A model of free sequential enzymes of the Michaelis–Menten type is represented in Fig. 1, in the case of a two-enzyme system catalysing the transformation of an initial substrate S₁ into a final product S₃. This model system is made of two reaction circuits, where the first and second circuits correspond to the activity of the first and second enzyme, E₁ and E₂, respectively. Clearly, it would be easy to model three-, four-, …, n-enzyme systems by adding a third, fourth, …, nth reaction circuit corresponding to enzymes $E_3,E_4,\dots ,E_n$, respectively (not shown).

The numerical simulations have consisted of studying the dependence of the rate, v, of the overall reaction of transformation of S₁ into S₃, on the concentration of S₁ for various values of the forward and reverse rate constants, $h_{i\mathrm{f}}$ and $h_{i\mathrm{r}}$ (see Fig. 1). In practise, we have used dimensionless expressions of all the variables and parameters. For the definition of these dimensionless quantities, see Appendix A. Independent steady-state equations have been obtained by writing down the mass balance of the species involved, i.e.

In these equations, $k_{i\mathrm{f}}$, $k_{i\mathrm{r}}$, τ, $e_1$, $e_2$, $s_1$, $s_2$, $s_3$, $e_1{s}_1$, $e_1{s}_2$, $e_2{s}_2$ and $e_2{s}_3$ are the dimensionless expressions of the forward and reverse rate constants, $h_{i\mathrm{f}}$ and $h_{i\mathrm{r}}$, of the time, t, and of the concentrations of E₁, E₂, S₁, S₂, S₃, E₁S₁, E₁S₂, E₂S₂, and E₂S₃.

To write down the steady-state conditions of functioning of the system, we have assumed that external mechanisms supply S₁ and remove S₃ as and when they are consumed and produced, respectively, such that S₁ is maintained at a constant concentration $(s_1=\text{constant})$ and S₃ at a zero concentration $(s_3=0)$. Under such steady-state conditions, v is measured indifferently by the rate of consumption of S₁ or the rate of production of S₃, i.e. for the dimensionless expression of v:

Moreover, two relationships have to be taken into account between the rate constants. One is imposed by how the dimensionless quantities have been defined (see Appendix A, Eq. (A.9)),

The numerical simulations of the dependence of v on $s_1$ (with $s_3=0$) for a variety of values of the rate constants have been carried out using the MAPLE software to solve the algebraic system (Eqs. (1) to (5)). We have always found, as expected from the calculation of the first derivative of v (not shown), that the $\{s_1,v\}$ curves were increasing monotonically up to a saturation plateau (see examples in Fig. 2A). Moreover, in these numerical simulations, we have never found any curve exhibiting one or several inflexion points. Sometimes, the shape of the curves was close to that of a hyperbola as shown by the fact that the corresponding curves in the system of coordinates $\{v/s_1,v\}$ (Fig. 2B) were close to straight lines (see curve b); however, most often this was not the case (see especially curves d and e). When adjusting straight lines to the computed curves in Fig. 2A, in the range of $s_1$ values from 0 to the abscissa, $s_{\mathrm{int}}$, of the point of intersection of the tangent at origin with the saturation plateau of each curve (see Eq. (B.5) in Appendix B for the determination of $s_{\mathrm{int}}$), the values of the regression coefficient, $r^2$, of these linear adjustments were 0.9608, 0.9682, 0.9843, 0.9991, and 0.9999 for curves (a) to (e), i.e. for $k_{2\mathrm{f}}$-values varying from 10 (curve a) to 0.01 (curve e). This means that decreasing the value of $k_{2\mathrm{f}}$ tends to render the reaction rate, v, proportional to the concentration of initial substrate, $s_1$; moreover, with a $k_{2\mathrm{f}}$ value as low as 0.01, it appears that proportionality remains valid (curve e practically undistinguishable from its tangent at origin) almost up to the saturation plateau. It is also visible in the figure that the linear approximation thus observed at a low $k_{2\mathrm{f}}$ value extends far beyond the quasi-linear zone that is known to exist at the beginning ($s_1$ close to zero) of a hyperbolic $\{s_1,v\}$ curve (compare curve (e) with the hyperbolic curve drawn in dotted line in Fig. 2A).

Increasing $k_{1\mathrm{f}}/k_{1\mathrm{r}}$ or $k_{4\mathrm{f}}/k_{4\mathrm{r}}$ and/or decreasing $k_{3\mathrm{f}}/k_{3\mathrm{r}}$ have the same effect as decreasing $k_{2\mathrm{f}}/k_{2\mathrm{r}}$, i.e. this tends to favour the proportional response of v as a function of $s_1$. Increasing $k_{10\mathrm{f}}/k_{10\mathrm{r}}$ tends to increase the value of the saturation plateau. In our present approach, the equilibrium constant, K, of the overall reaction of transformation of S₁ into S₃ is involved only in the calculation of $k_{4\mathrm{f}}$ from the other rate constants (Eq. (8)); since we have imposed $s_3=0$, the value of $k_{4\mathrm{f}}$, and consequently that of K, has no effect on the result of the numerical simulations.

As an example of an n-enzyme system, we have studied the case of a system made of four free enzymes. We have always found $\{s_1,v\}$ curves (i) monotonically increasing up to a saturation plateau and (ii) sometimes exhibiting an extended region where v was proportional to $s_1$. On total, the results were never qualitatively very different from those observed with two-enzyme systems.

With a system of two free sequential enzymes of the Michaelis–Menten type in a metabolic pathway, we have always observed in our numerical simulations that the $\{s_1,v\}$ curves were monotonically increasing up to a saturation plateau. Depending on the values of the rate constants, the curve shape varied from quasi-hyperbolic to extremely non-hyperbolic, including cases in which v became quasi proportional to $s_1$ almost up to the saturation plateau (which never occurs with individual enzymes). Choices of values of the rate constants either increasing the efficiency of enzyme E₁ in the transformation of S₁ to S₂ (high values of $k_{1\mathrm{f}}/k_{1\mathrm{r}}$ and low values of $k_{2\mathrm{f}}/k_{2\mathrm{r}}$) or decreasing the efficiency of enzyme E₂ in the transformation of S₂ to S₃ (low values of $k_{3\mathrm{f}}/k_{3\mathrm{r}}$ and high values of $k_{4\mathrm{f}}/k_{4\mathrm{r}}$) favoured the appearance of this quasi-proportional response. The behaviour of four-enzyme systems was qualitatively not very different from that of two-enzyme systems.

Dimensionless quantities have been defined by normalising all concentrations (with [X] = concentration of X) to the sum of the total concentrations of E₁ and E₂, $[{\mathrm{E}}_{1\mathrm{t}}]+[{\mathrm{E}}_{2\mathrm{t}}]$, and all time values to $1/h_{1\mathrm{r}}$. As a consequence, the molar fractions of enzymes E₁ and E₂ are:

The steady-state rate of transformation of S₁ into S₃ at saturation, $v_{\max }$ $(s_1=\infty )$, and the slope at origin, $v^{\prime }(0)$ $(s_1=0)$, can be derived from Eqs. (1) to (5). Define Q by: