Enzyme Kinetics :¶

Our model: `E + S <-> ES` (with kinetic parameters k1_forward and k1_reverse), and `ES -> E + P` (k2_forward)¶

In experiment `enzyme_1_a`, we were given `k1_forward`, `k1_reverse` and `k2_forward`... But what to do if we're just given `kM` and `kcat` ?¶

Background: please see experiment enzyme_1_a

THE REACTION:¶

the enzyme Adenosinedeaminase,
with the substrate 2,6-Diamino-9-β-D-deoxyribofuranosyl-9-H-purine.

Source of kinetic parameters: page 16 of "Analysis of Enzyme Reaction Kinetics, Vol. 1", by F. Xavier Malcata, Wiley, 2023

TAGS : "uniform compartment", "chemistry", "numerical", "enzymes"¶

In [1]:

LAST_REVISED = "Dec. 15, 2024"
LIFE123_VERSION = "1.0-rc.1"      # Library version this experiment is based on

In [2]:

#import set_path                    # Using MyBinder?  Uncomment this before running the next cell!

In [3]:

#import sys
#sys.path.append("C:/some_path/my_env_or_install")   # CHANGE to the folder containing your venv or libraries installation!
# NOTE: If any of the imports below can't find a module, uncomment the lines above, or try:  import set_path   

import numpy as np
import plotly.express as px

from life123 import check_version, ReactionEnz

In [4]:

check_version(LIFE123_VERSION)

OK

In [ ]:

Assume we're only given values for `kM` and `kcat`¶

What values of `k1_forward`, `k1_reverse` and `k2_forward` are compatible with them?¶

In [5]:

# We'll use the following values, taken from experiment `enzyme_1_a`
kM = 8.27777777777777
kcat = 49

In [6]:

# k2_forward equals kcat ; not much to say here!
k2_forward = kcat
k2_forward

Out[6]:

By definition: kM = (k2_forward + k1_reverse) / k1_forward

We are given kM and k2_forward (same as kcat), as those are typical quantities measured experimentally, i.e.:

kM = (kcat + k1_reverse) / k1_forward

But how to solve for k1_forward and k1_reverse?? We have just 1 equation and 2 variables! The system of equations is "underdetermined" : what can we do?

We'll explore fixing a guess for k1_forward, and then computing the corresponding k1_reverse - or vice versa.

The Life123 class ReactionEnz conveniently provides the necessary transformations.

In [7]:

enz = ReactionEnz()

In [8]:

# Example, using the k1_forward=18. from experiment `enzyme_1_a`, to determine k1_reverse

k1_reverse = enz.compute_k1_reverse(kM=kM, kcat=kcat, k1_forward = 18.)
k1_reverse

Out[8]:

99.99999999999986

In [9]:

# Conversely, using the k1_reverse=100. from experiment `enzyme_1`, to determine k1_forward

k1_forward = enz.compute_k1_forward(kM=kM, kcat=kcat, k1_reverse = 100.)
k1_forward

Out[9]:

18.000000000000018

Naturally, we're getting the same values we had in experiment `enzyme_1_a`,¶

namely k1_forward = 18. and k1_reverse = 100.

But what if neither `k1_forward` nor `k1_reverse` are known?¶

In [ ]:

1. Let's try a variety of values for `k1_reverse`, and determine the corresponding values for `k1_forward`¶

`k1_reverse` must be non-negative because it's a reaction rate constant, but in other respects there's no conceptual restriction on its value, as plugged into our equation:¶

kM = (kcat + k1_reverse) / k1_forward

In [10]:

k1_reverse_choices = np.linspace(0., 300., 3)    # Even grid of values
k1_reverse_choices

Out[10]:

array([  0., 150., 300.])

In [11]:

k1_forward_choices = enz.compute_k1_forward(kM=kM, kcat=kcat, k1_reverse = k1_reverse_choices)
k1_forward_choices

Out[11]:

array([ 5.91946309, 24.04026846, 42.16107383])

In [12]:

fig = px.line(x=k1_reverse_choices, y=k1_forward_choices, title="k1_forward for given k1_reverse values")

fig.update_layout(xaxis_title='k1_reverse',
                  yaxis_title='k1_forward')

fig.add_vline(x=100., line_width=1, line_dash="dot", line_color="gray")
fig.add_hline(y=18.,  line_width=1, line_dash="dot", line_color="gray")
fig.add_scatter(x=[100.], y=[18.], 
                mode="markers", marker={"color": "red"}, name="actual values")

In [ ]:

2. Conversely, let's try a variety of values for `k1_forward`, and determine the corresponding values for `k1_reverse`¶

There's an extra consideration in pursuing the converse approach, namely we cannot simply give `k1_forward` any non-negative value as we please, because exessively small values would cause `k1_reverse` to become negative, as may be observed from our equation:¶

kM = (kcat + k1_reverse) / k1_forward
which can be re-written as:
kM * k1_forward = kcat + k1_reverse, i.e.
k1_reverse = kM * k1_forward - kcat

To insure that k1_reverse >= 0 we must enforce kM * k1_forward >= kcat, i.e. k1_forward >= kcat / kM

Hence, the required minimun value for k1_forward is kcat / kM. This may be conveniently computed with the following function call:

In [13]:

enz.min_k1_forward(kM=kM, kcat=kcat)

Out[13]:

5.919463087248328

We'll use the above min value as the start of the range of values that we'll explore for k1_forward

In [14]:

k1_forward_choices = np.linspace(5.92, 50., 3)    # Even grid of values; notice the start value just above the min required value
k1_forward_choices

Out[14]:

array([ 5.92, 27.96, 50.  ])

In [15]:

k1_reverse_choices = enz.compute_k1_reverse(kM=kM, kcat=kcat, k1_forward = k1_forward_choices)
k1_reverse_choices

Out[15]:

array([4.44444444e-03, 1.82446667e+02, 3.64888889e+02])

In [16]:

fig = px.line(x=k1_forward_choices, y=k1_reverse_choices, 
              title="k1_reverse for given k1_forward values<br>(dashed lines show actual values)")

fig.update_layout(xaxis_title='k1_forward',
                  yaxis_title='k1_reverse')

fig.add_vline(x=18., line_width=1, line_dash="dot", line_color="gray")
fig.add_hline(y=100., line_width=1, line_dash="dot", line_color="gray")
fig.add_scatter(x=[18.], y=[100.], 
                mode="markers", marker={"color": "red"}, name="actual values")

Note that smallest value of k1_forward is NOT zero, but rather about 5.92, as discussed earlier

In the continuation experiment, `enzyme_2_b`, we'll explore how variations of `k1_forward` and `k1_reverse` (guesses consistent with `kM` and `kcat`) affect the kinetics of our enzymatic reaction...¶

In [ ]:

Enzyme Kinetics :¶

Our model: E + S <-> ES (with kinetic parameters k1_forward and k1_reverse), and ES -> E + P (k2_forward)¶

In experiment enzyme_1_a, we were given k1_forward, k1_reverse and k2_forward... But what to do if we're just given kM and kcat ?¶

THE REACTION:¶

TAGS : "uniform compartment", "chemistry", "numerical", "enzymes"¶

Assume we're only given values for kM and kcat¶

What values of k1_forward, k1_reverse and k2_forward are compatible with them?¶

Naturally, we're getting the same values we had in experiment enzyme_1_a,¶

But what if neither k1_forward nor k1_reverse are known?¶

1. Let's try a variety of values for k1_reverse, and determine the corresponding values for k1_forward¶

k1_reverse must be non-negative because it's a reaction rate constant, but in other respects there's no conceptual restriction on its value, as plugged into our equation:¶

2. Conversely, let's try a variety of values for k1_forward, and determine the corresponding values for k1_reverse¶

There's an extra consideration in pursuing the converse approach, namely we cannot simply give k1_forward any non-negative value as we please, because exessively small values would cause k1_reverse to become negative, as may be observed from our equation:¶

In the continuation experiment, enzyme_2_b, we'll explore how variations of k1_forward and k1_reverse (guesses consistent with kM and kcat) affect the kinetics of our enzymatic reaction...¶

Our model: `E + S <-> ES` (with kinetic parameters k1_forward and k1_reverse), and `ES -> E + P` (k2_forward)¶

In experiment `enzyme_1_a`, we were given `k1_forward`, `k1_reverse` and `k2_forward`... But what to do if we're just given `kM` and `kcat` ?¶

Assume we're only given values for `kM` and `kcat`¶

What values of `k1_forward`, `k1_reverse` and `k2_forward` are compatible with them?¶

Naturally, we're getting the same values we had in experiment `enzyme_1_a`,¶

But what if neither `k1_forward` nor `k1_reverse` are known?¶

1. Let's try a variety of values for `k1_reverse`, and determine the corresponding values for `k1_forward`¶

`k1_reverse` must be non-negative because it's a reaction rate constant, but in other respects there's no conceptual restriction on its value, as plugged into our equation:¶

2. Conversely, let's try a variety of values for `k1_forward`, and determine the corresponding values for `k1_reverse`¶

There's an extra consideration in pursuing the converse approach, namely we cannot simply give `k1_forward` any non-negative value as we please, because exessively small values would cause `k1_reverse` to become negative, as may be observed from our equation:¶

In the continuation experiment, `enzyme_2_b`, we'll explore how variations of `k1_forward` and `k1_reverse` (guesses consistent with `kM` and `kcat`) affect the kinetics of our enzymatic reaction...¶