`A` down-regulates `B` ,¶

by being the limiting reagent in reaction `A + 2 B <-> Y` (mostly forward)¶

1st-order kinetics.
If [A] is low and [B] is high, then [B] remains high. If [A] goes high, [B] goes low. However, at that point, A can no longer bring B up to any substantial extent.

Initialize the system¶

In [4]:

# Initialize the system
chem_data = chem(names=["A", "B", "Y"])

# Reaction A + 2 B <-> Y , with 1st-order kinetics for all species
chem_data.add_reaction(reactants=[("A") , (2, "B")], products=[("Y")],
                       forward_rate=8., reverse_rate=2.)

chem_data.describe_reactions()

# Send the plot of the reaction network to the HTML log file
graph_data = chem_data.prepare_graph_network()
GraphicLog.export_plot(graph_data, "vue_cytoscape_1")

Number of reactions: 1 (at temp. 25 C)
0: A + 2 B <-> Y  (kF = 8 / kR = 2 / Delta_G = -3,436.56 / K = 4) | 1st order in all reactants & products
[GRAPHIC ELEMENT SENT TO LOG FILE `down_regulate_2.log.htm`]

Set the initial concentrations of all the chemicals¶

In [5]:

dynamics = ReactionDynamics(reaction_data=chem_data)
dynamics.set_conc(conc={"A": 5., "B": 100., "Y": 0.},
                  snapshot=True)      # A is scarce, B is plentiful, Y is absent
dynamics.describe_state()

SYSTEM STATE at Time t = 0:
3 species:
  Species 0 (A). Conc: 5.0
  Species 1 (B). Conc: 100.0
  Species 2 (Y). Conc: 0.0

1. Take the initial system to equilibrium¶

In [6]:

# All of these settings are currently close to the default values... but subject to change; set for repeatability
dynamics.set_thresholds(norm="norm_A", low=0.5, high=1.0, abort=1.44)
dynamics.set_thresholds(norm="norm_B", low=0.2, high=0.5, abort=1.5)
dynamics.set_step_factors(upshift=1.4, downshift=0.5, abort=0.5)
dynamics.set_error_step_factor(0.333)

dynamics.single_compartment_react(initial_step=0.0005, reaction_duration=0.015,
                                  variable_steps=True, explain_variable_steps=False)

INFO: the tentative time step (0.0005) leads to a least one norm value > its ABORT threshold:
      -> will backtrack, and re-do step with a SMALLER delta time, multiplied by 0.5 (set to 0.00025) [Step started at t=0, and will rewind there]
28 total step(s) taken

A, as the scarse limiting reagent, stops the reaction.
When A is low, B is also low.

In [7]:

dynamics.plot_curves(colors=['red', 'darkorange', 'green'],
                     title="Changes in concentrations (reaction A + 2 B <-> Y)")

In [8]:

dynamics.get_history()

Out[8]:

	SYSTEM TIME	A	B	Y	caption
0	0.000000	5.000000	100.000000	0.000000	Initial state
1	0.000250	4.000000	98.000000	1.000000
2	0.000500	3.216500	96.433000	1.783500
3	0.000625	2.906769	95.813538	2.093231
4	0.000800	2.517591	95.035182	2.482409
5	0.001045	2.049858	94.099716	2.950142
6	0.001388	1.522589	93.045178	3.477411
7	0.001731	1.136233	92.272466	3.863767
8	0.002074	0.851194	91.702389	4.148806
9	0.002417	0.639853	91.279707	4.360147
10	0.002760	0.482579	90.965159	4.517421
11	0.003103	0.365222	90.730445	4.634778
12	0.003446	0.277475	90.554949	4.722525
13	0.003789	0.211767	90.423533	4.788233
14	0.004132	0.162507	90.325015	4.837493
15	0.004475	0.125548	90.251096	4.874452
16	0.004818	0.097800	90.195600	4.902200
17	0.005161	0.076958	90.153916	4.923042
18	0.005504	0.061297	90.122594	4.938703
19	0.005847	0.049526	90.099053	4.950474
20	0.006327	0.037139	90.074277	4.962861
21	0.006807	0.029054	90.058108	4.970946
22	0.007288	0.023776	90.047553	4.976224
23	0.007960	0.018952	90.037905	4.981048
24	0.008632	0.016472	90.032944	4.983528
25	0.009573	0.014686	90.029373	4.985314
26	0.010891	0.013887	90.027773	4.986113
27	0.012736	0.013833	90.027665	4.986167
28	0.015318	0.013858	90.027716	4.986142

In [9]:

dynamics.explain_time_advance(use_history=True)

From time 0 to 0.0005, in 2 steps of 0.00025
From time 0.0005 to 0.000625, in 1 step of 0.000125
From time 0.000625 to 0.0008, in 1 step of 0.000175
From time 0.0008 to 0.001045, in 1 step of 0.000245
From time 0.001045 to 0.005847, in 14 steps of 0.000343
From time 0.005847 to 0.007288, in 3 steps of 0.00048
From time 0.007288 to 0.008632, in 2 steps of 0.000672
From time 0.008632 to 0.009573, in 1 step of 0.000941
From time 0.009573 to 0.01089, in 1 step of 0.00132
From time 0.01089 to 0.01274, in 1 step of 0.00184
From time 0.01274 to 0.01532, in 1 step of 0.00258
(28 steps total)

Equilibrium¶

In [10]:

# Verify that the reaction has reached equilibrium
dynamics.is_in_equilibrium()

A + 2 B <-> Y
Final concentrations:  [Y] = 4.986 ; [A] = 0.01386 ; [B] = 90.03
1. Ratio of reactant/product concentrations, adjusted for reaction orders: 3.99663
    Formula used:  [Y] / ([A][B])
2. Ratio of forward/reverse reaction rates: 4.0
Discrepancy between the two values: 0.08418 %
Reaction IS in equilibrium (within 1% tolerance)

Out[10]:

True

In [ ]:

2. Now, let's suddenly increase [A]¶

In [11]:

dynamics.set_chem_conc(species_name="A", conc=40., snapshot=True)
dynamics.describe_state()

SYSTEM STATE at Time t = 0.015318388:
3 species:
  Species 0 (A). Conc: 40.0
  Species 1 (B). Conc: 90.02771559198881
  Species 2 (Y). Conc: 4.98614220400561

In [12]:

dynamics.history.get_dataframe(tail=5)

Out[12]:

	SYSTEM TIME	A	B	Y	caption
25	0.009573	0.014686	90.029373	4.985314
26	0.010891	0.013887	90.027773	4.986113
27	0.012736	0.013833	90.027665	4.986167
28	0.015318	0.013858	90.027716	4.986142
29	0.015318	40.000000	90.027716	4.986142	Set concentration of `A`

Again, take the system to equilibrium¶

In [13]:

dynamics.single_compartment_react(initial_step=0.0005, target_end_time=0.055,
                                  variable_steps=True, explain_variable_steps=False)

INFO: the tentative time step (0.0005) leads to a least one norm value > its ABORT threshold:
      -> will backtrack, and re-do step with a SMALLER delta time, multiplied by 0.5 (set to 0.00025) [Step started at t=0.015318, and will rewind there]
INFO: the tentative time step (0.00025) leads to a least one norm value > its ABORT threshold:
      -> will backtrack, and re-do step with a SMALLER delta time, multiplied by 0.5 (set to 0.000125) [Step started at t=0.015318, and will rewind there]
INFO: the tentative time step (0.000125) leads to a least one norm value > its ABORT threshold:
      -> will backtrack, and re-do step with a SMALLER delta time, multiplied by 0.5 (set to 6.25e-05) [Step started at t=0.015318, and will rewind there]
INFO: the tentative time step (6.25e-05) leads to a least one norm value > its ABORT threshold:
      -> will backtrack, and re-do step with a SMALLER delta time, multiplied by 0.5 (set to 3.125e-05) [Step started at t=0.015318, and will rewind there]
47 total step(s) taken

In [14]:

dynamics.plot_curves(colors=['red', 'darkorange', 'green'],
                     title="Changes in concentrations (reaction A + 2 B <-> Y)")

A, still the limiting reagent, is again stopping the reaction.
The (transiently) high value of [A] led to a high value of [B]

In [15]:

dynamics.get_history()

Out[15]:

	SYSTEM TIME	A	B	Y	caption
0	0.000000	5.000000	100.000000	0.000000	Initial state
1	0.000250	4.000000	98.000000	1.000000
2	0.000500	3.216500	96.433000	1.783500
3	0.000625	2.906769	95.813538	2.093231
4	0.000800	2.517591	95.035182	2.482409
...	...	...	...	...	...
72	0.034954	1.383073	12.793862	43.603069
73	0.038427	1.194334	12.416383	43.791809
74	0.043288	1.043380	12.114476	43.942762
75	0.050094	0.953305	11.934325	44.032838
76	0.059623	0.925189	11.878093	44.060953

77 rows × 5 columns

In [16]:

# Verify that the reaction has reached equilibrium
dynamics.is_in_equilibrium()

A + 2 B <-> Y
Final concentrations:  [Y] = 44.06 ; [A] = 0.9252 ; [B] = 11.88
1. Ratio of reactant/product concentrations, adjusted for reaction orders: 4.00938
    Formula used:  [Y] / ([A][B])
2. Ratio of forward/reverse reaction rates: 4.0
Discrepancy between the two values: 0.2344 %
Reaction IS in equilibrium (within 1% tolerance)

Out[16]:

True

In [ ]:

3. Let's again suddenly increase [A]¶

In [17]:

dynamics.set_chem_conc(species_name="A", conc=30., snapshot=True)
dynamics.describe_state()

SYSTEM STATE at Time t = 0.059622836:
3 species:
  Species 0 (A). Conc: 30.0
  Species 1 (B). Conc: 11.878093044952234
  Species 2 (Y). Conc: 44.06095347752391

In [18]:

dynamics.history.get_dataframe(tail=5)

Out[18]:

	SYSTEM TIME	A	B	Y	caption
73	0.038427	1.194334	12.416383	43.791809
74	0.043288	1.043380	12.114476	43.942762
75	0.050094	0.953305	11.934325	44.032838
76	0.059623	0.925189	11.878093	44.060953
77	0.059623	30.000000	11.878093	44.060953	Set concentration of `A`

Yet again, take the system to equilibrium¶

In [19]:

dynamics.single_compartment_react(initial_step=0.001, target_end_time=0.09,
                                  variable_steps=True, explain_variable_steps=False)

INFO: the tentative time step (0.001) leads to a least one norm value > its ABORT threshold:
      -> will backtrack, and re-do step with a SMALLER delta time, multiplied by 0.5 (set to 0.0005) [Step started at t=0.059623, and will rewind there]
19 total step(s) taken

In [20]:

dynamics.plot_curves(colors=['red', 'darkorange', 'green'],
                     title="Changes in concentrations (reaction A + 2 B <-> Y)")

In [21]:

dynamics.get_history()

Out[21]:

	SYSTEM TIME	A	B	Y	caption
0	0.000000	5.000000	100.000000	0.000000	Initial state
1	0.000250	4.000000	98.000000	1.000000
2	0.000500	3.216500	96.433000	1.783500
3	0.000625	2.906769	95.813538	2.093231
4	0.000800	2.517591	95.035182	2.482409
...	...	...	...	...	...
92	0.072916	24.315963	0.510018	49.744991
93	0.076605	24.316986	0.512064	49.743968
94	0.081771	24.316330	0.510753	49.744624
95	0.089002	24.317286	0.512664	49.743668
96	0.099126	24.314800	0.507693	49.746154

97 rows × 5 columns

In [22]:

# Verify that the reaction has reached equilibrium
dynamics.is_in_equilibrium()

A + 2 B <-> Y
Final concentrations:  [Y] = 49.75 ; [A] = 24.31 ; [B] = 0.5077
1. Ratio of reactant/product concentrations, adjusted for reaction orders: 4.02984
    Formula used:  [Y] / ([A][B])
2. Ratio of forward/reverse reaction rates: 4.0
Discrepancy between the two values: 0.746 %
Reaction IS in equilibrium (within 1% tolerance)

Out[22]:

True

A, again the scarse limiting reagent, stops the reaction yet again

In [ ]:

4. A can down-regulate B, but it cannot bring it up to any significant amount¶

Even if A is completely taken out (i.e., [A] set to 0), [B] can only slightly increase, from the reverse reaction ("Le Chatelier's principle".)¶

Let's try it:

In [23]:

dynamics.set_chem_conc(species_name="A", conc=0., snapshot=True)   # Completely eliminate A
dynamics.describe_state()

SYSTEM STATE at Time t = 0.099125931:
3 species:
  Species 0 (A). Conc: 0.0
  Species 1 (B). Conc: 0.5076929235350717
  Species 2 (Y). Conc: 49.74615353823248

In [24]:

dynamics.single_compartment_react(initial_step=0.001, target_end_time=0.16,
                                  variable_steps=True, explain_variable_steps=False)

19 total step(s) taken

In [25]:

dynamics.plot_curves(colors=['red', 'darkorange', 'green'],
                     title="Changes in concentrations (reaction A + 2 B <-> Y)")

In [26]:

dynamics.get_history()

Out[26]:

	SYSTEM TIME	A	B	Y	caption
0	0.000000	5.000000	100.000000	0.000000	Initial state
1	0.000250	4.000000	98.000000	1.000000
2	0.000500	3.216500	96.433000	1.783500
3	0.000625	2.906769	95.813538	2.093231
4	0.000800	2.517591	95.035182	2.482409
...	...	...	...	...	...
112	0.124564	1.876146	4.259985	47.870007
113	0.131943	2.110805	4.729304	47.635348
114	0.142274	2.269994	5.047682	47.476159
115	0.156737	2.317528	5.142749	47.428625
116	0.176984	2.307597	5.122887	47.438557

117 rows × 5 columns

In [27]:

# Verify that the reaction has reached equilibrium
dynamics.is_in_equilibrium()

A + 2 B <-> Y
Final concentrations:  [Y] = 47.44 ; [A] = 2.308 ; [B] = 5.123
1. Ratio of reactant/product concentrations, adjusted for reaction orders: 4.01289
    Formula used:  [Y] / ([A][B])
2. Ratio of forward/reverse reaction rates: 4.0
Discrepancy between the two values: 0.3221 %
Reaction IS in equilibrium (within 1% tolerance)

Out[27]:

True

As expected, even the complete withdrawal of A (red), brings about only a modest increase of B's concentration, from the reverse reaction (i.e. [B] slightly increases at the expense of [Y].)¶

The change is modest because our reaction A + 2 B <-> Y is mostly in the forward direction (K = 4)¶

Le Chatelier's principle in action: "A change in one of the variables that describe a system at equilibrium produces a shift in the position of the equilibrium that counteracts the effect of this change."

In [ ]:

A down-regulates B ,¶

by being the limiting reagent in reaction A + 2 B <-> Y (mostly forward)¶

Initialize the system¶

Set the initial concentrations of all the chemicals¶

1. Take the initial system to equilibrium¶

Equilibrium¶

2. Now, let's suddenly increase [A]¶

Again, take the system to equilibrium¶

3. Let's again suddenly increase [A]¶

Yet again, take the system to equilibrium¶

4. A can down-regulate B, but it cannot bring it up to any significant amount¶

Even if A is completely taken out (i.e., [A] set to 0), [B] can only slightly increase, from the reverse reaction ("Le Chatelier's principle".)¶

As expected, even the complete withdrawal of A (red), brings about only a modest increase of B's concentration, from the reverse reaction (i.e. [B] slightly increases at the expense of [Y].)¶

The change is modest because our reaction A + 2 B <-> Y is mostly in the forward direction (K = 4)¶

`A` down-regulates `B` ,¶

by being the limiting reagent in reaction `A + 2 B <-> Y` (mostly forward)¶