`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

Take the initial system to equilibrium¶

In [6]:

dynamics.single_compartment_react(time_step=0.0005, n_steps=30,
                                  dynamic_substeps=2, rel_fast_threshold=15)

df = dynamics.get_history()
df

single_compartment_react(): setting abs_fast_threshold to 300.0
30 total step(s) taken

Out[6]:

	SYSTEM TIME	A	B	Y	caption
0	0.00000	5.000000	100.000000	0.000000	Initial state
1	0.00025	4.000000	98.000000	1.000000	Interm. step, due to the fast rxns: [0]
2	0.00050	3.216500	96.433000	1.783500
3	0.00075	2.597038	95.194077	2.402962	Interm. step, due to the fast rxns: [0]
4	0.00100	2.103794	94.207589	2.896206
5	0.00125	1.708856	93.417711	3.291144	Interm. step, due to the fast rxns: [0]
6	0.00150	1.391227	92.782453	3.608773
7	0.00175	1.134868	92.269736	3.865132	Interm. step, due to the fast rxns: [0]
8	0.00200	0.927373	91.854745	4.072627
9	0.00225	0.759042	91.518084	4.240958	Interm. step, due to the fast rxns: [0]
10	0.00250	0.622230	91.244460	4.377770
11	0.00275	0.510869	91.021738	4.489131	Interm. step, due to the fast rxns: [0]
12	0.00300	0.420113	90.840226	4.579887
13	0.00325	0.346077	90.692154	4.653923	Interm. step, due to the fast rxns: [0]
14	0.00350	0.285631	90.571262	4.714369
15	0.00375	0.236248	90.472496	4.763752	Interm. step, due to the fast rxns: [0]
16	0.00400	0.195882	90.391764	4.804118
17	0.00425	0.162872	90.325744	4.837128	Interm. step, due to the fast rxns: [0]
18	0.00450	0.135867	90.271735	4.864133
19	0.00500	0.091672	90.183343	4.908328
20	0.00550	0.063511	90.127022	4.936489
21	0.00600	0.045551	90.091102	4.954449
22	0.00650	0.034091	90.068181	4.965909
23	0.00700	0.026775	90.053549	4.973225
24	0.00750	0.022103	90.044206	4.977897
25	0.00800	0.019120	90.038240	4.980880
26	0.00850	0.017215	90.034430	4.982785
27	0.00900	0.015998	90.031996	4.984002
28	0.00950	0.015221	90.030441	4.984779
29	0.01000	0.014724	90.029448	4.985276
30	0.01050	0.014407	90.028814	4.985593
31	0.01100	0.014204	90.028409	4.985796
32	0.01150	0.014075	90.028150	4.985925
33	0.01200	0.013992	90.027985	4.986008
34	0.01250	0.013940	90.027879	4.986060
35	0.01300	0.013906	90.027812	4.986094
36	0.01350	0.013884	90.027769	4.986116
37	0.01400	0.013871	90.027741	4.986129
38	0.01450	0.013862	90.027723	4.986138
39	0.01500	0.013856	90.027712	4.986144

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

Equilibrium ¶

In [7]:

# 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: 3.99712
    Formula used:  [Y] / ([A][B])
2. Ratio of forward/reverse reaction rates: 4.0
Discrepancy between the two values: 0.07189 %
Reaction IS in equilibrium (within 1% tolerance)

Out[7]:

True

Plots of changes of concentration with time¶

In [8]:

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

Now, let's suddenly increase [A]¶

In [9]:

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

SYSTEM STATE at Time t = 0.015:
3 species:
  Species 0 (A). Conc: 40.0
  Species 1 (B). Conc: 90.0277121942094
  Species 2 (Y). Conc: 4.986143902895314

In [10]:

dynamics.history.get(tail=5)

Out[10]:

	SYSTEM TIME	A	B	Y	caption
36	0.0135	0.013884	90.027769	4.986116
37	0.0140	0.013871	90.027741	4.986129
38	0.0145	0.013862	90.027723	4.986138
39	0.0150	0.013856	90.027712	4.986144
40	0.0150	40.000000	90.027712	4.986144	Set concentration of `A`

Again, take the system to equilibrium¶

In [11]:

dynamics.single_compartment_react(time_step=0.001, n_steps=40,
                                  dynamic_substeps=2, rel_fast_threshold=15)

df = dynamics.history.get()
df

single_compartment_react(): setting abs_fast_threshold to 150.0
40 total step(s) taken

Out[11]:

	SYSTEM TIME	A	B	Y	caption
0	0.00000	5.000000	100.000000	0.000000	Initial state
1	0.00025	4.000000	98.000000	1.000000	Interm. step, due to the fast rxns: [0]
2	0.00050	3.216500	96.433000	1.783500
3	0.00075	2.597038	95.194077	2.402962	Interm. step, due to the fast rxns: [0]
4	0.00100	2.103794	94.207589	2.896206
...	...	...	...	...	...
96	0.05100	1.006306	12.040324	43.979838
97	0.05200	0.997335	12.022383	43.988808
98	0.05300	0.989390	12.006493	43.996754
99	0.05400	0.982351	11.992414	44.003793
100	0.05500	0.976112	11.979937	44.010031

101 rows × 5 columns

In [12]:

# Verify that the reaction has reached equilibrium
dynamics.is_in_equilibrium(tolerance=6)

A + 2 B <-> Y
Final concentrations:  [Y] = 44.01 ; [A] = 0.9761 ; [B] = 11.98
1. Ratio of reactant/product concentrations, adjusted for reaction orders: 3.76355
    Formula used:  [Y] / ([A][B])
2. Ratio of forward/reverse reaction rates: 4.0
Discrepancy between the two values: 5.911 %
Reaction IS in equilibrium (within 6% tolerance)

Out[12]:

True

In [13]:

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]

Let's again suddenly increase [A]¶

In [14]:

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

SYSTEM STATE at Time t = 0.055:
3 species:
  Species 0 (A). Conc: 30.0
  Species 1 (B). Conc: 11.979937031950595
  Species 2 (Y). Conc: 44.010031484024715

In [15]:

dynamics.history.get(tail=5)

Out[15]:

	SYSTEM TIME	A	B	Y	caption
97	0.052	0.997335	12.022383	43.988808
98	0.053	0.989390	12.006493	43.996754
99	0.054	0.982351	11.992414	44.003793
100	0.055	0.976112	11.979937	44.010031
101	0.055	30.000000	11.979937	44.010031	Set concentration of `A`

Yet again, take the system to equilibrium¶

In [16]:

dynamics.single_compartment_react(time_step=0.001, n_steps=35,
                                  dynamic_substeps=2, rel_fast_threshold=15)

df = dynamics.get_history()
df

single_compartment_react(): setting abs_fast_threshold to 150.0
35 total step(s) taken

Out[16]:

	SYSTEM TIME	A	B	Y	caption
0	0.00000	5.000000	100.000000	0.000000	Initial state
1	0.00025	4.000000	98.000000	1.000000	Interm. step, due to the fast rxns: [0]
2	0.00050	3.216500	96.433000	1.783500
3	0.00075	2.597038	95.194077	2.402962	Interm. step, due to the fast rxns: [0]
4	0.00100	2.103794	94.207589	2.896206
...	...	...	...	...	...
140	0.08600	24.266272	0.512481	49.743760
141	0.08700	24.266271	0.512480	49.743760
142	0.08800	24.266271	0.512479	49.743760
143	0.08900	24.266271	0.512479	49.743761
144	0.09000	24.266271	0.512479	49.743761

145 rows × 5 columns

In [17]:

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

A + 2 B <-> Y
Final concentrations:  [Y] = 49.74 ; [A] = 24.27 ; [B] = 0.5125
1. Ratio of reactant/product concentrations, adjusted for reaction orders: 4
    Formula used:  [Y] / ([A][B])
2. Ratio of forward/reverse reaction rates: 4.0
Discrepancy between the two values: 6.852e-05 %
Reaction IS in equilibrium (within 1% tolerance)

Out[17]:

True

In [18]:

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

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

In [ ]:

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 [19]:

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

SYSTEM STATE at Time t = 0.09:
3 species:
  Species 0 (A). Conc: 0.0
  Species 1 (B). Conc: 0.5124787710943272
  Species 2 (Y). Conc: 49.74376061445283

In [20]:

dynamics.single_compartment_react(time_step=0.001, n_steps=70,
                                  dynamic_substeps=2, rel_fast_threshold=15)

dynamics.get_history()

single_compartment_react(): setting abs_fast_threshold to 150.0
70 total step(s) taken

Out[20]:

	SYSTEM TIME	A	B	Y	caption
0	0.00000	5.000000	100.000000	0.000000	Initial state
1	0.00025	4.000000	98.000000	1.000000	Interm. step, due to the fast rxns: [0]
2	0.00050	3.216500	96.433000	1.783500
3	0.00075	2.597038	95.194077	2.402962	Interm. step, due to the fast rxns: [0]
4	0.00100	2.103794	94.207589	2.896206
...	...	...	...	...	...
224	0.15600	2.291779	5.096036	47.451982
225	0.15700	2.293251	5.098980	47.450510
226	0.15800	2.294606	5.101691	47.449155
227	0.15900	2.295853	5.104185	47.447907
228	0.16000	2.297001	5.106481	47.446759

229 rows × 5 columns

In [21]:

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

A + 2 B <-> Y
Final concentrations:  [Y] = 47.45 ; [A] = 2.297 ; [B] = 5.106
1. Ratio of reactant/product concentrations, adjusted for reaction orders: 4.04505
    Formula used:  [Y] / ([A][B])
2. Ratio of forward/reverse reaction rates: 4.0
Discrepancy between the two values: 1.126 %
Reaction IS in equilibrium (within 2% tolerance)

Out[21]:

True

In [22]:

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

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."