Enzyme Kinetics :¶

Our model: `E + S <-> ES` (with kinetic parameters k1_forward and k1_reverse), and `ES -> E + P` (k2_forward)¶

In experiment `enzyme_1_a`, we were given `k1_forward`, `k1_reverse` and `k2_forward`... But what to do if we're just given `kM` and `kcat` ?¶

In Part 1, we'll "cheat" and use the actual value of k1_forward
In Part 2, we'll explore what happens if, lacking an actual value, we under-estimate k1_forward
In Part 3, we'll explore what happens if we over-estimate k1_forward

Background: please see experiment enzyme_1_a

THE REACTION:¶

the enzyme Adenosinedeaminase 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

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   

from life123 import check_version, ChemData, ReactionRegistry, UniformCompartment, ReactionEnzyme, PlotlyHelper

In [4]:

check_version(LIFE123_VERSION)    # To check compatibility

OK

In [ ]:

Assume we're only given values for `kM` and `kcat`¶

In [5]:

# The following values are taken from experiment `enzyme_1`
kM = 8.27777777777777
kcat = 49

What values of k1_forward, k1_reverse and k2_forward are compatible with them?
That question was explored in experiment enzyme2_a. Here we'll explore the dynamic consequences of selecting various combinations of values.

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

In [6]:

S0 = 20.
E0 = 1.

In [ ]:

PART 1. Using the known, exact values for `k1_forward` and `k1_reverse`¶

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

In [7]:

k1_forward = 18.
k1_reverse = 100.

In [8]:

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

In [9]:

# Our Enzyme
chem_data.add_chemical(name="Adenosinedeaminase", 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[9]:

	name	label	plot_color
0	P	P	green
1	ES	ES	red
2	Adenosinedeaminase	E	violet
3	2,6-Diamino-9-β-D-deoxyribofuranosyl-9-H-purine	S	darkturquoise

Specify the Kinetic Parameters¶

In [10]:

rxns = ReactionRegistry(chem_data=chem_data)

In [11]:

# Enzymatic reaction `E + S <-> ES -> E + P` 
rxns.add_reaction(reactants="S", products="P", enzyme="E",
                  k1_F=k1_forward, k1_R=k1_reverse, k2_F=kcat)

rxns.describe_reactions()

Number of reactions: 1
0: E + S <-> ES -> E + P  (Enzymatic reaction)  (k1_F = 18 / k1_R = 100 / k2_F = 49)
Chemicals involved in the above reactions: {"P" (green), "ES" (red), "E" (violet), "S" (darkturquoise)}

In [ ]:

In [12]:

# Here we use the "slower" preset for the variable steps, a conservative option prioritizing accuracy over speed
uc = UniformCompartment(reactions=rxns, 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
Chemicals involved in reactions: ['ES', 'P', 'S', 'E']

In [13]:

# Perform the reactions
uc.single_compartment_react(duration=1.2, initial_step=0.05)

784 total variable step(s) taken in 1.548 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: 1.2018

In [14]:

history_exact = uc.get_history()    # Let's save the historical data for a comparison at the very end

In [15]:

uc.plot_history(show_intervals=True, 
                title_prefix="Using exact values for k1_forward and k1_reverse")

plot_pandas() NOTICE: Excessive number of vertical lines (785) - only showing 1 every 6 lines

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_exact = uc.get_rate_history()
rates_exact

Out[16]:

	SYSTEM TIME	rxn0_rate_1	rxn0_rate_2	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
...	...	...	...	...
779	1.166644	1.308959	1.462999	779
780	1.173671	1.261193	1.409959	780
781	1.180698	1.215086	1.358736	781
782	1.187725	1.170584	1.309274	782
783	1.194752	1.127638	1.261520	783

784 rows × 4 columns

In [17]:

# Let's take a look at how the reaction rate varies with time
PlotlyHelper.plot_pandas(df=rates_exact, colors="blue",
                         title="Reaction rate, dP/dt, over time",
                         x_var="SYSTEM TIME", fields="rxn0_rate_2", 
                         x_label="time", y_label="dP/dt")

In [ ]:

PART 2. Guessing a SMALLER value of `k1_forward` (UNDER-ESTIMATED)¶

In [18]:

enz = uc.get_single_reaction(0)

In [19]:

enz.min_k1_forward(kM=kM, kcat=kcat)   # The smallest possible value

Out[19]:

5.919463087248328

In [20]:

k1_forward = 6.5    # Close to the smallest possible value

In [21]:

k1_reverse = enz.compute_k1_reverse(kM, kcat, k1_forward=k1_forward)
k1_reverse

Out[21]:

4.8055555555555

In [ ]:

In [22]:

chem_data.all_chemicals()   # Nothing changed here; same chemicals as before

Out[22]:

	name	label	plot_color
0	P	P	green
1	ES	ES	red
2	Adenosinedeaminase	E	violet
3	2,6-Diamino-9-β-D-deoxyribofuranosyl-9-H-purine	S	darkturquoise

In [ ]:

In [23]:

rxns_underest = ReactionRegistry(chem_data=chem_data)   # New set of reactions

In [24]:

# Enzymatic reaction `E + S <-> ES -> E + P` 
# In first part of the reaction, a forward rate that is much faster than its revers one
rxns_underest.add_reaction(reactants="S", products="P", enzyme="E",
                           k1_F=k1_forward, k1_R=k1_reverse, k2_F=kcat)
                  
rxns_underest.describe_reactions()

Number of reactions: 1
0: E + S <-> ES -> E + P  (Enzymatic reaction)  (k1_F = 6.5 / k1_R = 4.8056 / k2_F = 49)
Chemicals involved in the above reactions: {"P" (green), "ES" (red), "E" (violet), "S" (darkturquoise)}

In [25]:

# Here we use the "slower" preset for the variable steps, a conservative option prioritizing accuracy over speed
uc = UniformCompartment(reactions=rxns_underest, 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
Chemicals involved in reactions: ['ES', 'P', 'S', 'E']

In [26]:

# Perform the reactions
uc.single_compartment_react(duration=1.2, initial_step=0.05)

788 total variable step(s) taken in 1.839 sec
Number of step re-do's because of negative concentrations: 1
Number of step re-do's because of elective soft aborts: 2
Norm usage: {'norm_A': 704, 'norm_B': 709, 'norm_C': 704, 'norm_D': 704}
System Time is now: 1.2057

In [27]:

history_underest = uc.get_history()    # Let's save the historical data for a comparison at the very end

In [28]:

uc.plot_history(show_intervals=True, 
                title_prefix="Using SMALLER (under-estimated) value for k1_forward")

plot_pandas() NOTICE: Excessive number of vertical lines (789) - only showing 1 every 6 lines

At first glance, not too different from before, overall, but let's take a closer look

In [29]:

rates_underest = uc.get_rate_history()
rates_underest

Out[29]:

	SYSTEM TIME	rxn0_rate_1	rxn0_rate_2	step
0	0.000000	130.000000	0.000000	0
1	0.000500	120.842601	3.185000	1
2	0.000750	116.711975	4.626306	2
3	0.000763	116.515396	4.694958	3
4	0.000769	116.417346	4.729203	4
...	...	...	...	...
783	1.171227	1.169882	1.318201	783
784	1.178115	1.125129	1.268138	784
785	1.185004	1.082003	1.219867	785
786	1.191892	1.040450	1.173332	786
787	1.198781	1.000418	1.128479	787

788 rows × 4 columns

In [30]:

# Let's take a look at how the reaction rate varies with time
PlotlyHelper.plot_pandas(df=rates_underest, colors="blue",
                         title="Reaction rate, dP/dt, over time",
                         x_var="SYSTEM TIME", fields="rxn0_rate_2", 
                         x_label="time", y_label="dP/dt")

In [ ]:

PART 3. Guessing a LARGER value of `k1_forward` (OVER-ESTIMATED)¶

In [31]:

k1_forward = 40.    # Well above the known value of 18.

In [32]:

k1_reverse = enz.compute_k1_reverse(kM, kcat, k1_forward=k1_forward)
k1_reverse

Out[32]:

282.1111111111108

In [ ]:

In [33]:

chem_data.all_chemicals()   # Nothing changed here; same chemicals as before

Out[33]:

	name	label	plot_color
0	P	P	green
1	ES	ES	red
2	Adenosinedeaminase	E	violet
3	2,6-Diamino-9-β-D-deoxyribofuranosyl-9-H-purine	S	darkturquoise

In [ ]:

In [34]:

rxns_overest = ReactionRegistry(chem_data=chem_data)   # New set of reactions

In [35]:

# Enzymatic reaction `E + S <-> ES -> E + P` 
# In first part of the reaction, a forward rate that is much faster than its reverse one
rxns_overest.add_reaction(reactants="S", products="P", enzyme="E",
                          k1_F=k1_forward, k1_R=k1_reverse, k2_F=kcat)

rxns_overest.describe_reactions()

Number of reactions: 1
0: E + S <-> ES -> E + P  (Enzymatic reaction)  (k1_F = 40 / k1_R = 282.11 / k2_F = 49)
Chemicals involved in the above reactions: {"P" (green), "ES" (red), "E" (violet), "S" (darkturquoise)}

In [36]:

# Here we use the "slower" preset for the variable steps, a conservative option prioritizing accuracy over speed
uc = UniformCompartment(reactions=rxns_overest, 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
Chemicals involved in reactions: ['ES', 'P', 'S', 'E']

In [37]:

# Perform the reactions
uc.single_compartment_react(duration=1.2, initial_step=0.05)

864 total variable step(s) taken in 1.882 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': 829, 'norm_B': 835, 'norm_C': 829, 'norm_D': 829}
System Time is now: 1.2001

In [38]:

history_overest = uc.get_history()    # Let's save the historical data for a comparison at the very end

In [39]:

uc.plot_history(show_intervals=True, 
                title_prefix="Using LARGER (over-estimated) value for k1_forward")

plot_pandas() NOTICE: Excessive number of vertical lines (865) - only showing 1 every 6 lines

At first glance, not too different from before, overall, but let's take a closer look

In [40]:

rates_overest = uc.get_rate_history()
rates_overest

Out[40]:

	SYSTEM TIME	rxn0_rate_1	rxn0_rate_2	step
0	0.000000	800.000000	0.000000	0
1	0.000500	357.555556	19.600000	1
2	0.000750	265.637774	23.739956	2
3	0.000763	262.356893	23.888118	3
4	0.000769	260.739940	23.961149	4
...	...	...	...	...
859	1.170211	1.335790	1.490884	859
860	1.176184	1.296873	1.445487	860
861	1.182158	1.255105	1.401986	861
862	1.188131	1.219475	1.358993	862
863	1.194105	1.178798	1.318155	863

864 rows × 4 columns

In [41]:

# Let's take a look at how the reaction rate varies with time
PlotlyHelper.plot_pandas(df=rates_overest, colors="blue",
                         title="Reaction rate, dP/dt, over time",
                         x_var="SYSTEM TIME", fields="rxn0_rate_2", 
                         x_label="time", y_label="dP/dt")

In [ ]:

In [42]:

history_overest = uc.get_history()
history_overest

Out[42]:

	SYSTEM TIME	P	ES	E	S	step	caption
0	0.000000	0.000000	0.000000	1.000000	20.000000		Set concentration
1	0.000500	0.000000	0.400000	0.600000	19.600000	1	1st reaction step
2	0.000750	0.004900	0.484489	0.515511	19.510611	2
3	0.000763	0.005197	0.487513	0.512487	19.507291	3
4	0.000769	0.005346	0.489003	0.510997	19.505651	4
...	...	...	...	...	...	...	...
860	1.176184	19.722714	0.029500	0.970500	0.247787	860
861	1.182158	19.731349	0.028612	0.971388	0.240039	861
862	1.188131	19.739724	0.027735	0.972265	0.232542	862
863	1.194105	19.747842	0.026901	0.973099	0.225257	863
864	1.200079	19.755716	0.026069	0.973931	0.218216	864	last reaction step

865 rows × 7 columns

In [ ]:

Comparing the 3 scenarios¶

The concentrations of the intermediate `ES` over time¶

In [43]:

es_under = PlotlyHelper.plot_pandas(df=history_underest, 
                         title="underestimated k1_forward",
                         x_var="SYSTEM TIME", fields="ES", 
                         x_label="time", y_label="ES", colors = "yellow")

In [44]:

es_exact = PlotlyHelper.plot_pandas(df=history_exact, 
                         title="exact k1_forward",
                         x_var="SYSTEM TIME", fields="ES", 
                         x_label="time", y_label="ES", colors = "green")

In [45]:

es_over = PlotlyHelper.plot_pandas(df=history_overest, 
                         title="overestimated k1_forward",
                         x_var="SYSTEM TIME", fields="ES", 
                         x_label="time", y_label="ES", colors = "purple")

In [46]:

PlotlyHelper.combine_plots([es_under, es_exact, es_over],
                            title="ES")

The concentrations of the product `P` over time¶

In [47]:

p_under = PlotlyHelper.plot_pandas(df=history_underest, 
                         title="underestimated k1_forward",
                         x_var="SYSTEM TIME", fields="P", 
                         x_label="time", y_label="P", colors = "yellow")

In [48]:

p_exact = PlotlyHelper.plot_pandas(df=history_exact, 
                         title="exact k1_forward",
                         x_var="SYSTEM TIME", fields="P", 
                         x_label="time", y_label="P", colors = "cyan")

In [49]:

p_over = PlotlyHelper.plot_pandas(df=history_overest, 
                         title="overestimated k1_forward",
                         x_var="SYSTEM TIME", fields="P", 
                         x_label="time", y_label="P", colors = "purple")

In [50]:

PlotlyHelper.combine_plots([p_under, p_exact, p_over],
                            title="P")

In [ ]:

The Reaction Rate over time¶

In [51]:

r1_under = PlotlyHelper.plot_pandas(df=rates_underest, 
                         title="underestimated k1_forward",
                         x_var="SYSTEM TIME", fields="rxn0_rate_2", 
                         x_label="time", y_label="dP/dt", colors = "yellow")

In [52]:

r1_exact = PlotlyHelper.plot_pandas(df=rates_exact, 
                         title="exact k1_forward",
                         x_var="SYSTEM TIME", fields="rxn0_rate_2", 
                         x_label="time", y_label="dP/dt", colors = "blue")

In [53]:

r1_over = PlotlyHelper.plot_pandas(df=rates_overest, 
                         title="overestimated k1_forward",
                         x_var="SYSTEM TIME", fields="rxn0_rate_2", 
                         x_label="time", y_label="dP/dt", colors = "purple")

In [54]:

PlotlyHelper.combine_plots([r1_under, r1_exact, r1_over],
                            title="Reaction Rate")

In [ ]:

Enzyme Kinetics :¶

Our model: E + S <-> ES (with kinetic parameters k1_forward and k1_reverse), and ES -> E + P (k2_forward)¶

In experiment enzyme_1_a, we were given k1_forward, k1_reverse and k2_forward... But what to do if we're just given kM and kcat ?¶

THE REACTION:¶

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

Assume we're only given values for kM and kcat¶

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

PART 1. Using the known, exact values for k1_forward and k1_reverse¶

Specify the Kinetic Parameters¶

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

PART 2. Guessing a SMALLER value of k1_forward (UNDER-ESTIMATED)¶

PART 3. Guessing a LARGER value of k1_forward (OVER-ESTIMATED)¶

Comparing the 3 scenarios¶

The concentrations of the intermediate ES over time¶

The concentrations of the product P over time¶

The Reaction Rate over time¶

Our model: `E + S <-> ES` (with kinetic parameters k1_forward and k1_reverse), and `ES -> E + P` (k2_forward)¶

In experiment `enzyme_1_a`, we were given `k1_forward`, `k1_reverse` and `k2_forward`... But what to do if we're just given `kM` and `kcat` ?¶

Assume we're only given values for `kM` and `kcat`¶

PART 1. Using the known, exact values for `k1_forward` and `k1_reverse`¶

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

PART 2. Guessing a SMALLER value of `k1_forward` (UNDER-ESTIMATED)¶

PART 3. Guessing a LARGER value of `k1_forward` (OVER-ESTIMATED)¶

The concentrations of the intermediate `ES` over time¶

The concentrations of the product `P` over time¶