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

A hypothetical scenario with 1st-order kinetics in both directions.¶

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

Based on experiment 1D/reaction/reaction_2

Diffusion not applicable (just 1 bin).

This is the 1D version of the single-compartment reaction by the same name.

TAGS : "reactions 1D"¶

In [1]:

LAST_REVISED = "May 5, 2025"
LIFE123_VERSION = "1.0.0rc3"        # Library version this experiment is based on

In [2]:

#import set_path                    # Using MyBinder?  Uncomment this before running the next cell!

In [3]:

#import sys
#sys.path.append("C:/some_path/my_env_or_install")   # CHANGE to the folder containing your venv or libraries installation!
# NOTE: If any of the imports below can't find a module, uncomment the lines above, or try:  import set_path   
 
from experiments.get_notebook_info import get_notebook_basename

from life123 import check_version, UniformCompartment,  BioSim1D, PlotlyHelper, GraphicLog

import plotly.express as px

In [4]:

# 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 [5]:

# Initialize the system.  NOTE: Diffusion not applicable (just 1 bin)
uc = UniformCompartment(names=["A", "B"])


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

uc.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 [6]:

bio = BioSim1D(n_bins=1, reaction_handler=uc)

bio.set_uniform_concentration(chem_label="A", conc=10.)
bio.set_uniform_concentration(chem_label="B", conc=50.)

bio.describe_state()

SYSTEM STATE at Time t = 0:
1 bins and 2 chemical species:

Out[6]:

	Species	Diff rate	Bin 0
0	A	None	10.0
1	B	None	50.0

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In [7]:

# Let's enable history - by default for all chemicals and all bins (we're only using one)
bio.enable_history(take_snapshot=True, caption="Initial setup")

History enabled for bins None and chemicals None (None means 'all')

In [8]:

bio.get_bin_history(bin_address=0)

Out[8]:

	SYSTEM TIME	A	B	caption
0	0.0	10.0	50.0	Initial setup

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In [9]:

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

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

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To equilibrium¶

In [10]:

# Using smaller steps that in experiment reaction_2, to avoid the initial overshooting
bio.react(time_step=0.05, n_steps=10)

System Time is now: 0.5

In [11]:

bio.describe_state()

SYSTEM STATE at Time t = 0.5:
1 bins and 2 chemical species:

Out[11]:

	Species	Diff rate	Bin 0
0	A	None	14.543907
1	B	None	36.368280

In [12]:

bio.get_bin_history(bin_address=0)

Out[12]:

	SYSTEM TIME	A	B	caption
0	0.00	10.000000	50.000000	Initial setup
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 [13]:

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

A <-> 3 B
Current 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[13]:

True

Plots of changes of concentration with time¶

In [14]:

bio.plot_history_single_bin(bin_address=0, 
                            title_prefix="Reaction A <-> 3B")

In [ ]:

Same data, but shown differently¶

In [15]:

df = bio.get_bin_history(bin_address=0)

In [16]:

fig0 = px.line(data_frame=df, 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 [17]:

# 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:
    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 [18]:

# Combine the two above plots, 
# while using the layout of fig1 (which includes the title and annotations)

PlotlyHelper.combine_plots(fig_list = [fig0, fig1], layout_index=1)

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

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