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 experiments `enzyme_1_a` and `enzyme_1_b` but with a lavish amount of Enzyme relative to the initial Substrate concentration.¶

Experiment enzyme_1_b had a ten-fold increase in enzyme, relative to enzyme_1_a. Here, we do another x10 increase!

Real-life kinetic parameters.¶

Please refer to enzyme_1_a and enzyme_1_b 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 experiments enzyme_1 and enzyme_2

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 experiments `enzyme_1_a` and `enzyme_1_b`, the reaction is "awash" in enzyme, relative to initial substrate concentration¶

In [9]:

S0 = 20.
E0 = 100.        # A LAVISH AMOUNT!  10 times more than in experiment `enzyme_2` - and x100 as much as experiment`enzyme_1`

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: 100.0
  Species 3 (S). Conc: 20.0
Chemicals involved in reactions: ['S', 'E', 'ES', 'P']

Advance the reactions to equilibrium¶

In [11]:

# Perform the reactions
uc.single_compartment_react(duration=0.05, initial_step=0.01)  # A much-shorter duration than in previous experiments in the series

998 total variable step(s) taken in 2.010 sec
Number of step re-do's because of negative concentrations: 2
Number of step re-do's because of elective soft aborts: 3
Norm usage: {'norm_A': 997, 'norm_B': 1000, 'norm_C': 996, 'norm_D': 996}
System Time is now: 0.050362

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 (999) - only showing 1 every 7 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	100.000000	20.000000		Set concentration	0.000000
1	0.000010	0.000000	0.360000	99.640000	19.640000	1	1st reaction step	17.640000
2	0.000011	0.000018	0.395523	99.604477	19.604459	2		19.380617
3	0.000011	0.000019	0.397295	99.602705	19.602686	3		19.467446
4	0.000011	0.000019	0.398181	99.601819	19.601800	4		19.510856
...	...	...	...	...	...	...	...	...
993	0.047918	17.775498	2.101933	97.898067	0.122569	993		102.994725
994	0.048397	17.824842	2.055364	97.944636	0.119793	994		100.712859
995	0.048881	17.873576	2.009370	97.990630	0.117055	995		98.459109
996	0.049370	17.921695	1.963952	98.036048	0.114353	996		96.233637
997	0.049863	17.969197	1.919114	98.080886	0.111689	997		94.036596

998 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]:

4900.0

In [19]:

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

Out[19]:

3465.6188605108055

Both `vmax` and the initial reaction rate are 10x what they were in experiment `enzyme_1_a`, 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	100.000000	20.000000		Set concentration	0.000000	3465.618861
1	0.000010	0.000000	0.360000	99.640000	19.640000	1	1st reaction step	17.640000	3447.122503
2	0.000011	0.000018	0.395523	99.604477	19.604459	2		19.380617	3445.270565
3	0.000011	0.000019	0.397295	99.602705	19.602686	3		19.467446	3445.178054
4	0.000011	0.000019	0.398181	99.601819	19.601800	4		19.510856	3445.131800
...	...	...	...	...	...	...	...	...	...
993	0.047918	17.775498	2.101933	97.898067	0.122569	993		102.994725	71.495562
994	0.048397	17.824842	2.055364	97.944636	0.119793	994		100.712859	69.899661
995	0.048881	17.873576	2.009370	97.990630	0.117055	995		98.459109	68.323957
996	0.049370	17.921695	1.963952	98.036048	0.114353	996		96.233637	66.768545
997	0.049863	17.969197	1.919114	98.080886	0.111689	997		94.036596	65.233510

998 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=1.65,
                         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_a`¶

In this scenario, with a lot more initial enzyme `E` than in experiment `enzyme_1_a`, 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	100.000000	20.000000		Set concentration	0.000000	3465.618861	889.979162
1	0.000010	0.000000	0.360000	99.640000	19.640000	1	1st reaction step	17.640000	3447.122503	889.979162
2	0.000011	0.000018	0.395523	99.604477	19.604459	2		19.380617	3445.270565	889.978385
3	0.000011	0.000019	0.397295	99.602705	19.602686	3		19.467446	3445.178054	889.978342
4	0.000011	0.000019	0.398181	99.601819	19.601800	4		19.510856	3445.131800	889.978321
...	...	...	...	...	...	...	...	...	...	...
993	0.047918	17.775498	2.101933	97.898067	0.122569	993		102.994725	71.495562	100.506658
994	0.048397	17.824842	2.055364	97.944636	0.119793	994		100.712859	69.899661	98.280755
995	0.048881	17.873576	2.009370	97.990630	0.117055	995		98.459109	68.323957	96.082240
996	0.049370	17.921695	1.963952	98.036048	0.114353	996		96.233637	66.768545	93.911274
997	0.049863	17.969197	1.919114	98.080886	0.111689	997		94.036596	65.233510	91.768007

998 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=1.65,
                         colors=["blue", "yellow", "orange"])

The Morrison model is VASTLY better than the Michaelis-Menten one in this high-enzyme scenario!¶

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 experiments enzyme_1_a and enzyme_1_b but with a lavish amount 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 experiments enzyme_1_a and enzyme_1_b, the reaction is "awash" in 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_a, 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¶

We no longer have the striking overlap we had in experiment enzyme_1_a¶

In this scenario, with a lot more initial enzyme E than in experiment enzyme_1_a, 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 is VASTLY better than the Michaelis-Menten one in this high-enzyme 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 experiments `enzyme_1_a` and `enzyme_1_b` but with a lavish amount of Enzyme relative to the initial Substrate concentration.¶

Unlike in experiments `enzyme_1_a` and `enzyme_1_b`, the reaction is "awash" in 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_a`, because we started with an `E0` 10 times as large¶

We no longer have the striking overlap we had in experiment `enzyme_1_a`¶

In this scenario, with a lot more initial enzyme `E` than in experiment `enzyme_1_a`, 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]`¶