Coupled pair of reactions: `A <-> B` , and `A` + `E` <-> `B` + `E`¶

A direct reaction and same reaction, catalyzed

Enzyme `E` initially zero, and then suddenly added mid-reaction¶

LAST REVISED: Nov. 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.chemicals.chem_data import ChemData
from src.modules.reactions.reaction_dynamics import ReactionDynamics

In [3]:

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

# Reaction A <-> B , 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="A", products="B",
                       forward_rate=1., delta_G=-3989.73)

# 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: 2 (at temp. 25 C)
0: A <-> B  (kF = 1 / kR = 0.2 / delta_G = -3,989.7 / K = 5) | 1st order in all reactants & products
1: A + E <-> B + 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): {'B', 'A'}
Set of enzymes involved in the above reactions: {'E'}

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

In [4]:

dynamics = ReactionDynamics(chem_data=chem_data)
dynamics.set_conc(conc={"A": 20., "B": 0., "E": 0.},
                  snapshot=True)      # Initially, no 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: 0.0
Set of chemicals involved in reactions (not counting enzymes): {'B', 'A'}
Set of enzymes involved in reactions: {'E'}

Advance the reactions (for now without enzyme), but stopping well before 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=0.25,
                                  initial_step=0.05, variable_steps=True, explain_variable_steps=False)

Some steps were backtracked and re-done, to prevent negative concentrations or excessively large concentration changes
9 total step(s) taken

In [6]:

#dynamics.explain_time_advance()

In [7]:

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

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

Now suddently add a lot of enzyme¶

In [8]:

dynamics.set_single_conc(30., species_name="E", snapshot=True)    # Plenty of enzyme `E`

In [9]:

dynamics.describe_state()

SYSTEM STATE at Time t = 0.275:
3 species:
  Species 0 (A). Conc: 15.232227244425584
  Species 1 (B). Conc: 4.767772755574415
  Species 2 (E). Conc: 30.0
Set of chemicals involved in reactions (not counting enzymes): {'B', 'A'}
Set of enzymes involved in reactions: {'E'}

Take the system to equilibrium¶

In [10]:

dynamics.single_compartment_react(reaction_duration=0.04, 
                                  initial_step=0.005, variable_steps=True, explain_variable_steps=False)

*** CAUTION: negative concentration in chemical `A` in step starting at t=0.275.  It will be AUTOMATICALLY CORRECTED with a reduction in time step size, as follows:
    INFO: the tentative time step (0.005) leads to a NEGATIVE concentration of `A` from reaction A + E <-> B + E (rxn # 1): 
      Baseline value: 15.232 ; delta conc: -21.418
      -> will backtrack, and re-do step with a SMALLER delta time, multiplied by 0.25 (set to 0.00125) [Step started at t=0.275, and will rewind there]
* INFO: the tentative time step (0.00125) 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 0.0005) [Step started at t=0.275, and will rewind there]
* 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=0.275, and will rewind there]
Some steps were backtracked and re-done, to prevent negative concentrations or excessively large concentration changes
30 total step(s) taken

Note how the (proposed) initial step - in spite of having been reduced from the earlier run - is now appearing large, given the much-faster reaction dynamics. However, the variable-step engine intercepts and automatically corrects the problem!¶

In [11]:

#dynamics.explain_time_advance()

In [12]:

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

Notice the dramatic acceleration of the reaction as soon as the enzyme `E` is added at t = 0.275!¶

The reactions simulator automatically switches to small time steps is in order to handle the rapid amount of change

In [13]:

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

0: A <-> B
Final concentrations:  [B] = 16.67 ; [A] = 3.333
1. Ratio of reactant/product concentrations, adjusted for reaction orders: 5.00003
    Formula used:  [B] / [A]
2. Ratio of forward/reverse reaction rates: 5.00000519021548
Discrepancy between the two values: 0.0005779 %
Reaction IS in equilibrium (within 1% tolerance)

1: A + E <-> B + E
Final concentrations:  [B] = 16.67 ; [E] = 30 ; [A] = 3.333 ; [E] = 30
1. Ratio of reactant/product concentrations, adjusted for reaction orders: 5.00003
    Formula used:  ([B][E]) / ([A][E])
2. Ratio of forward/reverse reaction rates: 5.00000519021548
Discrepancy between the two values: 0.0005779 %
Reaction IS in equilibrium (within 1% tolerance)

Out[13]:

True

In [14]:

dynamics.get_history()

Out[14]:

	SYSTEM TIME	A	B	E	caption
0	0.000000	20.000000	0.000000	0.0	Initial state
1	0.050000	19.000000	1.000000	0.0
2	0.100000	18.060000	1.940000	0.0
3	0.125000	17.618200	2.381800	0.0
4	0.150000	17.189654	2.810346	0.0
5	0.175000	16.773964	3.226036	0.0
6	0.200000	16.370745	3.629255	0.0
7	0.225000	15.979623	4.020377	0.0
8	0.250000	15.600234	4.399766	0.0
9	0.275000	15.232227	4.767773	0.0
10	0.275000	15.232227	4.767773	30.0	Set concentration of `E`
11	0.275200	14.372651	5.627349	30.0
12	0.275400	13.575171	6.424829	30.0
13	0.275600	12.835300	7.164700	30.0
14	0.275800	12.148878	7.851122	30.0
15	0.276000	11.512043	8.487957	30.0
16	0.276200	10.921213	9.078787	30.0
17	0.276440	10.263435	9.736565	30.0
18	0.276728	9.542527	10.457473	30.0
19	0.277074	8.767428	11.232572	30.0
20	0.277419	8.089086	11.910914	30.0
21	0.277834	7.376689	12.623311	30.0
22	0.278249	6.771007	13.228993	30.0
23	0.278663	6.256054	13.743946	30.0
24	0.279161	5.730676	14.269324	30.0
25	0.279659	5.299738	14.700262	30.0
26	0.280256	4.875569	15.124431	30.0
27	0.280853	4.542897	15.457103	30.0
28	0.281570	4.229802	15.770198	30.0
29	0.282430	3.951341	16.048659	30.0
30	0.283462	3.720982	16.279018	30.0
31	0.284700	3.547589	16.452411	30.0
32	0.286186	3.432586	16.567414	30.0
33	0.287969	3.368656	16.631344	30.0
34	0.290109	3.341352	16.658648	30.0
35	0.292677	3.333912	16.666088	30.0
36	0.295758	3.333265	16.666735	30.0
37	0.299456	3.333353	16.666647	30.0
38	0.303893	3.333317	16.666683	30.0
39	0.309218	3.333343	16.666657	30.0
40	0.315608	3.333314	16.666686	30.0

In [ ]:

Coupled pair of reactions: A <-> B , and A + E <-> B + E¶

Enzyme E initially zero, and then suddenly added mid-reaction¶