.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/kinetics/enzyme/plot_01_michaelis_menten.py"
.. LINE NUMBERS ARE GIVEN BELOW.

.. only:: html

    .. note::
        :class: sphx-glr-download-link-note

        :ref:`Go to the end <sphx_glr_download_api_gallery_kinetics_enzyme_plot_01_michaelis_menten.py>`
        to download the full example code.

.. rst-class:: sphx-glr-example-title

.. _sphx_glr_api_gallery_kinetics_enzyme_plot_01_michaelis_menten.py:


Michaelis and Menten's saturating enzyme rate law
====================================================

Michaelis and Menten (1913) showed that an enzyme-catalyzed rate
saturates with substrate, :math:`v = V_{max}[S]/(K_m + [S])`, reaching
exactly half of :math:`V_{max}` at :math:`[S] = K_m`. This example draws
that saturation curve with its two limiting regimes, integrates the
substrate-depletion progress curve an assay actually records, and shows
how competitive and noncompetitive inhibitors distort the curve.

.. GENERATED FROM PYTHON SOURCE LINES 14-30

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from chemistrykit.kinetics.systems.enzyme import (
        MichaelisMentenProgress,
        competitive_inhibition_rate,
        michaelis_menten_rate,
        noncompetitive_inhibition_rate,
    )
    from chemistrykit.kinetics.visualizers.kinetics_plots import plot_concentration_vs_time

    Vmax, Km = 10.0, 2.0
    S = np.linspace(0.0, 20.0, 400)

    fig, axes = plt.subplots(1, 3, figsize=(16, 4.5))




.. image-sg:: /api/gallery/kinetics/enzyme/images/sphx_glr_plot_01_michaelis_menten_001.png
   :alt: plot 01 michaelis menten
   :srcset: /api/gallery/kinetics/enzyme/images/sphx_glr_plot_01_michaelis_menten_001.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 31-34

The saturation curve: first order in [S] well below :math:`K_m`
(:math:`v \approx (V_{max}/K_m)[S]`), zero order well above it
(:math:`v \approx V_{max}`), and exactly :math:`V_{max}/2` at :math:`K_m`.

.. GENERATED FROM PYTHON SOURCE LINES 34-46

.. code-block:: Python


    axes[0].plot(S, michaelis_menten_rate(S, Vmax, Km), color="steelblue", label="Michaelis-Menten")
    axes[0].plot(S[S < 4], Vmax / Km * S[S < 4], ":", color="gray", label=r"$(V_{max}/K_m)[S]$")
    axes[0].axhline(Vmax, color="gray", linestyle="--", linewidth=0.8, label=r"$V_{max}$")
    axes[0].plot([Km, Km, 0], [0, Vmax / 2, Vmax / 2], color="crimson", linewidth=0.8)
    axes[0].plot(Km, michaelis_menten_rate(Km, Vmax, Km), "o", color="crimson", label=r"$v(K_m) = V_{max}/2$")
    axes[0].set_xlabel("[S]")
    axes[0].set_ylabel("v")
    axes[0].set_title("Saturation of the enzyme")
    axes[0].legend()
    print(f"v(Km) / Vmax = {michaelis_menten_rate(Km, Vmax, Km) / Vmax:.3f}")





.. rst-class:: sphx-glr-script-out

 .. code-block:: none

    v(Km) / Vmax = 0.500




.. GENERATED FROM PYTHON SOURCE LINES 47-50

The progress curve: substrate first disappears at a nearly constant
rate (enzyme saturated), then decays exponentially once [S] falls
below :math:`K_m`.

.. GENERATED FROM PYTHON SOURCE LINES 50-58

.. code-block:: Python


    progress = MichaelisMentenProgress(S0=20.0, Vmax=Vmax, Km=Km)
    result = progress.integrate((0.0, 4.0), dt=1e-3, method="rk4")
    plot_concentration_vs_time(result, ax=axes[1])
    axes[1].axhline(Km, color="crimson", linestyle=":", linewidth=0.8, label="[S] = Km")
    axes[1].legend()
    axes[1].set_title("Substrate progress curve")





.. rst-class:: sphx-glr-script-out

 .. code-block:: none


    Text(0.5, 1.0, 'Substrate progress curve')



.. GENERATED FROM PYTHON SOURCE LINES 59-62

Inhibition: competitive inhibitors raise the apparent :math:`K_m` but
can be out-competed by enough substrate; noncompetitive inhibitors
lower :math:`V_{max}` itself.

.. GENERATED FROM PYTHON SOURCE LINES 62-74

.. code-block:: Python


    axes[2].plot(S, michaelis_menten_rate(S, Vmax, Km), label="uninhibited", color="steelblue")
    axes[2].plot(S, competitive_inhibition_rate(S, I=3.0, Vmax=Vmax, Km=Km, Ki=1.0), label="competitive (I=3)", color="darkorange")
    axes[2].plot(S, noncompetitive_inhibition_rate(S, I=3.0, Vmax=Vmax, Km=Km, Ki=1.0), label="noncompetitive (I=3)", color="seagreen")
    axes[2].axhline(Vmax, color="gray", linestyle="--", linewidth=0.8)
    axes[2].set_xlabel("[S]")
    axes[2].set_ylabel("v")
    axes[2].set_title("Competitive vs. noncompetitive inhibition")
    axes[2].legend()

    fig.tight_layout()
    plt.show()








.. rst-class:: sphx-glr-timing

   **Total running time of the script:** (0 minutes 0.895 seconds)


.. _sphx_glr_download_api_gallery_kinetics_enzyme_plot_01_michaelis_menten.py:

.. only:: html

  .. container:: sphx-glr-footer sphx-glr-footer-example

    .. container:: sphx-glr-download sphx-glr-download-jupyter

      :download:`Download Jupyter notebook: plot_01_michaelis_menten.ipynb <plot_01_michaelis_menten.ipynb>`

    .. container:: sphx-glr-download sphx-glr-download-python

      :download:`Download Python source code: plot_01_michaelis_menten.py <plot_01_michaelis_menten.py>`

    .. container:: sphx-glr-download sphx-glr-download-zip

      :download:`Download zipped: plot_01_michaelis_menten.zip <plot_01_michaelis_menten.zip>`


.. only:: html

 .. rst-class:: sphx-glr-signature

    `Gallery generated by Sphinx-Gallery <https://sphinx-gallery.github.io>`_