Comparing the reaction `A` <-> `B` with and without an enzyme¶

LAST REVISED: June 4, 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.reactions.reaction_data import ChemData
from src.modules.reactions.reaction_dynamics import ReactionDynamics

import numpy as np
import plotly.express as px

1. WITHOUT ENZYME¶

`A` <-> `B`¶

In [3]:

# Initialize the system
chem_data = ChemData(names=["A", "B"])

# Reaction A <-> B , with 1st-order kinetics, and a forward rate that is slower than it would be with the enzyme of part 2
chem_data.add_reaction(reactants="A", products="B",
                       forward_rate=1., delta_G=-3989.73)

chem_data.describe_reactions()

Number of reactions: 1 (at temp. 25 C)
0: A <-> B  (kF = 1 / kR = 0.2 / Delta_G = -3,989.73 / K = 5.00001) | 1st order in all reactants & products

Set the initial concentrations of all the chemicals¶

In [4]:

dynamics = ReactionDynamics(reaction_data=chem_data)
dynamics.set_conc(conc={"A": 20., "B": 0.},
                  snapshot=True)
dynamics.describe_state()

SYSTEM STATE at Time t = 0:
2 species:
  Species 0 (A). Conc: 20.0
  Species 1 (B). Conc: 0.0

Take the initial system 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=1.5, downshift=0.5, abort=0.5)
dynamics.set_error_step_factor(0.5)

dynamics.single_compartment_react(initial_step=0.1, reaction_duration=3.0,
                                  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 0.5 (set to 0.05) [Step started at t=0, and will rewind there]
30 total step(s) taken

In [6]:

#dynamics.explain_time_advance()

In [7]:

dynamics.plot_curves(colors=['darkorange', 'green'], show_intervals=True, title_prefix="WITHOUT enzyme")

Note how the time steps get automatically adjusted, as needed by the amount of change - include a complete step abort/redo at time=0¶

In [8]:

dynamics.curve_intersection("A", "B", t_start=0, t_end=1.0)

Min abs distance found at data row: 18

Out[8]:

(0.7406363068115296, 10.0)

In [9]:

# Verify that the reaction has reached equilibrium
dynamics.is_in_equilibrium(tolerance=3)

0: A <-> B
Final concentrations:  [B] = 16.6 ; [A] = 3.398
1. Ratio of reactant/product concentrations, adjusted for reaction orders: 4.88533
    Formula used:  [B] / [A]
2. Ratio of forward/reverse reaction rates: 5.000005788498923
Discrepancy between the two values: 2.293 %
Reaction IS in equilibrium (within 3% tolerance)

Out[9]:

True

In [ ]:

2. WITH ENZYME `E`¶

`A` + `E` <-> `B` + `E`¶

Note: for the sake of the demo, we'll completely ignore the concomitant reaction A <-> B¶

This in an approximation that we'll drop in later experiments

In [10]:

# Initialize the system
chem_data = ChemData(names=["A", "B", "E"])

# Reaction A + E <-> B + 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=["A", "E"], products=["B", "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: 1 (at temp. 25 C)
0: A + E <-> B + E  (kF = 10 / kR = 2 / Delta_G = -3,989.73 / K = 5.00001) | Enzyme: E | 1st order in all reactants & products

Notice how, while the ratio kF/kR is the same as it was without the enzyme (since it's dictated by the energy difference), the individual values of kF and kR are now each 10 times bigger¶

Set the initial concentrations of all the chemicals¶

In [11]:

dynamics = ReactionDynamics(reaction_data=chem_data)
dynamics.set_conc(conc={"A": 20., "B": 0., "E": 30.},
                  snapshot=True)      # Plenty of enzyme `E`
dynamics.describe_state()

SYSTEM STATE at Time t = 0:
3 species:
  Species 0 (A). Conc: 20.0
  Species 1 (B). Conc: 0.0
  Species 2 (E). Conc: 30.0

Take the initial system to equilibrium¶

In [12]:

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)

dynamics.single_compartment_react(initial_step=0.1, reaction_duration=0.1,
                                  variable_steps=True, explain_variable_steps=False)

*** CAUTION: negative concentration in chemical `A` in step starting at t=0.  It will be AUTOMATICALLY CORRECTED with a reduction in time step size, as follows:
    INFO: the tentative time step (0.1) leads to a NEGATIVE concentration of `A` from reaction A + E <-> B + E (rxn # 0): 
      Baseline value: 20 ; delta conc: -600
      -> will backtrack, and re-do step with a SMALLER delta time, multiplied by 0.25 (set to 0.025) [Step started at t=0, and will rewind there]

*** CAUTION: negative concentration in chemical `A` in step starting at t=0.  It will be AUTOMATICALLY CORRECTED with a reduction in time step size, as follows:
    INFO: the tentative time step (0.025) leads to a NEGATIVE concentration of `A` from reaction A + E <-> B + E (rxn # 0): 
      Baseline value: 20 ; delta conc: -150
      -> will backtrack, and re-do step with a SMALLER delta time, multiplied by 0.25 (set to 0.00625) [Step started at t=0, and will rewind there]

*** CAUTION: negative concentration in chemical `A` in step starting at t=0.  It will be AUTOMATICALLY CORRECTED with a reduction in time step size, as follows:
    INFO: the tentative time step (0.00625) leads to a NEGATIVE concentration of `A` from reaction A + E <-> B + E (rxn # 0): 
      Baseline value: 20 ; delta conc: -37.5
      -> will backtrack, and re-do step with a SMALLER delta time, multiplied by 0.25 (set to 0.0015625) [Step started at t=0, and will rewind there]
* INFO: the tentative time step (0.0015625) 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 0.000625) [Step started at t=0, and will rewind there]
* INFO: the tentative time step (0.000625) 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 0.00025) [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
42 total step(s) taken

Note how the (proposed) initial step - kept the same as the previous run - is now extravagantly large, given the much-faster reaction dynamics. However, the variable-step engine intercepts and automatically corrects the problem!¶

In [13]:

#dynamics.explain_time_advance()

In [14]:

dynamics.plot_curves(colors=['darkorange', 'green', 'violet'], show_intervals=True, title_prefix="WITH enzyme")

In [15]:

dynamics.curve_intersection("A", "B", t_start=0, t_end=0.02)

Min abs distance found at data row: 15

Out[15]:

(0.0024615346985334676, 10.0)

In [16]:

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

0: A + E <-> B + E
Final concentrations:  [B] = 16.67 ; [E] = 30 ; [A] = 3.334 ; [E] = 30
1. Ratio of reactant/product concentrations, adjusted for reaction orders: 4.99958
    Formula used:  ([B][E]) / ([A][E])
2. Ratio of forward/reverse reaction rates: 5.000005788498923
Discrepancy between the two values: 0.008527 %
Reaction IS in equilibrium (within 1% tolerance)

Out[16]:

True

Thanks to the (abundant) enzyme, the reaction reaches equilibrium roughtly around t=0.02, far sooner than the roughly t=3.5 without enzyme¶

The concentrations of A and B now become equal (cross-over) at t=0.00246 , rather than t=0.740

In [17]:

dynamics.get_history()

Out[17]:

	SYSTEM TIME	A	B	E	caption
0	0.000000	20.000000	0.000000	30.0	Initial state
1	0.000250	18.500000	1.500000	30.0
2	0.000500	17.135000	2.865000	30.0
3	0.000625	16.513925	3.486075	30.0
4	0.000750	15.920798	4.079202	30.0
5	0.000875	15.354362	4.645638	30.0
6	0.001000	14.813416	5.186584	30.0
7	0.001125	14.296812	5.703188	30.0
8	0.001250	13.803456	6.196544	30.0
9	0.001375	13.332300	6.667700	30.0
10	0.001525	12.792356	7.207644	30.0
11	0.001675	12.281569	7.718431	30.0
12	0.001855	11.701723	8.298277	30.0
13	0.002071	11.050997	8.949003	30.0
14	0.002330	10.330846	9.669154	30.0
15	0.002589	9.677894	10.322106	30.0
16	0.002900	8.967465	11.032535	30.0
17	0.003274	8.210411	11.789589	30.0
18	0.003647	7.555081	12.444919	30.0
19	0.004095	6.874353	13.125647	30.0
20	0.004543	6.303387	13.696613	30.0
21	0.004991	5.824486	14.175514	30.0
22	0.005528	5.342468	14.657532	30.0
23	0.006066	4.953716	15.046284	30.0
24	0.006711	4.577479	15.422521	30.0
25	0.007485	4.230824	15.769176	30.0
26	0.008413	3.930744	16.069256	30.0
27	0.009528	3.691047	16.308953	30.0
28	0.010865	3.518817	16.481183	30.0
29	0.012470	3.411650	16.588350	30.0
30	0.014396	3.357349	16.642651	30.0
31	0.016707	3.337366	16.662634	30.0
32	0.019480	3.333337	16.666663	30.0
33	0.022808	3.333329	16.666671	30.0
34	0.026802	3.333331	16.666669	30.0
35	0.031594	3.333330	16.666670	30.0
36	0.037345	3.333331	16.666669	30.0
37	0.044245	3.333329	16.666671	30.0
38	0.052526	3.333331	16.666669	30.0
39	0.062463	3.333327	16.666673	30.0
40	0.074388	3.333341	16.666659	30.0
41	0.088697	3.333284	16.666716	30.0
42	0.105869	3.333567	16.666433	30.0

In [ ]:

Comparing the reaction A <-> B with and without an enzyme¶

1. WITHOUT ENZYME¶

A <-> B¶

Set the initial concentrations of all the chemicals¶

Take the initial system to equilibrium¶

Note how the time steps get automatically adjusted, as needed by the amount of change - include a complete step abort/redo at time=0¶

2. WITH ENZYME E¶

A + E <-> B + E¶

Note: for the sake of the demo, we'll completely ignore the concomitant reaction A <-> B¶

Notice how, while the ratio kF/kR is the same as it was without the enzyme (since it's dictated by the energy difference), the individual values of kF and kR are now each 10 times bigger¶

Set the initial concentrations of all the chemicals¶

Take the initial system to equilibrium¶

Note how the (proposed) initial step - kept the same as the previous run - is now extravagantly large, given the much-faster reaction dynamics. However, the variable-step engine intercepts and automatically corrects the problem!¶

Thanks to the (abundant) enzyme, the reaction reaches equilibrium roughtly around t=0.02, far sooner than the roughly t=3.5 without enzyme¶

Comparing the reaction `A` <-> `B` with and without an enzyme¶

`A` <-> `B`¶

2. WITH ENZYME `E`¶

`A` + `E` <-> `B` + `E`¶