^{*}Alfonsas Juška;

The aim of the research is to analize and model the progress of product accumulation in relation with the kinetic properties of the enzyme and substrate concentration. Conventional methods of analysis are used. The results are as follows: `linear' decline of high substrate concentration (or rise of the product) is determined by the kinetic properties of the enzyme taking into account initial substrate concentration (Model 1); the exponential decline corresponds to Model 2; the deviations from linearity occurring either because of product inhibition of the reaction, the backward reaction or substrate exhaustion are modeled by algebraic sum of linear and exponential functions. Modification of the conventional scheme of the enzyme-catalyzed reaction, therefore, facilitates deriving the equations highlighting the role of the reaction product and initial substrate concentration in the progress of the reaction. The equations enable estimation of the kinetic parameters of the enzyme from the data of conventional experiments and an additional experiment corresponding to the `end' of the reaction.

The most essential property of enzymes is their ability to catalyze very specific chemical reactions of biological importance. Some enzymes can increase the rate of a chemical reaction by as much as 10^{12}-fold over the spontaneous rate of the uncatalyzed reaction [^{17}-fold [^{19}-fold [^{23}-fold [

The methods of data analysis suggested in biochemistry (enzyme) textbooks (see, e.g.,[

Equally important is the selectivity of the enzyme. That can be characterized by quasi-equilibrium dissociation constant or so-called Michaelis constant. Again, to determine this constant from the experimental data, inadequate methods are used, involving coordinate transformation, the most favourite being the double reciprocal or Lineweaver–Burk (linear) plot (see, e.g., [

It is important to relate the conversion of substrate into reaction product to the known properties of the enzyme. A great variety of the enzyme properties have been established up to date (see, e.g., [

In the present work, therefore, the simplest possible relationships of the progress of enzyme-catalyzed reaction with the kinetic properties of the enzyme has been analysed. It has been shown that the initial rate of enzymatic reaction (its efficacy) can be determined on the basis of a simple model making use of

Standard software was used. The algebraic and differential equations or their systems were solved with

The modeling is based on the scheme of interactions presented in Figure

Schematic representation of enzyme-catalyzed reaction. Top: a conventional scheme. Bottom: the scheme equivalent to the former but more convenient for shift to mathematical modeling of the progress of the reaction. The capital non-italic letters (E, ES, EP, S, P, S*, P*) denote the enzyme (free or bound) and the substrate or product (free or bound), the corresponding lowercase italic letters (c0, cS, cP, s, p), the probabilities of the enzyme states and the ligand concentrations,

For steady state, supposing the algebraic sum of absolute forward and backward rates of the transitions to be zero and the sum of the probabilities of the enzyme to reside in either state or corresponding relative concentrations being unity, it follows from the scheme that

Solutions of the above system are impractical to be presented here. For

Ignoring _{
S
}
_{
P
}

It is clear that the sum of concentrations of the substrate (_{0}) during the reaction. That makes one of the equations. At the same time, the rate of the enzymatic conversion of the substrate to product (the forward reaction) is proportional to the concentration of the enzyme (let it be _{0}) and the probability of the enzyme to be in state ES (3). The product of the forward reaction is a substrate for the backward one (ES ← EP) whose rate is proportional to the same _{0} and the probability of the enzyme to reside in EP state which is expressed by (4). The net rate makes another equation. Thus

For

and

where

Reaction product accumulation (rising) and substrate exhaustion (declining; high substrate concentration). A. No effect of product on the reaction progress being taken into account; inhibition of the reaction by the product and the backward reaction being taken into account. B. Various product dissociation constants, p. C. Various rates of the backward reaction, _{0} = 1, _{0} = 0.001), no inhibition by the reaction product (p →

The 1st term of the right-hand side of the 2nd equation of System (5) models the absolute rate of the forward reaction, while the 2nd term does that of the backward one, the fractions modeling the probabilities of the enzyme to be in states ES and EP (see Figure

Product concentration being low (_{0}) and p being finite, the differential equation of product accumulation is (see Appendix A):

and solution of the above equation under initial conditions (let _{0},

This equation models product accumulation under various conditions. The model is presented in Figure

In this case the scheme of the reaction is (see [

Here it is sufficient to analyse only the 1st step (S & E ↔ ES) of the process. The model being simple can be extended for any relationship between _{0} and _{0} (_{0}<_{0}<

and, consequently,

Solutions of this equation are depicted in Figure

Reaction product accumulation (rising) and substrate exhaustion (declining; low substrate concentration) at constant enzyme concentration, fixed substrate–enzyme dissociation constant (_{0} = 100, s = 100) and varying initial substrate concentrations, no inhibition by the reaction product (p →

At low initial substrate concentration (_{0}<< s) and in the absence of product action on the enzyme (p →

the solution of the above equation being exponential (see Figure

The scheme depicted in Figure _{1}-bound (EP), product_{2}-bound (EQ) and both products-bound (EPQ). Correspondingly, a system (analogous to System (1)) of 5 algebraic equations with regard to those states has to be composed. Solutions of this system with respect to states ES and EP (assuming

On the basis of (14) and (15) and taking into account that _{0}) a system of differential equations analogous to System (5) has been composed. Model curves corresponding to the solutions of the system are presented in Figure 5.

Simplified representation of enzyme-catalyzed reaction producing 2 products. Here s, p and q are dissociation constants of substrate and products with the enzyme.

Product accumulation and substrate exhaustion in the reaction resulting in 2 products. Enzyme concentration, substrate and enzyme dissociation constant and initial substrate concentration (_{0} = 0.01, s = 0.1 and _{0} = 1 arbitrary unit) are the same throughout. No effect of product on the reaction progress being taken into account (p →

Not using the subscripts in the notations of rate constants, arithmetic signs (+) in the scheme and avoiding to use the brackets (denoting the probabilities of the states and the relative concentrations by lowercase letters (_{0}, _{
S
}, _{
P
}, _{−1}, _{−2},..., i.e., with the “minus” signs) [

The dissociation constants used in the equations are related to corresponding ligands and have the same dimension as the ligands. The notations used (s, p, q), therefore, seem to be quite justified.

The assumption that _{0}, _{
S
} and _{
P
}) being determined by (2)–(4). The processes of interactions correspond to the

It is interesting to note that neither the above schemes nor System (5) incorporate the rate of chemical reaction proceeding spontaneously (without the enzyme), neither the specificity nor efficacy depending on this rate. Conversely, therefore, the enzyme-caused increase in the above rate occurs, presumably, due to high specificity (low dissociation constant, s), high efficacy (high rate of the conversion,

System (5) is equivalent to the scheme and adequately models the conversion of substrate to product (S* → P*) containing 4 independent parameters (s, p,

As seen from (9), reaction product accumulation (_{0} and _{0}). It is clear that the deviation can be ignored for a short period of time. Although (9) at first glance may look cumbersome, its parameters of similar meaning are denoted by similar symbols and grouped together making the equation clear and simple. Expression of the same relationship using the conventional notations would make the equation incomprehensible.

It should be noted that the reaction progress curves corresponding to high initial substrate concentrations (all the other parameters including the dissociation constant(s) being the same) are not convenient to present on the same graph because of the scale. For this reason, the initial substrate concentration is assumed to be the same (1 arbitrary unit) for all the curves, the dissociation constant(s) for different curves being different. Numerical solutions of system (5) for fixed initial conditions (_{0} and

In spite of the non-linearity (both theoretical and experimental), usually only the so-called initial velocity of the reaction is estimated from experimental data by drawing the straight line tangent to the curve approximating (rather arbitrarily) the data, valuable information (often most of it) contained in the data not being used. Use of (9) for comparison with experimental data of product accumulation to extract information concerning the unknown parameters is impractical, however, because of too many parameters (s, p,

It should be noted that presenting the enzyme-catalyzed reaction as in the above schemes (either the conventional one or its modification, Figure _{0}<<_{0}). The assumption that the reaction is not inhibited by the product (its dissociation constant, p, being high) enables to simplify the reaction scheme as shown in the scheme (10) and that in turn allows to extend the range of substrate concentration (0 <_{0} considering the binding rate to be proportional to _{0} (Model 2). (Similar extension of Model 1 would make it too cumbersome).

Modeling the non-linear product accumulation (the non-linearity being caused solely by product accumulation) is rather straightforward. Indeed, let the substrate concentration be high (_{0}>>_{0}) and the rate of substrate conversion into the product (without taking into account the effect of the product on the reaction) be proportional to _{0}/(_{0} + s), (the fraction considered as the degree of saturation of the enzyme by the substrate). It seems reasonable to assume the result of the product action on the deviation of product accumulation from the straight line to be proportional to its concentration,

The right-hand side of the above equation is of the same form as that of (8). Its solution under the same conditions is

i.e., the solution is of the same form as that of (9) and, as a matter of fact, equivalent to (9). It can be reminded that the latter follows from System (5) in which (3) and (4), i.e, approximations concerning the rate constants are used. Parameter

Initial substrate concentration being much lower than that of the enzyme (_{0}<<_{0}) and, as a consequence, the backward reaction being absent, corresponds to so-called _{0}, the substrate–enzyme dissociation constant, s, is of importance in the

(3) adequately models the selectivity. Product action on the reaction being negligible (1/p → 0), (2) and (3) are reduced to expressions which are usually referred to as Henri–Michaelis–Menten [_{
m
}) which in the present notations would be _{
m
} = (_{
m
} is less than, greater than, or equal to” s [_{
m
} for any 0 ≤_{
m
} for

The relationship between the initial rate of product accumulation and substrate concentration resulting from the reaction corresponds to the dependence of probability of the enzyme to be in state ES on substrate concentration ignoring any possible action of the product on the reaction (ignoring any affinity of the product to the enzyme, p →

Direct application of this model for analysis of experimental data (without any transformation), being very simple and clear (being no problem if any computer spreadsheet is used), enables the best accuracy of the parameter s estimation. For the best fit of the model to the data the least squares method can be applied [20–22]. If experimental data suggest a decline after the rise (probably caused by excess of substrate concentration (see [

Marangoni [_{0} and _{0}): at high substrate concentration (_{0}> s) its decline is with a clear break (and, therefore, most informative), deviating considerably from the exponential, but without a break at low substrate concentration (see Figs. 2D and 3). That makes the use of the information contained in the curve possible only at high substrate concentration. Besides, experimental data of the reaction progress “until its end” are necessary. Marangoni [

Parameter s can be estimated from comparison of (18) with the corresponding data, direct fitting enabling better estimation as stated above. From comparison of the solution of (14) with experimental data, parameter _{
S
}/_{
P
} =

where _{0} is experimentally established parameter corresponding to _{0} and _{0}; _{
∞
} and _{
∞
} are substrate and product concentrations resulting from the indefinite progress of the reaction. (19) and (20) enable, therefore, estimation of the constants of the backward reaction (p and

Numerical solving of differential equations is not necessary for parameter estimation (used here only to present the curves on the graphs). All the parameters of the system (i.e., the kinetic parameters of the enzyme, see Figure

In the case of the reaction resulting in 2 products (see Figure _{2}O) whose concentration is practically independent of the reaction, it can be considered as one product reaction analysed above.

Modification of the conventional scheme of the enzyme-catalyzed reaction facilitates the use of mathematics. The `linear' decline of the substrate concentration (or rise of product concentration) is modeled by the equation depending on the enzyme and initial substrate concentrations (Model 1); the exponential decline corresponds to Model 2. Numerical solving of differential equations is not necessary for parameter estimation.