Reversible Synthesis elementary reaction `A + B <-> C`¶

taken to equilibrium.¶

Comparison of 2 approximate solutions, and the exact solutions¶

TAGS : "uniform compartment", "numerical"¶

In [1]:

LAST_REVISED = "Jan. 4, 2026"
LIFE123_VERSION = "1.0.0rc7"        # Library version this experiment is based on

In [2]:

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

In [3]:

#import sys
#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   

import pandas as pd
from life123 import check_version, UniformCompartment,PlotlyHelper

In [4]:

check_version(LIFE123_VERSION)

OK

In [ ]:

1. Run a simulation with low accuracy¶

Initialize the System¶

Specify the chemicals, the reactions, and the initial concentrations

In [5]:

# Instantiate the simulator and specify the chemicals
# Here we use the "fast" preset for the variable steps, which leads to fewer steps, but generally less-accurate results
uc_fast = UniformCompartment(preset="fast")    

In [6]:

# Elementary reaction A + B <-> C
uc_fast.add_reaction(reactants=["A" , "B"], products="C", kF=5., kR=2.)

uc_fast.describe_reactions()

add_reaction(): detected reaction type `ReactionSynthesis`
Number of reactions: 1
0: A + B  <-> C  Elementary Synthesis reaction  (kF = 5 / kR = 2 / delta_G = -2,271.4 / K = 2.5 / Temp = 25 C)
Chemicals involved in the above reactions: ['A', 'B', 'C']

In [7]:

# Set the initial concentrations of all the chemicals
uc_fast.set_conc({"A": 10., "B": 50., "C": 20.})

In [ ]:

Run the reaction¶

In [8]:

uc_fast.single_compartment_react(initial_step=0.004, duration=0.06, variable_steps=True)

23 total variable step(s) taken in 0.046 sec
Number of step re-do's because of elective soft aborts: 3
Norm usage: {'norm_A': 13, 'norm_B': 9, 'norm_C': 9, 'norm_D': 9}
System Time is now: 0.0674

In [9]:

uc_fast.get_history()

Out[9]:

	SYSTEM TIME	A	B	C	step	caption
0	0.000000	10.000000	50.000000	20.000000		Set concentration
1	0.000864	7.874560	47.874560	22.125440	1	1st reaction step
2	0.001555	6.602265	46.602265	23.397735	2
3	0.002246	5.571266	45.571266	24.428734	3
4	0.002938	4.727594	44.727594	25.272406	4
5	0.003629	4.031746	44.031746	25.968254	5
6	0.004666	3.165305	43.165305	26.834695	6
7	0.005702	2.512652	42.512652	27.487348	7
8	0.006739	2.015898	42.015898	27.984102	8
9	0.007776	1.634842	41.634842	28.365158	9
10	0.008813	1.340804	41.340804	28.659196	10
11	0.009850	1.112883	41.112883	28.887117	11
12	0.010886	0.935595	40.935595	29.064405	12
13	0.011923	0.797321	40.797321	29.202679	13
14	0.013478	0.635211	40.635211	29.364789	14
15	0.015034	0.525833	40.525833	29.474167	15
16	0.016589	0.451805	40.451805	29.548195	16
17	0.018922	0.376490	40.376490	29.623510	17
18	0.021254	0.337393	40.337393	29.662607	18
19	0.024754	0.306871	40.306871	29.693129	19
20	0.030002	0.293965	40.293965	29.706035	20
21	0.037876	0.295437	40.295437	29.704563	21
22	0.049685	0.294082	40.294082	29.705918	22
23	0.067400	0.296969	40.296969	29.703031	23	last reaction step

Plots changes of concentrations with time¶

In [10]:

uc_fast.plot_history(show_intervals=True)

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2. Let's now repeat the simulation with the "slow" preset, which yields more data points¶

and, generally, more accuracy¶

In [11]:

# Instantiate the simulator and specify the chemicals
uc_slow = UniformCompartment(reactions=uc_fast.get_reactions(), preset="slow")
# Re-use the chemicals and reactions of part 1, but now with the "slow" preset

In [12]:

# Set the initial concentrations of all the chemicals
uc_slow.set_conc({"A": 10., "B": 50., "C": 20.})

In [13]:

uc_slow.single_compartment_react(initial_step=0.004, duration=0.06, variable_steps=True)

80 total variable step(s) taken in 0.166 sec
Number of step re-do's because of elective soft aborts: 2
Norm usage: {'norm_A': 38, 'norm_B': 36, 'norm_C': 36, 'norm_D': 36}
System Time is now: 0.061008

Note that 80 steps were now used, instead of the earlier 23¶

In [14]:

uc_slow.get_history()

Out[14]:

	SYSTEM TIME	A	B	C	step	caption
0	0.000000	10.000000	50.000000	20.000000		Set concentration
1	0.000160	9.606400	49.606400	20.393600	1	1st reaction step
2	0.000336	9.194224	49.194224	20.805776	2
3	0.000530	8.764451	48.764451	21.235549	3
4	0.000743	8.318407	48.318407	21.681593	4
...	...	...	...	...	...	...
76	0.045329	0.294997	40.294997	29.705003	76
77	0.048708	0.294914	40.294914	29.705086	77
78	0.052424	0.294886	40.294886	29.705114	78
79	0.056511	0.294879	40.294879	29.705121	79
80	0.061008	0.294878	40.294878	29.705122	80	last reaction step

81 rows × 6 columns

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3. And, finally, get the EXACT analytical solution¶

In [15]:

# Instantiate the simulator and specify the chemicals
uc_exact = UniformCompartment(preset="slow", reactions=uc_fast.get_reactions(),
                              exact=True)
# Re-use the chemicals and reactions of part 1; 
# the "slow" preset doesn't particularly matter now because it's just 1 reaction, and we'll be getting the analytical solutin.  
# Note the `exact` flag

In [16]:

# Set the initial concentrations of all the chemicals
uc_exact.set_conc({"A": 10., "B": 50., "C": 20.}, snapshot=True)

In [17]:

uc_exact.single_compartment_react(initial_step=0.004, duration=0.06, variable_steps=True)

79 total variable step(s) taken in 0.156 sec
Number of step re-do's because of elective soft aborts: 2
Norm usage: {'norm_A': 38, 'norm_B': 36, 'norm_C': 36, 'norm_D': 36}
System Time is now: 0.061037

In [18]:

uc_exact.get_history() 

Out[18]:

	SYSTEM TIME	A	B	C	step	caption
0	0.000000	10.000000	50.000000	20.000000		Set concentration
1	0.000160	9.615719	49.615719	20.384281	1	1st reaction step
2	0.000336	9.213647	49.213647	20.786353	2
3	0.000530	8.794750	48.794750	21.205250	3
4	0.000743	8.360318	48.360318	21.639682	4
...	...	...	...	...	...	...
75	0.045358	0.295598	40.295598	29.704402	75
76	0.048737	0.295238	40.295238	29.704762	76
77	0.052453	0.295046	40.295046	29.704954	77
78	0.056540	0.294950	40.294950	29.705050	78
79	0.061037	0.294906	40.294906	29.705094	79	last reaction step

80 rows × 6 columns

In [ ]:

4. Let's compare plots for the concentration of `C`, as a function of time, from the earlier 3 simulations¶

In [19]:

p1 = uc_fast.plot_history(chemicals="C", colors=['#F5B914'], title="fast (less precise)", show=False)
p2 = uc_slow.plot_history(chemicals="C", colors=['#CBE504'], title="slow (more precise)", show=False)
p3 = uc_exact.plot_history(chemicals="C", colors=['forestgreen'], title="exact",  show=False)

In [20]:

PlotlyHelper.combine_plots([p1, p2, p3], title="Comparison of simulation accuracies", xrange=[0, 0.015])

The comparison reveals a gradient "less precise" -> "more precise" -> "exact" solution¶

In [ ]:

Reversible Synthesis elementary reaction A + B <-> C¶

taken to equilibrium.¶

Comparison of 2 approximate solutions, and the exact solutions¶

TAGS : "uniform compartment", "numerical"¶

1. Run a simulation with low accuracy¶

Initialize the System¶

Run the reaction¶

Plots changes of concentrations with time¶

2. Let's now repeat the simulation with the "slow" preset, which yields more data points¶

and, generally, more accuracy¶

Note that 80 steps were now used, instead of the earlier 23¶

3. And, finally, get the EXACT analytical solution¶

4. Let's compare plots for the concentration of C, as a function of time, from the earlier 3 simulations¶

The comparison reveals a gradient "less precise" -> "more precise" -> "exact" solution¶

Reversible Synthesis elementary reaction `A + B <-> C`¶

4. Let's compare plots for the concentration of `C`, as a function of time, from the earlier 3 simulations¶