Diffusion of a bell-shaped initial concentration along an persistent gradient¶

Similar to the experiment diffusion_along_gradient_1, but now the gradient is NOT an initial condition; rather, it's a persistent dynamic condition (explored in experiment gradient_1).

The one-chemical system starts out with a uniform concentration. The persistent concentration gradient is attained by continuosly injecting and draining, at opposite ends.

After a stable gradient is established, a one-time injection is performed, to add a bell-shape concentration near one end of the system, on the "uphill" side of the gradient.

Just as seen in the case of diffusion_along_gradient_1, the concentration peak remains in place, and simply spreads out from there

LAST REVISED: June 4, 2023

In [1]:

import set_path      # Importing this module will add the project's home directory to sys.path

Added 'D:\Docs\- MY CODE\BioSimulations\life123-Win7' to sys.path

In [2]:

from src.life_1D.bio_sim_1d import BioSim1D

import plotly.express as px
from src.modules.reactions.reaction_data import ChemData as chem

In [3]:

# Initialize the system with a uniform concentration (of the only species)
chem_data = chem(names=["A"], diffusion_rates=[3.])
bio = BioSim1D(n_bins=200, chem_data=chem_data)

bio.set_uniform_concentration(species_index=0, conc=150.)

In [4]:

# Visualize the system state so far
bio.visualize_system()

In [5]:

# Show as heatmap
fig = px.imshow(bio.system_snapshot().T, 
                title= f"Diffusion. System snapshot as a heatmap at time t={bio.system_time}", 
                labels=dict(x="Bin number", y="Chem. species", color="Concentration"),
                color_continuous_scale="gray_r")

fig.data[0].xgap=0
fig.data[0].ygap=0

fig.show()

Start the simulation steps¶

In [6]:

delta_time = 1.

In [7]:

leftmost_bin = 0
rightmost_bin = bio.system_size() - 1
delta_injection = 2

print(f"Repeatedly injecting {delta_injection} at bin {leftmost_bin} and draining it at {rightmost_bin}")

for i in range(2001):
    # Inject to the leftmost bin
    bio.inject_conc_to_bin(bin_address=leftmost_bin, species_index=0, delta_conc=delta_injection, zero_clip = False)
    
    # Drain from the rightmost bin
    bio.inject_conc_to_bin(bin_address=rightmost_bin, species_index=0, delta_conc=-delta_injection, zero_clip = False)
    
    # Note: the NET GAIN of moles of A in the system is zero!
    
    # Diffuse for the time span delta_time
    status = bio.diffuse(total_duration=delta_time, time_step=0.1)

    
    if (i == 2) or (i == 1000) or (i == 2000):   # Display the early, middle and final state     
        bio.visualize_system()

Repeatedly injecting 2 at bin 0 and draining it at 199

In [8]:

(bio.bin_concentration(bin_address=0, species_name="A") ,
 bio.bin_concentration(bin_address=rightmost_bin, species_name="A"))

Out[8]:

(203.33945718438923, 96.66054281561514)

By now, the gradient has stabilized with¶

[A] = 203.34 on the left and [A] = 96.66 on the right¶

Their average is 150, equal to the initial uniform concentration, since the net injection/drain is balanced

Note: if the drain is too large, relative to the diffusion rate, the smaller concentration could "saturate" at zero

In [9]:

# Show as heatmap
fig = px.imshow(bio.system_snapshot().T, 
                title= f"Diffusion. System snapshot as a heatmap at time t={bio.system_time}", 
                labels=dict(x="Bin number", y="Chem. species", color="Concentration"),
                color_continuous_scale="gray_r")

fig.data[0].xgap=0
fig.data[0].ygap=0

fig.show()

In [10]:

# Now that the system is at a steady statem, 
# inject a bell-shape concentration, with the peak close to one end of the system
bio.inject_bell_curve(species_name="A", mean=0.25, sd=0.1, amplitude=30., bias=0)

In [11]:

# Visualize the system state at this stage
bio.visualize_system()

In [12]:

print(f"Resuming repeatedly injecting {delta_injection} at bin {leftmost_bin} and draining it at {rightmost_bin}")

for i in range(501):
    # Inject to the leftmost bin
    bio.inject_conc_to_bin(bin_address=leftmost_bin, species_index=0, delta_conc=delta_injection, zero_clip = False)
    
    # Drain from the rightmost bin
    bio.inject_conc_to_bin(bin_address=rightmost_bin, species_index=0, delta_conc=-delta_injection, zero_clip = False)
    
    # Note: the NET GAIN of moles of A in the system is zero!
    
    # Diffuse for the time span delta_time
    status = bio.diffuse(total_duration=delta_time, time_step=0.1)

    
    if (i%20 == 0):        # Display every so often     
        bio.visualize_system()

Resuming repeatedly injecting 2 at bin 0 and draining it at 199

In [13]:

(bio.bin_concentration(bin_address=0, species_name="A") ,
 bio.bin_concentration(bin_address=rightmost_bin, species_name="A"))

Out[13]:

(262.99621803202314, 95.88704136179823)

The one-time pulse injected into the system, gradually "melted into" the gradient.¶

The gradient finally re-stabilizes with
[A] = 263 on the left and [A] = 95.89 on the right.
Their average is about 179.4, which is equal to the initial uniform concentration of 150, plus most of the bell curve, which had amplitude=30. (i.e. total area under it of 30, minus a little that was clipped to the left.)

In [ ]: