Enzyme Kinetics :¶

Accurate numerical solution - 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.¶

Two Coupled Reactions: `E + S <-> ES`, and `ES -> E + P` , using 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 = "Nov. 4, 2024"
LIFE123_VERSION = "1.0.0.rc.0"      # 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 ipynbname
import pandas as pd

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

In [4]:

check_version(LIFE123_VERSION)

OK

In [ ]:

1. Accurate numerical solution¶

In [5]:

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

In [6]:

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

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

In [7]:

chem_data.all_chemicals()

Out[7]:

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

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 [8]:

# Reaction E + S <-> ES , with 1st-order kinetics, 
chem_data.add_reaction(reactants=["E", "S"], products=["ES"],
                       forward_rate=18., reverse_rate=100.) 

# Reaction ES <-> E + P , with 1st-order kinetics, ignoring the reverse reaction
chem_data.add_reaction(reactants=["ES"], products=["E", "P"],
                       forward_rate=49., reverse_rate=0)   # We are ignoring the reverse reaction

chem_data.describe_reactions()

Number of reactions: 2 (at temp. 25 C)
0: E + S <-> ES  (kF = 18 / kR = 100 / delta_G = 4,250.9 / K = 0.18) | 1st order in all reactants & products
1: ES <-> E + P  (kF = 49 / kR = 0) | 1st order in all reactants & products
Set of chemicals involved in the above reactions: {'P', 'E', 'ES', 'S'}

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]:

# 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")
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
Set of chemicals involved in reactions: {'P', 'E', 'ES', 'S'}

In [11]:

uc.enable_diagnostics()   # To save diagnostic information about the simulation - in particular, the REACTION RATES

Advance the reactions to equilibrium¶

In [12]:

# 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`

Some steps were backtracked and re-done, to prevent negative concentrations or overly large concentration changes
615 total step(s) taken
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}

In [13]:

uc.plot_history(colors=['green', 'red', 'violet', 'darkturquoise'], 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 [14]:

rates = uc.get_diagnostics().get_system_history_with_rxn_rates(rxn_index=1)   # We specify the reaction that involves `P`
rates

Reaction:  ES <-> E + P

Out[14]:

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

615 rows × 7 columns

In [15]:

# 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="TIME", fields="rate", 
                         x_label="time", y_label="dP/dt")

The initial transient phase is no longer miniscule in duration

In [ ]:

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 [16]:

rxn = ReactionEnz(enzyme="E", substrate="S", product="P",
                  k1_F=chem_data.get_forward_rate(0), k1_R=chem_data.get_reverse_rate(0), 
                  k2_F=chem_data.get_forward_rate(1))

In [17]:

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

Out[17]:

8.277777777777779

In [18]:

rxn.kcat

Out[18]:

49.0

In [19]:

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

Out[19]:

490.0

In [20]:

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

Out[20]:

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 [21]:

# 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[21]:

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

615 rows × 8 columns

In [22]:

# 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=["rate", "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 [ ]:

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 [23]:

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

Out[23]:

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

615 rows × 9 columns

In [24]:

PlotlyHelper.plot_pandas(df=rates, x_var="S", fields=["rate", "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_3, we'll see how the Morrison model is the absolute winner when the amount of enzyme is even larger

In [ ]:

Enzyme Kinetics :¶

Accurate numerical solution - 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.¶

Two Coupled Reactions: E + S <-> ES, and ES -> E + P , using real-life kinetic parameters.¶

THE REACTION:¶

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

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?¶

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]¶

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...¶

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

Two Coupled Reactions: `E + S <-> ES`, and `ES -> E + P` , using real-life kinetic parameters.¶

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]`¶