`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", 1)], products="Y",
                       forward_rate=8., reverse_rate=2.)

chem_data.describe_reactions()

# Send the plot of the reaction network to the HTML log file
chem_data.plot_reaction_network("vue_cytoscape_2")

Number of reactions: 1 (at temp. 25 C)
0: A + 2 B <-> Y  (kF = 8 / kR = 2 / delta_G = -3,436.6 / K = 4) | 1st order in all reactants & products
Set of chemicals involved in the above reactions: {'B', 'Y', 'A'}
[GRAPHIC ELEMENT SENT TO LOG FILE `down_regulate_2.log.htm`]

Set the initial concentrations of all the chemicals¶

In [5]:

dynamics = UniformCompartment(chem_data=chem_data, preset="fast")
dynamics.set_conc(conc={"A": 5., "B": 100.},
                  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
Set of chemicals involved in reactions: {'B', 'Y', 'A'}

In [ ]:

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.single_compartment_react(initial_step=0.0005, duration=0.015,
                                  variable_steps=True)

Some steps were backtracked and re-done, to prevent negative concentrations or excessively large concentration changes
31 total step(s) taken
Number of step re-do's because of negative concentrations: 0
Number of step re-do's because of elective soft aborts: 1
Norm usage: {'norm_A': 8, 'norm_B': 7, 'norm_C': 7, 'norm_D': 7}

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

In [7]:

dynamics.plot_history(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	Initialized state
1	0.000300	3.800000	97.600000	1.200000
2	0.000600	2.910608	95.821216	2.089392
3	0.000900	2.242506	94.485013	2.757494
4	0.001200	1.735641	93.471282	3.264359
5	0.001500	1.348242	92.696483	3.651758
6	0.001800	1.050487	92.100974	3.949513
7	0.002100	0.820655	91.641309	4.179345
8	0.002400	0.642668	91.285336	4.357332
9	0.002700	0.504484	91.008968	4.495516
10	0.003000	0.396991	90.793982	4.603009
11	0.003300	0.313246	90.626493	4.686754
12	0.003600	0.247926	90.495852	4.752074
13	0.003900	0.196930	90.393861	4.803070
14	0.004200	0.157089	90.314178	4.842911
15	0.004500	0.125945	90.251890	4.874055
16	0.004800	0.101589	90.203179	4.898411
17	0.005100	0.082535	90.165071	4.917465
18	0.005400	0.067626	90.135251	4.932374
19	0.005700	0.055956	90.111912	4.944044
20	0.006000	0.046821	90.093642	4.953179
21	0.006300	0.039669	90.079338	4.960331
22	0.006600	0.034069	90.068138	4.965931
23	0.007050	0.027492	90.054983	4.972508
24	0.007500	0.023054	90.046108	4.976946
25	0.007950	0.020060	90.040120	4.979940
26	0.008625	0.017029	90.034059	4.982971
27	0.009300	0.015477	90.030954	4.984523
28	0.010312	0.014284	90.028568	4.985716
29	0.011831	0.013804	90.027607	4.986196
30	0.014109	0.013874	90.027747	4.986126
31	0.017527	0.013806	90.027612	4.986194

In [9]:

dynamics.explain_time_advance(use_history=True)

From time 0 to 0.0066, in 22 steps of 0.0003
From time 0.0066 to 0.00795, in 3 steps of 0.00045
From time 0.00795 to 0.0093, in 2 steps of 0.000675
From time 0.0093 to 0.01031, in 1 step of 0.00101
From time 0.01031 to 0.01183, in 1 step of 0.00152
From time 0.01183 to 0.01411, in 1 step of 0.00228
From time 0.01411 to 0.01753, in 1 step of 0.00342
(31 steps total)

Equilibrium¶

In [10]:

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

0: A + 2 B <-> Y
Final concentrations: [A] = 0.01381 ; [B] = 90.03 ; [Y] = 4.986
1. Ratio of reactant/product concentrations, adjusted for reaction orders: 4.01174
    Formula used:  [Y] / ([A][B])
2. Ratio of forward/reverse reaction rates: 4
Discrepancy between the two values: 0.2934 %
Reaction IS in equilibrium (within 1% tolerance)

Out[10]:

True

In [ ]:

2. Now, let's suddenly increase [A]¶

In [11]:

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

SYSTEM STATE at Time t = 0.017526562:
3 species:
  Species 0 (A). Conc: 40.0
  Species 1 (B). Conc: 90.0276115740671
  Species 2 (Y). Conc: 4.986194212966452
Set of chemicals involved in reactions: {'B', 'Y', 'A'}

In [12]:

dynamics.history.get_dataframe(tail=5)

Out[12]:

	SYSTEM TIME	A	B	Y	caption
28	0.010312	0.014284	90.028568	4.985716
29	0.011831	0.013804	90.027607	4.986196
30	0.014109	0.013874	90.027747	4.986126
31	0.017527	0.013806	90.027612	4.986194
32	0.017527	40.000000	90.027612	4.986194	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)

40 total step(s) taken
Number of step re-do's because of negative concentrations: 0
Number of step re-do's because of elective soft aborts: 5
Norm usage: {'norm_A': 32, 'norm_B': 19, 'norm_C': 19, 'norm_D': 19}

In [14]:

dynamics.plot_history(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	Initialized state
1	0.000300	3.800000	97.600000	1.200000
2	0.000600	2.910608	95.821216	2.089392
3	0.000900	2.242506	94.485013	2.757494
4	0.001200	1.735641	93.471282	3.264359
...	...	...	...	...	...
68	0.036135	1.489987	13.007586	43.496207
69	0.038991	1.295592	12.618796	43.690602
70	0.043275	1.109602	12.246816	43.876592
71	0.049702	0.974898	11.977408	44.011296
72	0.059343	0.922923	11.873458	44.063271

73 rows × 5 columns

In [16]:

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

0: A + 2 B <-> Y
Final concentrations: [A] = 0.9229 ; [B] = 11.87 ; [Y] = 44.06
1. Ratio of reactant/product concentrations, adjusted for reaction orders: 4.021
    Formula used:  [Y] / ([A][B])
2. Ratio of forward/reverse reaction rates: 4
Discrepancy between the two values: 0.525 %
Reaction IS in equilibrium (within 1% tolerance)

Out[16]:

True

In [ ]:

3. Let's again suddenly increase [A]¶

In [17]:

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

SYSTEM STATE at Time t = 0.059342556:
3 species:
  Species 0 (A). Conc: 30.0
  Species 1 (B). Conc: 11.873458232380969
  Species 2 (Y). Conc: 44.06327088380949
Set of chemicals involved in reactions: {'B', 'Y', 'A'}

In [18]:

dynamics.history.get_dataframe(tail=5)

Out[18]:

	SYSTEM TIME	A	B	Y	caption
69	0.038991	1.295592	12.618796	43.690602
70	0.043275	1.109602	12.246816	43.876592
71	0.049702	0.974898	11.977408	44.011296
72	0.059343	0.922923	11.873458	44.063271
73	0.059343	30.000000	11.873458	44.063271	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)

21 total step(s) taken
Number of step re-do's because of negative concentrations: 0
Number of step re-do's because of elective soft aborts: 2
Norm usage: {'norm_A': 11, 'norm_B': 9, 'norm_C': 9, 'norm_D': 9}

In [20]:

dynamics.plot_history(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	Initialized state
1	0.000300	3.800000	97.600000	1.200000
2	0.000600	2.910608	95.821216	2.089392
3	0.000900	2.242506	94.485013	2.757494
4	0.001200	1.735641	93.471282	3.264359
...	...	...	...	...	...
90	0.070570	24.334850	0.543158	49.728421
91	0.073304	24.317670	0.508797	49.745601
92	0.077404	24.319757	0.512972	49.743514
93	0.083555	24.317814	0.509086	49.745457
94	0.092782	24.321984	0.517426	49.741287

95 rows × 5 columns

In [22]:

# Verify that the reaction has reached equilibrium
dynamics.is_in_equilibrium(tolerance=2)

0: A + 2 B <-> Y
Final concentrations: [A] = 24.32 ; [B] = 0.5174 ; [Y] = 49.74
1. Ratio of reactant/product concentrations, adjusted for reaction orders: 3.95248
    Formula used:  [Y] / ([A][B])
2. Ratio of forward/reverse reaction rates: 4
Discrepancy between the two values: 1.188 %
Reaction IS in equilibrium (within 2% 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_single_conc(species_name="A", conc=0., snapshot=True)   # Completely eliminate A
dynamics.describe_state()

SYSTEM STATE at Time t = 0.092781775:
3 species:
  Species 0 (A). Conc: 0.0
  Species 1 (B). Conc: 0.5174262547179348
  Species 2 (Y). Conc: 49.741286872641005
Set of chemicals involved in reactions: {'B', 'Y', 'A'}

In [24]:

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

19 total step(s) taken
Number of step re-do's because of negative concentrations: 0
Number of step re-do's because of elective soft aborts: 0
Norm usage: {'norm_A': 9, 'norm_B': 10, 'norm_C': 9, 'norm_D': 9}

In [25]:

dynamics.plot_history(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	Initialized state
1	0.000300	3.800000	97.600000	1.200000
2	0.000600	2.910608	95.821216	2.089392
3	0.000900	2.242506	94.485013	2.757494
4	0.001200	1.735641	93.471282	3.264359
...	...	...	...	...	...
110	0.115432	1.744222	4.005870	47.997065
111	0.121507	1.987812	4.493050	47.753475
112	0.130619	2.207024	4.931475	47.534262
113	0.144288	2.316339	5.150105	47.424947
114	0.164791	2.304340	5.126107	47.436947

115 rows × 5 columns

In [27]:

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

0: A + 2 B <-> Y
Final concentrations: [A] = 2.304 ; [B] = 5.126 ; [Y] = 47.44
1. Ratio of reactant/product concentrations, adjusted for reaction orders: 4.0159
    Formula used:  [Y] / ([A][B])
2. Ratio of forward/reverse reaction rates: 4
Discrepancy between the two values: 0.3974 %
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)¶