Enzyme Kinetics :¶

An accurate numerical solution of an enzymatic reaction `E + S <-> ES -> E + P` - compared to the Michaelis-Menten model approximation and to the alternative Morrison model.¶

Same as experiment `enzyme_1_a` but with much-larger amounts of Enzyme relative to the initial Substrate concentration.¶

Real-life kinetic parameters.¶

Please refer to enzyme_1_a for more details.

THE REACTION:¶

the enzyme Adenosine deaminase with the substrate 2,6-Diamino-9-β-D-deoxyribofuranosyl-9-H-purine (same as in experiment enzyme_1)
satisfies the customary Michaelis-Menten assumptions that k1_reverse >> k2_forward

However, the initial concentration values choosen below, do NOT satisfy the expected Michaelis-Menten assumptions that [E] << [S]

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 = "Sep. 2, 2025"
LIFE123_VERSION = "1.0.0rc6"         # 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 pandas as pd

from life123 import check_version, ChemData, UniformCompartment, ReactionEnzyme, PlotlyHelper

In [4]:

check_version(LIFE123_VERSION)    # To check compatibility

OK

In [ ]:

PART 1. Accurate numerical solution¶

In [5]:

chem_data = ChemData(names=["P", "ES"], plot_colors=["green", "red"])

In [6]:

# Our Enzyme
chem_data.add_chemical(name="Adenosine deaminase", label="E", plot_color="violet") 

# Our Substrate
chem_data.add_chemical(name="2,6-Diamino-9-β-D-deoxyribofuranosyl-9-H-purine", label="S", plot_color="darkturquoise")

chem_data.all_chemicals()

Out[6]:

	name	label	plot_color
0	P	P	green
1	ES	ES	red
2	Adenosine deaminase	E	violet
3	2,6-Diamino-9-β-D-deoxyribofuranosyl-9-H-purine	S	darkturquoise

Specify the Kinetic Parameters¶

This part is identical to experiment enzyme_1

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

In [7]:

# Here we use the "slower" preset for the variable steps, a conservative option prioritizing accuracy over speed
uc = UniformCompartment(chem_data=chem_data, preset="slower")

In [8]:

# Enzymatic reaction `E + S <-> ES -> E + P` 
uc.add_reaction(reactants="S", products="P", enzyme="E",
                k1_F=18., k1_R=100., k2_F=49.)

uc.describe_reactions()

Number of reactions: 1
0: E + S <-> ES -> E + P  (Enzymatic reaction)  (k1_F = 18.0 / k1_R = 100.0 / k2_F = 49.0 / Temp = 25 C)
Chemicals involved in the above reactions: {"P" (green), "ES" (red), "E" (violet), "S" (darkturquoise)}

In [ ]:

Set the initial concentrations of all the chemicals¶

Unlike in experiment `enzyme_1`, there's substantially more enzyme relative to initial substrate concentration¶

In [9]:

S0 = 20.
E0 = 10.        # 10 times more than in experiment `enzyme_1` ; almost as much enzyme as substrate!

In [10]:

uc.set_conc(conc={"S": S0, "E": E0})      # Small ampount of enzyme `E`, relative to substrate `S`
uc.describe_state()

SYSTEM STATE at Time t = 0:
4 species:
  Species 0 (P). Conc: 0.0
  Species 1 (ES). Conc: 0.0
  Species 2 (E). Conc: 10.0
  Species 3 (S). Conc: 20.0
Chemicals involved in reactions: ['E', 'ES', 'S', 'P']

Advance the reactions to equilibrium¶

In [11]:

# Perform the reactions
uc.single_compartment_react(duration=0.2, initial_step=0.05)  # A much-shorter duration than the 1.5 of experiment `enzyme_1`

615 total variable step(s) taken in 1.290 sec
Number of step re-do's because of negative concentrations: 2
Number of step re-do's because of elective soft aborts: 1
Norm usage: {'norm_A': 513, 'norm_B': 516, 'norm_C': 512, 'norm_D': 512}
System Time is now: 0.20067

In [12]:

uc.plot_history(show_intervals=True, 
                title_prefix="Almost as much E as S(0)")

plot_pandas() NOTICE: Excessive number of vertical lines (616) - only showing 1 every 5 lines

The general shapes of the various curves is reminiscent of their counterparts in experiment enzyme_1, but notice:

The product P is being produced on a much-faster timescale than before
The maximum amount of the bound enzyme ES is no longer puny; in fact, it's a good fraction of the initial [S]
The buildup of ES, at the very beginning, no longer looks instantaneous relatively to the main reaction: by the time ES reaches its maximum value, a non-trivial amount of product P has been produced

In short, the "transient" is not so "transient"!

In [ ]:

What is the rate of production of the final reaction product `P`?¶

In [13]:

# We specify the reaction that involves `P`
rates = uc.add_rate_to_conc_history(rate_name="rxn0_rate_2")
rates

Out[13]:

	SYSTEM TIME	P	ES	E	S	step	caption	rxn0_rate_2
0	0.000000	0.000000	0.000000	10.000000	20.000000		Set concentration	0.000000
1	0.000500	0.000000	1.800000	8.200000	18.200000	1	1st reaction step	88.200000
2	0.000750	0.022050	2.404530	7.595470	17.573420	2		117.821970
3	0.000763	0.023523	2.430084	7.569916	17.546393	3		119.074126
4	0.000769	0.024267	2.442764	7.557236	17.532969	4		119.695434
...	...	...	...	...	...	...	...	...
610	0.195573	19.863565	0.081649	9.918351	0.054786	610		4.000798
611	0.196593	19.867645	0.079217	9.920783	0.053138	611		3.881655
612	0.197612	19.871602	0.076858	9.923142	0.051540	612		3.766032
613	0.198632	19.875442	0.074568	9.925432	0.049990	613		3.653827
614	0.199652	19.879167	0.072346	9.927654	0.048487	614		3.544941

615 rows × 8 columns

In [14]:

# Let's take a look at how the reaction rate varies with time
PlotlyHelper.plot_pandas(df=rates, 
                         title="Reaction rate, dP/dt, over time",
                         x_var="SYSTEM TIME", fields="rxn0_rate_2", 
                         x_label="time", y_label="dP/dt")

The initial transient phase is no longer miniscule in duration

In [ ]:

PART 2. Comparing the results to the Michaelis-Menten model approximation¶

Let's compute some parameters used by the Michaelis-Menten model¶

for background reference, see: https://vallance.chem.ox.ac.uk/pdfs/KineticsLectureNotes.pdf (p. 20)

In [15]:

rxn = uc.get_single_reaction(0)

In [16]:

rxn.kM          #  For the data in this experiment, it comes out to (49. + 100.) / 18.

Out[16]:

8.277777777777779

In [17]:

rxn.kcat

Out[17]:

49.0

In [18]:

vmax = rxn.compute_vmax(E_tot=E0)          # kcat * E0
vmax

Out[18]:

490.0

In [19]:

initial_rxn_rate = rxn.compute_rate(S_conc=S0)    #  (vmax * S0) / (kM + S0)
initial_rxn_rate

Out[19]:

346.56188605108053

Both `vmax` and the initial reaction rate are 10x what they were in experiment `enzyme_1`, because we started with an `E0` 10 times as large¶

In [ ]:

Now, let's look at rate as a function of [S]; we'll compare what we computed earlier vs. as given by the approximation of the Michaelis-Menten model¶

In [20]:

# Let's add a column with the rate estimated by the Michaelis-Menten model
rates["Michaelis_rate"] = rxn.compute_rate(S_conc=rates["S"])
rates

Out[20]:

	SYSTEM TIME	P	ES	E	S	step	caption	rxn0_rate_2	Michaelis_rate
0	0.000000	0.000000	0.000000	10.000000	20.000000		Set concentration	0.000000	346.561886
1	0.000500	0.000000	1.800000	8.200000	18.200000	1	1st reaction step	88.200000	336.810743
2	0.000750	0.022050	2.404530	7.595470	17.573420	2		117.821970	333.097749
3	0.000763	0.023523	2.430084	7.569916	17.546393	3		119.074126	332.933539
4	0.000769	0.024267	2.442764	7.557236	17.532969	4		119.695434	332.851850
...	...	...	...	...	...	...	...	...	...
610	0.195573	19.863565	0.081649	9.918351	0.054786	610		4.000798	3.221698
611	0.196593	19.867645	0.079217	9.920783	0.053138	611		3.881655	3.125424
612	0.197612	19.871602	0.076858	9.923142	0.051540	612		3.766032	3.032014
613	0.198632	19.875442	0.074568	9.925432	0.049990	613		3.653827	2.941384
614	0.199652	19.879167	0.072346	9.927654	0.048487	614		3.544941	2.853451

615 rows × 9 columns

In [21]:

# Let's see how our computed rate compares with the approximations from the Michaelis-Menten model
PlotlyHelper.plot_pandas(df=rates, x_var="S", fields=["rxn0_rate_2", "Michaelis_rate"],
                         title="Reaction rate, dP/dt, as a function of Substrate concentration",
                         y_label="dP/dt", legend_header="Rates",
                         vertical_lines_to_add=12.1,
                         colors=["blue", "yellow"])

Let's recall that our reactions started out with [S]=20
The curve overlap is still passable at later times (left part of graph, when [S] drops below about 12), but is rather bad at earlier times (when [S] > 12)
We no longer have the striking overlap we had in experiment enzyme_1

In this scenario, with a lot more initial enzyme `E` than in experiment `enzyme_1`, the Michaelis-Menten model suffers from poor accuracy for an non-trivial early time, because the transient early phase (when `ES` builds up from zero) is no longer very brief¶

This is as expected, because we substantially deviated from the Michaelis-Menten assumptions that `[E] << [S]`¶

In [ ]:

PART 3. Comparing the results to the Morrison model¶

Following section 7.1 of "Analysis of Enzyme Reaction Kinetics, Vol. 1", by F. Xavier Malcata, Wiley, 2023 , we'll test out an the alternative Morrison approach, which is expected to perform better than the Michaelis-Menten model when the Enzyme concentration isn't so small¶

In [22]:

rates["Morrison_rate"] = rxn.compute_rate_morrison(E_tot=E0,
                                                   S_tot=rates["S"] + rates["ES"])
rates

Out[22]:

	SYSTEM TIME	P	ES	E	S	step	caption	rxn0_rate_2	Michaelis_rate	Morrison_rate
0	0.000000	0.000000	0.000000	10.000000	20.000000		Set concentration	0.000000	346.561886	305.920075
1	0.000500	0.000000	1.800000	8.200000	18.200000	1	1st reaction step	88.200000	336.810743	305.920075
2	0.000750	0.022050	2.404530	7.595470	17.573420	2		117.821970	333.097749	305.762583
3	0.000763	0.023523	2.430084	7.569916	17.546393	3		119.074126	332.933539	305.752055
4	0.000769	0.024267	2.442764	7.557236	17.532969	4		119.695434	332.851850	305.746735
...	...	...	...	...	...	...	...	...	...	...
610	0.195573	19.863565	0.081649	9.918351	0.054786	610		4.000798	3.221698	3.645236
611	0.196593	19.867645	0.079217	9.920783	0.053138	611		3.881655	3.125424	3.536609
612	0.197612	19.871602	0.076858	9.923142	0.051540	612		3.766032	3.032014	3.431196
613	0.198632	19.875442	0.074568	9.925432	0.049990	613		3.653827	2.941384	3.328903
614	0.199652	19.879167	0.072346	9.927654	0.048487	614		3.544941	2.853451	3.229639

615 rows × 10 columns

In [23]:

PlotlyHelper.plot_pandas(df=rates, x_var="S", fields=["rxn0_rate_2", "Michaelis_rate", "Morrison_rate"],
                         title="Reaction rate, dP/dt, as a function of Substrate concentration",
                         y_label="dP/dt", legend_header="Rates",
                         vertical_lines_to_add=12.1,
                         colors=["blue", "yellow", "orange"])

The Morrison model appears to be only mildly better than the Michaelis-Menten one in this scenario...¶

In the next experiment, enzyme_1_c, we'll see how the Morrison model is the absolute winner when the amount of enzyme is even larger

In [ ]:

Enzyme Kinetics :¶

An accurate numerical solution of an enzymatic reaction E + S <-> ES -> E + P - compared to the Michaelis-Menten model approximation and to the alternative Morrison model.¶

Same as experiment enzyme_1_a but with much-larger amounts of Enzyme relative to the initial Substrate concentration.¶

Real-life kinetic parameters.¶

THE REACTION:¶

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

PART 1. Accurate numerical solution¶

Specify the Kinetic Parameters¶

Set the initial concentrations of all the chemicals¶

Unlike in experiment enzyme_1, there's substantially more enzyme relative to initial substrate concentration¶

Advance the reactions to equilibrium¶

What is the rate of production of the final reaction product P?¶

PART 2. Comparing the results to the Michaelis-Menten model approximation¶

Let's compute some parameters used by the Michaelis-Menten model¶

Both vmax and the initial reaction rate are 10x what they were in experiment enzyme_1, because we started with an E0 10 times as large¶

Now, let's look at rate as a function of [S]; we'll compare what we computed earlier vs. as given by the approximation of the Michaelis-Menten model¶

In this scenario, with a lot more initial enzyme E than in experiment enzyme_1, the Michaelis-Menten model suffers from poor accuracy for an non-trivial early time, because the transient early phase (when ES builds up from zero) is no longer very brief¶

This is as expected, because we substantially deviated from the Michaelis-Menten assumptions that [E] << [S]¶

PART 3. Comparing the results to the Morrison model¶

Following section 7.1 of "Analysis of Enzyme Reaction Kinetics, Vol. 1", by F. Xavier Malcata, Wiley, 2023 , we'll test out an the alternative Morrison approach, which is expected to perform better than the Michaelis-Menten model when the Enzyme concentration isn't so small¶

The Morrison model appears to be only mildly better than the Michaelis-Menten one in this scenario...¶

An accurate numerical solution of an enzymatic reaction `E + S <-> ES -> E + P` - compared to the Michaelis-Menten model approximation and to the alternative Morrison model.¶

Same as experiment `enzyme_1_a` but with much-larger amounts of Enzyme relative to the initial Substrate concentration.¶

Unlike in experiment `enzyme_1`, there's substantially more enzyme relative to initial substrate concentration¶

What is the rate of production of the final reaction product `P`?¶

Both `vmax` and the initial reaction rate are 10x what they were in experiment `enzyme_1`, because we started with an `E0` 10 times as large¶

In this scenario, with a lot more initial enzyme `E` than in experiment `enzyme_1`, the Michaelis-Menten model suffers from poor accuracy for an non-trivial early time, because the transient early phase (when `ES` builds up from zero) is no longer very brief¶

This is as expected, because we substantially deviated from the Michaelis-Menten assumptions that `[E] << [S]`¶