Enzyme Kinetics :¶

Accurate numerical solution - compared to the Michaelis-Menten model approximation and to the alternative Morrison model.¶

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

Scenario with small amount of Enzyme, relative to the initial Substrate concentration.¶

In the follow-up experiments enzyme_1_b and enzyme_1_c, the enzyme concentration gets progressively increased.

Two other reactions being assumed negligible, and not included, are :

the reverse direction of the 2nd reaction
the non-catalyzed E + S <-> P reaction

THE REACTION:¶

the enzyme Adenosine deaminase with the substrate 2,6-Diamino-9-β-D-deoxyribofuranosyl-9-H-purine,
and the initial concentration values choosen below, all satisfy the customary Michaelis-Menten assumptions that [E] << [S] and that the reaction rate constants satisfy k1_reverse >> k2_forward

For this reaction: k1_forward = 18, k1_reverse = 100, k2_forward = 49

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. 17, 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, GraphicLog, PlotlyHelper

In [4]:

check_version(LIFE123_VERSION)

OK

In [5]:

# Initialize the HTML logging (for the graphics)
log_file = ipynbname.name() + ".log.htm"    # Use the notebook base filename for the log file
                                            # IN CASE OF PROBLEMS, set manually to any desired name

# Set up the use of some specified graphic (Vue) components
GraphicLog.config(filename=log_file,
                  components=["vue_cytoscape_2"],
                  extra_js="https://cdnjs.cloudflare.com/ajax/libs/cytoscape/3.21.2/cytoscape.umd.js")

-> Output will be LOGGED into the file 'enzyme_1_a.log.htm'

In [ ]:

1. Accurate numerical solution¶

In [6]:

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

In [7]:

# 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 [8]:

chem_data.all_chemicals()

Out[8]:

	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¶

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

In [9]:

# 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) 

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: {'E', 'ES', 'P', 'S'}

In [10]:

# Send a plot of the network of reactions to the HTML log file
chem_data.plot_reaction_network("vue_cytoscape_2")

[GRAPHIC ELEMENT SENT TO LOG FILE `enzyme_1_a.log.htm`]

In [ ]:

Set the initial concentrations of all the chemicals¶

in the scenario we're exploring, there's LITTLE ENZYME relative to initial substrate concentration¶

In the follow-up experiments enzyme_1_b and enzyme_1_c, the enzyme concentration gets progressively increased.

In [11]:

S0 = 20.
E0 = 1.

In [12]:

# Here we use the "slower" preset for the variable steps, a very 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: 1.0
  Species 3 (S). Conc: 20.0
Set of chemicals involved in reactions: {'E', 'ES', 'P', 'S'}

Advance the reactions to equilibrium¶

In [13]:

# Perform the reactions
uc.single_compartment_react(duration=1.5, initial_step=0.05)

827 total step(s) taken in 0.03 min
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': 706, 'norm_B': 710, 'norm_C': 706, 'norm_D': 706}
System Time is now: 1.5039

In [14]:

uc.plot_history(colors=['green', 'red', 'violet', 'darkturquoise'], show_intervals=True, 
                title_prefix="Small amout of E relative to S(0)")

plot_pandas() NOTICE: Excessive number of vertical lines (828) - only showing 1 every 6 lines

In [15]:

# Highlight a detail about the initial buildup of ES
uc.plot_history(colors=['green', 'red', 'violet', 'darkturquoise'], 
                title_prefix="DETAIL at early times",
                range_x=[0, 0.5], range_y=[0, 2.])

Notice how the bound enzyme `ES` (red) quickly builds up at the very beginning, from time 0 to roughly 0.01 ... and that, in the longer term, the enzyme returns to its unbound state `E`¶

In [ ]:

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

One could take the numerical derivative (gradient) of the time values of [P] - but no need to! Reaction rates are computed in the course of the simulation, and stored in a rate-history dataframe

In [16]:

rates = uc.get_rate_history()   # We'll be interested in rxn1_rate (the reaction that leads to `P`)
rates

Out[16]:

	SYSTEM TIME	rxn0_rate	rxn1_rate
0	0.000000	360.000000	0.000000
1	0.000500	274.543200	8.820000
2	0.000750	243.269275	12.075109
3	0.000763	241.911753	12.216716
4	0.000769	241.237493	12.287060
...	...	...	...
822	1.468805	0.253941	0.285397
823	1.475832	0.244291	0.274566
824	1.482859	0.235005	0.264142
825	1.489886	0.226068	0.254109
826	1.496913	0.217468	0.244454

827 rows × 3 columns

In [17]:

# 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="rxn1_rate", 
                         x_label="time", y_label="dP/dt")

In [18]:

# A closer peek at its maximum value
PlotlyHelper.plot_pandas(df=rates, 
                         title="Reaction rate, dP/dt, over time (DETAIL at early times)",
                         x_var="SYSTEM TIME", fields="rxn1_rate", 
                         range_x=[0,0.05], range_y=[33., 34.5])

In [ ]:

As we saw earlier, the time it took for ES to build up was about 0.01
Ignoring the brief initial transient phase, the reaction rate starts at about 34.1

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 [19]:

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 [20]:

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

Out[20]:

8.277777777777779

In [21]:

rxn.kcat

Out[21]:

49.0

In [22]:

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

Out[22]:

49.0

In [23]:

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

Out[23]:

34.656188605108056

Not too far from the value of about 34.1 we saw earlier...
To keep in mind that the initial part of the reaction is affected by transients

In [ ]:

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

First, we'll merge the concentration history and and the rate history into a single dataframe df

In [24]:

df = uc.add_rate_to_conc_history(rate_name="rxn1_rate", new_rate_name="P_rate")
df

Out[24]:

	SYSTEM TIME	P	ES	E	S	caption	P_rate
0	0.000000	0.000000	0.000000	1.000000	20.000000	Set concentration	0.000000
1	0.000500	0.000000	0.180000	0.820000	19.820000	1st reaction step	8.820000
2	0.000750	0.002205	0.246431	0.753569	19.751364		12.075109
3	0.000763	0.002356	0.249321	0.750679	19.748323		12.216716
4	0.000769	0.002432	0.250756	0.749244	19.746811		12.287060
...	...	...	...	...	...	...	...
822	1.468805	19.947437	0.005824	0.994176	0.046738		0.285397
823	1.475832	19.949443	0.005603	0.994397	0.044954		0.274566
824	1.482859	19.951372	0.005391	0.994609	0.043237		0.264142
825	1.489886	19.953228	0.005186	0.994814	0.041586		0.254109
826	1.496913	19.955014	0.004989	0.995011	0.039997		0.244454

827 rows × 7 columns

In [25]:

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

Out[25]:

	SYSTEM TIME	P	ES	E	S	caption	P_rate	Michaelis_rate
0	0.000000	0.000000	0.000000	1.000000	20.000000	Set concentration	0.000000	34.656189
1	0.000500	0.000000	0.180000	0.820000	19.820000	1st reaction step	8.820000	34.564299
2	0.000750	0.002205	0.246431	0.753569	19.751364		12.075109	34.528950
3	0.000763	0.002356	0.249321	0.750679	19.748323		12.216716	34.527380
4	0.000769	0.002432	0.250756	0.749244	19.746811		12.287060	34.526599
...	...	...	...	...	...	...	...	...
822	1.468805	19.947437	0.005824	0.994176	0.046738		0.285397	0.275111
823	1.475832	19.949443	0.005603	0.994397	0.044954		0.274566	0.264664
824	1.482859	19.951372	0.005391	0.994609	0.043237		0.264142	0.254610
825	1.489886	19.953228	0.005186	0.994814	0.041586		0.254109	0.244934
826	1.496913	19.955014	0.004989	0.995011	0.039997		0.244454	0.235622

827 rows × 8 columns

In [26]:

# Let's see how our computed rate compares with the approximations from the Michaelis-Menten model
PlotlyHelper.plot_pandas(df=df, x_var="S", fields=["P_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=18.9, colors=["blue", "yellow"])

Let's recall that our reactions started with [S]=20
Virtually overlapped plots, except at the very right! (very early times, when [S] is greater than about 19)

In this scenario, the Michaelis-Menten model is remarkable accurate EXCEPT at the very early times of the reaction (the large values of `S`), which is the brief initial transient phase when `ES` builds up from zero¶

This is as expected, because the kinetic parameters of this reaction satisfy the Michaelis-Menten assumptions that `[E] << [S]` and that the rates satisfy `k1_reverse >> k2_forward`¶

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. But, in this scenario, we only have small amounts of enzyme - and we'll see below that no significant improvement is gained by switching model.¶

In [27]:

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

Out[27]:

	SYSTEM TIME	P	ES	E	S	caption	P_rate	Michaelis_rate	Morrison_rate
0	0.000000	0.000000	0.000000	1.000000	20.000000	Set concentration	0.000000	34.656189	34.292188
1	0.000500	0.000000	0.180000	0.820000	19.820000	1st reaction step	8.820000	34.564299	34.292188
2	0.000750	0.002205	0.246431	0.753569	19.751364		12.075109	34.528950	34.291025
3	0.000763	0.002356	0.249321	0.750679	19.748323		12.216716	34.527380	34.290945
4	0.000769	0.002432	0.250756	0.749244	19.746811		12.287060	34.526599	34.290905
...	...	...	...	...	...	...	...	...	...
822	1.468805	19.947437	0.005824	0.994176	0.046738		0.285397	0.275111	0.276209
823	1.475832	19.949443	0.005603	0.994397	0.044954		0.274566	0.264664	0.265721
824	1.482859	19.951372	0.005391	0.994609	0.043237		0.264142	0.254610	0.255628
825	1.489886	19.953228	0.005186	0.994814	0.041586		0.254109	0.244934	0.245914
826	1.496913	19.955014	0.004989	0.995011	0.039997		0.244454	0.235622	0.236566

827 rows × 9 columns

In [28]:

PlotlyHelper.plot_pandas(df=df, x_var="S", fields=["P_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=18.9,
                         colors=["blue", "yellow", "orange"])

In [29]:

# Let's take a closer look at the rightmost portion
PlotlyHelper.plot_pandas(df=df, x_var="S", fields=["P_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=18.9,
                         colors=["blue", "yellow", "orange"],
                         range_x=[17, 20], range_y=[32,35])

In this scenario, the Morrison model appears to be only very mildly better than the Michaelis-Menten one at the early reaction times (large [S], close to the initial value of 20.) - and virtually identical everywhere else.¶

Then why even bother with it? In the next two experiments, enzyme_2 and enzyme_3, we'll see how the Morrison model becomes progressively better than Michaelis-Menten at increasingly higher amounts of enzyme.

In [ ]:

Enzyme Kinetics :¶

Accurate numerical solution - compared to the Michaelis-Menten model approximation and to the alternative Morrison model.¶

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

Scenario with small amount of Enzyme, relative to the initial Substrate concentration.¶

THE REACTION:¶

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

1. Accurate numerical solution¶

Specify the Kinetic Parameters¶

Set the initial concentrations of all the chemicals¶

in the scenario we're exploring, there's LITTLE ENZYME relative to initial substrate concentration¶

Advance the reactions to equilibrium¶

Notice how the bound enzyme ES (red) quickly builds up at the very beginning, from time 0 to roughly 0.01 ... and that, in the longer term, the enzyme returns to its unbound state E¶

What is the initial 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¶

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

In this scenario, the Michaelis-Menten model is remarkable accurate EXCEPT at the very early times of the reaction (the large values of S), which is the brief initial transient phase when ES builds up from zero¶

This is as expected, because the kinetic parameters of this reaction satisfy the Michaelis-Menten assumptions that [E] << [S] and that the rates satisfy k1_reverse >> k2_forward¶

3. Comparing the results to the Morrison model¶

In this scenario, the Morrison model appears to be only very mildly better than the Michaelis-Menten one at the early reaction times (large [S], close to the initial value of 20.) - and virtually identical everywhere else.¶

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

Notice how the bound enzyme `ES` (red) quickly builds up at the very beginning, from time 0 to roughly 0.01 ... and that, in the longer term, the enzyme returns to its unbound state `E`¶

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

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

In this scenario, the Michaelis-Menten model is remarkable accurate EXCEPT at the very early times of the reaction (the large values of `S`), which is the brief initial transient phase when `ES` builds up from zero¶

This is as expected, because the kinetic parameters of this reaction satisfy the Michaelis-Menten assumptions that `[E] << [S]` and that the rates satisfy `k1_reverse >> k2_forward`¶