The attenuation over time of a single-frequency in the initial concentration¶

The initial system state is a sine wave of frequency 2 (i.e. 2 cycles across the system's length), of amplitude 10, with a baseline (bias) of 30¶

Afterward, the process is restarted and repeated with a frequency 5 times larger¶

LAST REVISED: June 23, 2024 (using v. 1.0 beta34.1)

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 life123 import BioSim1D

import plotly.express as px
import plotly.graph_objects as go

from life123 import ChemData as chem

In [3]:

# Initialize the system.  We use a RELATIVELY LARGE NUMBER OF BINS, 
# to captures the finer changes in the frequency components of the concentration function
chem_data = chem(names=["A"], diffusion_rates=[0.5])
bio = BioSim1D(n_bins=500, chem_data=chem_data)

Initial Preparation -¶

Start with a sinusoidal concentration, with exactly 2 cycles over the length of the system¶

(of amplitude 10 and bias value of 30)¶

In [4]:

bio.inject_sine_conc(species_name="A", amplitude=10, bias=30, frequency=2)

In [5]:

bio.show_system_snapshot()

SYSTEM SNAPSHOT at time 0:
             A
0    30.000000
1    30.251301
2    30.502443
3    30.753268
4    31.003617
..         ...
495  28.746668
496  28.996383
497  29.246732
498  29.497557
499  29.748699

[500 rows x 1 columns]

In [6]:

fig = px.line(data_frame=bio.system_snapshot(), y=["A"], 
              title= "Initial System State",
              color_discrete_sequence = ['red'],
              labels={"value":"concentration", "variable":"Chemical", "index":"Bin number"})
fig.show()

In [7]:

# Show as heatmap
fig = px.imshow(bio.system_snapshot().T, 
                title= "Initial System State (as a heatmap)", 
                labels=dict(x="Bin number", y="Chem. species", color="Concentration"),
                text_auto=False, color_continuous_scale="gray_r") 

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

fig.show()

In [8]:

# Take a look at the frequency domain of the concentration values
frequency_data = bio.frequency_analysis(species_name="A")
frequency_data

Out[8]:

	Frequency	Relative Amplitude
0	0.0	3.0
1	2.0	1.0

In [9]:

ratio = frequency_data.loc[1, "Relative Amplitude"] / frequency_data.loc[0, "Relative Amplitude"]
ratio

Out[9]:

0.3333333333333333

In [10]:

bio.add_snapshot(data_snapshot={"ratio": ratio})
bio.get_history()

Out[10]:

	SYSTEM TIME	ratio	caption
0	0	0.333333

Start the diffusion - to time t=10¶

In [11]:

bio.diffuse(total_duration=10, n_steps=100)

Out[11]:

{'steps': 100}

In [12]:

# Take a look at the frequency domain of the concentration values
frequency_data = bio.frequency_analysis(species_name="A")
frequency_data

Out[12]:

	Frequency	Relative Amplitude
0	0.0	136826.215286
1	1.0	1.000000
2	2.0	45466.923974
3	3.0	2.992211
4	4.0	3.980559
...	...	...
246	246.0	11.415221
247	247.0	11.413727
248	248.0	11.412659
249	249.0	11.412018
250	250.0	11.411805

251 rows × 2 columns

In [13]:

ratio = frequency_data.loc[2, "Relative Amplitude"] / frequency_data.loc[0, "Relative Amplitude"]
ratio

Out[13]:

0.33229687657832013

In [14]:

bio.add_snapshot(data_snapshot={"ratio": ratio})
bio.get_history()

Out[14]:

	SYSTEM TIME	ratio	caption
0	0.0	0.333333
1	10.0	0.332297

Do 49 more rounds of diffusion - to time t = 500¶

In [15]:

for i in range(49):
    bio.diffuse(total_duration=10, n_steps=100)
    frequency_data = bio.frequency_analysis(species_name="A")
    ratio = frequency_data.loc[2, "Relative Amplitude"] / frequency_data.loc[0, "Relative Amplitude"]
    bio.add_snapshot(data_snapshot={"ratio": ratio})

In [16]:

bio.get_history()

Out[16]:

	SYSTEM TIME	ratio
0	0.0	0.333333
1	10.0	0.332297
2	20.0	0.331275
3	30.0	0.330264
4	40.0	0.329262
5	50.0	0.328268
6	60.0	0.327283
7	70.0	0.326304
8	80.0	0.325332
9	90.0	0.324367
10	100.0	0.323409
11	110.0	0.322456
12	120.0	0.321510
13	130.0	0.320569
14	140.0	0.319634
15	150.0	0.318705
16	160.0	0.317781
17	170.0	0.316862
18	180.0	0.315948
19	190.0	0.315040
20	200.0	0.314137
21	210.0	0.313238
22	220.0	0.312345
23	230.0	0.311456
24	240.0	0.310572
25	250.0	0.309692
26	260.0	0.308817
27	270.0	0.307947
28	280.0	0.307081
29	290.0	0.306219
30	300.0	0.305362
31	310.0	0.304509
32	320.0	0.303660
33	330.0	0.302816
34	340.0	0.301976
35	350.0	0.301139
36	360.0	0.300307
37	370.0	0.299479
38	380.0	0.298655
39	390.0	0.297835
40	400.0	0.297018
41	410.0	0.296206
42	420.0	0.295397
43	430.0	0.294592
44	440.0	0.293791
45	450.0	0.292994
46	460.0	0.292200
47	470.0	0.291410
48	480.0	0.290624
49	490.0	0.289841
50	500.0	0.289062

In [17]:

fig = px.line(data_frame=bio.system_snapshot(), y=["A"], 
              title= f"System State at time t={bio.system_time}",
              color_discrete_sequence = ['red'],
              labels={"value":"concentration", "variable":"Chemical", "index":"Bin number"})
fig.show()

Note how the curve is beginning to flatten; the peaks are no longer between 20 and 40¶

Do 150 more rounds of diffusion - to time t = 2000¶

In [18]:

for i in range(150):
    bio.diffuse(total_duration=10, n_steps=75)    # Notice the gradual decreas of the number of intermediate steps, given the smaller gradient
    frequency_data = bio.frequency_analysis(species_name="A")
    ratio = frequency_data.loc[2, "Relative Amplitude"] / frequency_data.loc[0, "Relative Amplitude"]
    bio.add_snapshot(data_snapshot={"ratio": ratio})

In [19]:

bio.get_history()

Out[19]:

	SYSTEM TIME	ratio	caption
0	0.0	0.333333
1	10.0	0.332297
2	20.0	0.331275
3	30.0	0.330264
4	40.0	0.329262
...	...	...	...
196	1960.0	0.203404
197	1970.0	0.202963
198	1980.0	0.202523
199	1990.0	0.202085
200	2000.0	0.201648

201 rows × 3 columns

In [20]:

fig = px.line(data_frame=bio.system_snapshot(), y=["A"], 
          title= f"Diffusion. System snapshot at time t={bio.system_time}",
          color_discrete_sequence = ['red'],
          labels={"value":"concentration", "variable":"Chemical", "index":"Bin number"})
fig.show()

Note how the curve is flatter still - and even beginning to lose shape at the boundary¶

Do 800 more rounds of diffusion - to time t = 10000¶

In [21]:

for i in range(800):
    bio.diffuse(total_duration=10, n_steps=20)   # Note how we're gradually increasing the time steps, because the gradiants are now smaller
    frequency_data = bio.frequency_analysis(species_name="A")
    ratio = frequency_data.loc[2, "Relative Amplitude"] / frequency_data.loc[0, "Relative Amplitude"]
    bio.add_snapshot(data_snapshot={"ratio": ratio})

In [22]:

bio.get_history()

Out[22]:

	SYSTEM TIME	ratio	caption
0	0.0	0.333333
1	10.0	0.332297
2	20.0	0.331275
3	30.0	0.330264
4	40.0	0.329262
...	...	...	...
996	9960.0	0.062422
997	9970.0	0.062358
998	9980.0	0.062295
999	9990.0	0.062232
1000	10000.0	0.062168

1001 rows × 3 columns

In [23]:

fig = px.line(data_frame=bio.system_snapshot(), y=["A"], 
          title= f"Diffusion. System snapshot at time t={bio.system_time}",
          color_discrete_sequence = ['red'],
          labels={"value":"concentration", "variable":"Chemical", "index":"Bin number"})
fig.show()

Getting decisively flatter, as it approaches equilibrium¶

Now, let's look how the amplitude of the sine signal, relative to the constant bias, changed over time¶

In [24]:

fig = px.line(data_frame=bio.get_history(), y=["ratio"], 
          title= "Component of frequency=2, relative to amplitude of constant bias",
          color_discrete_sequence = ['purple'],
          labels={"value":"ratio", "variable":"Freq. 2", "index":"Time (groups of 10 units)"})
fig.show()

In [25]:

fig = px.line(data_frame=bio.get_history(), y=["ratio"], 
              title= "Component of frequency=2, relative to amplitude of constant bias (log scale in y-axis)",
              color_discrete_sequence = ['purple'],
              labels={"value":"ratio", "variable":"Freq. 2", "index":"Time (groups of 10 units)"},
              log_y=True)
fig.show()

In [26]:

saved_freq_2_df = bio.get_history().copy(deep=True)   # Save the whole dataframe, for later use

Start a NEW system¶

Everything same as before EXCEPT the frequency is now 5 times bigger (10 cycles over the length of the system)¶

In [27]:

bio = BioSim1D(n_bins=500, chem_data=chem_data)

In [28]:

bio.inject_sine_conc(species_name="A", amplitude=10, bias=30, frequency=10)   # x5 higher frequency than before

In [29]:

fig = px.line(data_frame=bio.system_snapshot(), y=["A"], 
              title= "Initial System State",
              color_discrete_sequence = ['red'],
              labels={"value":"concentration", "variable":"Chemical", "index":"Bin number"})
fig.show()

In [30]:

# Show as heatmap
fig = px.imshow(bio.system_snapshot().T, 
                title= "Initial System State (as a heatmap)", 
                labels=dict(x="Bin number", y="Chem. species", color="Concentration"),
                text_auto=False, color_continuous_scale="gray_r") 

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

fig.show()

In [31]:

# Take a look at the frequency domain of the concentration values
frequency_data = bio.frequency_analysis(species_name="A")
frequency_data

Out[31]:

	Frequency	Relative Amplitude
0	0.0	3.0
1	10.0	1.0

In [32]:

ratio = frequency_data.loc[1, "Relative Amplitude"] / frequency_data.loc[0, "Relative Amplitude"]
ratio

Out[32]:

0.3333333333333333

In [33]:

bio.add_snapshot(data_snapshot={"ratio": ratio})
bio.get_history()

Out[33]:

	SYSTEM TIME	ratio	caption
0	0	0.333333

In [34]:

bio.diffuse(total_duration=10, n_steps=100)

Out[34]:

{'steps': 100}

In [35]:

# Take a look at the frequency domain of the concentration values
frequency_data = bio.frequency_analysis(species_name="A")
frequency_data.loc[10]

Out[35]:

Frequency               10.000000
Relative Amplitude    8721.103713
Name: 10, dtype: float64

In [36]:

ratio = frequency_data.loc[10, "Relative Amplitude"] / frequency_data.loc[0, "Relative Amplitude"]
ratio

Out[36]:

0.30839073632101904

In [37]:

bio.add_snapshot(data_snapshot={"ratio": ratio})
bio.get_history()

Out[37]:

	SYSTEM TIME	ratio	caption
0	0.0	0.333333
1	10.0	0.308391

In [38]:

fig = px.line(data_frame=bio.system_snapshot(), y=["A"], 
              title= "Initial System State",
              color_discrete_sequence = ['red'],
              labels={"value":"concentration", "variable":"Chemical", "index":"Bin number"})
fig.show()

In [39]:

for i in range(4):
    bio.diffuse(total_duration=10, n_steps=100)
    frequency_data = bio.frequency_analysis(species_name="A")
    ratio = frequency_data.loc[10, "Relative Amplitude"] / frequency_data.loc[0, "Relative Amplitude"]
    bio.add_snapshot(data_snapshot={"ratio": ratio})

In [40]:

fig = px.line(data_frame=bio.system_snapshot(), y=["A"], 
              title= f"System State at time t={bio.system_time}",
              color_discrete_sequence = ['red'],
              labels={"value":"concentration", "variable":"Chemical", "index":"Bin number"})
fig.show()

In [41]:

bio.get_history()

Out[41]:

	SYSTEM TIME	ratio
0	0.0	0.333333
1	10.0	0.308391
2	20.0	0.285576
3	30.0	0.264622
4	40.0	0.245345
5	50.0	0.227592

In [42]:

for i in range(10):
    bio.diffuse(total_duration=10, n_steps=100)
    frequency_data = bio.frequency_analysis(species_name="A")
    ratio = frequency_data.loc[10, "Relative Amplitude"] / frequency_data.loc[0, "Relative Amplitude"]
    bio.add_snapshot(data_snapshot={"ratio": ratio})

In [43]:

fig = px.line(data_frame=bio.system_snapshot(), y=["A"], 
              title= f"System State at time t={bio.system_time}",
              color_discrete_sequence = ['red'],
              labels={"value":"concentration", "variable":"Chemical", "index":"Bin number"})
fig.show()

In [44]:

bio.get_history()

Out[44]:

	SYSTEM TIME	ratio
0	0.0	0.333333
1	10.0	0.308391
2	20.0	0.285576
3	30.0	0.264622
4	40.0	0.245345
5	50.0	0.227592
6	60.0	0.211230
7	70.0	0.196140
8	80.0	0.182216
9	90.0	0.169363
10	100.0	0.157493
11	110.0	0.146526
12	120.0	0.136390
13	130.0	0.127020
14	140.0	0.118354
15	150.0	0.110337

In [45]:

for i in range(35):
    bio.diffuse(total_duration=10, n_steps=100)
    frequency_data = bio.frequency_analysis(species_name="A")
    ratio = frequency_data.loc[10, "Relative Amplitude"] / frequency_data.loc[0, "Relative Amplitude"]
    bio.add_snapshot(data_snapshot={"ratio": ratio})

In [46]:

fig = px.line(data_frame=bio.system_snapshot(), y=["A"], 
              title= f"System State at time t={bio.system_time}",
              color_discrete_sequence = ['red'],
              labels={"value":"concentration", "variable":"Chemical", "index":"Bin number"})
fig.show()

In [47]:

bio.get_history()

Out[47]:

	SYSTEM TIME	ratio
0	0.0	0.333333
1	10.0	0.308391
2	20.0	0.285576
3	30.0	0.264622
4	40.0	0.245345
5	50.0	0.227592
6	60.0	0.211230
7	70.0	0.196140
8	80.0	0.182216
9	90.0	0.169363
10	100.0	0.157493
11	110.0	0.146526
12	120.0	0.136390
13	130.0	0.127020
14	140.0	0.118354
15	150.0	0.110337
16	160.0	0.102919
17	170.0	0.096052
18	180.0	0.089695
19	190.0	0.083806
20	200.0	0.078351
21	210.0	0.073296
22	220.0	0.068611
23	230.0	0.064267
24	240.0	0.060238
25	250.0	0.056500
26	260.0	0.053032
27	270.0	0.049813
28	280.0	0.046824
29	290.0	0.044048
30	300.0	0.041469
31	310.0	0.039073
32	320.0	0.036846
33	330.0	0.034775
34	340.0	0.032849
35	350.0	0.031057
36	360.0	0.029389
37	370.0	0.027836
38	380.0	0.026390
39	390.0	0.025043
40	400.0	0.023788
41	410.0	0.022618
42	420.0	0.021526
43	430.0	0.020508
44	440.0	0.019558
45	450.0	0.018671
46	460.0	0.017842
47	470.0	0.017068
48	480.0	0.016345
49	490.0	0.015668
50	500.0	0.015035

In [48]:

for i in range(50):
    bio.diffuse(total_duration=10, n_steps=100)
    frequency_data = bio.frequency_analysis(species_name="A")
    ratio = frequency_data.loc[10, "Relative Amplitude"] / frequency_data.loc[0, "Relative Amplitude"]
    bio.add_snapshot(data_snapshot={"ratio": ratio})

In [49]:

fig = px.line(data_frame=bio.system_snapshot(), y=["A"], 
              title= f"System State at time t={bio.system_time}",
              color_discrete_sequence = ['red'],
              labels={"value":"concentration", "variable":"Chemical", "index":"Bin number"})
fig.show()

Most of the concentration graph is now flat as a board!

In [50]:

bio.get_history()

Out[50]:

	SYSTEM TIME	ratio	caption
0	0.0	0.333333
1	10.0	0.308391
2	20.0	0.285576
3	30.0	0.264622
4	40.0	0.245345
...	...	...	...
96	960.0	0.005431
97	970.0	0.005376
98	980.0	0.005324
99	990.0	0.005273
100	1000.0	0.005224

101 rows × 3 columns

Like done earlier for the smaller frequency, let's look how the amplitude of the sine signal, relative to the constant bias, changed over time¶

In [51]:

fig = px.line(data_frame=bio.get_history(), y=["ratio"], 
          title= "Component of frequency=10, relative to amplitude of constant bias",
          color_discrete_sequence = ['green'],
          labels={"value":"ratio", "variable":"Freq. 10", "index":"Time (groups of 10 units)"})
fig.show()

In [52]:

fig = px.line(data_frame=bio.get_history(), y=["ratio"], 
              title= "Component of frequency=10, relative to amplitude of constant bias (log scale in y-axis)",
              color_discrete_sequence = ['green'],
              labels={"value":"ratio", "variable":"Freq. 10", "index":"Time (groups of 10 units)"},
              log_y=True)
fig.show()

Resurrect the earlier plot for frequency=2 (using a saved dataset)¶

and then superpose them¶

In [53]:

fig_2 = px.line(data_frame=saved_freq_2_df, y=["ratio"], 
              title= "Component of frequency=2, relative to amplitude of constant bias (log scale in y-axis)",
              color_discrete_sequence = ['purple'],
              labels={"value":"ratio", "variable":"Freq. 2", "index":"Time (groups of 10 units)"},
              log_y=True)
fig_2.show()

In [54]:

# Combine the last 2 plots
all_fig = go.Figure(data=fig.data + fig_2.data)    # Note that the + is concatenating lists
all_fig.update_layout(title="Superposed plots for frequency 2 (purple) and 10 (green)")
all_fig.show()

Note how more more substantial the attenuation of the higher frequency (green) is!¶

In [ ]: