`A <-> 3B` reaction, taken to equilibrium.¶

Examine State Space trajectory, using [A] and [B] as state variables¶

1st-order kinetics in both directions

See also the experiment "1D/reaction/reaction_2"

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 experiments.get_notebook_info import get_notebook_basename

from life123 import ChemData
from life123 import UniformCompartment

import plotly.express as px
import plotly.graph_objects as go
from life123 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_2"],
                  extra_js="https://cdnjs.cloudflare.com/ajax/libs/cytoscape/3.21.2/cytoscape.umd.js")

-> Output will be LOGGED into the file 'state_space_1.log.htm'

In [4]:

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


# Reaction A <-> 3B , with 1st-order kinetics in both directions
chem_data.add_reaction(reactants="A", products=[(3,"B",1)], forward_rate=5., reverse_rate=2.)

chem_data.describe_reactions()

Number of reactions: 1 (at temp. 25 C)
0: A <-> 3 B  (kF = 5 / kR = 2 / delta_G = -2,271.4 / K = 2.5) | 1st order in all reactants & products
Set of chemicals involved in the above reactions: {'A', 'B'}

In [5]:

dynamics = UniformCompartment(chem_data=chem_data)
dynamics.set_conc(conc={"A": 10., "B": 50.},
                  snapshot=True)
dynamics.describe_state()

SYSTEM STATE at Time t = 0:
2 species:
  Species 0 (A). Conc: 10.0
  Species 1 (B). Conc: 50.0
Set of chemicals involved in reactions: {'A', 'B'}

In [6]:

# Send the plot to the HTML log file
chem_data.plot_reaction_network("vue_cytoscape_2")

[GRAPHIC ELEMENT SENT TO LOG FILE `state_space_1.log.htm`]

To equilibrium¶

In [7]:

dynamics.single_compartment_react(initial_step=0.05, n_steps=10, variable_steps=False)

10 total step(s) taken

In [8]:

df = dynamics.get_history()
df

Out[8]:

	SYSTEM TIME	A	B	caption
0	0.00	10.000000	50.000000	Initialized state
1	0.05	12.500000	42.500000
2	0.10	13.625000	39.125000
3	0.15	14.131250	37.606250
4	0.20	14.359063	36.922812
5	0.25	14.461578	36.615266
6	0.30	14.507710	36.476870
7	0.35	14.528470	36.414591
8	0.40	14.537811	36.386566
9	0.45	14.542015	36.373955
10	0.50	14.543907	36.368280

In [9]:

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

0: A <-> 3 B
Final concentrations: [A] = 14.54 ; [B] = 36.37
1. Ratio of reactant/product concentrations, adjusted for reaction orders: 2.50059
    Formula used:  [B] / [A]
2. Ratio of forward/reverse reaction rates: 2.5
Discrepancy between the two values: 0.02341 %
Reaction IS in equilibrium (within 1% tolerance)

Out[9]:

True

In [10]:

dynamics.plot_history(colors=['navy', 'orange'])

In [ ]:

Same data, but shown differently¶

In [11]:

fig0 = px.line(data_frame=dynamics.get_history(), x="A", y="B", 
              title="State space of reaction A <-> 3B : [A] vs. [B]",
              color_discrete_sequence = ['#C83778'],
              labels={"value":"concentration", "variable":"Chemical"})
fig0.show()

In [12]:

# Now show the individual data points

df['SYSTEM TIME'] = round(df['SYSTEM TIME'], 2)    # To avoid clutter from too many digits, in the column

fig1 = px.scatter(data_frame=df, x="A", y="B",
                  title="Trajectory in State space: [A] vs. [B]",
                  hover_data=['SYSTEM TIME'])

fig1.update_traces(marker={"size": 6, "color": "#2FAC74"})    # Modify the style of the dots

# Add annotations (showing the System Time value) to SOME of the points, to avoid clutter
for ind in df.index:     # for each row in the Pandas dataframe
    label = df["SYSTEM TIME"][ind]
    if ind == 0:
        label = f"t={label}"
        
    label_x = ind*16
    label_y = 20 + ind*8    # A greater y value here means further DOWN!!
        
    if (ind <= 3) or (ind%2 == 0):
        fig1.add_annotation(x=df["A"][ind], y=df["B"][ind],
                            text=label,
                            font=dict(
                                size=10,
                                color="grey"
                            ),
                            showarrow=True, arrowhead=0, ax=label_x, ay=label_y, arrowcolor="#b0b0b0",
                            bordercolor="#c7c7c7")

fig1.show()

In [13]:

# Combine the two above plots, while using the layout of fig1 (which includes the title and annotations)
all_fig = go.Figure(data=fig0.data + fig1.data, layout = fig1.layout)
all_fig.show()

Note how the trajectory is progressively slowing down towards the dynamical system's "attractor" (equilibrium state of the reaction)¶