One-bin `A <-> 2C + D`, with 1st-order kinetics for each species, taken to equilibrium¶

Diffusion not applicable (just 1 bin)

LAST REVISED: July 14, 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 experiments.get_notebook_info import get_notebook_basename

from src.modules.chemicals.chem_data import ChemData as chem
from src.life_1D.bio_sim_1d import BioSim1D

import plotly.express as px
from src.modules.visualization.graphic_log import GraphicLog

In [3]:

# Initialize the HTML logging
log_file = get_notebook_basename() + ".log.htm"    # Use the notebook base filename for the log file

# Set up the use of some specified graphic (Vue) components
GraphicLog.config(filename=log_file,
                  components=["vue_cytoscape_1"],
                  extra_js="https://cdnjs.cloudflare.com/ajax/libs/cytoscape/3.21.2/cytoscape.umd.js")

-> Output will be LOGGED into the file 'reaction_5.log.htm'

In [4]:

# Initialize the system
chem_data = chem(names=["A", "C", "D"])     # NOTE: Diffusion not applicable (just 1 bin)

# Reaction A <-> 2C + D , with 1st-order kinetics for each species
chem_data.add_reaction(reactants=[("A")], products=[(2, "C") , ("D")],
                 forward_rate=5., reverse_rate=2.)

bio = BioSim1D(n_bins=1, chem_data=chem_data)

bio.set_all_uniform_concentrations( [4., 7., 2.] )

bio.describe_state()

SYSTEM STATE at Time t = 0:
1 bins and 3 species:
  Species 0 (A). Diff rate: None. Conc: [4.]
  Species 1 (C). Diff rate: None. Conc: [7.]
  Species 2 (D). Diff rate: None. Conc: [2.]

In [5]:

# Save the state of the concentrations of all species at bin 0
bio.add_snapshot(bio.bin_snapshot(bin_address = 0), caption="Initial state")
bio.get_history()

Out[5]:

	SYSTEM TIME	A	C	D	caption
0	0	4.0	7.0	2.0	Initial state

In [6]:

chem_data.describe_reactions()

Number of reactions: 1 (at temp. 25 C)
0: A <-> 2 C + D  (kF = 5 / kR = 2 / Delta_G = -2,271.45 / K = 2.5) | 1st order in all reactants & products

In [7]:

# Send the plot to the HTML log file
graph_data = chem_data.prepare_graph_network()
GraphicLog.export_plot(graph_data, "vue_cytoscape_1")

[GRAPHIC ELEMENT SENT TO LOG FILE `reaction_5.log.htm`]

First step ¶

In [8]:

# First step
bio.react(time_step=0.2, n_steps=1)
bio.describe_state()

SYSTEM STATE at Time t = 0.2:
1 bins and 3 species:
  Species 0 (A). Diff rate: None. Conc: [5.6]
  Species 1 (C). Diff rate: None. Conc: [3.8]
  Species 2 (D). Diff rate: None. Conc: [0.4]

Note:  the above values are quite inaccurate because of the large time step 0.2

For example, the value for the concentration of D (0.4) is a wild overshot from the initial 2.0 to the equilibrium value of 1.68941267

A more precise calculation with bio.react(time_step=0.1, n_steps=2) gives conc_D(0.2) = 2.304

An even more precise calculation with bio.react(time_step=0.05, n_steps=4) gives conc_D(0.2) = 1.69037202

I.e. the system is almost at equilibrium already at t=0.2 !

TODO: explore the early dynamics of the system in a separate experiment

In [9]:

# Save the state of the concentrations of all species at bin 0
bio.add_snapshot(bio.bin_snapshot(bin_address = 0))
bio.get_history()

Out[9]:

	SYSTEM TIME	A	C	D	caption
0	0.0	4.0	7.0	2.0	Initial state
1	0.2	5.6	3.8	0.4

Numerous more steps ¶

In [10]:

# Numerous more steps
bio.react(time_step=0.05, n_steps=30, snapshots={"sample_bin": 0})

bio.describe_state()

SYSTEM STATE at Time t = 1.7:
1 bins and 3 species:
  Species 0 (A). Diff rate: None. Conc: [4.31058733]
  Species 1 (C). Diff rate: None. Conc: [6.37882534]
  Species 2 (D). Diff rate: None. Conc: [1.68941267]

Equilibrium ¶

Consistent with the 5/2 ratio of forward/reverse rates (and the 1st order reactions), the systems settles in the following equilibrium:
[A] = 4.31058733 , [C] = 6.37882534 , [D] = 1.68941267

In [11]:

# Verify that the reaction has reached equilibrium
bio.reaction_dynamics.is_in_equilibrium(rxn_index=0, conc=bio.bin_snapshot(bin_address = 0))

A <-> 2 C + D
Final concentrations:  [C] = 6.379 ; [D] = 1.689 ; [A] = 4.311
1. Ratio of reactant/product concentrations, adjusted for reaction orders: 2.5
    Formula used:  ([C][D]) / [A]
2. Ratio of forward/reverse reaction rates: 2.5
Discrepancy between the two values: 0 %
Reaction IS in equilibrium (within 1% tolerance)

Out[11]:

True

In [12]:

bio.get_history()

Out[12]:

	SYSTEM TIME	A	C	D	caption
0	0.00	4.000000	7.000000	2.000000	Initial state
1	0.20	5.600000	3.800000	0.400000
2	0.25	4.352000	6.296000	1.648000
3	0.30	4.301581	6.396838	1.698419
4	0.35	4.312637	6.374726	1.687363
5	0.40	4.310125	6.379749	1.689875
6	0.45	4.310692	6.378617	1.689308
7	0.50	4.310564	6.378872	1.689436
8	0.55	4.310593	6.378815	1.689407
9	0.60	4.310586	6.378828	1.689414
10	0.65	4.310588	6.378825	1.689412
11	0.70	4.310587	6.378825	1.689413
12	0.75	4.310587	6.378825	1.689413
13	0.80	4.310587	6.378825	1.689413
14	0.85	4.310587	6.378825	1.689413
15	0.90	4.310587	6.378825	1.689413
16	0.95	4.310587	6.378825	1.689413
17	1.00	4.310587	6.378825	1.689413
18	1.05	4.310587	6.378825	1.689413
19	1.10	4.310587	6.378825	1.689413
20	1.15	4.310587	6.378825	1.689413
21	1.20	4.310587	6.378825	1.689413
22	1.25	4.310587	6.378825	1.689413
23	1.30	4.310587	6.378825	1.689413
24	1.35	4.310587	6.378825	1.689413
25	1.40	4.310587	6.378825	1.689413
26	1.45	4.310587	6.378825	1.689413
27	1.50	4.310587	6.378825	1.689413
28	1.55	4.310587	6.378825	1.689413
29	1.60	4.310587	6.378825	1.689413
30	1.65	4.310587	6.378825	1.689413
31	1.70	4.310587	6.378825	1.689413

C and D get depleted, while A gets produced. A wild overshoot is present at t=0.2

Plots of changes of concentration with time¶

In [13]:

fig = px.line(data_frame=bio.get_history(), x="SYSTEM TIME", y=["A", "C", "D"], 
              title="Reaction A <-> 2C + D .  Changes in concentrations",
              color_discrete_sequence = ['navy', 'violet', 'red'],
              labels={"value":"concentration", "variable":"Chemical"})
fig.show()

Notice the wild overshoot present at t=0.2 ! (Too large a time step, early in the reaction!)¶

Variable, adaptive time steps are explored at length in the "reactions_single_compartment" experiments

In [ ]:

One-bin A <-> 2C + D, with 1st-order kinetics for each species, taken to equilibrium¶

First step¶

Numerous more steps¶

Equilibrium¶