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.¶

Unlike in experiment `enzyme_1_a`, we'll use data from a reaction that VIOLATES the customary Michaelis-Menten assumption that the rate constants satisfy `k1_reverse >> k2_forward`¶

THE REACTION:¶

the enzyme Aminopeptidase with the substrate Leu-Ala-DED,
and the initial concentration values choosen below, all satisfy the customary Michaelis-Menten assumptions that [E] << [S] BUT the reaction rate constants DON'T satisfy k1_reverse >> k2_forward

For this reaction: k1_forward = 160 , k1_reverse = 0.089 , k2_forward = 0.58

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 = "Jan. 12, 2025"
LIFE123_VERSION = "1.0.0rc2"        # 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"], plot_colors=["green", "red"])

In [6]:

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

# Our Substrate
chem_data.add_chemical(name="Leu-Ala-DED", label="S", plot_color="darkturquoise")

chem_data.all_chemicals()

Out[6]:

	name	label	plot_color
0	P	P	green
1	ES	ES	red
2	Aminopeptidase	E	violet
3	Leu-Ala-DED	S	darkturquoise

In [ ]:

Specify the Kinetic Parameters¶

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

Notice that the reaction rate constants below DON'T satisfy the customary `k1_reverse >> k2_forward`¶

In [7]:

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

In [8]:

# Reaction E + S <-> ES , with 1st-order kinetics, 
uc.add_reaction(reactants=["E", "S"], products="ES",
                forward_rate=160., reverse_rate=0.089) 

# Reaction ES <-> E + P , with 1st-order kinetics, ignoring the reverse reaction
uc.add_reaction(reactants="ES", products=["E", "P"],
                       forward_rate=0.58, reverse_rate=0)  

uc.describe_reactions()

Number of reactions: 2 (at temp. 25 C)
0: E + S <-> ES  (kF = 160 / kR = 0.089 / delta_G = -18,578 / K = 1,797.8) | 1st order in all reactants & products
1: ES <-> E + P  (kF = 0.58 / kR = 0) | 1st order in all reactants & products
Set of chemicals involved in the above reactions: {"ES" (red), "S" (darkturquoise), "P" (green), "E" (violet)}

In [ ]:

Set the initial concentrations of all the chemicals¶

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

just like we did in experiment enzyme_1_a

In [9]:

S0 = 20.
E0 = 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: 1.0
  Species 3 (S). Conc: 20.0
Set of chemicals involved in reactions: {'ES', 'S', 'P', 'E'}

In [ ]:

Simulate the very early part of the reaction¶

In [11]:

# Perform the reactions  
# (Note: by default, concentration and rate history is kept for each step; we'll later reduce this frequency)
uc.single_compartment_react(duration=0.0015, initial_step=0.00001)

173 total variable step(s) taken in 0.01 min
Norm usage: {'norm_A': 31, 'norm_B': 33, 'norm_C': 31, 'norm_D': 31}
System Time is now: 0.0015061

In [12]:

uc.plot_history(show_intervals=True, 
                title_prefix="Small amout of E relative to S(0)")

plot_pandas() NOTICE: Excessive number of vertical lines (174) - only showing 1 every 2 lines

In [13]:

# Highlight a detail about the initial buildup of ES
uc.plot_history(title_prefix="DETAIL at early times",
                range_y=[0, 1])

Notice the EXTREMELY fast kinetics involving the production of the intermediate `ES`¶

If one wishes to exam the history (of concentration or reaction-rate data) in tabular form, it can be done as follows:

In [14]:

uc.get_history()

Out[14]:

	SYSTEM TIME	P	ES	E	S	step	caption
0	0.000000	0.000000e+00	0.000000	1.000000	20.000000		Set concentration
1	0.000010	0.000000e+00	0.032000	0.968000	19.968000	1	1st reaction step
2	0.000021	2.041600e-07	0.066019	0.933981	19.933981	2
3	0.000024	3.305201e-07	0.075849	0.924151	19.924151	3
4	0.000025	3.740726e-07	0.078766	0.921234	19.921234	4
...	...	...	...	...	...	...	...
169	0.001437	6.495498e-04	0.988861	0.011139	19.010489	169
170	0.001454	6.594576e-04	0.989435	0.010565	19.009906	170
171	0.001472	6.693712e-04	0.989979	0.010021	19.009352	171
172	0.001489	6.792902e-04	0.990494	0.009506	19.008827	172
173	0.001506	6.892144e-04	0.990982	0.009018	19.008329	173	last reaction step

174 rows × 7 columns

In [15]:

uc.get_rate_history()

Out[15]:

	SYSTEM TIME	rxn0_rate	rxn1_rate	step
0	0.000000	3200.000000	0.000000	0
1	0.000010	3092.640992	0.018560	1
2	0.000021	2978.868127	0.038291	2
3	0.000024	2946.061101	0.043992	3
4	0.000025	2936.333355	0.045684	4
...	...	...	...	...
168	0.001420	35.636312	0.573188	168
169	0.001437	33.792720	0.573539	169
170	0.001454	32.046189	0.573872	170
171	0.001472	30.391600	0.574188	171
172	0.001489	28.824102	0.574486	172

173 rows × 4 columns

Note that rates are associated with the START times of the intervals. So, at the end time of the simulation, there's no rate.

In [ ]:

Advance the reactions to equilibrium¶

In [16]:

# From now on, the data from 1 of every 10 computation steps will be saved in the history (for later use in plots, etc). 
# Until this point, we've used the default of 1
uc.enable_history(frequency=10)

# Continue the reactions
uc.single_compartment_react(duration=40., initial_step=0.00001)

... running : currently at System Time 16.81 (doing step size 0.00012) after running for 0.5 min
110938 total variable step(s) taken in 0.67 min
Number of step re-do's because of negative concentrations: 4571
Number of step re-do's because of elective soft aborts: 6
Norm usage: {'norm_A': 110781, 'norm_B': 110800, 'norm_C': 110781, 'norm_D': 110781}
System Time is now: 40.002

In [17]:

uc.plot_history(title_prefix="Small amout of E relative to S(0)")

In [18]:

# Highlight a detail about the initial buildup of ES
uc.plot_history(title_prefix="DETAIL of ES and E",
                range_y=[0, 1.5])

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 (as we also saw earlier)

In [19]:

rates = uc.get_rate_history()
rates

Out[19]:

	SYSTEM TIME	rxn0_rate	rxn1_rate	step
0	0.000000	3200.000000	0.000000	0
1	0.000010	3092.640992	0.018560	1
2	0.000021	2978.868127	0.038291	2
3	0.000024	2946.061101	0.043992	3
4	0.000025	2936.333355	0.045684	4
...	...	...	...	...
11262	39.665419	-0.000790	0.010425	110889
11263	39.670501	-0.000339	0.010393	110899
11264	39.683681	-0.000027	0.010313	110909
11265	39.717866	0.000006	0.010110	110919
11266	39.806534	0.000006	0.009602	110929

11267 rows × 4 columns

Note that initially we were saving historical data at every simulation step, but later switched to every 10 steps

In [20]:

# 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")

Notice how very different is it from the Reaction rate over time seen in experiment `enzyme_1_a`¶

In [21]:

# A closer peek at its very quick buildup
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, 0.0015])

As we saw earlier, the time it took for ES to build up was about 0.001
Ignoring the very brief initial transient phase, the reaction rate starts at about 0.57, and stays at that value until the substrate gets depleted, arount t=33

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 [22]:

reactions = uc.get_reactions()

In [23]:

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

In [24]:

rxn.kM          #  For the data in this experiment, it comes out to (0.089 + 0.58) / 160.

Out[24]:

0.004181249999999999

In [25]:

rxn.kcat

Out[25]:

0.58

In [26]:

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

Out[26]:

0.58

In [27]:

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

Out[27]:

0.5798787690948362

Pretty close the value of about 0.57 we saw earlier, after the initial transient...

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 [28]:

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

Out[28]:

	SYSTEM TIME	P	ES	E	S	step	caption	P_rate
0	0.000000	0.000000e+00	0.000000	1.000000	20.000000		Set concentration	0.000000
1	0.000010	0.000000e+00	0.032000	0.968000	19.968000	1	1st reaction step	0.018560
2	0.000021	2.041600e-07	0.066019	0.933981	19.933981	2		0.038291
3	0.000024	3.305201e-07	0.075849	0.924151	19.924151	3		0.043992
4	0.000025	3.740726e-07	0.078766	0.921234	19.921234	4		0.045684
...	...	...	...	...	...	...	...	...
11262	39.665419	1.998202e+01	0.017975	0.982025	0.000005	110889		0.010425
11263	39.670501	1.998207e+01	0.017919	0.982081	0.000008	110899		0.010393
11264	39.683681	1.998221e+01	0.017781	0.982219	0.000010	110909		0.010313
11265	39.717866	1.998256e+01	0.017431	0.982569	0.000010	110919		0.010110
11266	39.806534	1.998344e+01	0.016556	0.983444	0.000009	110929		0.009602

11267 rows × 8 columns

In [29]:

# 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[29]:

	SYSTEM TIME	P	ES	E	S	step	caption	P_rate	Michaelis_rate
0	0.000000	0.000000e+00	0.000000	1.000000	20.000000		Set concentration	0.000000	0.579879
1	0.000010	0.000000e+00	0.032000	0.968000	19.968000	1	1st reaction step	0.018560	0.579879
2	0.000021	2.041600e-07	0.066019	0.933981	19.933981	2		0.038291	0.579878
3	0.000024	3.305201e-07	0.075849	0.924151	19.924151	3		0.043992	0.579878
4	0.000025	3.740726e-07	0.078766	0.921234	19.921234	4		0.045684	0.579878
...	...	...	...	...	...	...	...	...	...
11262	39.665419	1.998202e+01	0.017975	0.982025	0.000005	110889		0.010425	0.000714
11263	39.670501	1.998207e+01	0.017919	0.982081	0.000008	110899		0.010393	0.001106
11264	39.683681	1.998221e+01	0.017781	0.982219	0.000010	110909		0.010313	0.001370
11265	39.717866	1.998256e+01	0.017431	0.982569	0.000010	110919		0.010110	0.001371
11266	39.806534	1.998344e+01	0.016556	0.983444	0.000009	110929		0.009602	0.001301

11267 rows × 9 columns

In [30]:

# 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=19, 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)

The overlap at the very left (small [S], late in the reaction) has a few issues, but not too bad; let's magnify the previous plot:

In [31]:

PlotlyHelper.plot_pandas(df=df, x_var="S", fields=["P_rate", "Michaelis_rate"],
                         title="EARLY DETAIL of Reaction rate, dP/dt, as a function of Substrate concentration",
                         y_label="dP/dt", legend_header="Rates",
                         vertical_lines_to_add=19, colors=["blue", "yellow"],
                         range_x=[0, .2])

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.¶

Some discrepancy also at the end of the reaction (when `S` drops to zero)¶

This is as expected, because the kinetic parameters of this reaction satisfy the key Michaelis-Menten assumptions that `[E] << [S]` , but NOT the other assumption 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 [32]:

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

Out[32]:

	SYSTEM TIME	P	ES	E	S	step	caption	P_rate	Michaelis_rate	Morrison_rate
0	0.000000	0.000000e+00	0.000000	1.000000	20.000000		Set concentration	0.000000	0.579879	0.579872
1	0.000010	0.000000e+00	0.032000	0.968000	19.968000	1	1st reaction step	0.018560	0.579879	0.579872
2	0.000021	2.041600e-07	0.066019	0.933981	19.933981	2		0.038291	0.579878	0.579872
3	0.000024	3.305201e-07	0.075849	0.924151	19.924151	3		0.043992	0.579878	0.579872
4	0.000025	3.740726e-07	0.078766	0.921234	19.921234	4		0.045684	0.579878	0.579872
...	...	...	...	...	...	...	...	...	...	...
11262	39.665419	1.998202e+01	0.017975	0.982025	0.000005	110889		0.010425	0.000714	0.010384
11263	39.670501	1.998207e+01	0.017919	0.982081	0.000008	110899		0.010393	0.001106	0.010354
11264	39.683681	1.998221e+01	0.017781	0.982219	0.000010	110909		0.010313	0.001370	0.010275
11265	39.717866	1.998256e+01	0.017431	0.982569	0.000010	110919		0.010110	0.001371	0.010073
11266	39.806534	1.998344e+01	0.016556	0.983444	0.000009	110929		0.009602	0.001301	0.009567

11267 rows × 10 columns

In [33]:

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=19,
                         colors=["blue", "yellow", "orange"])

In [34]:

# Detail at the very left
PlotlyHelper.plot_pandas(df=df, x_var="S", fields=["P_rate", "Michaelis_rate", "Morrison_rate"],
                         title="EARLY DETAIL of reaction rate, dP/dt, as a function of Substrate concentration",
                         y_label="dP/dt", legend_header="Rates",
                         vertical_lines_to_add=19,
                         colors=["blue", "yellow", "orange"],
                         range_x=[0, 0.02])

In this scenario, the Morrison model appears better than the Michaelis-Menten one at late reaction times (small `S`, close to zero) - and virtually identical elsewhere¶

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.¶

Unlike in experiment enzyme_1_a, we'll use data from a reaction that VIOLATES the customary Michaelis-Menten assumption that the rate constants satisfy k1_reverse >> k2_forward¶

THE REACTION:¶

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

1. Accurate numerical solution¶

Specify the Kinetic Parameters¶

Notice that the reaction rate constants below DON'T satisfy the customary k1_reverse >> k2_forward¶

Set the initial concentrations of all the chemicals¶

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

Simulate the very early part of the reaction¶

Notice the EXTREMELY fast kinetics involving the production of the intermediate ES¶

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?¶

Notice how very different is it from the Reaction rate over time seen in experiment enzyme_1_a¶

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.¶

Some discrepancy also at the end of the reaction (when S drops to zero)¶

This is as expected, because the kinetic parameters of this reaction satisfy the key Michaelis-Menten assumptions that [E] << [S] , but NOT the other assumption that the rates satisfy k1_reverse >> k2_forward¶

3. Comparing the results to the Morrison model¶

In this scenario, the Morrison model appears better than the Michaelis-Menten one at late reaction times (small S, close to zero) - and virtually identical elsewhere¶

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

Unlike in experiment `enzyme_1_a`, we'll use data from a reaction that VIOLATES the customary Michaelis-Menten assumption that the rate constants satisfy `k1_reverse >> k2_forward`¶

Notice that the reaction rate constants below DON'T satisfy the customary `k1_reverse >> k2_forward`¶

Notice the EXTREMELY fast kinetics involving the production of the intermediate `ES`¶

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`?¶

Notice how very different is it from the Reaction rate over time seen in experiment `enzyme_1_a`¶

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.¶

Some discrepancy also at the end of the reaction (when `S` drops to zero)¶

This is as expected, because the kinetic parameters of this reaction satisfy the key Michaelis-Menten assumptions that `[E] << [S]` , but NOT the other assumption that the rates satisfy `k1_reverse >> k2_forward`¶

In this scenario, the Morrison model appears better than the Michaelis-Menten one at late reaction times (small `S`, close to zero) - and virtually identical elsewhere¶