Reaction-Diffusion in 2-D : `A + B <-> C`, taken to equilibrium¶

Mostly forward reaction, with 1st-order kinetics for each species¶

Initial concentrations of A and B are spatially separated to the opposite ends of the system; as a result, no C is being generated.

But, as soon as A and B, from their respective distant originating points, diffuse into the middle - and into each other - the reaction starts, consuming most of A and B, until an equilibrium is reached in both diffusion and reactions.

Note: This is a 2D version of the 1D experiment by the same name.

TAGS : "reactions 2D", "diffusion 2D", "quick-start"¶

In [1]:

LAST_REVISED = "Jan. 20, 2025"
LIFE123_VERSION = "1.0.0rc2"        # 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 check_version, BioSim2D, ChemData, UniformCompartment

In [4]:

check_version(LIFE123_VERSION)

OK

In [ ]:

In [5]:

# Initialize the system
chem_data = ChemData(names=["A", "B", "C"], diffusion_rates=[50., 50., 1.],   # `A` and `B` diffuse fast; `C` diffuses slowly
                     plot_colors=["red", "blue", "purple"])                   # Color choice is a reminder that red + blue = purple

In [6]:

bio = BioSim2D(x_bins=7, y_bins=7, chem_data=chem_data)

In [7]:

reactions = bio.get_reactions()

# Reaction A + B <-> C , with 1st-order kinetics for each species; note that it's mostly in the forward direction
reactions.add_reaction(reactants=["A", "B"], products="C", forward_rate=20., reverse_rate=2.)
reactions.describe_reactions()

Number of reactions: 1 (at temp. 25 C)
0: A + B <-> C  (kF = 20 / kR = 2 / delta_G = -5,708 / K = 10) | 1st order in all reactants & products
Set of chemicals involved in the above reactions: {"B" (blue), "C" (purple), "A" (red)}

In [ ]:

TIME 0 : Inject initial concentrations of `A` and `B` at opposite ends of the system¶

In [8]:

bio.set_bin_conc(bin_address = (0,0), chem_label="A", conc=20.)
bio.set_bin_conc(bin_address = (6,6), chem_label="B", conc=20.)

bio.describe_state()   # A minimalist view of all the chemical concentrations

SYSTEM STATE at Time t = 0:
Species `A`:
      0    1    2    3    4    5    6
6   0.0  0.0  0.0  0.0  0.0  0.0  0.0
5   0.0  0.0  0.0  0.0  0.0  0.0  0.0
4   0.0  0.0  0.0  0.0  0.0  0.0  0.0
3   0.0  0.0  0.0  0.0  0.0  0.0  0.0
2   0.0  0.0  0.0  0.0  0.0  0.0  0.0
1   0.0  0.0  0.0  0.0  0.0  0.0  0.0
0  20.0  0.0  0.0  0.0  0.0  0.0  0.0
Species `B`:
     0    1    2    3    4    5     6
6  0.0  0.0  0.0  0.0  0.0  0.0  20.0
5  0.0  0.0  0.0  0.0  0.0  0.0   0.0
4  0.0  0.0  0.0  0.0  0.0  0.0   0.0
3  0.0  0.0  0.0  0.0  0.0  0.0   0.0
2  0.0  0.0  0.0  0.0  0.0  0.0   0.0
1  0.0  0.0  0.0  0.0  0.0  0.0   0.0
0  0.0  0.0  0.0  0.0  0.0  0.0   0.0
Species `C`:
     0    1    2    3    4    5    6
6  0.0  0.0  0.0  0.0  0.0  0.0  0.0
5  0.0  0.0  0.0  0.0  0.0  0.0  0.0
4  0.0  0.0  0.0  0.0  0.0  0.0  0.0
3  0.0  0.0  0.0  0.0  0.0  0.0  0.0
2  0.0  0.0  0.0  0.0  0.0  0.0  0.0
1  0.0  0.0  0.0  0.0  0.0  0.0  0.0
0  0.0  0.0  0.0  0.0  0.0  0.0  0.0

In [9]:

bio.system_snapshot(chem_label="A")

Out[9]:

	0	1	2	3	4	5	6
6	0.0	0.0	0.0	0.0	0.0	0.0	0.0
5	0.0	0.0	0.0	0.0	0.0	0.0	0.0
4	0.0	0.0	0.0	0.0	0.0	0.0	0.0
3	0.0	0.0	0.0	0.0	0.0	0.0	0.0
2	0.0	0.0	0.0	0.0	0.0	0.0	0.0
1	0.0	0.0	0.0	0.0	0.0	0.0	0.0
0	20.0	0.0	0.0	0.0	0.0	0.0	0.0

In [10]:

bio.system_snapshot(chem_label="B")

Out[10]:

	0	1	2	3	4	5	6
6	0.0	0.0	0.0	0.0	0.0	0.0	20.0
5	0.0	0.0	0.0	0.0	0.0	0.0	0.0
4	0.0	0.0	0.0	0.0	0.0	0.0	0.0
3	0.0	0.0	0.0	0.0	0.0	0.0	0.0
2	0.0	0.0	0.0	0.0	0.0	0.0	0.0
1	0.0	0.0	0.0	0.0	0.0	0.0	0.0
0	0.0	0.0	0.0	0.0	0.0	0.0	0.0

In [11]:

bio.system_heatmaps()

In [ ]:

Enable History¶

In [12]:

# Let's take a peek at the current concentrations of all chemicals in the bins with the initial concentration injections, as well as at the bin in the very center
bio.selected_concentrations(bins=[(0,0), (6,6), (3,3)])

Out[12]:

{(0, 0): {'A': 20.0, 'B': 0.0, 'C': 0.0},
 (6, 6): {'A': 0.0, 'B': 20.0, 'C': 0.0},
 (3, 3): {'A': 0.0, 'B': 0.0, 'C': 0.0}}

In [13]:

# Let's enable history for those same 3 bins
bio.enable_history(bins=[(0,0), (6,6), (3,3)], frequency=2, take_snapshot=True)     # Taking a snapshot to include the current initial state in the history

History enabled for bins [(0, 0), (6, 6), (3, 3)] and chemicals None (None means 'all')

In [ ]:

Part 1 : advance to time t=0.002 (with smaller fixed steps)¶

In [ ]:

In [14]:

bio.react_diffuse(total_duration=0.002, n_steps=10)
bio.describe_state()

SYSTEM STATE at Time t = 0.002:
Species `A`:
              0             1             2             3             4             5             6
6  3.765173e-09  1.523634e-10  2.352681e-12  1.621799e-14  4.200000e-17  0.000000e+00  0.000000e+00
5  4.371320e-07  2.201919e-08  4.530362e-10  4.685646e-12  2.428436e-14  5.040000e-17  0.000000e+00
4  3.549559e-05  2.133275e-06  5.474009e-08  7.538720e-10  5.855018e-12  2.428436e-14  4.200000e-17
3  1.980644e-03  1.380506e-04  4.240362e-06  7.288416e-08  7.538720e-10  4.685646e-12  1.621799e-14
2  7.282844e-02  5.764607e-03  2.059840e-04  4.240362e-06  5.474009e-08  4.530362e-10  2.352681e-12
1  1.602863e+00  1.416999e-01  5.764607e-03  1.380506e-04  2.133275e-06  2.201919e-08  1.523634e-10
0  1.649086e+01  1.602863e+00  7.282844e-02  1.980644e-03  3.549559e-05  4.371320e-07  3.765173e-09
Species `B`:
              0             1             2             3             4             5             6
6  3.765173e-09  4.371320e-07  3.549559e-05  1.980644e-03  7.282844e-02  1.602863e+00  1.649086e+01
5  1.523634e-10  2.201919e-08  2.133275e-06  1.380506e-04  5.764607e-03  1.416999e-01  1.602863e+00
4  2.352681e-12  4.530362e-10  5.474009e-08  4.240362e-06  2.059840e-04  5.764607e-03  7.282844e-02
3  1.621799e-14  4.685646e-12  7.538720e-10  7.288416e-08  4.240362e-06  1.380506e-04  1.980644e-03
2  4.200000e-17  2.428436e-14  5.855018e-12  7.538720e-10  5.474009e-08  2.133275e-06  3.549559e-05
1  0.000000e+00  5.040000e-17  2.428436e-14  4.685646e-12  4.530362e-10  2.201919e-08  4.371320e-07
0  0.000000e+00  0.000000e+00  4.200000e-17  1.621799e-14  2.352681e-12  1.523634e-10  3.765173e-09
Species `C`:
              0             1             2             3             4             5             6
6  1.077954e-20  4.714463e-20  4.546293e-20  9.576810e-21  0.000000e+00  0.000000e+00  0.000000e+00
5  4.714463e-20  3.739578e-19  6.899800e-19  3.546464e-19  4.198306e-20  0.000000e+00  0.000000e+00
4  4.546293e-20  6.899800e-19  2.318474e-18  2.286661e-18  6.619601e-19  4.198306e-20  0.000000e+00
3  9.576810e-21  3.546464e-19  2.286661e-18  4.113414e-18  2.286661e-18  3.546464e-19  9.576810e-21
2  0.000000e+00  4.198306e-20  6.619601e-19  2.286661e-18  2.318474e-18  6.899800e-19  4.546293e-20
1  0.000000e+00  0.000000e+00  4.198306e-20  3.546464e-19  6.899800e-19  3.739578e-19  4.714463e-20
0  0.000000e+00  0.000000e+00  0.000000e+00  9.576810e-21  4.546293e-20  4.714463e-20  1.077954e-20

In [15]:

bio.system_heatmaps()

In [16]:

# Let's take a peek at the history saved so far, for the bins we requested history-keeping; we'll plot it at the end
bio.conc_history.bin_history(bin_address = (0,0))

Out[16]:

	SYSTEM TIME	A
0	0.0000	20.000000
1	0.0004	19.212000
2	0.0008	18.470258
3	0.0012	17.771471
4	0.0016	17.112602
5	0.0020	16.490859

In [17]:

bio.conc_history.bin_history(bin_address = (6,6))  # Notice the symmetry between `B` at bin (6,6) and `A` at bin (0,0)

Out[17]:

	SYSTEM TIME	B
0	0.0000	20.000000
1	0.0004	19.212000
2	0.0008	18.470258
3	0.0012	17.771471
4	0.0016	17.112602
5	0.0020	16.490859

In [18]:

# And in the central bin
bio.conc_history.bin_history(bin_address = (3,3))   # Notice how `A` and `B` are just beginning to diffuse into this bin, and the reaction `A + B <-> C` is just beginning to produce `C`

Out[18]:

	SYSTEM TIME	A	B	C
0	0.0000	0.000000e+00	0.000000e+00	0.000000e+00
1	0.0004	0.000000e+00	0.000000e+00	0.000000e+00
2	0.0008	0.000000e+00	0.000000e+00	0.000000e+00
3	0.0012	4.000000e-10	4.000000e-10	0.000000e+00
4	0.0016	1.043070e-08	1.043070e-08	2.984245e-20
5	0.0020	7.288416e-08	7.288416e-08	4.113414e-18

In [ ]:

Part 2 : continue advancing the simulation, with occasional visualization of system snapshots as heatmaps¶

In [19]:

# Continue with a few larger steps
for _ in range(5):
    bio.react_diffuse(total_duration=0.03, n_steps=50)
    fig = bio.system_heatmaps()
    fig.show()

Notice how `C` begins to get produced when `A` and `B` diffuse into each other, starting at the center bin (3,3)¶

In [ ]:

Part 3 : advance the reaction/diffusion to equilibrium¶

In [20]:

# Continue with more, and larger, steps
for _ in range(4):
    bio.react_diffuse(total_duration=0.3, n_steps=300)
    fig = bio.system_heatmaps()
    fig.show()

In [ ]:

Visualization of concentration changes with time at particular bins¶

In [21]:

bio.plot_history_single_bin(bin_address = (0,0))    # The bin with the initial injection of 'A'

To separate the curves for B and C, one needs to substantially magnify the y-axis, because their concentrations at that bin are very similar throughout

In [ ]:

In [22]:

bio.plot_history_single_bin(bin_address = (6,6))    # The bin with the initial injection of 'B'

To separate the curves for A and C, one needs to substantially magnify the y-axis, because their concentrations at that bin are very similar throughout

In [ ]:

In [23]:

bio.plot_history_single_bin(bin_address = (3,3))      # The midpoint bin

A and B are completely superposed, from the perfect symmetry

In [ ]:

Reaction-Diffusion in 2-D : A + B <-> C, taken to equilibrium¶

Mostly forward reaction, with 1st-order kinetics for each species¶

TAGS : "reactions 2D", "diffusion 2D", "quick-start"¶

TIME 0 : Inject initial concentrations of A and B at opposite ends of the system¶

Enable History¶

Part 1 : advance to time t=0.002 (with smaller fixed steps)¶

Part 2 : continue advancing the simulation, with occasional visualization of system snapshots as heatmaps¶

Notice how C begins to get produced when A and B diffuse into each other, starting at the center bin (3,3)¶

Part 3 : advance the reaction/diffusion to equilibrium¶

Visualization of concentration changes with time at particular bins¶

Reaction-Diffusion in 2-D : `A + B <-> C`, taken to equilibrium¶

TIME 0 : Inject initial concentrations of `A` and `B` at opposite ends of the system¶

Notice how `C` begins to get produced when `A` and `B` diffuse into each other, starting at the center bin (3,3)¶