Table 2, respectively. The data can be divided into two parts and be assumed as the outset and near-end data. However, there is no data for studying the effects of changing enzyme or catalyst concentration. To cope with, the other model used is the reaction of acetone with bromine, catalyzed by H. Because there are two substrates, then the reaction occurred in conditions of excessive [Acetone] to [Bromine]. It is believed that in these conditions, the reaction has the rate law as presented in Eq. (11).
Where kobs = k [Acetone] and κ are the observed rates constant and catalytic constant, respectively. When Eq. (12-13) was assumed, Eq. (11) is similar to Eq. (8). The absorbance unit data (measured at 400 nm) is presented in Table 3 [8,9]. All data were analyzed using Eq. (9) and Eq. (10) and using the new equation. With slight changes in the notion of the maximum velocity concept, a new method, called Separate Rate-law Method or SRlM, for vmax determination is derived from Eq. (5).
The equation for K M determination is derived from Eq. (7).  Table 4 The variant of SRlM uses IRM (then called SRlMIRM) and IM (then called SRlMIM). Like SRlM, both SRlMIRM and SRlMIM can only be used when the reaction has a different rate law at the outset and near-end.

Data analysis
All methods are then used to determine the vmax and KM of the five reaction models. The results are presented in Table 4 through Table 8.

Data interpretations
A method is valid if vmax is determined when the reaction is zeroth-order and KM when the reaction is first-order. The vmax value is relative to the enzyme (catalyst) concentrations. However, KM value must be the same. A method will be ambiguously notified if it combines the outset and near-end data, regardless of the same or different reaction order.

First case
SRlM is an integral method; therefore, it is interesting to compare SRlM and SRlMIM. together with SRlMIRM. The advantage of IRM is that it directly gives the correct reaction order. However, the rate constant, k, value is uncertain. Save for zeroth-order reactions, k values are always different from IM. The correct value should be consulted with the Integral Method (IM) [10]. However, it is difficult to differentiate between integers and half-integers and even between sequences of integer order with IM. The actual value obtained from this method can be equated using the correction factor in Eq. (20-21). However, both of these equations will only be valid at a fixed β condition [11].
With Eq. (20), for zeroth-order, kIRM = kIM, and so on, vmax-IRM = vmax-IM. However, in Table 8, for [H + ] = 0.05 M, vmax-IRM ≠ vmax-IM, the percentage difference is 14.53 %. The percentage difference of vmax values between SRlM and SRlM IM relative to SRlM IM is 7.67%; this is slightly higher than the maximum allowable of 7%. The reason is that although SRlMIRM gives the correct order, the regression coefficient is 0.0265, which means there almost no correlation. For the same case, SRlMIM gives a regression coefficient of 0.9996 (Table 6.).
In this study, SRlM is taken as the standard method. It is only because the SRlM is directly derived from Eq. (5) and Eq. (7). Actually, SRlM is impractical to use. The order must determine in advance by IM. However, from these steps, vmax and KM can be directly determined by SRlMIM. As shown in Table 8, the percentage difference between SRlMIM and SRlM relative to SRlM for vmax was 3.99% and for KM was 2.49%. Therefore, it is better to use SRlMIM as the standard. The only drawback is to obtain a definite order of the reactions, and two different integral equations must be tested; zeroth-order and first-order. However, this can be overcome by using Eq. (22). Where β, tβ, ti, i, and n are the remaining reactant fractions (must be kept constant), first time observation, following observations, observation, and reaction order, respectively [12,13]. For example if n = 0 and β = 0.5 then t1 = tβ, t2 = 1/2 tβ, and t3 = 1/4 tβ. Then, if observations are made simultaneously, the rate will not be the same; it is faster. Thus, for n = 1, ti will always be the same with tβ.

Second case
There are three misconceptions in the L-B and E-H methods. First, both equations are based on IRM. IRM is best for determining reaction order but not for rate constants. There are two different laws.
Second, they do not consistently adhere to the basic concept of determining vmax and KM. The value of vmax must be determined when the reaction is zeroth-order (at the outset) and KM when the reaction is first-order (at near-end). However, neither method requires an examination of the real rate law. Instead, they use the combination of the outset and near-end data. There are values for vmax and KM, even if the reaction is half-order at the outset and first-order at the end (Table 4) or always first-order (Table 7). In Table 8, the vmax value obtained by all methods is almost the same. Theoretically, the value obtained by L-B and E-H should be lower compared to the new method. With the new method, vmax is determined when the reaction is zeroth-order, the reaction at its maximum velocity. This is because both methods are basically based on IRM (or differential method). There may be problems in using IRM for constant time interval observation data. In Table 5, the vmax value by the SRlMIRM is higher. As expected, the KM values obtained by the new methods are relatively higher than those of L-B and E-H. The KM value depends on vmax. Third, however, the main misconception of the L-B and E-H methods concerns their equations. If , v = vmax then both Eq. (15) and Eq. (16) must give: Eq. (20) states that v max and KM cannot be determined using a combination of outset and near-end data [11]. In actual, the used combined data is against Eq.

Third case.
Unusually, in Table 4, the order of [S] is half-order. Such cases occur when the enzyme or substrate is a weak acid or base. According to Eq. 7, [S] is first-order; however, the estimation using the modified method in the present study results in half-order. Therefore, the Michaelis reaction rate-law presented in Eq. 7 should be re-considered. In Table 8