6a. Michaelis-Menten enzyme kinetics
[1]:
# Colab setup ------------------
import os, sys, subprocess
if "google.colab" in sys.modules:
cmd = "pip install --upgrade colorcet biocircuits watermark"
process = subprocess.Popen(cmd.split(), stdout=subprocess.PIPE, stderr=subprocess.PIPE)
stdout, stderr = process.communicate()
# ------------------------------
import biocircuits
import bokeh.io
import colorcet
bokeh.io.output_notebook()
At the heart of a Michaelis-Menten description of enzyme kinetics is the following set of chemical reactions between the enzyme E and its substrate S to give product P.
\begin{align} \require{mhchem} \ce{E + S <=>[k_1][k_{-1}] ES ->[k_2] P + E}. \end{align}
The mechanism assumes that the enzyme reversibly binds the substrate with binding rate constant \(k_1\) and unbinding rate constant \(k_{-1}\). When bound, the enzyme can convert the substrate to product with rate constant \(k_2\). Applying mass action kinetics, we can write the dynamics as a system of ordinary differential equations.
\begin{align} \frac{\mathrm{d}c_\mathrm{e}}{\mathrm{d}t} &= -\frac{\mathrm{d}c_\mathrm{es}}{\mathrm{d}t} = -k_1c_\mathrm{e} c_\mathrm{s} + (k_{-1} + k_2)c_\mathrm{es},\\[1em] \frac{\mathrm{d}c_\mathrm{s}}{\mathrm{d}t} &= -k_1c_\mathrm{s} c_\mathrm{e} + k_{-1}c_\mathrm{es},\\[1em] \frac{\mathrm{d}c_p}{\mathrm{d}t} &= k_2 c_\mathrm{es}, \end{align}
where \(c_i\) denotes the concentration of species i. Note that
\begin{align} \frac{\mathrm{d}c_\mathrm{e}}{\mathrm{d}t} + \frac{\mathrm{d}c_\mathrm{es}}{\mathrm{d}t} = 0, \end{align}
which is a statement of conservation of enzyme. This means that we need to specify a total enzyme amount to fully determine the problem. We define this to be \(c_\mathrm{e}^0\) such that \(c_\mathrm{e}^0 = c_\mathrm{e} + c_\mathrm{es}\). With this conservation law, we can write the ODEs as
\begin{align} \frac{\mathrm{d}c_\mathrm{es}}{\mathrm{d}t} &= k_1(c_\mathrm{e}^0-c_\mathrm{es}) c_\mathrm{s} - (k_{-1} + k_2)c_\mathrm{es},\\[1em] \frac{\mathrm{d}c_\mathrm{s}}{\mathrm{d}t} &= -k_1(c_\mathrm{e}^0-c_\mathrm{es})c_\mathrm{s} + k_{-1}c_\mathrm{es},\\[1em] \frac{\mathrm{d}c_p}{\mathrm{d}t} &= k_2 c_\mathrm{es}. \end{align}
These equations describe the full dynamics of the enzyme catalyzed system. To simplify the analysis, we often make the quasi-steady state approximation that the bound substrate intermediate ES does not appreciably change in its concentration on the time scale of the production of the product P. That is,
\begin{align} \frac{\mathrm{d}c_\mathrm{es}}{\mathrm{d}t} = k_1(c_\mathrm{e}^0-c_\mathrm{es}) c_\mathrm{s} - (k_{-1} + k_2)c_\mathrm{es} \approx 0. \end{align}
This enables us to solve for the quasi-steady state fraction of enzyme that is bound to substrate.
\begin{align} \frac{c_\mathrm{es}}{c_\mathrm{e}^0} \approx \frac{c_\mathrm{s}/K_\mathrm{M}}{1 + c_\mathrm{s}/K_\mathrm{M}}, \end{align}
where we have defined the Michaelis constant
\begin{align} K_\mathrm{M} = \frac{k_{-1} + k_2}{k_1}. \end{align}
The Michaelis constant has dimension of concentration. It is analogous to a dissociation constant, in that it is the ratio of the total rate constants for leaving the bound state to the rate constant for enzyme-substrate binding.
Substitution of this expression gives
\begin{align} \frac{\mathrm{d}c_p}{\mathrm{d}t} \approx k_2\,c_\mathrm{e}^0\, \frac{c_\mathrm{s}/K_\mathrm{M}}{1 + c_\mathrm{s}/K_\mathrm{M}}. \end{align}
By conservation of mass, if \(\mathrm{d}c_\mathrm{es}/\mathrm{d}t \approx 0\), then
\begin{align} \frac{\mathrm{d}c_\mathrm{s}}{\mathrm{d}t} \approx -\frac{\mathrm{d}c_p}{\mathrm{d}t} \approx -k_2 c_\mathrm{e}^0\, \frac{c_\mathrm{s}/K_\mathrm{M}}{1 + c_\mathrm{s}/K_\mathrm{M}}. \end{align}
Testing the accuracy of the quasi-steady state approximation
To test the accuracy of the quasi-steady state approximation, it helps, as usual, to nondimensionalize the variables, as follows:
\begin{align} t &\leftarrow \frac{k_2c_\mathrm{e}^0}{K_\mathrm{M}}\, t,\\[1em] c_\mathrm{es} &\leftarrow \frac{c_\mathrm{es}}{c_\mathrm{e}^0},\\[1em] c_\mathrm{s} &\leftarrow \frac{c_\mathrm{s}}{K_\mathrm{M}},\\[1em] c_p &\leftarrow \frac{c_p}{K_\mathrm{M}}. \end{align}
The dimensionless dynamical equations are then
\begin{align} \kappa\,&\frac{\mathrm{d}\tilde{c}_\mathrm{s}}{\mathrm{d}\tilde{t}} = -(1 - \tilde{c}_\mathrm{es})\tilde{c}_\mathrm{s} + (1-\kappa)\tilde{c}_\mathrm{es},\\[1em] \kappa\zeta\,&\frac{\mathrm{d}\tilde{c}_\mathrm{es}}{\mathrm{d}\tilde{t}} = (1 - \tilde{c}_\mathrm{es})\tilde{c}_\mathrm{s} - \tilde{c}_\mathrm{es},\\[1em] &\frac{\mathrm{d}\tilde{c}_\mathrm{P}}{\mathrm{d}\tilde{t}} = \tilde{c}_\mathrm{es}. \end{align}
We have defined two dimensionless parameters. First, we have
\begin{align} \kappa = \frac{k_2}{k_{-1} + k_2}. \end{align}
This is the probability that a given enzyme-substrate complex will result in formation of a product molecule as opposed to dissociating. Note that \(0 < \kappa < 1\). Second, we have
\begin{align} \zeta = \frac{k_1 c_\mathrm{e}^0}{k_{-1} + k_2} = \frac{c_\mathrm{e}^0}{K_\mathrm{M}}. \end{align}
This is the ratio of the rate of transitions into the enzyme-substrate complex to the rate of transitions out of the complex.
It is apparent in looking at the dimensionless equations that the quasi-steady state approximation is valid when \(\zeta \kappa\) is small. Because \(0 < \kappa < 1\), it is sufficient to have \(\zeta \ll 1\) for the quasi-steady state approximation to hold. Thus, the approximation holds when the amount of enzyme is small compared to the Michaelis constant.
To investigate this, we can make plots and compare the solution to the full dynamical equations and those using the quasi-steady state approximation. In the plot below, we show the approximate dynamics of the substrate and the product as fainter colored lines and the exact solution as darker, thinner lines. All concentrations and times are dimensionless.
[2]:
bokeh.io.show(biocircuits.jsplots.michaelis_menten_approx())
By adjusting the parameters \(\zeta\) and \(\kappa\), we find that provided \(\zeta \ll 1\), the solution of the Michaelis-Menten equation very closely matches the exact solution When \(\zeta\) is large and the approximation breaks down, we see obvious problems with the approximate solution—the concentration of product goes negative because mass is not conserved. We also find that in the small \(\zeta\) regime, the parameter \(\kappa\) has no bearing on the solution. This is also evident in the dimensionless equations. When \(\zeta\) is small, we have
\begin{align} \frac{\mathrm{d}\tilde{c}_\mathrm{es}}{\mathrm{d}\tilde{t}} \approx 0 \end{align}
such that
\begin{align} (1 - \tilde{c}_\mathrm{es})\tilde{c}_\mathrm{s} \approx \tilde{c}_\mathrm{es}. \end{align}
The, the ODE describing substrate dynamics is
\begin{align} \kappa\,&\frac{\mathrm{d}\tilde{c}_\mathrm{s}}{\mathrm{d}\tilde{t}} = -(1 - \tilde{c}_\mathrm{es})\tilde{c}_\mathrm{s} + (1-\kappa)\tilde{c}_\mathrm{es} \approx -\tilde{c}_\mathrm{es} + (1-\kappa)\tilde{c}_\mathrm{es} = -\kappa \tilde{c}_\mathrm{es} \approx -\kappa \,\frac{\tilde{c}_\mathrm{s}}{1+\tilde{c}_\mathrm{s}}, \end{align}
giving
\begin{align} \frac{\mathrm{d}\tilde{c}_\mathrm{s}}{\mathrm{d}\tilde{t}} \approx \frac{\tilde{c}_\mathrm{s}}{1+\tilde{c}_\mathrm{s}}, \end{align}
such that the dynamics are independent of \(\kappa\).
Operating regimes
Now that we know when the quasi-steady state approximation is valid such that the rate of production of product is
\begin{align} \frac{\mathrm{d}c_p}{\mathrm{d}t} \approx k_2 c_\mathrm{e}^0\, \frac{c_\mathrm{s}/K_\mathrm{M}}{1 + c_\mathrm{s}/K_\mathrm{M}}, \end{align}
we can investigate limits of small and large substrate concentrations on the dynamics. First, in the limit of small substrate concentration where \(c_\mathrm{s}/K_\mathrm{M} \ll 1\), we have
\begin{align} \frac{\mathrm{d}c_\mathrm{p}}{\mathrm{d}t} \approx \frac{k_2 c_\mathrm{e}^0}{K_\mathrm{M}}\,c_\mathrm{s}. \end{align}
In this case, we have pseudo-first order kinetics in which the rate of production of product is linear in the substrate concentration with an effective rate constant of \(k_2c_\mathrm{e}^0/K_\mathrm{M}\).
Second, in the limit of large substrate concentration where \(c_\mathrm{s}/K_\mathrm{M} \gg 1\), we have
\begin{align} \frac{\mathrm{d}c_\mathrm{p}}{\mathrm{d}t} \approx k_2 c_\mathrm{e}^0, \end{align}
which is apparently zero-order in the substrate concentration. An enzyme operating in this high substrate concentration regime is said to be operating at saturation. In this sense, the Michaelis constant sets the concentration scale of the substrate for which maximal conversion rate may be achieved.
Reaction velocity and \(k_\mathrm{cat}\)
In many schemes for catalyzed conversion of substrate to product, we can write the approximate dynamics for the production rate of product as
\begin{align} \frac{\mathrm{d}c_\mathrm{p}}{\mathrm{d}t} = k_\mathrm{cat}\,c_\mathrm{e}^0\,\frac{c_\mathrm{s}/K_\mathrm{M}}{1 + c_\mathrm{s}/K_\mathrm{M}}. \end{align}
Some authors refer to \(\mathrm{d}c_\mathrm{p}/\mathrm{d}t\) as the reaction velocity, and the product \(k_\mathrm{cat}\,c_\mathrm{e}^0\) as the maximum reaction velocity, denoted \(v_\mathrm{max}\) or \(V_\mathrm{max}\).
For the standard Michaelis-Menten chemical reaction scheme we have considered here, \(k_\mathrm{cat} = k_2\) and \(K_\mathrm{M} = (k_{-1} + k_2) / k_{1})\). Other reaction schemes may have different expressions. For example, if an inhibitor is present that can bind teh enzyme, the reaction scheme is
\begin{align} \require{mhchem} &\ce{E + S <=>[k_1][k_{-1}] ES ->[k_2] E + P},\\ &\ce{E + I <=>[k_3][k_{-3}] EI}. \end{align}
It can be derived that
\begin{align} \frac{\mathrm{d}c_\mathrm{p}}{\mathrm{d}t} = k_\mathrm{cat}\,c_\mathrm{e}^0\,\frac{c_\mathrm{s}/K_\mathrm{M}}{1 + c_\mathrm{s}/K_\mathrm{M}}, \end{align}
where, like standard Michaelis-Menten kinetics, \(k_\mathrm{cat} = k_2\), but the Michaelis constant is adjusted to account for inhibition,
\begin{align} K_\mathrm{M} = \frac{k_{-1} + k_2}{k_1}\left(1+\frac{k_3}{k_{-3}}\,c_\mathrm{i}\right), \end{align}
where \(c_\mathrm{i}\) is the concentration of inhibitor. In the inhibited case, the Michaelis constant is larger, meaning that more substrate is necessary to get a faster production rate, as we would expect.
Computing environment
[3]:
%load_ext watermark
%watermark -v -p numpy,scipy,bokeh,biocircuits,jupyterlab
Python implementation: CPython
Python version : 3.10.10
IPython version : 8.10.0
numpy : 1.23.5
scipy : 1.10.0
bokeh : 3.1.0
biocircuits: 0.1.10
jupyterlab : 3.5.3