The authors have declared that no competing interests exist.

The aim of present study was to describe the graphical technique how to go from Hill concentration constant to Michaelis constant.

To compare enzymatic processes, the kinetics of which is subjected to different regularities, it is possible to use constants that characterize catalytic activity (_{max}) and concentration constants that are the substrate concentration at which the rate of the enzymatic process is equal to a half of maximum permissible rate. Concentration constants are _{0.5} for Hill equation and _{m} for Michaelis-Menton equation. The graphical approach was proposed in order to go from concentration constant of Hill equation to Michaelis concentration of the process that could be characterized by the same catalytic activity (the same values of minimum and maximum rates) similar to that observed in the process described by Hill equation.

There exist enormous amounts of research findings on the activity of various enzymes. As a result, these days a greater attention is focused on determination of the mechanisms of enzymatic reactions and kinetic constants. To find the constants for monomeric enzymes for which the dependence of the rate of a reaction, catalyzed by these enzymes, on substrate concentration is subjected to the hyperbolic kinetics

V_{max}_{m}^{+ S),}

Where

Vis the rate of a reaction catalyzed by the enzyme;

Sis a substrate concentration;

V_{max} is maximum rate of enzymatic reaction (i.e. maximum value of the rate when S→∞);

K_{m} is the Michaelis constant (Michaelis concentration), which is determined graphically as substrate (S) concentration at which V = 0.5V_{max} and used for characterization of the affinity of the enzyme to a substrate.

The enzymes, the molecules of which consist of several identical monomers (subunits), are allosteric enzymes. A sigmoid (S-shape) but not Michaelis-Menten hyperbolic plot of the dependence of the rate of enzymatic reaction on the initial substrate concentration is typical of allosteric enzymes

V = V_{max}·S^{n}/_{0.5}^{n }^{n}

where

S_{0.5} is substrate (S) concentration at which V=0.5V_{max} (Hill concentration constant);

n is Hill coefficient used for assessment of S-shape functions.

As to stable soluble enzymes of simple structure, the enzymatic activity can be determined in cell-free extracts of microorganisms. But some enzyme complexes can not remain active during cell destruction. So, the activity of these enzymes of multi-subunit structure cannot be determined in cell-free extract. The activity of these enzymes is assessed indirectly by the response of intact cells to enzyme substrate. This refers, for instance, to benzoate 1,2-dioxygenase. The activity of this enzyme is estimated by change in oxygen consumption by whole cells after injection of enzyme substrate (polarographic determination)

How to compare the enzymes the kinetics of which is different? In a series of papers for the comparison of enzyme activity, the investigators went from Hill constant to Michaelis constant using the calculation formula. Hence, the use of the constants of hyperbolic dependency is necessary to calculate the inhibition constants

Reasonable comparison facilitates understanding the mechanism of the process, but the unmotivated comparison has no results. For the comparison of two processes with different mechanisms the catalytic activities (V_{max}) have to be compared. Concentration constants can also be compared: they are Michaelis constant (Michaelis concentration), which is determined graphically as substrate concentration at which the rate of enzyme reaction is half of V_{max}, and substrate concentration S_{0.5} of Hill equation at which V=0.5V_{max}.

Even having not estimated calculation formulas described in literature to go from S_{0.5} to K_{m} it should be noted that complex mathematical manipulations are difficult for understanding. So, in this paper, a simple in use approach is suggested to go from Hill constant to Michaelis constant.

The coefficient n (Hill coefficient) is present in the Hill equation. The Hill coefficient is dimensionless. It should be taken into account, that the Hill coefficient is a parameter of an empirical formula and has no physical sense. Hill coefficient is not larger (≤) than the amount of active sites, although in terms of rather strong enzyme-substrate interactions this coefficient can be close proximity to the amount of active sites _{0.5} of Hill equation to Michaelis constant, K_{m}.

The Hill coefficient value (<1 or >1) characterizes a type of deviation from the Michaelis-Menten hyperbolic dependence: “negative” or “positive kinetic cooperativity” by substrate for allosteric enzyme. This coefficient is > 1 under “positive kinetic cooperativity” by a substrate and <1 under “negative kinetic cooperativity” (n = 1 for “classical” Michaelis-Menten hyperbolic dependency).

Negative kinetic cooperativity was observed, for example, for heterogeneous enzymatic preparations. Examples of this type of deviation from hyperbolic dependency are shown in

The physiological significance of positive kinetic cooperativity lies in the fact that it is sufficient to have traces of a substrate to activate metabolism. Moreover, saturation is achieved at lower substrate concentrations; this is important for bacteria, for which toxic substances often serve as substrates

Under positive cooperativity, the catalytic efficiency of active sites (and the affinity to a substrate) of the allosteric enzyme increases as active sites are filled by a substrate. The greater is the affinity to a substrate; the lower is the value of a constant that characterizes the affinity to a substrate.

If V vs. S dependence of the process is depicted by S-shape curve, it is necessary to decide what parameters of the process (Michaelis-Menten kinetics), for which K_{m} has to be calculated, should be. If for the process of Michaelis-Menten kinetics the maximum rate of enzyme-substrate reaction and minimum substrate concentration, to which the enzyme is sensible, should be similar to those of S-shape curve (positive kinetic cooperativity), then curves of V vs. S dependence for Michaelis-Menten kinetics and positive kinetic cooperativity (Hill equation) will be the same as shown in

Non-linearity of the dependency of 1/V on 1/S confirms that the V vs. S dependency is not hyperbolic. But if the 1/V on 1/S dependency is linear, V vs. S dependency is hyperbolic (

To go from S_{0.5} to K_{m}, it is enough to plot a line similar to line 2 in _{min}) of enzyme-substrate reaction, which can be detected.

As a matter of fact, the line should pass through a cluster of points which are reciprocal to V_{max}: this is a saturation area of the curve V vs. S. The line has to cross the axis _{max}, where V_{max} is a known value. V_{max} is the constant of the sigmoidal dependency (Hill equation). The equation of the plotted line is _{max}, and the required Michaelis concentration will be obtained when _{m} = b/a. So, the estimated value of the Michaelis constant will be calculated for the process with the catalytic activity which is the same for the process with Hill kinetics. The example how to go from S_{0.5} to K_{m} is given below.

With the use of experimental data we plot a curve of V vs. S (_{max}·S^{n}/_{0.5}^{n }^{n}_{0.5} = 69.789x10^{-6} mM = 69.789 µM, V_{max} = 15.653 pA/s, n = 2.00. Then, the curve in reciprocal coordinates, _{max} = 1/15.653 = 0.06389. The program calculates the equation of the line curve plotted ^{-5 }x_{m}. Hence, K_{m} = 2.3722/0.0639x10^{-5} = 371.2 µM.

With the graphical approach it is rather easy to go from concentration constant of Hill equation to Michaelis concentration.