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

A direct reaction and the same reaction, catalyzed

NOTE: we're exploring a very HYPOTHETICAL scenario where the latter reaction follows the kinetics of a 2nd-order elementary reaction!¶

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

LAST REVISED: June 23, 2024 (using v. 1.0 beta36)

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 life123 import ChemData
from life123 import UniformCompartment

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): {'A', 'B'}
Set of enzymes involved in the above reactions: {'E'}

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

In [4]:

dynamics = UniformCompartment(chem_data=chem_data, preset="mid")
dynamics.set_conc(conc={"A": 20.},
                  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): {'A', 'B'}
Set of enzymes involved in reactions: {'E'}

Advance the reactions (for now without enzyme), but stopping well before equilibrium¶

In [5]:

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

# Perform the reactions
dynamics.single_compartment_react(duration=0.25,
                                  initial_step=0.05, variable_steps=True)

Some steps were backtracked and re-done, to prevent negative concentrations or excessively large concentration changes
7 total step(s) taken
Number of step re-do's because of negative concentrations: 0
Number of step re-do's because of elective soft aborts: 0
Norm usage: {'norm_A': 1, 'norm_B': 2, 'norm_C': 1, 'norm_D': 1}

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"¶

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 = 0.26:
3 species:
  Species 0 (A). Conc: 15.43679907511829
  Species 1 (B). Conc: 4.56320092488171
  Species 2 (E). Conc: 30.0
Set of chemicals involved in reactions (not counting enzymes): {'A', 'B'}
Set of enzymes involved in reactions: {'E'}

Take the system to equilibrium¶

In [10]:

dynamics.single_compartment_react(duration=0.04,
                                  initial_step=0.005, variable_steps=True)

Some steps were backtracked and re-done, to prevent negative concentrations or excessively large concentration changes
30 total step(s) taken
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': 22, 'norm_B': 20, 'norm_C': 20, 'norm_D': 20}

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_history(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: [A] = 3.333 ; [B] = 16.67
1. Ratio of reactant/product concentrations, adjusted for reaction orders: 5.00003
    Formula used:  [B] / [A]
2. Ratio of forward/reverse reaction rates: 5.00001
Discrepancy between the two values: 0.0005878 %
Reaction IS in equilibrium (within 1% tolerance)

1: A + E <-> B + E
Final concentrations: [A] = 3.333 ; [E] = 30 ; [B] = 16.67
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: 5.00001
Discrepancy between the two values: 0.0005878 %
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	Initialized state
1	0.050000	19.000000	1.000000	0.0
2	0.110000	17.872000	2.128000	0.0
3	0.140000	17.348608	2.651392	0.0
4	0.170000	16.844058	3.155942	0.0
5	0.200000	16.357672	3.642328	0.0
6	0.230000	15.888796	4.111204	0.0
7	0.260000	15.436799	4.563201	0.0
8	0.260000	15.436799	4.563201	30.0	Set concentration of `E`
9	0.260200	14.562445	5.437555	30.0
10	0.260400	13.751254	6.248746	30.0
11	0.260600	12.998663	7.001337	30.0
12	0.260800	12.300439	7.699561	30.0
13	0.261000	11.652656	8.347344	30.0
14	0.261200	11.051668	8.948332	30.0
15	0.261440	10.382581	9.617419	30.0
16	0.261728	9.649278	10.350722	30.0
17	0.262074	8.860854	11.139146	30.0
18	0.262419	8.170849	11.829151	30.0
19	0.262834	7.446204	12.553796	30.0
20	0.263249	6.830109	13.169891	30.0
21	0.263663	6.306303	13.693697	30.0
22	0.264161	5.771892	14.228108	30.0
23	0.264659	5.333546	14.666454	30.0
24	0.265256	4.902084	15.097916	30.0
25	0.265853	4.563692	15.436308	30.0
26	0.266570	4.245215	15.754785	30.0
27	0.267430	3.961966	16.038034	30.0
28	0.268462	3.727647	16.272353	30.0
29	0.269700	3.551273	16.448727	30.0
30	0.271186	3.434292	16.565708	30.0
31	0.272969	3.369263	16.630737	30.0
32	0.275109	3.341490	16.658510	30.0
33	0.277677	3.333922	16.666078	30.0
34	0.280758	3.333264	16.666736	30.0
35	0.284456	3.333353	16.666647	30.0
36	0.288893	3.333317	16.666683	30.0
37	0.294218	3.333343	16.666657	30.0
38	0.300608	3.333314	16.666686	30.0

In [ ]:

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