Attaining a Concentration Gradient¶

by continuosly injecting and draining, at opposite ends¶

The system starts out with a uniform concentration.
Then identical concentrations are repeatedly injected to the left and drained from the right
Diffusion turn the forced concentration imbalance into a smooth gradient.

TAGS : "diffusion 1D"¶

In [1]:

LAST_REVISED = "Apr. 29, 2025"
LIFE123_VERSION = "1.0.0rc3"       # Library version this experiment is based on

In [2]:

#import set_path                    # Using MyBinder?  Uncomment this before running the next cell!

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#import sys, os
#os.getcwd()
#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, check_version

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check_version(LIFE123_VERSION)

OK

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In [5]:

# Initialize the system with a uniform concentration (of the only species)
chem_data = ChemData(names="A", diffusion_rates=0.6)
bio = BioSim1D(n_bins=9, chem_data=chem_data)

bio.set_uniform_concentration(chem_label="A", conc=100.)

bio.describe_state()

SYSTEM STATE at Time t = 0:
9 bins and 1 chemical species:

Out[5]:

	Species	Diff rate	Bin 0	Bin 1	Bin 2	Bin 3	Bin 4	Bin 5	Bin 6	Bin 7	Bin 8
0	A	0.6	100.0	100.0	100.0	100.0	100.0	100.0	100.0	100.0	100.0

In [6]:

bio.show_system_snapshot()

SYSTEM SNAPSHOT at time 0:

Out[6]:

	A
0	100.0
1	100.0
2	100.0
3	100.0
4	100.0
5	100.0
6	100.0
7	100.0
8	100.0

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# Visualize the system's initial state
bio.visualize_system(title_prefix="Diffusion")

In [8]:

bio.system_heatmaps(title_prefix="Diffusion", text_format=".0f")

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Start the simulation steps¶

In [9]:

delta_time = 1.

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for i in range(501):
    # Inject to the leftmost bin
    bio.inject_conc_to_bin(bin_address=0, chem_index=0, delta_conc=4, zero_clip = False)
    
    # Drain from the rightmost bin
    bio.inject_conc_to_bin(bin_address=8, chem_index=0, delta_conc=-4, 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 <= 12 and i%3 == 0) or (i%100 == 0):   # Display more frequently initially
        print()
        bio.describe_state(concise=True)
        
        # Show the system state as a line plot
        fig = bio.visualize_system(title_prefix="Diffusion")
        fig.show()
        
        # Show as heatmap
        fig = bio.system_heatmaps(title_prefix="Diffusion", text_format=".3g")
        fig.show()

SYSTEM STATE at Time t = 1:
[[102.50172719 101.14444998 100.29805415 100.04921258 100.
   99.95078742  99.70194585  98.85555002  97.49827281]]

SYSTEM STATE at Time t = 4:
[[107.42285216 104.74375636 102.29241276 100.85388504 100.
   99.14611496  97.70758724  95.25624364  92.57714784]]

SYSTEM STATE at Time t = 7:
[[110.88507422 107.69013213 104.36899081 101.90793997 100.
   98.09206003  95.63100919  92.30986787  89.11492578]]

SYSTEM STATE at Time t = 10:
[[113.59743191 110.06014065 106.11080705 102.82666877 100.
   97.17333123  93.88919295  89.93985935  86.40256809]]

SYSTEM STATE at Time t = 13:
[[115.76754    111.96612839 107.52278434 103.57670355 100.
   96.42329645  92.47721566  88.03387161  84.23246   ]]

SYSTEM STATE at Time t = 101:
[[124.65573484 119.78183626 113.32331552 106.66286788 100.
   93.33713212  86.67668448  80.21816374  75.34426516]]

SYSTEM STATE at Time t = 201:
[[124.67064114 119.79494464 113.33304492 106.66804479 100.
   93.33195521  86.66695508  80.20505536  75.32935886]]

SYSTEM STATE at Time t = 301:
[[124.67065158 119.79495382 113.33305173 106.66804841 100.
   93.33195159  86.66694827  80.20504618  75.32934842]]

SYSTEM STATE at Time t = 401:
[[124.67065159 119.79495383 113.33305174 106.66804842 100.
   93.33195158  86.66694826  80.20504617  75.32934841]]

SYSTEM STATE at Time t = 501:
[[124.67065159 119.79495383 113.33305174 106.66804842 100.
   93.33195158  86.66694826  80.20504617  75.32934841]]

By now, the gradient has stabilized with¶

[A] = 124.67065159 on the left and [A] = 75.32934841 on the right¶

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

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