Enzyme Kinetics¶

This is a re-visit of the first part of experiment `enzyme_1_a`¶

An under-the-hood look at how the enzymatic reaction is internally modeled with 2 coupled elementary reactions:¶

E + S <-> ES (reversible), and ES -> E + P (irreversible).

The results are identical, because the algorithm (including the choice of adaptive variable time steps) is the same. The only difference here is that we aren't making use of the native support for enzymatic reactions.

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

the reverse direction of the 2nd reaction
the non-catalyzed 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", "technical"¶

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 ipynbname
import pandas as pd

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

In [4]:

check_version(LIFE123_VERSION)    # To check compatibility

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_technical.log.htm'

In [ ]:

Revisit Part 1 of experiment `enzyme_1_a`, but WITHOUT making use of native support for enzymatic reactions¶

In [6]:

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

In [7]:

# 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[7]:

	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

In [ ]:

Specify the Kinetic Parameters¶

In [8]:

# 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")

Here's where we start to diverge from experiment `enzyme_1_a` ...¶

In [9]:

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

# Reaction ES <-> E + P , with 1st-order kinetics, ignoring the reverse reaction
uc.add_reaction(reactants="ES", products=["E", "P"],
                kF=49., kR=0) 

uc.describe_reactions()

Number of reactions: 2
0: E + S  <-> ES  (Elementary Synthesis reaction)  (kF = 18 / kR = 100 / delta_G = 4,250.9 / K = 0.18 / Temp = 25 C)
1: ES <-> E + P  (Elementary Decomposition reaction)  (kF = 49 / kR = 0 / Temp = 25 C)
Chemicals involved in the above reactions: {"P" (green), "ES" (red), "E" (violet), "S" (darkturquoise)}

In [10]:

# Send a plot of the network of reactions to the HTML log file
# NOTE: 2 reactions, instead of the 1 reaction of experiment `enzyme_1_a` will now show up
uc.plot_reaction_network("vue_cytoscape_2")

[GRAPHIC ELEMENT SENT TO LOG FILE `enzyme_1_a_technical.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]:

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
Chemicals involved in reactions: ['S', 'ES', 'P', 'E']

Advance the reactions to equilibrium¶

In [13]:

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

827 total variable step(s) taken in 1.711 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': 706, 'norm_B': 710, 'norm_C': 706, 'norm_D': 706}
System Time is now: 1.5039

In [14]:

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

827 rows × 4 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])

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

ALL THE RESULTS ARE IDENTICAL TO WHAT WE HAD IN EXPERIMENT enzyme_1_a

(For additional analysis, see that experiment)

In [ ]:

Enzyme Kinetics¶

This is a re-visit of the first part of experiment enzyme_1_a¶

An under-the-hood look at how the enzymatic reaction is internally modeled with 2 coupled elementary reactions:¶

THE REACTION:¶

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

Revisit Part 1 of experiment enzyme_1_a, but WITHOUT making use of native support for enzymatic reactions¶

Specify the Kinetic Parameters¶

Here's where we start to diverge from experiment enzyme_1_a ...¶

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?¶

This is a re-visit of the first part of experiment `enzyme_1_a`¶

Revisit Part 1 of experiment `enzyme_1_a`, but WITHOUT making use of native support for enzymatic reactions¶

Here's where we start to diverge from experiment `enzyme_1_a` ...¶

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`?¶