.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/kinetics/enzyme/plot_02_lineweaver_burk.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_02_lineweaver_burk.py>`
        to download the full example code.

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

.. _sphx_glr_api_gallery_kinetics_enzyme_plot_02_lineweaver_burk.py:


Lineweaver and Burk's double-reciprocal plot
===============================================

Lineweaver and Burk (1934) inverted the Michaelis-Menten equation into a
straight line,
:math:`1/v = (K_m/V_{max})(1/[S]) + 1/V_{max}`, so :math:`V_{max}` and
:math:`K_m` could be read off the intercept and slope.
:func:`~chemistrykit.kinetics.systems.enzyme.fit_lineweaver_burk` does
exactly that fit on noisy initial-rate data. The second panel shows the
method's well-known weakness: the same small relative noise becomes a
large scatter at the low-[S] (large :math:`1/[S]`) end, which dominates
the fitted line.

.. GENERATED FROM PYTHON SOURCE LINES 17-37

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from chemistrykit.kinetics.systems.enzyme import fit_lineweaver_burk, michaelis_menten_rate
    from chemistrykit.kinetics.visualizers.kinetics_plots import plot_lineweaver_burk

    Vmax, Km = 10.0, 2.0
    S_samples = np.array([0.5, 1.0, 2.0, 4.0, 8.0, 16.0])

    rng = np.random.default_rng(1)
    v_exact = michaelis_menten_rate(S_samples, Vmax, Km)
    v_noisy = v_exact * (1.0 + rng.normal(0.0, 0.03, size=S_samples.shape))
    fit = fit_lineweaver_burk(S_samples, v_noisy)
    print(f"true:   Vmax={Vmax}, Km={Km}")
    print(f"fitted: Vmax={fit.Vmax:.3f}, Km={fit.Km:.3f}, R^2={fit.r_squared:.5f}")

    fig, axes = plt.subplots(1, 2, figsize=(11, 4.5))
    plot_lineweaver_burk(S_samples, v_noisy, fit=fit, ax=axes[0])
    axes[0].set_title(f"Lineweaver-Burk fit: Vmax={fit.Vmax:.2f}, Km={fit.Km:.2f}")




.. image-sg:: /api/gallery/kinetics/enzyme/images/sphx_glr_plot_02_lineweaver_burk_001.png
   :alt: Lineweaver-Burk fit: Vmax=10.00, Km=1.97
   :srcset: /api/gallery/kinetics/enzyme/images/sphx_glr_plot_02_lineweaver_burk_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    true:   Vmax=10.0, Km=2.0
    fitted: Vmax=10.004, Km=1.968, R^2=0.99929

    Text(0.5, 1.0, 'Lineweaver-Burk fit: Vmax=10.00, Km=1.97')



.. GENERATED FROM PYTHON SOURCE LINES 38-40

Error distortion: repeat the "experiment" many times with 3% noise and
look at the spread of each point in reciprocal coordinates.

.. GENERATED FROM PYTHON SOURCE LINES 40-51

.. code-block:: Python


    trials = np.array([1.0 / (v_exact * (1.0 + rng.normal(0.0, 0.03, size=S_samples.shape))) for _ in range(200)])
    axes[1].errorbar(1.0 / S_samples, trials.mean(axis=0), yerr=trials.std(axis=0), fmt="o", capsize=4, color="steelblue")
    axes[1].set_xlabel("1/[S]")
    axes[1].set_ylabel("1/v  (mean +/- std over 200 trials)")
    axes[1].set_title("Same 3% noise, very unequal reciprocal error bars")
    for inv_s, spread in zip(1.0 / S_samples, trials.std(axis=0)):
        print(f"1/[S] = {inv_s:5.3f}: std of 1/v = {spread:.4f}")

    fig.tight_layout()
    plt.show()




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

 .. code-block:: none

    1/[S] = 2.000: std of 1/v = 0.0152
    1/[S] = 1.000: std of 1/v = 0.0090
    1/[S] = 0.500: std of 1/v = 0.0059
    1/[S] = 0.250: std of 1/v = 0.0044
    1/[S] = 0.125: std of 1/v = 0.0036
    1/[S] = 0.062: std of 1/v = 0.0035





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

   **Total running time of the script:** (0 minutes 0.061 seconds)


.. _sphx_glr_download_api_gallery_kinetics_enzyme_plot_02_lineweaver_burk.py:

.. only:: html

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

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

      :download:`Download Jupyter notebook: plot_02_lineweaver_burk.ipynb <plot_02_lineweaver_burk.ipynb>`

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

      :download:`Download Python source code: plot_02_lineweaver_burk.py <plot_02_lineweaver_burk.py>`

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

      :download:`Download zipped: plot_02_lineweaver_burk.zip <plot_02_lineweaver_burk.zip>`


.. only:: html

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

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