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

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

1. WITHOUT ENZYME¶

`A` <-> `B`¶

In [3]:

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

# 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 part 2, below
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 = -3,989.7 / K = 5) | 1st order in all reactants & products
Set of chemicals involved in the above reactions: {'A', 'B'}

Set the initial concentrations of all the chemicals¶

In [4]:

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

Take the initial system to equilibrium¶

In [5]:

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

dynamics.single_compartment_react(duration=3.0,
                                  initial_step=0.1, variable_steps=True)

16 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: 1
Norm usage: {'norm_A': 10, 'norm_B': 9, 'norm_C': 8, 'norm_D': 8}

In [6]:

#dynamics.explain_time_advance()

In [7]:

dynamics.plot_history(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 - including a complete step abort/redo at time=0¶

In [8]:

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

Out[8]:

(0.7243697199909254, 10.000000000000002)

In [9]:

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

0: A <-> B
Final concentrations: [A] = 3.337 ; [B] = 16.66
1. Ratio of reactant/product concentrations, adjusted for reaction orders: 4.99265
    Formula used:  [B] / [A]
2. Ratio of forward/reverse reaction rates: 5.00001
Discrepancy between the two values: 0.1471 %
Reaction IS in equilibrium (within 1% 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 (much slower) direct 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 = -3,989.7 / 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'}

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 now are each 10 times bigger than before¶

Set the initial concentrations of all the chemicals (to what they originally were)¶

In [11]:

dynamics = UniformCompartment(chem_data=chem_data, preset="mid")
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
Set of chemicals involved in reactions (not counting enzymes): {'A', 'B'}
Set of enzymes involved in reactions: {'E'}

Take the initial system to equilibrium¶

In [12]:

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

dynamics.single_compartment_react(duration=0.1,
                                  initial_step=0.1, variable_steps=True)

Some steps were backtracked and re-done, to prevent negative concentrations or excessively large concentration changes
43 total step(s) taken
Number of step re-do's because of negative concentrations: 3
Number of step re-do's because of elective soft aborts: 2
Norm usage: {'norm_A': 31, 'norm_B': 28, 'norm_C': 28, 'norm_D': 28}

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

In [15]:

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

Out[15]:

(0.0024674107137523833, 10.000000000000002)

In [16]:

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

0: A + E <-> B + E
Final concentrations: [A] = 3.334 ; [E] = 30 ; [B] = 16.67
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: 5.00001
Discrepancy between the two values: 0.008546 %
Reaction IS in equilibrium (within 1% tolerance)

Out[16]:

True

Thanks to the (abundant) enzyme, the reaction reaches equilibrium roughly 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	Initialized state
1	0.000250	18.500000	1.500000	30.0
2	0.000375	17.817500	2.182500	30.0
3	0.000500	17.165712	2.834288	30.0
4	0.000625	16.543255	3.456745	30.0
5	0.000750	15.948809	4.051191	30.0
6	0.000875	15.381112	4.618888	30.0
7	0.001000	14.838962	5.161038	30.0
8	0.001125	14.321209	5.678791	30.0
9	0.001250	13.826755	6.173245	30.0
10	0.001375	13.354551	6.645449	30.0
11	0.001525	12.813405	7.186595	30.0
12	0.001675	12.301481	7.698519	30.0
13	0.001855	11.720345	8.279655	30.0
14	0.002071	11.068171	8.931829	30.0
15	0.002330	10.346417	9.653583	30.0
16	0.002589	9.692012	10.307988	30.0
17	0.002900	8.980003	11.019997	30.0
18	0.003274	8.221264	11.778736	30.0
19	0.003647	7.564476	12.435524	30.0
20	0.004095	6.882233	13.117767	30.0
21	0.004543	6.309997	13.690003	30.0
22	0.004991	5.830030	14.169970	30.0
23	0.005528	5.346939	14.653061	30.0
24	0.006066	4.957322	15.042678	30.0
25	0.006711	4.580248	15.419752	30.0
26	0.007485	4.232821	15.767179	30.0
27	0.008413	3.932073	16.067927	30.0
28	0.009528	3.691843	16.308157	30.0
29	0.010865	3.519230	16.480770	30.0
30	0.012470	3.411824	16.588176	30.0
31	0.014396	3.357403	16.642597	30.0
32	0.016707	3.337375	16.662625	30.0
33	0.019480	3.333337	16.666663	30.0
34	0.022808	3.333329	16.666671	30.0
35	0.026802	3.333331	16.666669	30.0
36	0.031594	3.333330	16.666670	30.0
37	0.037345	3.333331	16.666669	30.0
38	0.044245	3.333330	16.666670	30.0
39	0.052526	3.333332	16.666668	30.0
40	0.062463	3.333327	16.666673	30.0
41	0.074388	3.333341	16.666659	30.0
42	0.088697	3.333285	16.666715	30.0
43	0.105869	3.333568	16.666432	30.0

In [ ]:

Comparing the dynamics of 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 - including 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 (much slower) direct 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 now are each 10 times bigger than before¶

Set the initial concentrations of all the chemicals (to what they originally were)¶

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 roughly around t=0.02, far sooner than the roughly t=3.5 without enzyme¶

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

`A` <-> `B`¶

2. WITH ENZYME `E`¶

`A` + `E` <-> `B` + `E`¶