A chemical that can cross membranes, if not diffusing fast enough thru the length of a compartment, will get converted by a compartment enzyme into a form that cannot cross membranes, and thus gets trapped inside.¶

As in experiment membranes_racing_condition_1, a racing condition between diffusion and reaction.
Like before, the "window of opportunity to escape" closes fast!
This time, however, the initial concentration of A is outside the compartment, and the reaction is an enzyme-catalyed unimolecular A + E -> B + E . The enzyme E cannot cross membranes.

If, metaphorically speaking, we think of A as a "traveler", then:

SCENARIO 1 ("The Hotel California") - `A` diffuses so slowly that, in its journey across the compartment, it "falls under the spell of the enzyme" (gets converted into a form `B` that cannot cross membranes), and thus "can never leave"...¶

SCENARIO 2 ("Ulysses' escape") - `A` diffuses so fast that it "stays a step ahead of trouble", and is largely out of the compartment prior to getting entangled into it by the "magic spell" (reaction) that would have "trapped it into the island"¶

Recommended background:

experiment 1D/reaction_diffusion/membranes_racing_condition_1

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

In [1]:

LAST_REVISED = "Sep. 29, 2025"
LIFE123_VERSION = "1.0.0rc7"       # 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, ReactionEnzyme, PlotlyHelper, check_version

In [4]:

check_version(LIFE123_VERSION)

OK

In [ ]:

Initialize the Chemical Data and the Reactions. They will be RE-USED (except for a change in the diffusion rate of `A`) in both scenarios¶

In [5]:

# Initialize the chemical data.  
# The enzyme `E`, typically a large molecule, is given a relatively sall diffusion rate (not too important,
# because of our initial condition later, of `E` already uniformly diffused)
# We'll let the simulator estimate a diffusion rate constant for `EU` (so, passing None for now)
chem_data = ChemData(          names=["U", "L",  "E", "EU", "P",  "UP"], 
                     diffusion_rates=[200., 250., 80., None, 120., 90.],        # The diffusion rate of `U` will later be increased in scenario 2
                     plot_colors=["red", "green", "violet", "darkturquoise", "pink", "black"]) 

rxns = ReactionRegistry(chem_data=chem_data)

# Enzymatic reaction A + E -> B + E
r = ReactionEnzyme(substrate="U", product="L", enzyme="E", 
                   k1_F=10., k1_R=2., k2_F=50.)

rxns.register_reaction(r)

rxns.add_reaction(reactants=["U", "P"], products="UP",
                  kF=20., kR=1.)
    
rxns.describe_reactions()

register_reaction() INFO: diffusion rate for the reaction intermediates (`EU`), not yet specified, roughly estimated as 72.0
Number of reactions: 2
0: E + U <-> EU -> E + L  (Enzymatic reaction)  (k1_F = 10 / k1_R = 2 / k2_F = 50 / kM = 5.2 / kcat = 50)
1: U + P  <-> UP  (Elementary Synthesis reaction)  (kF = 20 / kR = 1 / K = 20)
Chemicals involved in the above reactions (not counting enzymes): {"U" (red), "L" (green), "EU" (darkturquoise), "P" (pink), "UP" (black)}
Enzymes involved in the above reactions: 
  "E" (violet)

In [6]:

chem_data.all_chemicals()

Out[6]:

	name	label	diff rate	plot_color
0	U	U	200.0	red
1	L	L	250.0	green
2	E	E	80.0	violet
3	EU	EU	72.0	darkturquoise
4	P	P	120.0	pink
5	UP	UP	90.0	black

In [ ]:

SCENARIO 1 - `U` diffuses slowly, relatively to the Enzymatic reaction `U + E -> L + E`¶

Initialize the 1D System, including Membranes¶

In [7]:

bio = BioSim1D(n_bins=30, chem_data=chem_data, reactions=rxns)

In [8]:

bio.membranes().set_membranes(membranes=[ (11, 21) ])
bio.membranes().membrane_list

Out[8]:

[(11, 21)]

In [9]:

# We'll use 1/2 of the diffusion rate of `A` as its membrane permeability (by passive transport)
# The conversion product `B` and the enzyne `E`, by constrast, keep their default 0 permeability (i.e., can't cross membranes)
bio.membranes().change_permeability("U", 50.)

In [10]:

bio.membranes().show_permeability()   # Values not shown mean 0

Out[10]:

{'U': 50.0}

In [ ]:

Initialize the initial concentrations¶

In [11]:

# Turning into into a function, for convenience of later re-using

def initialize_initial_concs():
    # Set up the initial bell-shape concentration of `U` ("Ulysses"), 
    # with a narrow peak close to the left end of the system,
    # centered at bin 5
    bio.inject_bell_curve(chem_label="U", center_bin=5, sd=0.05, max_amplitude=200., bias=0., clip=(0,9))

    # The enzyme `E`, by contrast, is uniformly distributed within the membranes of the 1st (and only) compartment
    bio.set_compartment_uniform_concentration(compartment_id=0, chem_label="E", conc=15.)

    # Set up the initial bell-shape concentration of `P` ("Penelope"), 
    # with a narrow peak close to the right end of the system, at bin 25
    bio.inject_bell_curve(chem_label="P", center_bin=25, sd=0.05, max_amplitude=30., bias=0., clip=(21,29))
    #for bin in range(21, 30):
        #bio.set_bin_conc(bin_address=bin, conc=10, chem_label="P")

In [12]:

initialize_initial_concs()

In [ ]:

In [13]:

# Show as heatmap (including the membranes, shown in brown)
bio.system_heatmaps(title_prefix="Initial strong, localized transient of chemical `U` (membranes shown in brown)")

In [ ]:

The initial transient of `U` is localized to the left of the compartment that spans the space between bins 10 and 20¶

The enzyme E is uniformly localized in that compartment.
L, UP and the reaction intermerdiate EU are not present anywhere.
P is localized to the right of the compartment (but, unlike U can't cross membranes)

In [14]:

# Visualize the system state so far
bio.visualize_system(title_prefix=["Initial strong, localized transient of chemical `U` (membranes shown in brown).", 
                                   "Notice that the enzyme `E` is found inside the compartment"])

In [15]:

df = bio.describe_state()
df

SYSTEM STATE at Time t = 0:
30 bins and 6 chemical species
Membranes present:  [(11, 21)]

Out[15]:

	Species	Diff rate	Bin 0	Bin 1	Bin 2	Bin 3	Bin 4	Bin 5	Bin 6	Bin 7	...	Bin 20	Bin 21	Bin 22	Bin 23	Bin 24	Bin 25	Bin 26	Bin 27	Bin 28	Bin 29
0	U	200.0	0.523622	4.451866	23.523695	77.251695	157.670157	200.0	157.670157	77.251695	...	0.0	0.00000	0.000000	0.000000	0.000000	0.0	0.000000	0.000000	0.000000	0.00000
1	L	250.0	0.000000	0.000000	0.000000	0.000000	0.000000	0.0	0.000000	0.000000	...	0.0	0.00000	0.000000	0.000000	0.000000	0.0	0.000000	0.000000	0.000000	0.00000
2	E	80.0	0.000000	0.000000	0.000000	0.000000	0.000000	0.0	0.000000	0.000000	...	15.0	0.00000	0.000000	0.000000	0.000000	0.0	0.000000	0.000000	0.000000	0.00000
3	EU	72.0	0.000000	0.000000	0.000000	0.000000	0.000000	0.0	0.000000	0.000000	...	0.0	0.00000	0.000000	0.000000	0.000000	0.0	0.000000	0.000000	0.000000	0.00000
4	P	120.0	0.000000	0.000000	0.000000	0.000000	0.000000	0.0	0.000000	0.000000	...	0.0	0.66778	3.528554	11.587754	23.650524	30.0	23.650524	11.587754	3.528554	0.66778
5	UP	90.0	0.000000	0.000000	0.000000	0.000000	0.000000	0.0	0.000000	0.000000	...	0.0	0.00000	0.000000	0.000000	0.000000	0.0	0.000000	0.000000	0.000000	0.00000

6 rows × 32 columns

In [16]:

df[df.columns[4:24]]  # Zoom in where the action is

Out[16]:

	Bin 2	Bin 3	Bin 4	Bin 5	Bin 6	Bin 7	Bin 8	Bin 9	Bin 11	Bin 12	Bin 13	Bin 14	Bin 15	Bin 16	Bin 17	Bin 18	Bin 19	Bin 20	Bin 21
0	23.523695	77.251695	157.670157	200.0	157.670157	77.251695	23.523695	4.451866	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.00000
1	0.000000	0.000000	0.000000	0.0	0.000000	0.000000	0.000000	0.000000	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.00000
2	0.000000	0.000000	0.000000	0.0	0.000000	0.000000	0.000000	0.000000	15.0	15.0	15.0	15.0	15.0	15.0	15.0	15.0	15.0	15.0	0.00000
3	0.000000	0.000000	0.000000	0.0	0.000000	0.000000	0.000000	0.000000	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.00000
4	0.000000	0.000000	0.000000	0.0	0.000000	0.000000	0.000000	0.000000	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.66778
5	0.000000	0.000000	0.000000	0.0	0.000000	0.000000	0.000000	0.000000	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.00000

In [ ]:

Request history-keeping for some bins¶

In [17]:

# Request to save the concentration history 
# at the bin with the initial concentration injection, on the left of the system, 
# plus 2 bins inside the compartment, and one on the far side on the right
bio.enable_history(bins=[5,11,15,25], frequency=30, take_snapshot=True)     

History enabled for bins [5, 11, 15, 25], for ALL chemicals

In [ ]:

Start the simulation of the reaction-diffusion¶

In [18]:

# The first round of reaction-diffusion, over a small time duration
bio.react_diffuse(total_duration=0.05, fraction_max_step=0.5, show_status=True)
bio.visualize_system(title_prefix=["The transient `U` starts diffusing into the compartment, and turning into `L` by `U + E -> L + E` ",
                                   "Notice the production of `L`, which can't cross the membranes"])

{'steps': 76, 'system time': '0.050666', 'time_step': 0.00066666}

At this point, some of the enzyme E is in the form of the intermediate EU, especially in the left of the compartment (near bin 11), where U is diffusing in from the left, and initiating the reaction

In [19]:

# SAME IN HEATMAP VIEW
bio.system_heatmaps(title_prefix=["The transient `U` starts diffusing into the compartment, and turning into `B` by `A + E -> B + E`"])

In [ ]:

Let's continue the reaction-diffusion¶

In [20]:

bio.react_diffuse(total_duration=0.05, fraction_max_step=0.5, show_status=True)
bio.visualize_system()

{'steps': 76, 'system time': '0.10133', 'time_step': 0.00066666}

In [21]:

bio.react_diffuse(total_duration=0.1, fraction_max_step=0.5, show_status=True)
bio.visualize_system(title_prefix="Here we turned off the curve smoothing in the plot!",smoothed=False)

{'steps': 151, 'system time': '0.202', 'time_step': 0.00066666}

In [22]:

bio.react_diffuse(total_duration=0.2, fraction_max_step=0.5, show_status=True)
bio.visualize_system(smoothed=False)

{'steps': 301, 'system time': '0.40266', 'time_step': 0.00066666}

In [23]:

bio.react_diffuse(total_duration=0.2, fraction_max_step=0.5, show_status=True)
bio.visualize_system(smoothed=False)

{'steps': 301, 'system time': '0.60333', 'time_step': 0.00066666}

In [24]:

bio.react_diffuse(total_duration=1, fraction_max_step=0.5, show_status=True)
bio.visualize_system(smoothed=False)

{'steps': 1501, 'system time': '1.604', 'time_step': 0.00066666}

In [25]:

bio.react_diffuse(total_duration=0.9, fraction_max_step=0.5, show_status=True)
bio.visualize_system(smoothed=False)

{'steps': 1351, 'system time': '2.5046', 'time_step': 0.00066666}

`U` is crossing to some extent the nearby left membrane, but not making it in time to reach the right membrane, before getting consumed¶

In [26]:

bio.system_heatmaps()

In [ ]:

Now, let's look at a few individual bins, and their concentration change with time¶

In [27]:

bio.plot_history_single_bin(title_prefix=["Diffusion, membrane passive transport, and reaction `U + E -> L + E`",
                                          "Time evolution at bin where the transient originates."], 
                             bin_address=5)

In [28]:

bio.plot_history_single_bin(title_prefix=["Time evolution inside the compartment, at the very left of it"], 
                            bin_address=11)

Notice the arrival of `U`, albeit short-lasting. Also notice the transient buildup of the intermediary `EU` (and its corresponding transient dip in `E`)¶

In [29]:

bio.plot_history_single_bin(title_prefix=["Time evolution at a bin in the middle of the compartment"], 
                            bin_address=15)

`U` enters the compartment from the left (bin 11), thru passive transport across the membrane), but barely has a chance to diffuse even to the middle of the compartment (bin 15) because of its quick conversion to `L`¶

In [30]:

bio.plot_history_single_bin(title_prefix=["Time evolution several bins to the right of the compartment"], 
                            bin_address=25)

Only PUNY amounts of `U` ever reach the faraway bins to the right of the compartment!¶

In [ ]:

SCENARIO 2 - `U` diffuses quickly¶

Let's repeat, starting like before, but with a much higher diffusion rate of U

In [31]:

# Initial conditioned just like before, with same reaction, membrane layout and concentrations

bio = BioSim1D(n_bins=30, chem_data=chem_data, reactions=rxns)

bio.membranes().set_membranes(membranes=[ (11, 21) ])

In [32]:

# All same as in scenario 1
initialize_initial_concs()

In [33]:

# 15 times faster diffusion rate of 'A', and of its membrane permeability, than before

chem_data.set_diffusion_rate(chem_label="U", diff_rate = 3000)  # **** x15 from scenario 1

bio.membranes().change_permeability("U", 750.)                 # **** x15 from scenario 1 

In [34]:

chem_data.all_chemicals()   # Notice that the plot_color for `EA` was automatically-assigned earlier

Out[34]:

	name	label	diff rate	plot_color
0	U	U	3000.0	red
1	L	L	250.0	green
2	E	E	80.0	violet
3	EU	EU	72.0	darkturquoise
4	P	P	120.0	pink
5	UP	UP	90.0	black

In [35]:

bio.membranes().membrane_list

Out[35]:

[(11, 21)]

In [36]:

bio.membranes().show_permeability()   # Values not shown mean 0

Out[36]:

{'U': 750.0}

In [ ]:

In [37]:

# Request to save the concentration history 
# at the bin with the initial concentration injection, on the left of the system, 
# plus 2 bins inside the compartment, and one on the far side on the right
bio.enable_history(bins=[5,11,15,25], frequency=30, take_snapshot=True)    

History enabled for bins [5, 11, 15, 25], for ALL chemicals

In [ ]:

In [38]:

# Visualize the system state so far
bio.visualize_system(title_prefix=["Initial strong, localized transient of chemical `U` (membranes shown in brown).", 
                                   "Notice that the enzyme `E` is found inside the compartment"])

In [ ]:

Repeat the simulation¶

In [39]:

# The first round of reaction-diffusion, over a small time duration
bio.react_diffuse(total_duration=0.05, fraction_max_step=0.5, show_status=True)
bio.visualize_system(title_prefix=["The transient `U` starts diffusing into the compartment, and turning into `L` by `U + E -> L + E` ",
                                   "Notice the production of `L`, which can't cross the membranes"])

{'steps': 901, 'system time': '0.050055', 'time_step': 5.5555e-05}

In [40]:

# SAME IN HEATMAP VIEW
bio.system_heatmaps(title_prefix=["The transient `A` starts diffusing into the compartment, and turning into `B` by `A + E -> B + E`"])

In [ ]:

Let's continue the reaction-diffusion¶

In [41]:

bio.react_diffuse(total_duration=0.05, fraction_max_step=0.5, show_status=True)
bio.visualize_system()

{'steps': 901, 'system time': '0.10011', 'time_step': 5.5555e-05}

In [42]:

bio.react_diffuse(total_duration=0.1, fraction_max_step=0.5, show_status=True)
bio.visualize_system(title_prefix="Here we turned off the curve smoothing in the plot!",smoothed=False)

{'steps': 1801, 'system time': '0.20016', 'time_step': 5.5555e-05}

In [43]:

bio.react_diffuse(total_duration=2.3, fraction_max_step=0.5, show_status=True)
bio.visualize_system(smoothed=False)

{'steps': 41401, 'system time': '2.5002', 'time_step': 5.5555e-05}

In [44]:

bio.system_heatmaps()

In [ ]:

Now, let's look at a few individual bins, and their concentration change with time¶

In [45]:

bio.plot_history_single_bin(title_prefix=["Diffusion, membrane passive transport, and reaction `U + E -> L + E`",
                                          "Time evolution at bin where the transient originates."], 
                             bin_address=5)

In [46]:

bio.plot_history_single_bin(title_prefix=["Time evolution inside the compartment, at the very left of it"], 
                            bin_address=11)

In [47]:

bio.plot_history_single_bin(title_prefix=["Time evolution at a bin in the middle of the compartment"], 
                            bin_address=15)

In [48]:

bio.plot_history_single_bin(title_prefix=["Time evolution several bins to the right of the compartment"], 
                            bin_address=25)

In [ ]:

A chemical that can cross membranes, if not diffusing fast enough thru the length of a compartment, will get converted by a compartment enzyme into a form that cannot cross membranes, and thus gets trapped inside.¶

SCENARIO 1 ("The Hotel California") - A diffuses so slowly that, in its journey across the compartment, it "falls under the spell of the enzyme" (gets converted into a form B that cannot cross membranes), and thus "can never leave"...¶

SCENARIO 2 ("Ulysses' escape") - A diffuses so fast that it "stays a step ahead of trouble", and is largely out of the compartment prior to getting entangled into it by the "magic spell" (reaction) that would have "trapped it into the island"¶

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

Initialize the Chemical Data and the Reactions. They will be RE-USED (except for a change in the diffusion rate of A) in both scenarios¶

SCENARIO 1 - U diffuses slowly, relatively to the Enzymatic reaction U + E -> L + E¶

Initialize the 1D System, including Membranes¶

Initialize the initial concentrations¶

The initial transient of U is localized to the left of the compartment that spans the space between bins 10 and 20¶

Request history-keeping for some bins¶

Start the simulation of the reaction-diffusion¶

Let's continue the reaction-diffusion¶

U is crossing to some extent the nearby left membrane, but not making it in time to reach the right membrane, before getting consumed¶

Now, let's look at a few individual bins, and their concentration change with time¶

Notice the arrival of U, albeit short-lasting. Also notice the transient buildup of the intermediary EU (and its corresponding transient dip in E)¶

U enters the compartment from the left (bin 11), thru passive transport across the membrane), but barely has a chance to diffuse even to the middle of the compartment (bin 15) because of its quick conversion to L¶

Only PUNY amounts of U ever reach the faraway bins to the right of the compartment!¶

SCENARIO 2 - U diffuses quickly¶

Repeat the simulation¶

Let's continue the reaction-diffusion¶

Now, let's look at a few individual bins, and their concentration change with time¶

SCENARIO 1 ("The Hotel California") - `A` diffuses so slowly that, in its journey across the compartment, it "falls under the spell of the enzyme" (gets converted into a form `B` that cannot cross membranes), and thus "can never leave"...¶

SCENARIO 2 ("Ulysses' escape") - `A` diffuses so fast that it "stays a step ahead of trouble", and is largely out of the compartment prior to getting entangled into it by the "magic spell" (reaction) that would have "trapped it into the island"¶

Initialize the Chemical Data and the Reactions. They will be RE-USED (except for a change in the diffusion rate of `A`) in both scenarios¶

SCENARIO 1 - `U` diffuses slowly, relatively to the Enzymatic reaction `U + E -> L + E`¶

The initial transient of `U` is localized to the left of the compartment that spans the space between bins 10 and 20¶

`U` is crossing to some extent the nearby left membrane, but not making it in time to reach the right membrane, before getting consumed¶

Notice the arrival of `U`, albeit short-lasting. Also notice the transient buildup of the intermediary `EU` (and its corresponding transient dip in `E`)¶

`U` enters the compartment from the left (bin 11), thru passive transport across the membrane), but barely has a chance to diffuse even to the middle of the compartment (bin 15) because of its quick conversion to `L`¶

Only PUNY amounts of `U` ever reach the faraway bins to the right of the compartment!¶

SCENARIO 2 - `U` diffuses quickly¶