A strong, localized transient concentration gets mopped up by a fast chemical reaction before it has a chance to diffuse far¶

A sizable transient injection of a chemical A near one end of the system, normally diffuses to the opposite end...
But if it reacts on a faster timescale, virtually none of it will make it to the far end.
It's a racing condition between the rate of diffusion of A, and the rate at which it's consumed by the plentiful B in the reaction A + B <-> C
When the reaction is fast, relative to A's diffusion, then only C (rather than A) will make it in substantial amounts to the far end of the system.

PART 1 explores the diffusion without the concomitant reaction.
PART 2 rewinds to the original initial states and carries out the reaction-diffusion.

Recommended background: experiment 1D/diffusion/localized_transient

TAGS : "reactions 1D", "diffusion 1D"¶

In [1]:

LAST_REVISED = "Aug. 28, 2025"
LIFE123_VERSION = "1.0.0rc6"       # 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 life123 import BioSim1D, ChemData, ReactionRegistry, PlotlyHelper, check_version

In [4]:

check_version(LIFE123_VERSION)

OK

In [ ]:

Initialize the Chemical Data and the Reactions; they will be used in both Part 1 and Part 2¶

In [5]:

# Initialize the chemical data
chem_data = ChemData(names=["A", "B", "C"], diffusion_rates=[100., 80., 120.],  
                     plot_colors=["red", "turquoise", "green"]) 

rxns = ReactionRegistry(chem_data=chem_data)

# Reaction A + B <-> C , with 1st-order kinetics for each species; note that it's mostly in the forward direction
rxns.add_reaction(reactants=["A", "B"], products="C", kF=0.1, kR=0.02)
rxns.describe_reactions()

Number of reactions: 1
0: A + B  <-> C  (Elementary Synthesis reaction)  (kF = 0.1 / kR = 0.02 / K = 5)
Chemicals involved in the above reactions: {"A" (red), "B" (turquoise), "C" (green)}

In [ ]:

PART 1 - JUST DIFFUSION, no reactions¶

In [6]:

bio1 = BioSim1D(n_bins=50, chem_data=chem_data, reactions=rxns)

In [7]:

# Set up the initial bell-shape concentration of `A`, with the very narrow peak close to one end of the system,
# centered at 1/10 of the width of the system, i.e. at bin 30
bio1.inject_bell_curve(chem_label="A", center=0.2, sd=0.06, max_amplitude=100., bias=0.)

In [8]:

# Chemical `B`, by contrast, is uniformly distributed
bio1.set_uniform_concentration(chem_label="B", conc=100.)

In [ ]:

In [9]:

df = bio1.describe_state()
df

SYSTEM STATE at Time t = 0:
50 bins and 3 chemical species

Out[9]:

	Species	Diff rate	Bin 0	Bin 1	Bin 2	Bin 3	Bin 4	Bin 5	Bin 6	Bin 7	...	Bin 40	Bin 41	Bin 42	Bin 43	Bin 44	Bin 45	Bin 46	Bin 47	Bin 48	Bin 49
0	A	100.0	0.386592	1.133778	2.961818	6.891982	14.285167	26.374356	43.374429	63.539099	...	1.223049e-21	3.506984e-23	8.957339e-25	2.037885e-26	4.129863e-28	7.454988e-30	1.198709e-31	1.716863e-33	2.190349e-35	2.489121e-37
1	B	80.0	100.000000	100.000000	100.000000	100.000000	100.000000	100.000000	100.000000	100.000000	...	1.000000e+02	1.000000e+02	1.000000e+02	1.000000e+02	1.000000e+02	1.000000e+02	1.000000e+02	1.000000e+02	1.000000e+02	1.000000e+02
2	C	120.0	0.000000	0.000000	0.000000	0.000000	0.000000	0.000000	0.000000	0.000000	...	0.000000e+00	0.000000e+00	0.000000e+00	0.000000e+00	0.000000e+00	0.000000e+00	0.000000e+00	0.000000e+00	0.000000e+00	0.000000e+00

3 rows × 52 columns

In [10]:

df[df.columns[3:22]]  # Zoom in where the action is

Out[10]:

	Bin 1	Bin 2	Bin 3	Bin 4	Bin 5	Bin 6	Bin 7	Bin 8	Bin 9	Bin 10	Bin 11	Bin 12	Bin 13	Bin 14	Bin 15	Bin 16	Bin 17	Bin 18	Bin 19
0	1.133778	2.961818	6.891982	14.285167	26.374356	43.374429	63.539099	82.909386	96.365531	99.768882	92.007635	75.580193	55.302848	36.044779	20.926308	10.821768	4.984932	2.045387	0.747562
1	100.000000	100.000000	100.000000	100.000000	100.000000	100.000000	100.000000	100.000000	100.000000	100.000000	100.000000	100.000000	100.000000	100.000000	100.000000	100.000000	100.000000	100.000000	100.000000
2	0.000000	0.000000	0.000000	0.000000	0.000000	0.000000	0.000000	0.000000	0.000000	0.000000	0.000000	0.000000	0.000000	0.000000	0.000000	0.000000	0.000000	0.000000	0.000000

In [11]:

tot_A = bio1.total_chem_mass(chem_label="A")    # We'll use this value for a mass-conservation check, later on
tot_A

Out[11]:

736.7898530810851

In [ ]:

Request history-keeping for some bins¶

In [12]:

# Request to save the concentration history at the bins with the initial concentration injection, 
# and the bins at, or near, both ends of the system
bio1.enable_history(bins=[0, 10, 20, 49], frequency=15, take_snapshot=True)    

History enabled for bins [0, 10, 20, 49], for ALL chemicals

In [ ]:

In [13]:

# Visualize the system state so far
bio1.visualize_system(title_prefix="Initial strong, localized transient of chemical `A`")

In [14]:

# Show as heatmap
bio1.system_heatmaps(title_prefix="Initial strong, localized transient")

In [15]:

# The first round of diffusion, over a small time duration
bio1.diffuse(total_duration=0.05, time_step=0.002)
diff_only = bio1.visualize_system(title_prefix="Diffusion only", show=True)  # SAVE this plot

In [16]:

# Do several more rounds of diffusion, each over the same small time duration
for _ in range(3):
    bio1.diffuse(total_duration=0.05, time_step=0.002)
    bio1.visualize_system(title_prefix="Diffusion only.", show=True)

In [17]:

# Advance the diffusion
bio1.diffuse(total_duration=0.5, time_step=0.002)
bio1.visualize_system(title_prefix="Diffusion only.")

In [18]:

# Continue advancing the diffusion
bio1.diffuse(total_duration=0.7, time_step=0.002)
bio1.visualize_system(title_prefix="Diffusion only.")

In [19]:

# Final rounds of diffusion, sampled over larger time spans
for _ in range(2):
    bio1.diffuse(total_duration=1.8, time_step=0.002)
    bio1.visualize_system(title_prefix="Diffusion only.", show=True)

In [20]:

bio1.check_mass_conservation(expected=tot_A, chem_label="A")    # `A` diffused away, but is still present in the same amount

Out[20]:

True

In [21]:

bio1.plot_history_single_bin(title_prefix="Diffusion only. Bin where the transient originates.", 
                             bin_address=10)

In [22]:

bio1.plot_history_single_bin(title_prefix="Diffusion only.", bin_address=0)

In [23]:

bio1.plot_history_single_bin(title_prefix="Diffusion only.", bin_address=20)

In [24]:

bio1.plot_history_single_bin(title_prefix="Diffusion only.  Faraway bin.", 
                             bin_address=49)

In [ ]:

PART 2 - Diffusion AND Reaction¶

In [ ]:

In [25]:

# Set up just like before
bio2 = BioSim1D(n_bins=50, chem_data=chem_data, reactions=rxns)
bio2.inject_bell_curve(chem_label="A", center=0.2, sd=0.06, amplitude=15.25, bias=0.)
bio2.set_uniform_concentration(chem_label="B", conc=100.)

In [26]:

bio2.enable_history(bins=[0, 10, 20, 49], frequency=15, take_snapshot=True)

History enabled for bins [0, 10, 20, 49], for ALL chemicals

In [27]:

bio2.visualize_system(title_prefix="Initial strong, localized transient of chemical `A`")

In [28]:

# The first round of REACTION-diffusion, over a small time duration
bio2.react_diffuse(total_duration=0.05, time_step=0.002)
react_diff = bio2.visualize_system(title_prefix=["The localized transient `A` starts getting consumed by the reaction `A + B <-> C`, ",
                                                 "before it has a chance to diffuse away much.  Notice the production of `C`"])
react_diff

In [29]:

# SAME IN HEATMAP VIEW
bio2.system_heatmaps(title_prefix=["HEATMAP VIEW.  The initial localized concentration of `A` starts getting consumed by the reaction `A + B <-> C`, ",
                                    "before it has a chance to diffuse away much.  Notice the production of `C`"])

Contrast-and-compare with the earlier diffusion-only¶

In [30]:

PlotlyHelper.combine_plots([diff_only, react_diff], 
                           title="Diffusion-only (dotted) vs. React-diffuse (solid)<br>Reaction `A + B <-> C`<br>Early system snapshot", 
                           layout_index=0, modify={0: "dot"})

In [ ]:

Let's continue the reaction-diffusion¶

In [31]:

# Do several more rounds of reaction-diffusion, each over the same small time duration
for _ in range(3):
    bio2.react_diffuse(total_duration=0.05, time_step=0.002)
    bio2.visualize_system(title_prefix="Reaction-Diffusion.", show=True)

In [32]:

# Advance the reaction-diffusion
bio2.react_diffuse(total_duration=0.5, time_step=0.002)
bio2.visualize_system(title_prefix="Reaction-Diffusion.")

Notice how `A` has been largely consumed by now... and what is diffusing away is `C` (as well as the dip in `B`) rather than `A`¶

In [33]:

# Continue advancing the reaction-diffusion
bio2.react_diffuse(total_duration=0.7, time_step=0.002)
bio2.visualize_system(title_prefix="Reaction-Diffusion.")

In [34]:

# Final rounds of reaction-diffusion, sampled over larger time spans
for _ in range(2):
    bio2.react_diffuse(total_duration=1.8, time_step=0.002)
    bio2.visualize_system(title_prefix="Reaction-Diffusion.", show=True)

In [ ]:

In [35]:

bio2.plot_history_single_bin(title_prefix="Reaction-Diffusion. Bin where the transient originates.", 
                             bin_address=10)

In [36]:

bio2.plot_history_single_bin(title_prefix="Reaction-Diffusion.", bin_address=0)

In [37]:

bio2.plot_history_single_bin(title_prefix="Reaction-Diffusion.", bin_address=20)

In [38]:

bio2.plot_history_single_bin(title_prefix="Reaction-Diffusion.  Faraway bin.", 
                             bin_address=49)

Virtually no `A` ever reaches the faraway bins!¶

In [ ]:

A strong, localized transient concentration gets mopped up by a fast chemical reaction before it has a chance to diffuse far¶

TAGS : "reactions 1D", "diffusion 1D"¶

Initialize the Chemical Data and the Reactions; they will be used in both Part 1 and Part 2¶

PART 1 - JUST DIFFUSION, no reactions¶

Request history-keeping for some bins¶

PART 2 - Diffusion AND Reaction¶

Contrast-and-compare with the earlier diffusion-only¶

Let's continue the reaction-diffusion¶

Notice how A has been largely consumed by now... and what is diffusing away is C (as well as the dip in B) rather than A¶

Virtually no A ever reaches the faraway bins!¶

Notice how `A` has been largely consumed by now... and what is diffusing away is `C` (as well as the dip in `B`) rather than `A`¶

Virtually no `A` ever reaches the faraway bins!¶