Cooperative Binding Kinetics

4.4.8. Cooperative Binding Kinetics#

4.4.8.1. Motivation#

Some processes involving enzymes do not follow Michaelis-Menten kinetics. One such example is cooperative binding in which the binding of one molecule affects the binding of another molecule. In this notebook we will look at the ability of an enzyme to bind multiple copies of the same molecule.

4.4.8.2. Learning Goals#

After working through this notebook, you will be able to

Write out the cooperative binding mechanism
Derive the cooperative binding rate law
Identify cooperative binding behavior in a plot of $v_0$ vs $[S]_0$
Identify negative and postive cooperative binding behavior.

4.4.8.3. Coding Concepts#

The following coding concepts are used in this notebook:

4.4.8.4. Cooperative Binding#

One well-known example of a process that does not follow Michaelis-Menten kinetics is the binding of oxygen to hemoglobin. The binding of one oxygen molecule to Hemoglobin enhances the binding rate of subsequent oxygen molecules. Hemoglobins are actually heterotetramers can bind up to four oxygen molecules.

An example mechanism for this process is the simultaneous binding of $n$ substrate molecules given by

(4.629)#\[\begin{align} E + nS &\overset{k_1}{\underset{k_{-1}}{\overset{\Longrightarrow}{\Longleftarrow}}} ES_n \\ ES_n &\overset{k_2}{\underset{k_{-2}}{\overset{\Longrightarrow}{\Longleftarrow}}} E + nP \end{align}\]

where $n$ is the number of substrates and $ES_n$ denotes the enzyme substrate complex with $n$ substrates bound.

This type of behavior can be observed in a $v_0$ vs $[S]_0$ plot by distinctly sigmoidal behavior and poor ability to fit the data with the Michaelis-Menten equation.

../../_images/206ef4791d15511ec4a32566f8a6b3e418f97c6b54f3005dad3c038ccb965d11.png

4.4.8.5. Derivation of the Hill Equation#

The rate law stemming from the above mechanism can be derived in a similar fashion to the derivation for Michaelis-Menten. To make the derivation slightly easier, we will assume the second step is irreversible

(4.630)#\[\begin{align} E + nS &\overset{k_1}{\underset{k_{-1}}{\overset{\Longrightarrow}{\Longleftarrow}}} ES_n \\ ES_n &\overset{k_2}{\Longrightarrow} E + nP \end{align}\]

We start by writing the differential expressions for substrate and enzyme-substrate complex

(4.631)#\[\begin{align} -\frac{1}{n}\frac{d[S]}{dt} &= k_1[E][S]^n - k_{-1}[ES_n] \\ -\frac{d[ES_n]}{dt} &= (k_2+k_{-1})[ES_n] - k_1[E][S]^n \end{align}\]

we can again plug in $[E] = [E]_0 - [ES_n]$ to get

(4.632)#\[\begin{align} -\frac{1}{n}\frac{d[S]}{dt} &= k_1[E]_0[S]^n - (k_{-1}+k_1[S]^n)[ES_n] \\ -\frac{d[ES_n]}{dt} &= (k_1[S]^n + k_2+k_{-1})[ES_n] - k_1[E]_0[S]^n \end{align}\]

Using the steady-state approximation for $[ES_n]$ we get

(4.633)#\[\begin{equation} [ES_n] \overset{s.s.}{=} \frac{k_1[E]_0[S]^n}{k_1[S]^n + k_2+k_{-1}} \end{equation}\]

Plugging this into the rate equation

(4.634)#\[\begin{align} v_0 = -\frac{1}{n}\frac{d[S]_0}{dt} &= k_1[E]_0[S]_0^n - (k_{-1}+k_1[S]_0^n)\frac{k_1[E]_0[S]_0^n}{k_1[S]_0^n + k_2+k_{-1}}\\ &= \frac{k_1[E]_0[S]_0^n\cdot(k_1[S]_0^n + k_2+k_{-1})-(k_{-1}+k_1[S]_0^n)k_1[E]_0[S]_0^n}{k_1[S]_0^n + k_2+k_{-1}} \\ &= \frac{k_1[E]_0[S]_0^n\cdot(k_1[S]_0^n + k_2+k_{-1})-k_{-1}k_1[E]_0[S]_0^n - k_1^2[E]_0[S]_0^{2n}}{k_1[S]_0^n + k_2+k_{-1}} \\ &= \frac{k_1k_2[E]_0[S]_0^n}{k_1[S]_0^n + k_2+k_{-1}} \\ &= \frac{k_2[E]_0[S]_0^n}{[S]_0^n + K_m} \end{align}\]

where $K_m = \frac{k_2+k_{-1}}{k_1}$ is defined similar to the Michaelis-Menten constant.

We see that the resulting initial rate law is very similar to the Michaelis-Menten rate law. Indeed, they are identical for $n=1$ as they should be (the mechanism is then the same).

The final Hill equation is given as

(4.635)#\[\begin{equation} v_0 = \frac{v_{max}[S]_0^n}{[S]_0^n + K_m} \end{equation}\]

where $n$ is referred to as the Hill number. While it is appealing to interpret $n$ as the number of substrates/ligands that bind to the enzyme, the Hill mechanism is so unphysical (simultaneous binding) that this interpretation is not necessarily true. Nonetheless, this mechanism and equation are used to fit enzyme kinetics and determine whether an enzyme displays cooperativity (negative or positive) or not.

../../_images/ce8d68d84cd6248c2dc3e5ed52392c2b5d36dae2fa4dba87601c801bac492512.png

4.4.8.6. Positive and Negative Cooperativity#

Both positive and negative cooperativity can be observed. Negative cooperativity simply means that binding of one ligand negatively impacts the binding of the second ligand. There is still cooperativity but the impact is to negate the binding of the second ligand. The Hill equation can be used to fit this type of behavior and it will be observed for values of $n<1$. Let’s look at a plot of different $n$ values.

Error in callback <function flush_figures at 0x7fdb60409b80> (for post_execute):

---------------------------------------------------------------------------
KeyboardInterrupt                         Traceback (most recent call last)
~/opt/anaconda3/lib/python3.9/site-packages/matplotlib_inline/backend_inline.py in flush_figures()
    124             # ignore the tracking, just draw and close all figures
    125             try:
--> 126                 return show(True)
    127             except Exception as e:
    128                 # safely show traceback if in IPython, else raise

~/opt/anaconda3/lib/python3.9/site-packages/matplotlib_inline/backend_inline.py in show(close, block)
     88     try:
     89         for figure_manager in Gcf.get_all_fig_managers():
---> 90             display(
     91                 figure_manager.canvas.figure,
     92                 metadata=_fetch_figure_metadata(figure_manager.canvas.figure)

~/opt/anaconda3/lib/python3.9/site-packages/IPython/core/display.py in display(include, exclude, metadata, transient, display_id, *objs, **kwargs)
    318             publish_display_data(data=obj, metadata=metadata, **kwargs)
    319         else:
--> 320             format_dict, md_dict = format(obj, include=include, exclude=exclude)
    321             if not format_dict:
    322                 # nothing to display (e.g. _ipython_display_ took over)

~/opt/anaconda3/lib/python3.9/site-packages/IPython/core/formatters.py in format(self, obj, include, exclude)
    178             md = None
    179             try:
--> 180                 data = formatter(obj)
    181             except:
    182                 # FIXME: log the exception

~/opt/anaconda3/lib/python3.9/site-packages/decorator.py in fun(*args, **kw)
    230             if not kwsyntax:
    231                 args, kw = fix(args, kw, sig)
--> 232             return caller(func, *(extras + args), **kw)
    233     fun.__name__ = func.__name__
    234     fun.__doc__ = func.__doc__

~/opt/anaconda3/lib/python3.9/site-packages/IPython/core/formatters.py in catch_format_error(method, self, *args, **kwargs)
    222     """show traceback on failed format call"""
    223     try:
--> 224         r = method(self, *args, **kwargs)
    225     except NotImplementedError:
    226         # don't warn on NotImplementedErrors

~/opt/anaconda3/lib/python3.9/site-packages/IPython/core/formatters.py in __call__(self, obj)
    339                 pass
    340             else:
--> 341                 return printer(obj)
    342             # Finally look for special method names
    343             method = get_real_method(obj, self.print_method)

~/opt/anaconda3/lib/python3.9/site-packages/IPython/core/pylabtools.py in print_figure(fig, fmt, bbox_inches, base64, **kwargs)
    149         FigureCanvasBase(fig)
    150 
--> 151     fig.canvas.print_figure(bytes_io, **kw)
    152     data = bytes_io.getvalue()
    153     if fmt == 'svg':

~/opt/anaconda3/lib/python3.9/site-packages/matplotlib/backend_bases.py in print_figure(self, filename, dpi, facecolor, edgecolor, orientation, format, bbox_inches, pad_inches, bbox_extra_artists, backend, **kwargs)
   2297             if bbox_inches:
   2298                 if bbox_inches == "tight":
-> 2299                     bbox_inches = self.figure.get_tightbbox(
   2300                         renderer, bbox_extra_artists=bbox_extra_artists)
   2301                     if pad_inches is None:

~/opt/anaconda3/lib/python3.9/site-packages/matplotlib/figure.py in get_tightbbox(self, renderer, bbox_extra_artists)
   1682 
   1683         for a in artists:
-> 1684             bbox = a.get_tightbbox(renderer)
   1685             if bbox is not None and (bbox.width != 0 or bbox.height != 0):
   1686                 bb.append(bbox)

~/opt/anaconda3/lib/python3.9/site-packages/matplotlib/axes/_base.py in get_tightbbox(self, renderer, call_axes_locator, bbox_extra_artists, for_layout_only)
   4673                     # this artist
   4674                     continue
-> 4675             bbox = a.get_tightbbox(renderer)
   4676             if (bbox is not None
   4677                     and 0 < bbox.width < np.inf

~/opt/anaconda3/lib/python3.9/site-packages/matplotlib/legend.py in get_tightbbox(self, renderer)
    913     def get_tightbbox(self, renderer):
    914         # docstring inherited
--> 915         return self._legend_box.get_window_extent(renderer)
    916 
    917     def get_frame_on(self):

~/opt/anaconda3/lib/python3.9/site-packages/matplotlib/offsetbox.py in get_window_extent(self, renderer)
    349         # docstring inherited
    350         w, h, xd, yd, offsets = self.get_extent_offsets(renderer)
--> 351         px, py = self.get_offset(w, h, xd, yd, renderer)
    352         return mtransforms.Bbox.from_bounds(px - xd, py - yd, w, h)
    353 

~/opt/anaconda3/lib/python3.9/site-packages/matplotlib/offsetbox.py in get_offset(self, width, height, xdescent, ydescent, renderer)
    288 
    289         """
--> 290         return (self._offset(width, height, xdescent, ydescent, renderer)
    291                 if callable(self._offset)
    292                 else self._offset)

~/opt/anaconda3/lib/python3.9/site-packages/matplotlib/legend.py in _findoffset(self, width, height, xdescent, ydescent, renderer)
    587 
    588         if self._loc == 0:  # "best".
--> 589             x, y = self._find_best_position(width, height, renderer)
    590         elif self._loc in Legend.codes.values():  # Fixed location.
    591             bbox = Bbox.from_bounds(0, 0, width, height)

~/opt/anaconda3/lib/python3.9/site-packages/matplotlib/legend.py in _find_best_position(self, width, height, renderer, consider)
   1028             # XXX TODO: If markers are present, it would be good to take them
   1029             # into account when checking vertex overlaps in the next line.
-> 1030             badness = (sum(legendBox.count_contains(line.vertices)
   1031                            for line in lines)
   1032                        + legendBox.count_contains(offsets)

~/opt/anaconda3/lib/python3.9/site-packages/matplotlib/legend.py in <genexpr>(.0)
   1028             # XXX TODO: If markers are present, it would be good to take them
   1029             # into account when checking vertex overlaps in the next line.
-> 1030             badness = (sum(legendBox.count_contains(line.vertices)
   1031                            for line in lines)
   1032                        + legendBox.count_contains(offsets)

~/opt/anaconda3/lib/python3.9/site-packages/matplotlib/transforms.py in count_contains(self, vertices)
    604         vertices = np.asarray(vertices)
    605         with np.errstate(invalid='ignore'):
--> 606             return (((self.min < vertices) &
    607                      (vertices < self.max)).all(axis=1).sum())
    608 

~/opt/anaconda3/lib/python3.9/site-packages/numpy-1.25.0-py3.9-macosx-10.9-x86_64.egg/numpy/core/_methods.py in _all(a, axis, dtype, out, keepdims, where)
     62     # Parsing keyword arguments is currently fairly slow, so avoid it for now
     63     if where is True:
---> 64         return umr_all(a, axis, dtype, out, keepdims)
     65     return umr_all(a, axis, dtype, out, keepdims, where=where)
     66 

KeyboardInterrupt: 

From the above plot we observe a number of differences between curves displaying positive cooperativity ($n>1$) and negative cooperativity ($n<1).

Negative cooperativity displays a more rapid initial increase in $v_0$ as compared to MM or positive cooperative behavior
The rate of all value is equivalent at $[S]_0 = 1$ $\mu$M.
The rate of negative cooperative enzymes is lower than MM or positive cooperative behvaior for $[S]_0 > 1$
Postive cooperative and MM curves all approach $v_{max}$ asymptotically.
Negative cooperative curves do not approach $v_{max}$ asymptotically.

4.4.8.7. Fitting the Hill Equation#

When fitting the Hill equation, we must consider the Michaelis-Menten like parameters ($v_{max}$ and $K_m$) as well as the Hill number, $n$. To fit these values, we have options:

Use non-linear fitting of the equation
Linearize in a Lineweaver-Burk style equation

Consider the second case, we must write the reciprocal Hill equation

(4.636)#\[\begin{equation} \frac{1}{v_0} = \frac{K_m}{v_{max}}\frac{1}{[S]_0^n} + \frac{1}{v_{max}} \end{equation}\]

Observe that this equation suggests that $\frac{1}{v_0}$ vs $\frac{1}{[S]_0^n}$ will be linear with slope $\frac{K_m}{v_{max}}$ and intercept $\frac{1}{v_{max}}$. To use this equation in a linear least-squares fitting procedure, we will need to plot $\frac{1}{v_0}$ vs $\frac{1}{[S]_0}$, $\frac{1}{v_0}$ vs $\frac{1}{[S]_0^2}$, $\frac{1}{v_0}$ vs $\frac{1}{[S]_0^3}$, $...$, fit lines to each one and determine the best fit by comparing $R^2$ values. This is not ideal, and, additionally, $n$ can take on non-integer values. We see that non-linear fitting is the appropriate way to go here.

Let’s see how it is done.

4.4.8.7.1. Example: Fitting the Hill Equation for $n=1$#

In this example, we will use the same data as the example from the Michaelis-Menten notes. Thus, we expect to get a fit with $n\approx1$. The data is as follows

$[S]_0$ (mM)	$v_0$ ($\mu$M/s)
1	2.5
2	4.0
5	6.3
10	7.6
20	9.0

# Put the datat into numpy arrays
s0 = np.array([1,2,5,10,20.0])
v0 = np.array([2.5,4.0,6.3,7.6,9.0])

Perform the non-linear fit and get the parameters:

v_max =  10.8 micro M/s
Km =  3.2 mM
n =  0.9 unitless

../../_images/aa18dabf0e7798e8d4118433876d9efa8d97d15e72a39ba7e2df71d9c806cf5d.png

4.4.8.7.2. Example: Fitting the Hill Equation for $n$ Different than 1#

Consider the following data, fit the Hill equation

$[S]_0$ (mM)	$v_0$ ($\mu$M/s)
0.05	0.10
1	1.26
2	4.55
5	7.18
10	7.52
20	7.42

# Put the data into numpy arrays
import numpy as np
s0 = np.array([0.5,1,2,5,10,20.0])
v0 = np.array([0.19,1.26,4.55,7.18,7.52,7.42])

../../_images/3ce475a3c39e8a061a19de93170a8b5998a539f669ef856fb8bd3c751c51f3d2.png

../../_images/1da5b578388b6fdbe679d7944519893b41515be28fe238e5392fe093e7ca0731.png

Perform the non-linear fit to the Hill equation and get the parameters:

v_max =  7.49 +/- 0.04 muM/s
Km =  5.0 +/- 0.2 mM
n =  2.9 +/- 0.1

Perform a non-linear fit the MM equation for reference:

v_max =  9.4 +/- 1.5 muM/s
Km =  2.8 +/- 1.4 mM

Plot the results:

<matplotlib.legend.Legend at 0x7fd400bc9eb0>

../../_images/b67b6fdac8608549dddf9f5e40c79536bc39cd5e20203b39d018c9c18a1df341.png

4.4.8.8. Example: Multiple Data Sets#

  [S]0    Trial 1
------  ---------
  1.5454
  2.0766
  2.75409
  3.23132
  3.74453

# perform non-linear fit
# import least squares function from scipy library
from scipy.optimize import curve_fit
# define Michaelis-Menten function
def hill(s,vmax,Km,n):  
    return vmax*s**n/(Km + s**n)
# make an initial guess of parameters
popt, pcov = curve_fit(hill, s0_total.flatten(), data.flatten())
err = np.sqrt(np.diag(pcov))
print("v_max = ", np.round(popt[0],1),"+/-", np.round(err[0],1), "muM/s")
print("Km = ", np.round(popt[1],1),"+/-", np.round(err[1],1), "mM")
print("n = ", np.round(popt[2],1),"+/-", np.round(err[2],1))

v_max =  5.2 +/- 0.4 muM/s
Km =  2.3 +/- 0.2 mM
n =  0.6 +/- 0.1