Coupled pair of reactions: `S <-> P` , and `S + E <-> P + E`¶

A direct reaction and the same reaction, catalyzed

Re-run from same initial concentrations of S ("Substrate") and P ("Product"), for various concentations of the enzyme `E`: from zero to hugely abundant¶

LAST REVISED: Dec. 3, 2023

In [1]:

import set_path      # Importing this module will add the project's home directory to sys.path

Added 'D:\Docs\- MY CODE\BioSimulations\life123-Win7' to sys.path

In [2]:

from src.modules.chemicals.chem_data import ChemData
from src.modules.reactions.reaction_dynamics import ReactionDynamics
from src.modules.movies.movies import MovieTabular

import pandas as pd
import plotly.express as px

In [3]:

# Initialize the system
chem_data = ChemData(names=["S", "P", "E"])

# Reaction S <-> P , with 1st-order kinetics, favorable thermodynamics in the forward direction, 
# and a forward rate that is much slower than it would be with the enzyme - as seen in the next reaction, below
chem_data.add_reaction(reactants="S", products="P",
                       forward_rate=1., delta_G=-3989.73)

# Reaction S + E <-> P + E , with 1st-order kinetics, and a forward rate that is faster than it was without the enzyme
# Thermodynamically, there's no change from the reaction without the enzyme
chem_data.add_reaction(reactants=["S", "E"], products=["P", "E"],
                       forward_rate=10., delta_G=-3989.73)

chem_data.describe_reactions()     # Notice how the enzyme `E` is noted in the printout below

Number of reactions: 2 (at temp. 25 C)
0: S <-> P  (kF = 1 / kR = 0.2 / delta_G = -3,989.7 / K = 5) | 1st order in all reactants & products
1: S + E <-> P + E  (kF = 10 / kR = 2 / delta_G = -3,989.7 / K = 5) | Enzyme: E | 1st order in all reactants & products
Set of chemicals involved in the above reactions (not counting enzymes): {'S', 'P'}
Set of enzymes involved in the above reactions: {'E'}

In [ ]:

1. Set the initial concentrations of all the chemicals - starting with no enzyme¶

In [4]:

dynamics = ReactionDynamics(chem_data=chem_data)
dynamics.set_conc(conc={"S": 20., "P": 0., "E": 0.},
                  snapshot=True)      # Initially, no enzyme `E`
dynamics.describe_state()

SYSTEM STATE at Time t = 0:
3 species:
  Species 0 (S). Conc: 20.0
  Species 1 (P). Conc: 0.0
  Species 2 (E). Conc: 0.0
Set of chemicals involved in reactions (not counting enzymes): {'S', 'P'}
Set of enzymes involved in reactions: {'E'}

Advance the reactions (for now without enzyme) to equilibrium¶

In [5]:

dynamics.set_diagnostics()       # To save diagnostic information about the call to single_compartment_react()

# All of these settings are currently close to the default values... but subject to change; set for repeatability
dynamics.set_thresholds(norm="norm_A", low=0.5, high=0.8, abort=1.44)
dynamics.set_thresholds(norm="norm_B", low=0.08, high=0.5, abort=1.5)
dynamics.set_step_factors(upshift=1.2, downshift=0.5, abort=0.4)
dynamics.set_error_step_factor(0.25)

# Perform the reactions
dynamics.single_compartment_react(reaction_duration=4.0,
                                  initial_step=0.1, variable_steps=True, explain_variable_steps=False)

* INFO: the tentative time step (0.1) leads to a least one norm value > its ABORT threshold:
      -> will backtrack, and re-do step with a SMALLER delta time, multiplied by 0.4 (set to 0.04) [Step started at t=0, and will rewind there]
35 total step(s) taken

In [6]:

#dynamics.explain_time_advance()

In [7]:

dynamics.plot_history(colors=['darkorange', 'green', 'violet'], show_intervals=True, title_prefix="With ZERO enzyme")

The reactions, lacking enzyme, are proceeding slowly towards equilibrium, just like the reaction that was discussed in part 1 of the experiment "enzyme_1"¶

In [8]:

# To save up snapshots of crossover times at various Enzyme concentrations
crossover_points = MovieTabular(parameter_name="Enzyme concentration")

In [9]:

new_crossover = dynamics.curve_intersection("S", "P", t_start=0, t_end=1.0)
crossover_points.store(par=dynamics.get_chem_conc("E"), 
                       data_snapshot = {"crossover time": new_crossover[0]})
new_crossover

Min abs distance found at data row: 20

Out[9]:

(0.7435259497128707, 9.999999999999996)

In [10]:

# Verify that the reaction has reached equilibrium
dynamics.is_in_equilibrium(tolerance=2)

0: S <-> P
Final concentrations: [S] = 3.372 ; [P] = 16.63
1. Ratio of reactant/product concentrations, adjusted for reaction orders: 4.93068
    Formula used:  [P] / [S]
2. Ratio of forward/reverse reaction rates: 5.00000519021548
Discrepancy between the two values: 1.387 %
Reaction IS in equilibrium (within 2% tolerance)

1: S + E <-> P + E
Final concentrations: [S] = 3.372 ; [P] = 16.63 ; [E] = 0
Reaction IS in equilibrium because it can't proceed in either direction due to zero concentrations in both some reactants and products!

D:\Docs\- MY CODE\BioSimulations\life123-Win7\src\modules\reactions\reaction.py:468: RuntimeWarning:

invalid value encountered in double_scalars

Out[10]:

True

In [ ]:

2. Re-start all the reactions from the same initial concentrations - except for now having a tiny amount of enzyme (two orders of magnitude less than the starting [S])¶

In [11]:

dynamics = ReactionDynamics(chem_data=chem_data)   # A brand-new simulation  

In [12]:

dynamics.set_conc(conc={"S": 20., "P": 0., "E": 0.2},
                  snapshot=True)      # A tiny bit of enzyme `E`

In [13]:

dynamics.describe_state()

SYSTEM STATE at Time t = 0:
3 species:
  Species 0 (S). Conc: 20.0
  Species 1 (P). Conc: 0.0
  Species 2 (E). Conc: 0.2
Set of chemicals involved in reactions (not counting enzymes): {'S', 'P'}
Set of enzymes involved in reactions: {'E'}

Re-take the new system (now with a tiny amount of enzyme) to equilibrium¶

In [14]:

dynamics.single_compartment_react(reaction_duration=1.5, 
                                  initial_step=0.05, variable_steps=True, explain_variable_steps=False)

* INFO: the tentative time step (0.05) leads to a least one norm value > its ABORT threshold:
      -> will backtrack, and re-do step with a SMALLER delta time, multiplied by 0.4 (set to 0.02) [Step started at t=0, and will rewind there]
Some steps were backtracked and re-done, to prevent negative concentrations or excessively large concentration changes
33 total step(s) taken

In [15]:

#dynamics.explain_time_advance()

In [16]:

dynamics.plot_history(colors=['darkorange', 'green', 'violet'], show_intervals=True, title_prefix="With a tiny amount of enzyme")

In [17]:

new_crossover = dynamics.curve_intersection("S", "P", t_start=0, t_end=0.5)
crossover_points.store(par=dynamics.get_chem_conc("E"), 
                       data_snapshot = {"crossover time": new_crossover[0]})
new_crossover

Min abs distance found at data row: 17

Out[17]:

(0.24710707675961718, 10.0)

Notice how even a tiny amount of enzyme (1/100 of the initial [A]) makes a very pronounced difference!¶

In [18]:

# Verify that the reaction has reached equilibrium
dynamics.is_in_equilibrium()

0: S <-> P
Final concentrations: [S] = 3.34 ; [P] = 16.66
1. Ratio of reactant/product concentrations, adjusted for reaction orders: 4.98745
    Formula used:  [P] / [S]
2. Ratio of forward/reverse reaction rates: 5.00000519021548
Discrepancy between the two values: 0.2511 %
Reaction IS in equilibrium (within 1% tolerance)

1: S + E <-> P + E
Final concentrations: [S] = 3.34 ; [P] = 16.66 ; [E] = 0.2
1. Ratio of reactant/product concentrations, adjusted for reaction orders: 4.98745
    Formula used:  ([P][E]) / ([S][E])
2. Ratio of forward/reverse reaction rates: 5.00000519021548
Discrepancy between the two values: 0.2511 %
Reaction IS in equilibrium (within 1% tolerance)

Out[18]:

True

In [ ]:

3. Re-start all the reactions from the same initial concentrations - except for now having a more substantial amount of enzyme¶

In [19]:

dynamics = ReactionDynamics(chem_data=chem_data)   # A brand-new simulation  

In [20]:

dynamics.set_conc(conc={"S": 20., "P": 0., "E": 1.},
                  snapshot=True)      # A more substantial amount of enzyme `E`

In [21]:

dynamics.describe_state()

SYSTEM STATE at Time t = 0:
3 species:
  Species 0 (S). Conc: 20.0
  Species 1 (P). Conc: 0.0
  Species 2 (E). Conc: 1.0
Set of chemicals involved in reactions (not counting enzymes): {'S', 'P'}
Set of enzymes involved in reactions: {'E'}

Re-take the new system (now with a more substantial amount of enzyme) to equilibrium¶

In [22]:

dynamics.single_compartment_react(reaction_duration=0.5, 
                                  initial_step=0.02, variable_steps=True, explain_variable_steps=False)

* INFO: the tentative time step (0.02) leads to a least one norm value > its ABORT threshold:
      -> will backtrack, and re-do step with a SMALLER delta time, multiplied by 0.4 (set to 0.008) [Step started at t=0, and will rewind there]
* INFO: the tentative time step (0.008) leads to a least one norm value > its ABORT threshold:
      -> will backtrack, and re-do step with a SMALLER delta time, multiplied by 0.4 (set to 0.0032) [Step started at t=0, and will rewind there]
Some steps were backtracked and re-done, to prevent negative concentrations or excessively large concentration changes
39 total step(s) taken

In [23]:

#dynamics.explain_time_advance()

In [24]:

dynamics.plot_history(colors=['darkorange', 'green', 'violet'], show_intervals=True, title_prefix="With a more substantial amount of enzyme")

In [25]:

new_crossover = dynamics.curve_intersection("S", "P", t_start=0, t_end=0.5)
crossover_points.store(par=dynamics.get_chem_conc("E"), 
                       data_snapshot = {"crossover time": new_crossover[0]})
new_crossover

Min abs distance found at data row: 21

Out[25]:

(0.06769827987210066, 10.0)

Notice the continued - and substantial - speedup of the reaction, over the earlier runs¶

In [26]:

# Verify that the reaction has reached equilibrium
dynamics.is_in_equilibrium(explain=False)

Out[26]:

True

In [ ]:

4. Re-start all the reactions from the same initial concentrations - except for now having a good amount of enzyme¶

In [27]:

dynamics = ReactionDynamics(chem_data=chem_data)   # A brand-new simulation  

In [28]:

dynamics.set_conc(conc={"S": 20., "P": 0., "E": 5.},
                  snapshot=True)      # A good amount of enzyme `E`

In [29]:

dynamics.describe_state()

SYSTEM STATE at Time t = 0:
3 species:
  Species 0 (S). Conc: 20.0
  Species 1 (P). Conc: 0.0
  Species 2 (E). Conc: 5.0
Set of chemicals involved in reactions (not counting enzymes): {'S', 'P'}
Set of enzymes involved in reactions: {'E'}

Re-take the new system to equilibrium¶

In [30]:

dynamics.single_compartment_react(reaction_duration=0.2, 
                                  initial_step=0.01, variable_steps=True, explain_variable_steps=False)

* INFO: the tentative time step (0.01) leads to a least one norm value > its ABORT threshold:
      -> will backtrack, and re-do step with a SMALLER delta time, multiplied by 0.4 (set to 0.004) [Step started at t=0, and will rewind there]
* INFO: the tentative time step (0.004) leads to a least one norm value > its ABORT threshold:
      -> will backtrack, and re-do step with a SMALLER delta time, multiplied by 0.4 (set to 0.0016) [Step started at t=0, and will rewind there]
Some steps were backtracked and re-done, to prevent negative concentrations or excessively large concentration changes
36 total step(s) taken

In [31]:

#dynamics.explain_time_advance()

In [32]:

dynamics.plot_history(colors=['darkorange', 'green', 'violet'], show_intervals=True, title_prefix="With a good amount of enzyme")

In [33]:

new_crossover = dynamics.curve_intersection("S", "P", t_start=0, t_end=0.5)
crossover_points.store(par=dynamics.get_chem_conc("E"), 
                       data_snapshot = {"crossover time": new_crossover[0]})
new_crossover

Min abs distance found at data row: 15

Out[33]:

(0.014494267066104415, 10.000000000000002)

Notice the continued - and substantial - speedup of the reaction, over the earlier runs¶

In [34]:

# Verify that the reaction has reached equilibrium
dynamics.is_in_equilibrium(explain=False)

Out[34]:

True

In [ ]:

5. Re-start all the reactions from the same initial concentrations - except for now having a lot of enzyme (same as the initial [A])¶

In [35]:

dynamics = ReactionDynamics(chem_data=chem_data)   # A brand-new simulation  

In [36]:

dynamics.set_conc(conc={"S": 20., "P": 0., "E": 20.},
                  snapshot=True)      # A lot of enzyme `E`

In [37]:

dynamics.describe_state()

SYSTEM STATE at Time t = 0:
3 species:
  Species 0 (S). Conc: 20.0
  Species 1 (P). Conc: 0.0
  Species 2 (E). Conc: 20.0
Set of chemicals involved in reactions (not counting enzymes): {'S', 'P'}
Set of enzymes involved in reactions: {'E'}

Re-take the new system (now with a lot of enzyme) to equilibrium¶

In [38]:

dynamics.single_compartment_react(reaction_duration=0.05, 
                                  initial_step=0.005, variable_steps=True, explain_variable_steps=False)

* INFO: the tentative time step (0.005) leads to a least one norm value > its ABORT threshold:
      -> will backtrack, and re-do step with a SMALLER delta time, multiplied by 0.4 (set to 0.002) [Step started at t=0, and will rewind there]
* INFO: the tentative time step (0.002) leads to a least one norm value > its ABORT threshold:
      -> will backtrack, and re-do step with a SMALLER delta time, multiplied by 0.4 (set to 0.0008) [Step started at t=0, and will rewind there]
* INFO: the tentative time step (0.0008) leads to a least one norm value > its ABORT threshold:
      -> will backtrack, and re-do step with a SMALLER delta time, multiplied by 0.4 (set to 0.00032) [Step started at t=0, and will rewind there]
Some steps were backtracked and re-done, to prevent negative concentrations or excessively large concentration changes
39 total step(s) taken

In [39]:

#dynamics.explain_time_advance()

In [40]:

dynamics.plot_history(colors=['darkorange', 'green', 'violet'], show_intervals=True, title_prefix="With a lot of enzyme")

In [41]:

new_crossover = dynamics.curve_intersection("S", "P", t_start=0, t_end=0.5)
crossover_points.store(par=dynamics.get_chem_conc("E"), 
                       data_snapshot = {"crossover time": new_crossover[0]})
new_crossover

Min abs distance found at data row: 17

Out[41]:

(0.003687634618431487, 9.999999999999996)

Notice the continued - and substantial - speedup of the reaction, over the earlier runs¶

In [42]:

# Verify that the reaction has reached equilibrium
dynamics.is_in_equilibrium(explain=False)

Out[42]:

True

In [ ]:

6. Re-start all the reactions from the same initial concentrations - except for now having a very large amount of enzyme (more than the initial substrate concentration [S])¶

In [43]:

dynamics = ReactionDynamics(chem_data=chem_data)   # A brand-new simulation  

In [44]:

dynamics.set_conc(conc={"S": 20., "P": 0., "E": 30.},
                  snapshot=True)      # A very large amount of enzyme `E`

In [45]:

dynamics.describe_state()

SYSTEM STATE at Time t = 0:
3 species:
  Species 0 (S). Conc: 20.0
  Species 1 (P). Conc: 0.0
  Species 2 (E). Conc: 30.0
Set of chemicals involved in reactions (not counting enzymes): {'S', 'P'}
Set of enzymes involved in reactions: {'E'}

Re-take the new system to equilibrium¶

In [46]:

dynamics.single_compartment_react(reaction_duration=0.02, 
                                  initial_step=0.001, variable_steps=True, explain_variable_steps=False)

* INFO: the tentative time step (0.001) leads to a least one norm value > its ABORT threshold:
      -> will backtrack, and re-do step with a SMALLER delta time, multiplied by 0.4 (set to 0.0004) [Step started at t=0, and will rewind there]
* INFO: the tentative time step (0.0004) leads to a least one norm value > its ABORT threshold:
      -> will backtrack, and re-do step with a SMALLER delta time, multiplied by 0.4 (set to 0.00016) [Step started at t=0, and will rewind there]
Some steps were backtracked and re-done, to prevent negative concentrations or excessively large concentration changes
36 total step(s) taken

In [47]:

#dynamics.explain_time_advance()

In [48]:

dynamics.plot_history(colors=['darkorange', 'green', 'violet'], show_intervals=True, title_prefix="With a very large amount of enzyme")

In [49]:

new_crossover = dynamics.curve_intersection("S", "P", t_start=0, t_end=0.5)
crossover_points.store(par=dynamics.get_chem_conc("E"), 
                       data_snapshot = {"crossover time": new_crossover[0]})
new_crossover

Min abs distance found at data row: 18

Out[49]:

(0.002465868124424963, 10.0)

Yet more speedup of the reaction, over the previous run¶

In [50]:

# Verify that the reaction has reached equilibrium
dynamics.is_in_equilibrium(explain=False)

Out[50]:

True

In [ ]:

7. Finally, re-start all the reactions from the same initial concentrations - except for now having a huge amount of enzyme (far more than the initial [A])¶

In [51]:

dynamics = ReactionDynamics(chem_data=chem_data)   # A brand-new simulation  

In [52]:

dynamics.set_conc(conc={"S": 20., "P": 0., "E": 100.},
                  snapshot=True)      # A lavish amount of enzyme `E`

In [53]:

dynamics.describe_state()

SYSTEM STATE at Time t = 0:
3 species:
  Species 0 (S). Conc: 20.0
  Species 1 (P). Conc: 0.0
  Species 2 (E). Conc: 100.0
Set of chemicals involved in reactions (not counting enzymes): {'S', 'P'}
Set of enzymes involved in reactions: {'E'}

Re-take the new system to equilibrium¶

In [54]:

dynamics.single_compartment_react(reaction_duration=0.02, 
                                  initial_step=0.0005, variable_steps=True, explain_variable_steps=False)

* INFO: the tentative time step (0.0005) leads to a least one norm value > its ABORT threshold:
      -> will backtrack, and re-do step with a SMALLER delta time, multiplied by 0.4 (set to 0.0002) [Step started at t=0, and will rewind there]
* INFO: the tentative time step (0.0002) leads to a least one norm value > its ABORT threshold:
      -> will backtrack, and re-do step with a SMALLER delta time, multiplied by 0.4 (set to 8e-05) [Step started at t=0, and will rewind there]
40 total step(s) taken

In [55]:

#dynamics.explain_time_advance()

In [56]:

dynamics.plot_history(colors=['darkorange', 'green', 'violet'], show_intervals=True, title_prefix="With a huge amount of enzyme")

In [57]:

new_crossover = dynamics.curve_intersection("S", "P", t_start=0, t_end=0.5)
crossover_points.store(par=dynamics.get_chem_conc("E"), 
                       data_snapshot = {"crossover time": new_crossover[0]})
new_crossover

Min abs distance found at data row: 15

Out[57]:

(0.0007383445224719487, 10.0)

Yet more speedup of the reaction, over the previous run¶

In [58]:

# Verify that the reaction has reached equilibrium
dynamics.is_in_equilibrium(explain=False)

Out[58]:

True

In [ ]:

8. Finally, re-start all the reactions from the same initial concentrations - except for now having a LAVISH amount of enzyme (two orders of magnitude more than the starting substrate concentration [S])¶

In [59]:

dynamics = ReactionDynamics(chem_data=chem_data)   # A brand-new simulation  

In [60]:

dynamics.set_conc(conc={"S": 20., "P": 0., "E": 2000.},
                  snapshot=True)      # A lavish amount of enzyme `E`

In [61]:

dynamics.describe_state()

SYSTEM STATE at Time t = 0:
3 species:
  Species 0 (S). Conc: 20.0
  Species 1 (P). Conc: 0.0
  Species 2 (E). Conc: 2000.0
Set of chemicals involved in reactions (not counting enzymes): {'S', 'P'}
Set of enzymes involved in reactions: {'E'}

Re-take the new system (now with a lavish amount of enzyme) to equilibrium¶

In [62]:

dynamics.single_compartment_react(reaction_duration=0.0015, 
                                  initial_step=0.000005, variable_steps=True, explain_variable_steps=False)

* INFO: the tentative time step (5e-06) leads to a least one norm value > its ABORT threshold:
      -> will backtrack, and re-do step with a SMALLER delta time, multiplied by 0.4 (set to 2e-06) [Step started at t=0, and will rewind there]
48 total step(s) taken

In [63]:

#dynamics.explain_time_advance()

In [64]:

dynamics.plot_history(chemicals=['S', 'P'],
                      colors=['darkorange', 'green', 'violet'], show_intervals=True, title_prefix="With a LAVISH amount of enzyme (NOT shown)")

Note: NOT showing the enzyme (concentration 2,000) in the graph, to avoid squishing down the other curves!

In [65]:

new_crossover = dynamics.curve_intersection("S", "P", t_start=0, t_end=0.5)
crossover_points.store(par=dynamics.get_chem_conc("E"), 
                       data_snapshot = {"crossover time": new_crossover[0]})
new_crossover

Min abs distance found at data row: 20

Out[65]:

(3.7174487975074336e-05, 10.0)

Yet more speedup of the reaction, over the previous run¶

In [66]:

# Verify that the reaction has reached equilibrium
dynamics.is_in_equilibrium()

0: S <-> P
Final concentrations: [S] = 3.313 ; [P] = 16.69
1. Ratio of reactant/product concentrations, adjusted for reaction orders: 5.03748
    Formula used:  [P] / [S]
2. Ratio of forward/reverse reaction rates: 5.00000519021548
Discrepancy between the two values: 0.7496 %
Reaction IS in equilibrium (within 1% tolerance)

1: S + E <-> P + E
Final concentrations: [S] = 3.313 ; [P] = 16.69 ; [E] = 2,000
1. Ratio of reactant/product concentrations, adjusted for reaction orders: 5.03748
    Formula used:  ([P][E]) / ([S][E])
2. Ratio of forward/reverse reaction rates: 5.00000519021548
Discrepancy between the two values: 0.7496 %
Reaction IS in equilibrium (within 1% tolerance)

Out[66]:

True

In [ ]:

Now, let's look at the time of the crossover points (for the [S] and [P] curves to intersect), as a function of the Enzyme concentration¶

In [67]:

df = crossover_points.get_dataframe()
df

Out[67]:

	Enzyme concentration	crossover time
0	0.0	0.743526
1	0.2	0.247107
2	1.0	0.067698
3	5.0	0.014494
4	20.0	0.003688
5	30.0	0.002466
6	100.0	0.000738
7	2000.0	0.000037

As we previously observed, as the Enzyme concentration is increased over repeated runs, the crossover happens earlier and earlier¶

In [68]:

# Let's plot just the first 6 data points, to avoid squishing the curve
px.line(data_frame=df.loc[0:5],
              x="Enzyme concentration", y=["crossover time"],
              title="Time of crossover of [S] and [P] , as a function of Enzyme concentration (first 6 points)",
              labels={"value":"Crossover time"},
              line_shape="spline")

In [69]:

# Same plot, but with log scales on both axes (all data points used this time, 
# but the value E = 0 is automatically dropped by the graphic function because of the log scale)
px.line(data_frame=df,
              x="Enzyme concentration", y=["crossover time"],
              log_x=True, log_y=True,
              title="Time of crossover of [S] and [P] , as a function of Enzyme conc (log scales on both axes)",
              labels={"value":"Crossover time"},
              line_shape="spline")

In [ ]:

Coupled pair of reactions: S <-> P , and S + E <-> P + E¶

Re-run from same initial concentrations of S ("Substrate") and P ("Product"), for various concentations of the enzyme E: from zero to hugely abundant¶

1. Set the initial concentrations of all the chemicals - starting with no enzyme¶

Advance the reactions (for now without enzyme) to equilibrium¶

The reactions, lacking enzyme, are proceeding slowly towards equilibrium, just like the reaction that was discussed in part 1 of the experiment "enzyme_1"¶

2. Re-start all the reactions from the same initial concentrations - except for now having a tiny amount of enzyme (two orders of magnitude less than the starting [S])¶

Re-take the new system (now with a tiny amount of enzyme) to equilibrium¶

Notice how even a tiny amount of enzyme (1/100 of the initial [A]) makes a very pronounced difference!¶

3. Re-start all the reactions from the same initial concentrations - except for now having a more substantial amount of enzyme¶

Re-take the new system (now with a more substantial amount of enzyme) to equilibrium¶

Notice the continued - and substantial - speedup of the reaction, over the earlier runs¶

4. Re-start all the reactions from the same initial concentrations - except for now having a good amount of enzyme¶

Re-take the new system to equilibrium¶

Notice the continued - and substantial - speedup of the reaction, over the earlier runs¶

5. Re-start all the reactions from the same initial concentrations - except for now having a lot of enzyme (same as the initial [A])¶

Re-take the new system (now with a lot of enzyme) to equilibrium¶

Notice the continued - and substantial - speedup of the reaction, over the earlier runs¶

6. Re-start all the reactions from the same initial concentrations - except for now having a very large amount of enzyme (more than the initial substrate concentration [S])¶

Re-take the new system to equilibrium¶

Yet more speedup of the reaction, over the previous run¶

7. Finally, re-start all the reactions from the same initial concentrations - except for now having a huge amount of enzyme (far more than the initial [A])¶

Re-take the new system to equilibrium¶

Yet more speedup of the reaction, over the previous run¶

8. Finally, re-start all the reactions from the same initial concentrations - except for now having a LAVISH amount of enzyme (two orders of magnitude more than the starting substrate concentration [S])¶

Re-take the new system (now with a lavish amount of enzyme) to equilibrium¶

Yet more speedup of the reaction, over the previous run¶

Now, let's look at the time of the crossover points (for the [S] and [P] curves to intersect), as a function of the Enzyme concentration¶

As we previously observed, as the Enzyme concentration is increased over repeated runs, the crossover happens earlier and earlier¶

Coupled pair of reactions: `S <-> P` , and `S + E <-> P + E`¶

Re-run from same initial concentrations of S ("Substrate") and P ("Product"), for various concentations of the enzyme `E`: from zero to hugely abundant¶