"Crossing the border in disguise" : a chemical crosses a membrane impermeable to it,¶

by first undergoing a reaction converting it into a different chemical, to which the membrane is permeable,¶

and then re-constituting the first chemical outside of the membrane, thru the reverse reaction.¶

Reaction-diffusion `A <-> B` in 1D, in the presence of membranes with selective permeability to passive transport.¶

Eventually, the reaction, the passive membrane transport and the diffusion all come to an equilibrium.

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

In [1]:

LAST_REVISED = "June 6, 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 check_version, BioSim1D, ChemData, UniformCompartment

In [4]:

check_version(LIFE123_VERSION)

OK

In [ ]:

Prepare the initial system, for starters with IM-PERMEABLE membranes¶

with two chemicals A and B

In [5]:

chem_data = ChemData(diffusion_rates=[2., 2.], plot_colors=["turquoise", "green"])   # Names "A", "B" automatically assigned

bio = BioSim1D(n_bins=9, chem_data=chem_data)

In [6]:

reactions = bio.get_reactions()

# Reaction A <-> B , 1st-order kinetics, mostly (but not hugely) in the forward direction
reactions.add_reaction(reactants="A", products="B", forward_rate=8., reverse_rate=2.)
reactions.describe_reactions()

Number of reactions: 1 (at temp. 25 C)
0: A <-> B  (kF = 8 / kR = 2 / delta_G = -3,436.6 / K = 4) | 1st order in all reactants & products
Set of chemicals involved in the above reactions: {"B" (green), "A" (turquoise)}

In [7]:

bio.set_bin_conc(bin_address=4, chem_label="A", conc=10.)   # Set the initial concentration of `A` in middle bin

bio.membranes().set_membranes(membranes=[ (4,5) ])    # By default impermeable

bio.describe_state()

SYSTEM STATE at Time t = 0:
9 bins and 2 chemical species
Membranes present:  <life123.bio_sim_1d.Membranes1D object at 0x000002B715646040>

Out[7]:

	Species	Diff rate	Bin 0	Bin 1	Bin 2	Bin 3	Bin 4	Bin 5	Bin 6	Bin 7	Bin 8
0	A	2.0	0.0	0.0	0.0	0.0	10.0	0.0	0.0	0.0	0.0
1	B	2.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0

In [8]:

bio.system_heatmaps()

In [ ]:

Request history-keeping for some bins¶

In [9]:

# Request to save the concentration history at the bin with the initial concentration injection, 
# and at a couple of other bins
bio.enable_history(bins=[0, 2, 4], frequency=10, take_snapshot=True)    

History enabled for bins [0, 2, 4] and chemicals None (None means 'all')

In [ ]:

Advance reaction to equilibrium¶

The membranes are impermeable for now; so, no transport across them occurs

In [10]:

delta_t = 0.002   # This will be our time "quantum" (fixed time step for reactions, passive transport and diffusion) for this experiment

In [11]:

bio.react_diffuse(time_step=delta_t, n_steps=350)
bio.describe_state()

SYSTEM STATE at Time t = 0.7:
9 bins and 2 chemical species
Membranes present:  <life123.bio_sim_1d.Membranes1D object at 0x000002B715646040>

Out[11]:

	Species	Diff rate	Bin 0	Bin 1	Bin 2	Bin 3	Bin 4	Bin 5	Bin 6	Bin 7	Bin 8
0	A	2.0	0.0	0.0	0.0	0.0	2.006795	0.0	0.0	0.0	0.0
1	B	2.0	0.0	0.0	0.0	0.0	7.993205	0.0	0.0	0.0	0.0

In [12]:

bio.system_heatmaps()

In [13]:

bio.plot_history_single_bin(bin_address=4)

In [14]:

# Check the reaction equilibrium
bio.reaction_in_equilibrium(bin_address=4, rxn_index=0, explain=True) 

A <-> B
Current concentrations: [A] = 2.007 ; [B] = 7.993
1. Ratio of reactant/product concentrations, adjusted for reaction orders: 3.98307
    Formula used:  [B] / [A]
2. Ratio of forward/reverse reaction rates: 4
Discrepancy between the two values: 0.4233 %
Reaction IS in equilibrium (within 1% tolerance)

Out[14]:

True

In [ ]:

Let's siphon off the product `B` from the central bin 4, by making the membrane permeable to it¶

while still remaining impermeable to A. Note the system time when this happens:

In [15]:

bio.get_system_time()

Out[15]:

0.7

In [16]:

bio.membranes().change_permeability("B", 1.)          # Make the membrane permeable to `B` (and only to `B`!)

In [ ]:

Advance the reaction - and now also the passive transport of `B` across the membrane, and its diffusion outside¶

In [17]:

bio.react_diffuse(time_step=delta_t, n_steps=30)
bio.describe_state()

SYSTEM STATE at Time t = 0.76:
9 bins and 2 chemical species
Membranes present:  <life123.bio_sim_1d.Membranes1D object at 0x000002B715646040>

Out[17]:

	Species	Diff rate	Bin 0	Bin 1	Bin 2	Bin 3	Bin 4	Bin 5	Bin 6	Bin 7	Bin 8
0	A	2.0	0.000002	0.000058	0.001457	0.020475	1.959723	0.020475	0.001457	0.000058	0.000002
1	B	2.0	0.000022	0.000815	0.022148	0.396922	7.156478	0.396922	0.022148	0.000815	0.000022

In [18]:

bio.system_heatmaps()

Notice what's happening:

B is diffusing away (because we made the membranes permeable to it)
By Le Chatelier's principle, the reaction A <-> B in bin 4 is moving forward, because we're siphoning off the product B
In the other bins, A is re-forming from B, from the reverse reaction

In [ ]:

Continue advancing the reaction, passive transport of `B`, and diffusion¶

In [19]:

bio.react_diffuse(time_step=delta_t, n_steps=50)
bio.describe_state()

SYSTEM STATE at Time t = 0.86:
9 bins and 2 chemical species
Membranes present:  <life123.bio_sim_1d.Membranes1D object at 0x000002B715646040>

Out[19]:

	Species	Diff rate	Bin 0	Bin 1	Bin 2	Bin 3	Bin 4	Bin 5	Bin 6	Bin 7	Bin 8
0	A	2.0	0.000145	0.001753	0.016262	0.091303	1.782424	0.091303	0.016262	0.001753	0.000145
1	B	2.0	0.000896	0.011279	0.114146	0.806955	6.132099	0.806955	0.114146	0.011279	0.000896

In [20]:

bio.system_heatmaps()

Notice how the concentration of A in the central bin 4 is continuing to drop

In [21]:

bio.plot_history_single_bin(bin_address=4, vertical_lines_to_add=[0.7], 
                            title_prefix="Dashed vertical line at time when the membranes were made permeable to `B`")

In [ ]:

In [22]:

bio.react_diffuse(time_step=delta_t, n_steps=150)
bio.describe_state()

SYSTEM STATE at Time t = 1.16:
9 bins and 2 chemical species
Membranes present:  <life123.bio_sim_1d.Membranes1D object at 0x000002B715646040>

Out[22]:

	Species	Diff rate	Bin 0	Bin 1	Bin 2	Bin 3	Bin 4	Bin 5	Bin 6	Bin 7	Bin 8
0	A	2.0	0.006541	0.026681	0.097462	0.246465	1.245765	0.246465	0.097462	0.026681	0.006541
1	B	2.0	0.027446	0.114477	0.439127	1.285782	4.266270	1.285782	0.439127	0.114477	0.027446

In [23]:

bio.system_heatmaps()

Notice how [A] in bin 4 is continuing to drop, as A keeps turns into B

In [ ]:

In [24]:

bio.react_diffuse(time_step=delta_t, n_steps=500)
bio.describe_state()

SYSTEM STATE at Time t = 2.16:
9 bins and 2 chemical species
Membranes present:  <life123.bio_sim_1d.Membranes1D object at 0x000002B715646040>

Out[24]:

	Species	Diff rate	Bin 0	Bin 1	Bin 2	Bin 3	Bin 4	Bin 5	Bin 6	Bin 7	Bin 8
0	A	2.0	0.088838	0.133297	0.213601	0.298979	0.530569	0.298979	0.213601	0.133297	0.088838
1	B	2.0	0.355683	0.534950	0.865235	1.262993	1.962279	1.262993	0.865235	0.534950	0.355683

In [25]:

bio.system_heatmaps()

In [ ]:

In [26]:

bio.react_diffuse(time_step=delta_t, n_steps=1500)
bio.describe_state()

SYSTEM STATE at Time t = 5.16:
9 bins and 2 chemical species
Membranes present:  <life123.bio_sim_1d.Membranes1D object at 0x000002B715646040>

Out[26]:

	Species	Diff rate	Bin 0	Bin 1	Bin 2	Bin 3	Bin 4	Bin 5	Bin 6	Bin 7	Bin 8
0	A	2.0	0.210019	0.215086	0.223046	0.230093	0.243514	0.230093	0.223046	0.215086	0.210019
1	B	2.0	0.840091	0.860435	0.892771	0.924150	0.965106	0.924150	0.892771	0.860435	0.840091

In [27]:

bio.system_heatmaps()

In [ ]:

Let's take the system to equilibrium, in reactions and diffusion¶

In [28]:

bio.react_diffuse(time_step=delta_t, n_steps=3000)
bio.describe_state()

SYSTEM STATE at Time t = 11.16:
9 bins and 2 chemical species
Membranes present:  <life123.bio_sim_1d.Membranes1D object at 0x000002B715646040>

Out[28]:

	Species	Diff rate	Bin 0	Bin 1	Bin 2	Bin 3	Bin 4	Bin 5	Bin 6	Bin 7	Bin 8
0	A	2.0	0.222142	0.222175	0.222228	0.222274	0.222362	0.222274	0.222228	0.222175	0.222142
1	B	2.0	0.888566	0.888701	0.888915	0.889122	0.889390	0.889122	0.888915	0.888701	0.888566

In [29]:

bio.system_heatmaps()

In [30]:

# Check the reaction equilibrium in one of the bins (doesn't matter which one, since all bin concentrations are now the same)
bio.reaction_in_equilibrium(bin_address=0, rxn_index=0, explain=True) 

A <-> B
Current concentrations: [A] = 0.2221 ; [B] = 0.8886
1. Ratio of reactant/product concentrations, adjusted for reaction orders: 4
    Formula used:  [B] / [A]
2. Ratio of forward/reverse reaction rates: 4
Discrepancy between the two values: 1.211e-05 %
Reaction IS in equilibrium (within 1% tolerance)

Out[30]:

True

From a casual glance at the past heatmaps, it might seem that `A` has diffused to equilibrium across all bins...¶

But the actual mechanism was that `A` (trapped in bin 4 by a membrane impermeable to it) has transformed into `B`, which in turn has passively crossed the membrane, and then diffused across all bins, while at the same time forming `A` by the reverse reaction in all bins outside the membrane.¶

All said and done, `A` has "found a way, thru its disguise as `B`" to indirectly pass thru an impermeable membrane!¶

In [31]:

# Let's look at the central bin 4
bio.plot_history_single_bin(bin_address=4, vertical_lines_to_add=[0.7], 
                            title_prefix="Dashed vertical line at time when the membranes were made permeable to `B`;" \
                            "<br>notice the syphoning off of `B` and the resumed advance of the reaction that consumes `A`.")

In [32]:

# Let's look at a bin CLOSE to the central bin 4
bio.plot_history_single_bin(bin_address=2, vertical_lines_to_add=[0.7], 
                            title_prefix="Dashed vertical line at time when the membranes were made permeable to `B`;" \
                            "<br>notice the QUICK arrival of `B` by diffusion, and the subsequent formation of `A` by reverse reaction.")

In [33]:

# Let's look at a bin FAR from the central bin 4
bio.plot_history_single_bin(bin_address=0, vertical_lines_to_add=[0.7], 
                            title_prefix="Dashed vertical line at time when the membranes were made permeable to `B`;" \
                            "<br>notice the SLOW arrival of `B` by diffusion, and the subsequent formation of `A` by reverse reaction.")

In [ ]:

"Crossing the border in disguise" : a chemical crosses a membrane impermeable to it,¶

by first undergoing a reaction converting it into a different chemical, to which the membrane is permeable,¶

and then re-constituting the first chemical outside of the membrane, thru the reverse reaction.¶

Reaction-diffusion A <-> B in 1D, in the presence of membranes with selective permeability to passive transport.¶

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

Prepare the initial system, for starters with IM-PERMEABLE membranes¶

Request history-keeping for some bins¶

Advance reaction to equilibrium¶

Let's siphon off the product B from the central bin 4, by making the membrane permeable to it¶

Advance the reaction - and now also the passive transport of B across the membrane, and its diffusion outside¶

Continue advancing the reaction, passive transport of B, and diffusion¶

Let's take the system to equilibrium, in reactions and diffusion¶

From a casual glance at the past heatmaps, it might seem that A has diffused to equilibrium across all bins...¶

But the actual mechanism was that A (trapped in bin 4 by a membrane impermeable to it) has transformed into B, which in turn has passively crossed the membrane, and then diffused across all bins, while at the same time forming A by the reverse reaction in all bins outside the membrane.¶

All said and done, A has "found a way, thru its disguise as B" to indirectly pass thru an impermeable membrane!¶

Reaction-diffusion `A <-> B` in 1D, in the presence of membranes with selective permeability to passive transport.¶

Let's siphon off the product `B` from the central bin 4, by making the membrane permeable to it¶

Advance the reaction - and now also the passive transport of `B` across the membrane, and its diffusion outside¶

Continue advancing the reaction, passive transport of `B`, and diffusion¶

From a casual glance at the past heatmaps, it might seem that `A` has diffused to equilibrium across all bins...¶

But the actual mechanism was that `A` (trapped in bin 4 by a membrane impermeable to it) has transformed into `B`, which in turn has passively crossed the membrane, and then diffused across all bins, while at the same time forming `A` by the reverse reaction in all bins outside the membrane.¶

All said and done, `A` has "found a way, thru its disguise as `B`" to indirectly pass thru an impermeable membrane!¶