Excessively large time steps - and remediation¶

Based on experiment `cycles_1`¶

A cycle of reactions `A <-> B <-> C <-> A` and `C + E_High <-> A + E_Low`¶

RUN 1 - Initially, very large time steps are taken - at the edge of pushing some concentrations into negative values.¶

The system automatic detects those problems, intercepts the problematic steps and re-runs them with 1/2 the time step.
Negative concentrations are automatically avoided, but nonetheless the plots are ragged... and the solutions are eventually unstable.

RUN 2 - Same primary steps as for run #1, but with the option of using 1/2 substeps as needed,¶

with thresholds that lead to those substeps being actually utilized a fair part of the time.
The raggedness and instabilities are now eliminated. (Note: the 1/2 substeps are on a per-reaction basis)

LAST REVISED: Dec. 3, 2023 THIS IS AN ARCHIVED EXPERIMENT

IMPORTANT: DO NOT ATTEMPT TO RUN THIS NOTEBOOK!¶

This is a "frozen run" that depends on an old version of Life123, for demonstration purposes¶

(newer versions of Life123 tend to recover from instability more gracefully!!)¶

If you bypass the execution exit in the first cell, and run the other cells, you WILL NOT REPLICATE the results below!

In [1]:

# To stop the current and subsequent cells: USED TO PREVENT ACCIDENTAL RUNS OF THIS NOTEBOOK!

class StopExecution(Exception):
    def _render_traceback_(self):
        return []

raise StopExecution     # See: https://stackoverflow.com/a/56953105/5478830

In [ ]:

from experiments.get_notebook_info import get_notebook_basename

from src.modules.chemicals.chem_data import ChemData as chem
from src.modules.reactions.reaction_dynamics import ReactionDynamics

from src.modules.visualization.graphic_log import GraphicLog

In [3]:

# Initialize the HTML logging
log_file = get_notebook_basename() + ".log.htm"    # Use the notebook base filename for the log file

# Set up the use of some specified graphic (Vue) components
GraphicLog.config(filename=log_file,
                  components=["vue_cytoscape_1"],
                  extra_js="https://cdnjs.cloudflare.com/ajax/libs/cytoscape/3.21.2/cytoscape.umd.js")

-> Output will be LOGGED into the file 'large_time_steps_1.log.htm'

Initialize the system¶

In [4]:

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

# Reaction A <-> B, mostly in forward direction (favored energetically)
# Note: all reactions in this experiment have 1st-order kinetics for all species
chem_data.add_reaction(reactants="A", products="B",
                       forward_rate=9., reverse_rate=3.)

# Reaction B <-> C, also favored energetically
chem_data.add_reaction(reactants="B", products="C",
                       forward_rate=8., reverse_rate=4.)

# Reaction C + E_High <-> A + E_Low, also favored energetically, but kinetically slow
# Note that, thanks to the energy donation from E, we can go "upstream" from C, to the higher-energy level of "A"
chem_data.add_reaction(reactants=["C" , "E_high"], products=["A", "E_low"],
                       forward_rate=1., reverse_rate=0.2)

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: 3 (at temp. 25 C)
0: A <-> B  (kF = 9 / kR = 3 / Delta_G = -2,723.41 / K = 3) | 1st order in all reactants & products
1: B <-> C  (kF = 8 / kR = 4 / Delta_G = -1,718.28 / K = 2) | 1st order in all reactants & products
2: C + E_high <-> A + E_low  (kF = 1 / kR = 0.2 / Delta_G = -3,989.73 / K = 5) | 1st order in all reactants & products
[GRAPHIC ELEMENT SENT TO LOG FILE `large_time_steps_1.log.htm`]

Set the initial concentrations of all the chemicals¶

In [5]:

initial_conc = {"A": 100., "B": 0., "C": 0., "E_high": 1000., "E_low": 0.}  # Note the abundant energy source "E_high"
initial_conc

Out[5]:

{'A': 100.0, 'B': 0.0, 'C': 0.0, 'E_high': 1000.0, 'E_low': 0.0}

We'll split each simulation in three segments (apparent from the graphs of the runs, further down):¶

Time [0-0.03] fast changes
Time [0.03-5.] medium changes
Time [5.-8.] slow changes, as we approach equilibrium

RUN 1 - Initially, very large time steps are taken - at the edge of pushing some concentrations into negative values.¶

The system automatic detects those problems, intercepts the problematic steps and re-runs them with 1/2 the time step.
Negative concentrations are automatically avoided, but nonetheless the plots are ragged... and the solutions are eventually unstable.

In [6]:

dynamics = ReactionDynamics(chem_data=chem_data)
dynamics.set_conc(conc=initial_conc, snapshot=True)
dynamics.describe_state()

SYSTEM STATE at Time t = 0:
5 species:
  Species 0 (A). Conc: 100.0
  Species 1 (B). Conc: 0.0
  Species 2 (C). Conc: 0.0
  Species 3 (E_high). Conc: 1000.0
  Species 4 (E_low). Conc: 0.0

In [19]:

dynamics.set_diagnostics()       # To save diagnostic information about the call to single_compartment_react()

In [20]:

#dynamics.verbose_list = ["substeps"]     # Uncomment for debugging information

In [9]:

dynamics.single_compartment_react(time_step=0.0012, stop_time=0.03)
#dynamics.get_history()

The computation took 36 extra step(s) - automatically added to prevent negative concentrations
44 total step(s) taken

In [22]:

dynamics.explain_time_advance()

From time 0 to 0.0012, in 2 substeps of 0.0006 (each 1/2 of full step)
From time 0.0012 to 0.0162, in 50 substeps of 0.0003 (each 1/4 of full step)
From time 0.0162 to 0.024, in 13 substeps of 0.0006 (each 1/2 of full step)
From time 0.024 to 0.0312, in 6 FULL steps of 0.0012

Notice how the system automatically intervened and reduced some steps in 1/2, to prevent negative concentrations¶

In [11]:

dynamics.single_compartment_react(time_step=0.0025, stop_time=5.)
#dynamics.get_history()

The computation took 130 extra step(s) - automatically added to prevent negative concentrations
2053 total step(s) taken

In [24]:

dynamics.explain_time_advance()

From time 0 to 0.0012, in 2 substeps of 0.0006 (each 1/2 of full step)
From time 0.0012 to 0.0162, in 50 substeps of 0.0003 (each 1/4 of full step)
From time 0.0162 to 0.024, in 13 substeps of 0.0006 (each 1/2 of full step)
From time 0.024 to 0.0312, in 6 FULL steps of 0.0012
From time 0.0312 to 0.0337, in 2 substeps of 0.00125 (each 1/2 of full step)
From time 0.0337 to 0.17, in 218 substeps of 0.000625 (each 1/4 of full step)
From time 0.17 to 0.172, in 1 FULL step of 0.0025
From time 0.172 to 2.28, in 1686 substeps of 0.00125 (each 1/2 of full step)
From time 2.28 to 5, in 1089 FULL steps of 0.0025

Notice how the system automatically periodically intervened and reduced some steps in 1/2, to prevent negative concentrations.¶

The time step that was originally requested hovers near the cusp of being so large as to lead to negative concentrations

In [13]:

dynamics.single_compartment_react(time_step=0.008, stop_time=8.)

The computation took 49 extra step(s) - automatically added to prevent negative concentrations
400 total step(s) taken

In [26]:

df = dynamics.get_history()
df

Out[26]:

	SYSTEM TIME	A	B	C	E_high	E_low	caption
0	0.00000	100.000000	0.000000	0.000000	1000.000000	0.000000	Initial state
1	0.00060	99.460000	0.540000	0.000000	1000.000000	0.000000	Interm. step, due to the fast rxns: [0, 1, 2]
2	0.00120	98.923888	1.073520	0.002592	1000.000000	0.000000
3	0.00150	98.658537	1.337075	0.004388	999.999222	0.000778	Interm. step, due to the fast rxns: [0]
4	0.00180	98.394140	1.599676	0.006183	999.998445	0.001555
...	...	...	...	...	...	...	...
3623	7.97045	10.018391	30.006875	59.974735	32.330220	967.669780
3624	7.97845	10.018012	30.006785	59.975202	32.329439	967.670561
3625	7.98645	10.017978	30.006691	59.975330	32.328339	967.671661
3626	7.99445	10.017485	30.006608	59.975908	32.327699	967.672301
3627	8.00245	10.017621	30.006514	59.975865	32.326462	967.673538

3628 rows × 7 columns

Notice that with our aggressively-large time steps, we ended up with far fewer data points than in any of the runs in the experiment `cycles_1`, which this analysis is based on¶

In [27]:

dynamics.explain_time_advance()

From time 0 to 0.0012, in 2 substeps of 0.0006 (each 1/2 of full step)
From time 0.0012 to 0.0162, in 50 substeps of 0.0003 (each 1/4 of full step)
From time 0.0162 to 0.024, in 13 substeps of 0.0006 (each 1/2 of full step)
From time 0.024 to 0.0312, in 6 FULL steps of 0.0012
From time 0.0312 to 0.0337, in 2 substeps of 0.00125 (each 1/2 of full step)
From time 0.0337 to 0.17, in 218 substeps of 0.000625 (each 1/4 of full step)
From time 0.17 to 0.172, in 1 FULL step of 0.0025
From time 0.172 to 2.28, in 1686 substeps of 0.00125 (each 1/2 of full step)
From time 2.28 to 5, in 1089 FULL steps of 0.0025
From time 5 to 6.42, in 354 substeps of 0.004 (each 1/2 of full step)
From time 6.42 to 6.5, in 10 FULL steps of 0.008
From time 6.5 to 6.51, in 2 substeps of 0.004 (each 1/2 of full step)
From time 6.51 to 6.64, in 17 FULL steps of 0.008
From time 6.64 to 6.65, in 2 substeps of 0.004 (each 1/2 of full step)
From time 6.65 to 6.8, in 19 FULL steps of 0.008
From time 6.8 to 6.81, in 2 substeps of 0.004 (each 1/2 of full step)
From time 6.81 to 6.98, in 21 FULL steps of 0.008
From time 6.98 to 6.99, in 2 substeps of 0.004 (each 1/2 of full step)
From time 6.99 to 7.17, in 23 FULL steps of 0.008
From time 7.17 to 7.18, in 2 substeps of 0.004 (each 1/2 of full step)
From time 7.18 to 7.38, in 25 FULL steps of 0.008
From time 7.38 to 7.39, in 2 substeps of 0.004 (each 1/2 of full step)
From time 7.39 to 7.6, in 27 FULL steps of 0.008
From time 7.6 to 7.61, in 2 substeps of 0.004 (each 1/2 of full step)
From time 7.61 to 7.84, in 29 FULL steps of 0.008
From time 7.84 to 7.85, in 2 substeps of 0.004 (each 1/2 of full step)
From time 7.85 to 8, in 19 FULL steps of 0.008

Yet again, the system automatically keeps periodically throttling down the time steps, to prevent negative concentrations.¶

At times, especially near the end, the throttling gets very frequent ("frantic"!)

Plots of changes of concentration with time¶

In [16]:

dynamics.plot_history(chemicals=["E_high", "E_low"], colors=["red", "grey"])

In [17]:

dynamics.plot_history(chemicals=["A", "B", "C"])

The negative concentrations were automatically avoided, but nonetheless the plots are ragged... and the solutions display instabilities¶

In [ ]:

RUN 2 - Same primary steps as for run #1, but with the option of using 1/2 substeps as needed,¶

with thresholds that lead to those substeps being actually utilized a fair part of the time.
The raggedness and instabilities are now eliminated. (Note: the 1/2 substeps are on a per-reaction basis)

In [18]:

dynamics = ReactionDynamics(chem_data=chem_data)   # Note: OVER-WRITING the "dynamics" object
dynamics.set_conc(conc=initial_conc, snapshot=True) 
dynamics.describe_state()

SYSTEM STATE at Time t = 0:
5 species:
  Species 0 (A). Conc: 100.0
  Species 1 (B). Conc: 0.0
  Species 2 (C). Conc: 0.0
  Species 3 (E_high). Conc: 1000.0
  Species 4 (E_low). Conc: 0.0

In [7]:

dynamics.set_diagnostics()       # To save diagnostic information about the call to single_compartment_react()

In [8]:

#dynamics.verbose_list = ["substeps"]     # Uncomment for debugging information

In [21]:

dynamics.single_compartment_react(time_step=0.0012, stop_time=0.03,
                                  dynamic_substeps=2, abs_fast_threshold=750.)
#dynamics.get_history()

single_compartment_react(): setting rel_fast_threshold to 89.99999999999999
The computation took 38 extra step(s) - automatically added to prevent negative concentrations
(an extra dynamic substeps that might have been taken, are NOT part of the above count)
45 total step(s) taken

In [10]:

dynamics.explain_time_advance()

From time 0 to 0.0024, in 2 FULL steps of 0.0012
From time 0.0024 to 0.024, in 36 substeps of 0.0006 (each 1/2 of full step)
From time 0.024 to 0.0312, in 6 FULL steps of 0.0012

Compared to the earlier run, the system is now at times reaching for smaller 1/4 steps¶

(1/2 factors arise from either the negative concentrations or the optional substeps triggered by large changes; sometime both occur)

In [23]:

dynamics.single_compartment_react(time_step=0.0025, stop_time=5.,
                                  dynamic_substeps=2, abs_fast_threshold=250.)
#dynamics.get_history()

single_compartment_react(): setting rel_fast_threshold to 62.5
The computation took 109 extra step(s) - automatically added to prevent negative concentrations
(an extra dynamic substeps that might have been taken, are NOT part of the above count)
2043 total step(s) taken

In [12]:

dynamics.explain_time_advance()

From time 0 to 0.0024, in 2 FULL steps of 0.0012
From time 0.0024 to 0.024, in 36 substeps of 0.0006 (each 1/2 of full step)
From time 0.024 to 0.0312, in 6 FULL steps of 0.0012
From time 0.0312 to 0.17, in 111 substeps of 0.00125 (each 1/2 of full step)
From time 0.17 to 0.257, in 35 FULL steps of 0.0025
From time 0.257 to 0.259, in 1 substep of 0.00125 (1/2 of full step)
From time 0.259 to 0.276, in 7 FULL steps of 0.0025
From time 0.276 to 0.277, in 1 substep of 0.00125 (1/2 of full step)
From time 0.277 to 0.29, in 5 FULL steps of 0.0025
From time 0.29 to 0.291, in 1 substep of 0.00125 (1/2 of full step)
From time 0.291 to 0.304, in 5 FULL steps of 0.0025
From time 0.304 to 0.305, in 1 substep of 0.00125 (1/2 of full step)
From time 0.305 to 0.322, in 7 FULL steps of 0.0025
From time 0.322 to 0.324, in 1 substep of 0.00125 (1/2 of full step)
From time 0.324 to 0.341, in 7 FULL steps of 0.0025
From time 0.341 to 0.342, in 1 substep of 0.00125 (1/2 of full step)
From time 0.342 to 0.355, in 5 FULL steps of 0.0025
From time 0.355 to 0.356, in 1 substep of 0.00125 (1/2 of full step)
From time 0.356 to 0.374, in 7 FULL steps of 0.0025
From time 0.374 to 0.375, in 1 substep of 0.00125 (1/2 of full step)
From time 0.375 to 0.392, in 7 FULL steps of 0.0025
From time 0.392 to 0.394, in 1 substep of 0.00125 (1/2 of full step)
From time 0.394 to 0.416, in 9 FULL steps of 0.0025
From time 0.416 to 0.417, in 1 substep of 0.00125 (1/2 of full step)
From time 0.417 to 0.435, in 7 FULL steps of 0.0025
From time 0.435 to 0.436, in 1 substep of 0.00125 (1/2 of full step)
From time 0.436 to 0.459, in 9 FULL steps of 0.0025
From time 0.459 to 0.46, in 1 substep of 0.00125 (1/2 of full step)
From time 0.46 to 0.482, in 9 FULL steps of 0.0025
From time 0.482 to 0.484, in 1 substep of 0.00125 (1/2 of full step)
From time 0.484 to 0.511, in 11 FULL steps of 0.0025
From time 0.511 to 0.512, in 1 substep of 0.00125 (1/2 of full step)
From time 0.512 to 0.545, in 13 FULL steps of 0.0025
From time 0.545 to 0.546, in 1 substep of 0.00125 (1/2 of full step)
From time 0.546 to 0.579, in 13 FULL steps of 0.0025
From time 0.579 to 0.58, in 1 substep of 0.00125 (1/2 of full step)
From time 0.58 to 0.622, in 17 FULL steps of 0.0025
From time 0.622 to 0.624, in 1 substep of 0.00125 (1/2 of full step)
From time 0.624 to 0.676, in 21 FULL steps of 0.0025
From time 0.676 to 0.677, in 1 substep of 0.00125 (1/2 of full step)
From time 0.677 to 0.78, in 41 FULL steps of 0.0025
From time 0.78 to 0.781, in 1 substep of 0.00125 (1/2 of full step)
From time 0.781 to 5, in 1688 FULL steps of 0.0025

Again, we see many of those smaller 1/4 steps¶

In [25]:

dynamics.single_compartment_react(time_step=0.008, stop_time=8.,
                                  dynamic_substeps=2, abs_fast_threshold=2.)

single_compartment_react(): setting rel_fast_threshold to 1.6
375 total step(s) taken

In [14]:

df = dynamics.get_history()
df

Out[14]:

	SYSTEM TIME	A	B	C	E_high	E_low	caption
0	0.0000	100.000000	0.000000	0.000000	1000.000000	0.000000	Initial state
1	0.0012	98.920000	1.080000	0.000000	1000.000000	0.000000
2	0.0024	97.855552	2.134080	0.010368	1000.000000	0.000000
3	0.0030	97.337194	2.648440	0.014366	999.993779	0.006221
4	0.0036	96.824887	3.156616	0.018497	999.985232	0.014768
...	...	...	...	...	...	...	...
2493	7.9732	8.663600	30.034016	61.302385	33.266197	966.733803
2494	7.9812	11.674383	29.976478	58.349140	30.352451	969.647549
2495	7.9892	7.609519	30.046276	62.344205	34.296194	965.703806
2496	7.9972	13.130430	29.945103	56.924467	28.948509	971.051491
2497	8.0052	5.686231	30.076908	64.236860	36.165999	963.834001

2498 rows × 7 columns

In [15]:

dynamics.explain_time_advance()

From time 0 to 0.0024, in 2 FULL steps of 0.0012
From time 0.0024 to 0.024, in 36 substeps of 0.0006 (each 1/2 of full step)
From time 0.024 to 0.0312, in 6 FULL steps of 0.0012
From time 0.0312 to 0.17, in 111 substeps of 0.00125 (each 1/2 of full step)
From time 0.17 to 0.257, in 35 FULL steps of 0.0025
From time 0.257 to 0.259, in 1 substep of 0.00125 (1/2 of full step)
From time 0.259 to 0.276, in 7 FULL steps of 0.0025
From time 0.276 to 0.277, in 1 substep of 0.00125 (1/2 of full step)
From time 0.277 to 0.29, in 5 FULL steps of 0.0025
From time 0.29 to 0.291, in 1 substep of 0.00125 (1/2 of full step)
From time 0.291 to 0.304, in 5 FULL steps of 0.0025
From time 0.304 to 0.305, in 1 substep of 0.00125 (1/2 of full step)
From time 0.305 to 0.322, in 7 FULL steps of 0.0025
From time 0.322 to 0.324, in 1 substep of 0.00125 (1/2 of full step)
From time 0.324 to 0.341, in 7 FULL steps of 0.0025
From time 0.341 to 0.342, in 1 substep of 0.00125 (1/2 of full step)
From time 0.342 to 0.355, in 5 FULL steps of 0.0025
From time 0.355 to 0.356, in 1 substep of 0.00125 (1/2 of full step)
From time 0.356 to 0.374, in 7 FULL steps of 0.0025
From time 0.374 to 0.375, in 1 substep of 0.00125 (1/2 of full step)
From time 0.375 to 0.392, in 7 FULL steps of 0.0025
From time 0.392 to 0.394, in 1 substep of 0.00125 (1/2 of full step)
From time 0.394 to 0.416, in 9 FULL steps of 0.0025
From time 0.416 to 0.417, in 1 substep of 0.00125 (1/2 of full step)
From time 0.417 to 0.435, in 7 FULL steps of 0.0025
From time 0.435 to 0.436, in 1 substep of 0.00125 (1/2 of full step)
From time 0.436 to 0.459, in 9 FULL steps of 0.0025
From time 0.459 to 0.46, in 1 substep of 0.00125 (1/2 of full step)
From time 0.46 to 0.482, in 9 FULL steps of 0.0025
From time 0.482 to 0.484, in 1 substep of 0.00125 (1/2 of full step)
From time 0.484 to 0.511, in 11 FULL steps of 0.0025
From time 0.511 to 0.512, in 1 substep of 0.00125 (1/2 of full step)
From time 0.512 to 0.545, in 13 FULL steps of 0.0025
From time 0.545 to 0.546, in 1 substep of 0.00125 (1/2 of full step)
From time 0.546 to 0.579, in 13 FULL steps of 0.0025
From time 0.579 to 0.58, in 1 substep of 0.00125 (1/2 of full step)
From time 0.58 to 0.622, in 17 FULL steps of 0.0025
From time 0.622 to 0.624, in 1 substep of 0.00125 (1/2 of full step)
From time 0.624 to 0.676, in 21 FULL steps of 0.0025
From time 0.676 to 0.677, in 1 substep of 0.00125 (1/2 of full step)
From time 0.677 to 0.78, in 41 FULL steps of 0.0025
From time 0.78 to 0.781, in 1 substep of 0.00125 (1/2 of full step)
From time 0.781 to 5, in 1688 FULL steps of 0.0025
From time 5 to 5.38, in 47 FULL steps of 0.008
From time 5.38 to 5.38, in 1 substep of 0.004 (1/2 of full step)
From time 5.38 to 5.44, in 7 FULL steps of 0.008
From time 5.44 to 5.44, in 1 substep of 0.004 (1/2 of full step)
From time 5.44 to 5.48, in 5 FULL steps of 0.008
From time 5.48 to 5.49, in 1 substep of 0.004 (1/2 of full step)
From time 5.49 to 5.54, in 7 FULL steps of 0.008
From time 5.54 to 5.55, in 1 substep of 0.004 (1/2 of full step)
From time 5.55 to 5.59, in 5 FULL steps of 0.008
From time 5.59 to 5.59, in 1 substep of 0.004 (1/2 of full step)
From time 5.59 to 5.65, in 7 FULL steps of 0.008
From time 5.65 to 5.65, in 1 substep of 0.004 (1/2 of full step)
From time 5.65 to 5.69, in 5 FULL steps of 0.008
From time 5.69 to 5.69, in 1 substep of 0.004 (1/2 of full step)
From time 5.69 to 5.75, in 7 FULL steps of 0.008
From time 5.75 to 5.75, in 1 substep of 0.004 (1/2 of full step)
From time 5.75 to 5.79, in 5 FULL steps of 0.008
From time 5.79 to 5.8, in 1 substep of 0.004 (1/2 of full step)
From time 5.8 to 5.85, in 7 FULL steps of 0.008
From time 5.85 to 5.86, in 1 substep of 0.004 (1/2 of full step)
From time 5.86 to 5.9, in 5 FULL steps of 0.008
From time 5.9 to 5.9, in 1 substep of 0.004 (1/2 of full step)
From time 5.9 to 5.96, in 7 FULL steps of 0.008
From time 5.96 to 5.96, in 1 substep of 0.004 (1/2 of full step)
From time 5.96 to 6.02, in 7 FULL steps of 0.008
From time 6.02 to 6.02, in 1 substep of 0.004 (1/2 of full step)
From time 6.02 to 6.06, in 5 FULL steps of 0.008
From time 6.06 to 6.07, in 1 substep of 0.004 (1/2 of full step)
From time 6.07 to 6.12, in 7 FULL steps of 0.008
From time 6.12 to 6.13, in 1 substep of 0.004 (1/2 of full step)
From time 6.13 to 6.17, in 5 FULL steps of 0.008
From time 6.17 to 6.17, in 1 substep of 0.004 (1/2 of full step)
From time 6.17 to 6.23, in 7 FULL steps of 0.008
From time 6.23 to 6.23, in 1 substep of 0.004 (1/2 of full step)
From time 6.23 to 6.29, in 7 FULL steps of 0.008
From time 6.29 to 6.29, in 1 substep of 0.004 (1/2 of full step)
From time 6.29 to 6.33, in 5 FULL steps of 0.008
From time 6.33 to 6.33, in 1 substep of 0.004 (1/2 of full step)
From time 6.33 to 6.39, in 7 FULL steps of 0.008
From time 6.39 to 6.39, in 1 substep of 0.004 (1/2 of full step)
From time 6.39 to 6.45, in 7 FULL steps of 0.008
From time 6.45 to 6.45, in 1 substep of 0.004 (1/2 of full step)
From time 6.45 to 6.49, in 5 FULL steps of 0.008
From time 6.49 to 6.5, in 1 substep of 0.004 (1/2 of full step)
From time 6.5 to 6.55, in 7 FULL steps of 0.008
From time 6.55 to 6.56, in 1 substep of 0.004 (1/2 of full step)
From time 6.56 to 6.61, in 7 FULL steps of 0.008
From time 6.61 to 6.62, in 1 substep of 0.004 (1/2 of full step)
From time 6.62 to 6.66, in 5 FULL steps of 0.008
From time 6.66 to 6.66, in 1 substep of 0.004 (1/2 of full step)
From time 6.66 to 6.72, in 7 FULL steps of 0.008
From time 6.72 to 6.72, in 1 substep of 0.004 (1/2 of full step)
From time 6.72 to 6.78, in 7 FULL steps of 0.008
From time 6.78 to 6.78, in 1 substep of 0.004 (1/2 of full step)
From time 6.78 to 6.82, in 5 FULL steps of 0.008
From time 6.82 to 6.83, in 1 substep of 0.004 (1/2 of full step)
From time 6.83 to 6.88, in 7 FULL steps of 0.008
From time 6.88 to 6.89, in 1 substep of 0.004 (1/2 of full step)
From time 6.89 to 6.94, in 7 FULL steps of 0.008
From time 6.94 to 6.95, in 1 substep of 0.004 (1/2 of full step)
From time 6.95 to 6.99, in 5 FULL steps of 0.008
From time 6.99 to 6.99, in 1 substep of 0.004 (1/2 of full step)
From time 6.99 to 7.05, in 7 FULL steps of 0.008
From time 7.05 to 7.05, in 1 substep of 0.004 (1/2 of full step)
From time 7.05 to 7.11, in 7 FULL steps of 0.008
From time 7.11 to 7.11, in 1 substep of 0.004 (1/2 of full step)
From time 7.11 to 7.15, in 5 FULL steps of 0.008
From time 7.15 to 7.15, in 1 substep of 0.004 (1/2 of full step)
From time 7.15 to 7.21, in 7 FULL steps of 0.008
From time 7.21 to 7.21, in 1 substep of 0.004 (1/2 of full step)
From time 7.21 to 7.27, in 7 FULL steps of 0.008
From time 7.27 to 7.27, in 1 substep of 0.004 (1/2 of full step)
From time 7.27 to 7.31, in 5 FULL steps of 0.008
From time 7.31 to 7.32, in 1 substep of 0.004 (1/2 of full step)
From time 7.32 to 7.37, in 7 FULL steps of 0.008
From time 7.37 to 7.38, in 1 substep of 0.004 (1/2 of full step)
From time 7.38 to 7.43, in 7 FULL steps of 0.008
From time 7.43 to 7.44, in 1 substep of 0.004 (1/2 of full step)
From time 7.44 to 7.48, in 5 FULL steps of 0.008
From time 7.48 to 7.48, in 1 substep of 0.004 (1/2 of full step)
From time 7.48 to 7.54, in 7 FULL steps of 0.008
From time 7.54 to 7.54, in 1 substep of 0.004 (1/2 of full step)
From time 7.54 to 7.6, in 7 FULL steps of 0.008
From time 7.6 to 7.6, in 1 substep of 0.004 (1/2 of full step)
From time 7.6 to 7.64, in 5 FULL steps of 0.008
From time 7.64 to 7.65, in 1 substep of 0.004 (1/2 of full step)
From time 7.65 to 7.7, in 7 FULL steps of 0.008
From time 7.7 to 7.71, in 1 substep of 0.004 (1/2 of full step)
From time 7.71 to 7.76, in 7 FULL steps of 0.008
From time 7.76 to 7.77, in 1 substep of 0.004 (1/2 of full step)
From time 7.77 to 7.81, in 5 FULL steps of 0.008
From time 7.81 to 7.81, in 1 substep of 0.004 (1/2 of full step)
From time 7.81 to 7.87, in 7 FULL steps of 0.008
From time 7.87 to 7.87, in 1 substep of 0.004 (1/2 of full step)
From time 7.87 to 7.93, in 7 FULL steps of 0.008
From time 7.93 to 7.93, in 1 substep of 0.004 (1/2 of full step)
From time 7.93 to 7.97, in 5 FULL steps of 0.008
From time 7.97 to 7.97, in 1 substep of 0.004 (1/2 of full step)
From time 7.97 to 8.01, in 4 FULL steps of 0.008

In the later times (the 5.-8. range) there isn't that "frantic throttling up and down" that we had before¶

Plots of changes of concentration with time¶

In [28]:

dynamics.plot_history(chemicals=["E_high", "E_low"], colors=["red", "grey"])

In [29]:

dynamics.plot_history(chemicals=["A", "B", "C"])

The raggedness, and instability in the solutions, is now gone :)¶

In [ ]:

Excessively large time steps - and remediation¶

Based on experiment cycles_1¶

A cycle of reactions A <-> B <-> C <-> A and C + E_High <-> A + E_Low¶

RUN 1 - Initially, very large time steps are taken - at the edge of pushing some concentrations into negative values.¶

RUN 2 - Same primary steps as for run #1, but with the option of using 1/2 substeps as needed,¶

IMPORTANT: DO NOT ATTEMPT TO RUN THIS NOTEBOOK!¶

This is a "frozen run" that depends on an old version of Life123, for demonstration purposes¶

(newer versions of Life123 tend to recover from instability more gracefully!!)¶

Initialize the system¶

Set the initial concentrations of all the chemicals¶

We'll split each simulation in three segments (apparent from the graphs of the runs, further down):¶

RUN 1 - Initially, very large time steps are taken - at the edge of pushing some concentrations into negative values.¶

Notice how the system automatically intervened and reduced some steps in 1/2, to prevent negative concentrations¶

Notice how the system automatically periodically intervened and reduced some steps in 1/2, to prevent negative concentrations.¶

Notice that with our aggressively-large time steps, we ended up with far fewer data points than in any of the runs in the experiment cycles_1, which this analysis is based on¶

Yet again, the system automatically keeps periodically throttling down the time steps, to prevent negative concentrations.¶

Plots of changes of concentration with time¶

The negative concentrations were automatically avoided, but nonetheless the plots are ragged... and the solutions display instabilities¶

RUN 2 - Same primary steps as for run #1, but with the option of using 1/2 substeps as needed,¶

Compared to the earlier run, the system is now at times reaching for smaller 1/4 steps¶

Again, we see many of those smaller 1/4 steps¶

In the later times (the 5.-8. range) there isn't that "frantic throttling up and down" that we had before¶

Plots of changes of concentration with time¶

The raggedness, and instability in the solutions, is now gone :)¶

Based on experiment `cycles_1`¶

A cycle of reactions `A <-> B <-> C <-> A` and `C + E_High <-> A + E_Low`¶

Notice that with our aggressively-large time steps, we ended up with far fewer data points than in any of the runs in the experiment `cycles_1`, which this analysis is based on¶