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"## 2 COUPLED reactions of different speeds, forming a \"cascade\": \n",
"### `A <-> B` (fast) and `B <-> C` (slow)\n",
"Taken to equilibrium. Both reactions are mostly forward. All 1st order. \n",
"**The concentration of the intermediate product B manifests 1 oscillation (transient \"overshoot\")**\n",
"\n",
"Adaptive variable time resolution is used, with extensive diagnostics, \n",
"and finally compared to a new run using fixed time intervals, with the same initial data.\n",
"\n",
"In part2, some diagnostic insight is explored. \n",
"\n",
"In part3, two identical runs (\"adaptive variable steps\" vs. \"fixed small steps\") are compared. "
]
},
{
"cell_type": "markdown",
"id": "c8763f25-9fab-4b36-89cc-388321a16be0",
"metadata": {},
"source": [
"## Bathtub analogy:\n",
"A is initially full, while B and C are empty. \n",
"Tubs are progressively lower (reactions are mostly forward.) \n",
"A BIG pipe connects A and B: fast kinetics. A small pipe connects B and C: slow kinetics. \n",
"\n",
"INTUITION: B, unable to quickly drain into C while at the same time being blasted by a hefty inflow from A, \n",
"will experience a transient surge, in excess of its final equilibrium level.\n",
"\n",
"* **[Compare with the final reaction plot (the orange line is B)](#cascade_1_plot)**"
]
},
{
"cell_type": "markdown",
"id": "b11f68d6-d3bc-4908-93c0-eaee580acbc1",
"metadata": {},
"source": [
""
]
},
{
"cell_type": "code",
"execution_count": 1,
"id": "f5f1e6a4-2d9c-4638-bde7-09cb56e995a3",
"metadata": {},
"outputs": [],
"source": [
"LAST_REVISED = \"July 26, 2024\"\n",
"LIFE123_VERSION = \"1.0.0.beta.38\" # Version this experiment is based on"
]
},
{
"cell_type": "code",
"execution_count": 2,
"id": "8ef47cb2-812a-4cd1-87d3-0a6573ab0a5c",
"metadata": {},
"outputs": [],
"source": [
"#import set_path # Using MyBinder? Uncomment this before running the next cell!\n",
" # Importing this local file will add the project's home directory to sys.path"
]
},
{
"cell_type": "code",
"execution_count": 3,
"id": "3924c013",
"metadata": {
"tags": []
},
"outputs": [],
"source": [
"#import sys\n",
"#sys.path.append(\"C:/some_path/my_env_or_install\") # CHANGE to the folder containing your venv or libraries installation!\n",
"# NOTE: If any of the imports below can't find a module, uncomment the lines above, or try: import set_path\n",
"\n",
"import ipynbname\n",
"\n",
"from life123 import check_version, ChemData, UniformCompartment, PlotlyHelper, GraphicLog"
]
},
{
"cell_type": "code",
"execution_count": 4,
"id": "e96579d6-e07a-4d62-9655-aab230e6f186",
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"OK\n"
]
}
],
"source": [
"check_version(LIFE123_VERSION)"
]
},
{
"cell_type": "code",
"execution_count": 5,
"id": "83c3cc5f-de21-4f66-9988-2806fbf0666d",
"metadata": {
"tags": []
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"-> Output will be LOGGED into the file 'cascade_1.log.htm'\n"
]
}
],
"source": [
"# Initialize the HTML logging (for the graphics)\n",
"log_file = ipynbname.name() + \".log.htm\" # Use the notebook base filename for the log file\n",
"\n",
"# Set up the use of some specified graphic (Vue) components\n",
"GraphicLog.config(filename=log_file,\n",
" components=[\"vue_cytoscape_2\"],\n",
" extra_js=\"https://cdnjs.cloudflare.com/ajax/libs/cytoscape/3.21.2/cytoscape.umd.js\")"
]
},
{
"cell_type": "markdown",
"id": "9329208b-070f-4902-8f37-0f11ddf75ed6",
"metadata": {},
"source": [
"# Initialize the System\n",
"Specify the chemicals and the reactions"
]
},
{
"cell_type": "code",
"execution_count": 6,
"id": "72b4245c-de4e-480d-a501-3495b7ed8bc4",
"metadata": {
"tags": []
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"name": "stdout",
"output_type": "stream",
"text": [
"Number of reactions: 2\n"
]
}
],
"source": [
"# Specify the chemicals and the reactions; this data structure will get re-used in \n",
"# the various simulations below\n",
"chem = ChemData()\n",
"\n",
"# Reaction A <-> B (fast)\n",
"chem.add_reaction(reactants=\"A\", products=\"B\",\n",
" forward_rate=64., reverse_rate=8.) \n",
"\n",
"# Reaction B <-> C (slow)\n",
"chem.add_reaction(reactants=\"B\", products=\"C\",\n",
" forward_rate=12., reverse_rate=2.) \n",
"\n",
"print(\"Number of reactions: \", chem.number_of_reactions())"
]
},
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"cell_type": "code",
"execution_count": 7,
"id": "00ea560d-9a49-4041-b119-6de11bfcc7af",
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"name": "stdout",
"output_type": "stream",
"text": [
"Number of reactions: 2 (at temp. 25 C)\n",
"0: A <-> B (kF = 64 / kR = 8 / delta_G = -5,154.8 / K = 8) | 1st order in all reactants & products\n",
"1: B <-> C (kF = 12 / kR = 2 / delta_G = -4,441.7 / K = 6) | 1st order in all reactants & products\n",
"Set of chemicals involved in the above reactions: {'A', 'C', 'B'}\n"
]
}
],
"source": [
"chem.describe_reactions()"
]
},
{
"cell_type": "code",
"execution_count": 8,
"id": "cb582868-431c-4022-aa0e-a2f554f80d6c",
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"[GRAPHIC ELEMENT SENT TO LOG FILE `cascade_1.log.htm`]\n"
]
}
],
"source": [
"# Send a plot of the network of reactions to the HTML log file\n",
"chem.plot_reaction_network(\"vue_cytoscape_2\")"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "74a1e2bd-4e82-4aaa-a68f-7cee52b2ebbf",
"metadata": {},
"outputs": [],
"source": []
},
{
"cell_type": "markdown",
"id": "98a9fbe5-2090-4d38-9c5f-94fbf7c3eae2",
"metadata": {},
"source": [
"## Run the simulation"
]
},
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"cell_type": "code",
"execution_count": 9,
"id": "ae304704-c8d9-4cef-9e0b-2587bb3909ef",
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"name": "stdout",
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"text": [
"SYSTEM STATE at Time t = 0:\n",
"3 species:\n",
" Species 0 (A). Conc: 50.0\n",
" Species 1 (B). Conc: 0.0\n",
" Species 2 (C). Conc: 0.0\n",
"Set of chemicals involved in reactions: {'A', 'C', 'B'}\n"
]
}
],
"source": [
"dynamics = UniformCompartment(chem_data=chem, preset=\"fast\")\n",
"\n",
"dynamics.set_conc({\"A\": 50.}, snapshot=True) # Set the initial concentrations of all the chemicals, in their index order\n",
"dynamics.describe_state()"
]
},
{
"cell_type": "code",
"execution_count": 10,
"id": "0ff2c242-a15b-456d-ad56-0ba1041c0b4c",
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"
\n",
"\n",
"
\n",
" \n",
" \n",
" | \n",
" SYSTEM TIME | \n",
" A | \n",
" B | \n",
" C | \n",
" caption | \n",
"
\n",
" \n",
" \n",
" \n",
" | 0 | \n",
" 0.0 | \n",
" 50.0 | \n",
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\n",
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\n",
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"## Run the reaction"
]
},
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"Some steps were backtracked and re-done, to prevent negative concentrations or excessively large concentration changes\n",
"48 total step(s) taken\n",
"Number of step re-do's because of negative concentrations: 1\n",
"Number of step re-do's because of elective soft aborts: 5\n",
"Norm usage: {'norm_A': 25, 'norm_B': 17, 'norm_C': 16, 'norm_D': 16}\n"
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"dynamics.enable_diagnostics() # To save diagnostic information about the call to single_compartment_react()\n",
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"dynamics.single_compartment_react(initial_step=0.02, duration=0.4,\n",
" snapshots={\"initial_caption\": \"1st reaction step\",\n",
" \"final_caption\": \"last reaction step\"},\n",
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"cell_type": "markdown",
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"### Plots of changes of concentration with time\n",
"Notice the variable time steps (vertical dashed lines)"
]
},
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\n",
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" \n",
" | \n",
" SYSTEM TIME | \n",
" A | \n",
" B | \n",
" C | \n",
" caption | \n",
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\n",
" \n",
" \n",
" \n",
" | 0 | \n",
" 0.000000 | \n",
" 50.000000 | \n",
" 0.000000 | \n",
" 0.000000 | \n",
" Initialized state | \n",
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" | 1 | \n",
" 0.000778 | \n",
" 47.511680 | \n",
" 2.488320 | \n",
" 0.000000 | \n",
" 1st reaction step | \n",
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" | 2 | \n",
" 0.001400 | \n",
" 45.632475 | \n",
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" 0.002519 | \n",
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" | 11 | \n",
" 0.006003 | \n",
" 34.138767 | \n",
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" | 12 | \n",
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" | 15 | \n",
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" | 17 | \n",
" 0.009736 | \n",
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" 1.346531 | \n",
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" | 18 | \n",
" 0.010482 | \n",
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" | 19 | \n",
" 0.011602 | \n",
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" | 20 | \n",
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" \n",
" | 22 | \n",
" 0.014961 | \n",
" 20.144428 | \n",
" 27.030704 | \n",
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" | \n",
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" | 23 | \n",
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" 28.297603 | \n",
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" | 24 | \n",
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" 16.750734 | \n",
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" | \n",
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" \n",
" | 25 | \n",
" 0.020000 | \n",
" 15.344212 | \n",
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" | \n",
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" | 26 | \n",
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" | 27 | \n",
" 0.025039 | \n",
" 11.932250 | \n",
" 31.767190 | \n",
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" | 28 | \n",
" 0.027558 | \n",
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" \n",
" | 29 | \n",
" 0.031337 | \n",
" 9.044186 | \n",
" 32.324491 | \n",
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" | 30 | \n",
" 0.035116 | \n",
" 7.833986 | \n",
" 32.134026 | \n",
" 10.031989 | \n",
" | \n",
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" | 31 | \n",
" 0.038896 | \n",
" 6.910732 | \n",
" 31.675838 | \n",
" 11.413430 | \n",
" | \n",
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" | 32 | \n",
" 0.044564 | \n",
" 5.840026 | \n",
" 30.721211 | \n",
" 13.438763 | \n",
" | \n",
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" \n",
" | 33 | \n",
" 0.050233 | \n",
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" | 34 | \n",
" 0.058736 | \n",
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" | 35 | \n",
" 0.065538 | \n",
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\n",
" \n",
" | 36 | \n",
" 0.075742 | \n",
" 3.486129 | \n",
" 23.623393 | \n",
" 22.890478 | \n",
" | \n",
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\n",
" \n",
" | 37 | \n",
" 0.083905 | \n",
" 3.207568 | \n",
" 21.961627 | \n",
" 24.830805 | \n",
" | \n",
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\n",
" \n",
" | 38 | \n",
" 0.096149 | \n",
" 2.845240 | \n",
" 19.705148 | \n",
" 27.449612 | \n",
" | \n",
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\n",
" \n",
" | 39 | \n",
" 0.105945 | \n",
" 2.605699 | \n",
" 18.166190 | \n",
" 29.228112 | \n",
" | \n",
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\n",
" \n",
" | 40 | \n",
" 0.120638 | \n",
" 2.290745 | \n",
" 16.137006 | \n",
" 31.572249 | \n",
" | \n",
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\n",
" \n",
" | 41 | \n",
" 0.135332 | \n",
" 2.033442 | \n",
" 14.476842 | \n",
" 33.489716 | \n",
" | \n",
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\n",
" \n",
" | 42 | \n",
" 0.157371 | \n",
" 1.717708 | \n",
" 12.439976 | \n",
" 35.842316 | \n",
" | \n",
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\n",
" \n",
" | 43 | \n",
" 0.179411 | \n",
" 1.488195 | \n",
" 10.959299 | \n",
" 37.552505 | \n",
" | \n",
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\n",
" \n",
" | 44 | \n",
" 0.212471 | \n",
" 1.237929 | \n",
" 9.344771 | \n",
" 39.417300 | \n",
" | \n",
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\n",
" \n",
" | 45 | \n",
" 0.245531 | \n",
" 1.090175 | \n",
" 8.391543 | \n",
" 40.518282 | \n",
" | \n",
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\n",
" \n",
" | 46 | \n",
" 0.295121 | \n",
" 0.959315 | \n",
" 7.547371 | \n",
" 41.493315 | \n",
" | \n",
"
\n",
" \n",
" | 47 | \n",
" 0.369506 | \n",
" 0.883653 | \n",
" 7.059061 | \n",
" 42.057286 | \n",
" | \n",
"
\n",
" \n",
" | 48 | \n",
" 0.481083 | \n",
" 0.874584 | \n",
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" 42.123581 | \n",
" last reaction step | \n",
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\n",
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\n",
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"text/plain": [
" SYSTEM TIME A B C caption\n",
"0 0.000000 50.000000 0.000000 0.000000 Initialized state\n",
"1 0.000778 47.511680 2.488320 0.000000 1st reaction step\n",
"2 0.001400 45.632475 4.348950 0.018575 \n",
"3 0.002022 43.837347 6.111636 0.051017 \n",
"4 0.002519 42.465438 7.447097 0.087465 \n",
"5 0.003017 41.142542 8.725606 0.131851 \n",
"6 0.003515 39.866871 9.949300 0.183829 \n",
"7 0.004012 38.636703 11.120234 0.243063 \n",
"8 0.004510 37.450378 12.240391 0.309231 \n",
"9 0.005008 36.306298 13.311680 0.382022 \n",
"10 0.005505 35.202922 14.335939 0.461139 \n",
"11 0.006003 34.138767 15.314939 0.546294 \n",
"12 0.006501 33.112404 16.250386 0.637210 \n",
"13 0.006998 32.122455 17.143922 0.733623 \n",
"14 0.007496 31.167595 17.997130 0.835276 \n",
"15 0.008243 29.786018 19.218736 0.995246 \n",
"16 0.008989 28.477742 20.356337 1.165921 \n",
"17 0.009736 27.238764 21.414705 1.346531 \n",
"18 0.010482 26.065300 22.398347 1.536353 \n",
"19 0.011602 24.398010 23.768113 1.833877 \n",
"20 0.012722 22.862474 24.988386 2.149140 \n",
"21 0.013841 21.447911 26.071994 2.480095 \n",
"22 0.014961 20.144428 27.030704 2.824868 \n",
"23 0.016641 18.342204 28.297603 3.360193 \n",
"24 0.018320 16.750734 29.330012 3.919255 \n",
"25 0.020000 15.344212 30.158542 4.497247 \n",
"26 0.022519 13.477920 31.135708 5.386372 \n",
"27 0.025039 11.932250 31.767190 6.300559 \n",
"28 0.027558 10.648537 32.122231 7.229232 \n",
"29 0.031337 9.044186 32.324491 8.631323 \n",
"30 0.035116 7.833986 32.134026 10.031989 \n",
"31 0.038896 6.910732 31.675838 11.413430 \n",
"32 0.044564 5.840026 30.721211 13.438763 \n",
"33 0.050233 5.114478 29.509327 15.376196 \n",
"34 0.058736 4.338557 27.535703 18.125740 \n",
"35 0.065538 3.948219 25.924919 20.126862 \n",
"36 0.075742 3.486129 23.623393 22.890478 \n",
"37 0.083905 3.207568 21.961627 24.830805 \n",
"38 0.096149 2.845240 19.705148 27.449612 \n",
"39 0.105945 2.605699 18.166190 29.228112 \n",
"40 0.120638 2.290745 16.137006 31.572249 \n",
"41 0.135332 2.033442 14.476842 33.489716 \n",
"42 0.157371 1.717708 12.439976 35.842316 \n",
"43 0.179411 1.488195 10.959299 37.552505 \n",
"44 0.212471 1.237929 9.344771 39.417300 \n",
"45 0.245531 1.090175 8.391543 40.518282 \n",
"46 0.295121 0.959315 7.547371 41.493315 \n",
"47 0.369506 0.883653 7.059061 42.057286 \n",
"48 0.481083 0.874584 7.001835 42.123581 last reaction step"
]
},
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"metadata": {},
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],
"source": [
"dynamics.get_history()"
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},
{
"cell_type": "code",
"execution_count": 17,
"id": "144d0388-a632-4165-a9c5-50bc7943fc88",
"metadata": {},
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{
"name": "stdout",
"output_type": "stream",
"text": [
"From time 0 to 0.0007776, in 1 step of 0.000778\n",
"From time 0.0007776 to 0.002022, in 2 steps of 0.000622\n",
"From time 0.002022 to 0.007496, in 11 steps of 0.000498\n",
"From time 0.007496 to 0.01048, in 4 steps of 0.000746\n",
"From time 0.01048 to 0.01496, in 4 steps of 0.00112\n",
"From time 0.01496 to 0.02, in 3 steps of 0.00168\n",
"From time 0.02 to 0.02756, in 3 steps of 0.00252\n",
"From time 0.02756 to 0.0389, in 3 steps of 0.00378\n",
"From time 0.0389 to 0.05023, in 2 steps of 0.00567\n",
"From time 0.05023 to 0.05874, in 1 step of 0.0085\n",
"From time 0.05874 to 0.06554, in 1 step of 0.0068\n",
"From time 0.06554 to 0.07574, in 1 step of 0.0102\n",
"From time 0.07574 to 0.08391, in 1 step of 0.00816\n",
"From time 0.08391 to 0.09615, in 1 step of 0.0122\n",
"From time 0.09615 to 0.1059, in 1 step of 0.0098\n",
"From time 0.1059 to 0.1353, in 2 steps of 0.0147\n",
"From time 0.1353 to 0.1794, in 2 steps of 0.022\n",
"From time 0.1794 to 0.2455, in 2 steps of 0.0331\n",
"From time 0.2455 to 0.2951, in 1 step of 0.0496\n",
"From time 0.2951 to 0.3695, in 1 step of 0.0744\n",
"From time 0.3695 to 0.4811, in 1 step of 0.112\n",
"(48 steps total)\n"
]
}
],
"source": [
"dynamics.diagnostics.explain_time_advance()"
]
},
{
"cell_type": "markdown",
"id": "c02a8f55-a671-4771-86c9-fc4d1b126bf8",
"metadata": {},
"source": [
"### Check the final equilibrium"
]
},
{
"cell_type": "code",
"execution_count": 18,
"id": "b139f5e4-625f-4a5e-8f57-8f00244dced4",
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"0: A <-> B\n",
"Final concentrations: [A] = 0.8746 ; [B] = 7.002\n",
"1. Ratio of reactant/product concentrations, adjusted for reaction orders: 8.0059\n",
" Formula used: [B] / [A]\n",
"2. Ratio of forward/reverse reaction rates: 8\n",
"Discrepancy between the two values: 0.07378 %\n",
"Reaction IS in equilibrium (within 1% tolerance)\n",
"\n",
"1: B <-> C\n",
"Final concentrations: [B] = 7.002 ; [C] = 42.12\n",
"1. Ratio of reactant/product concentrations, adjusted for reaction orders: 6.01608\n",
" Formula used: [C] / [B]\n",
"2. Ratio of forward/reverse reaction rates: 6\n",
"Discrepancy between the two values: 0.268 %\n",
"Reaction IS in equilibrium (within 1% tolerance)\n",
"\n"
]
},
{
"data": {
"text/plain": [
"True"
]
},
"execution_count": 18,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# Verify that all the reactions have reached equilibrium\n",
"dynamics.is_in_equilibrium()"
]
},
{
"cell_type": "markdown",
"id": "c279c0b2-754e-4574-95a6-ffbe8e162463",
"metadata": {},
"source": [
"### Let's look at the final concentrations of `A` and `C` (i.e., the reactant and product of the composite reaction)"
]
},
{
"cell_type": "code",
"execution_count": 19,
"id": "c093910f-62f2-43cf-88ef-d5df935a44b8",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"0.8745841244150234"
]
},
"execution_count": 19,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"A_final = dynamics.get_chem_conc(\"A\")\n",
"A_final"
]
},
{
"cell_type": "code",
"execution_count": 20,
"id": "22636dc7-fd18-468f-8ddb-8152891eb9b5",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"42.123581082850514"
]
},
"execution_count": 20,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"C_final = dynamics.get_chem_conc(\"C\")\n",
"C_final"
]
},
{
"cell_type": "markdown",
"id": "420e0075-6b81-4836-8641-b0ccdeacf550",
"metadata": {},
"source": [
"#### Their ratio:"
]
},
{
"cell_type": "code",
"execution_count": 21,
"id": "8280c0be-2f8d-4f4c-adc0-9ca7f5d5060e",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"48.16412727709344"
]
},
"execution_count": 21,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"C_final / A_final"
]
},
{
"cell_type": "markdown",
"id": "07fe7c6c-f8dd-4460-aa08-d7c61d60bd62",
"metadata": {},
"source": [
"### As expected the equilibrium constant for the overall reaction `A <-> C` (approx. 48) is indeed the product of the equilibrium constants of the two elementary reactions (K = 8 and K = 6, respectively) that we saw earlier."
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "587c2018-e781-43c1-aee8-e78b7b6dc9bf",
"metadata": {},
"outputs": [],
"source": []
},
{
"cell_type": "code",
"execution_count": null,
"id": "520b3b44-7da9-49d3-9cfd-685607169f01",
"metadata": {},
"outputs": [],
"source": []
},
{
"cell_type": "markdown",
"id": "5c3f8b4f-3a75-4a21-8579-13550bcebb3c",
"metadata": {},
"source": [
"# PART 2 - DIAGNOSTIC INSIGHT\n",
"\n",
"Perform some verification"
]
},
{
"cell_type": "markdown",
"id": "07dd5aa4-9e71-40c1-9f15-a16a9333ed10",
"metadata": {},
"source": [
"### Take a peek at the diagnostic data saved during the earlier reaction simulation"
]
},
{
"cell_type": "code",
"execution_count": 22,
"id": "5ad64dc5-ba89-4f09-b7db-da40de1df4ed",
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Reaction: A <-> B\n"
]
},
{
"data": {
"text/html": [
"\n",
"\n",
"
\n",
" \n",
" \n",
" | \n",
" START_TIME | \n",
" time_step | \n",
" Delta A | \n",
" Delta B | \n",
" caption | \n",
" rate | \n",
"
\n",
" \n",
" \n",
" \n",
" | 0 | \n",
" 0.000000 | \n",
" 0.020000 | \n",
" NaN | \n",
" NaN | \n",
" aborted: neg. conc. in `A` | \n",
" NaN | \n",
"
\n",
" \n",
" | 1 | \n",
" 0.000000 | \n",
" 0.010000 | \n",
" -32.000000 | \n",
" 32.000000 | \n",
" aborted: excessive norm value(s) | \n",
" 3200.000000 | \n",
"
\n",
" \n",
" | 2 | \n",
" 0.000000 | \n",
" 0.006000 | \n",
" -19.200000 | \n",
" 19.200000 | \n",
" aborted: excessive norm value(s) | \n",
" 3200.000000 | \n",
"
\n",
" \n",
" | 3 | \n",
" 0.000000 | \n",
" 0.003600 | \n",
" -11.520000 | \n",
" 11.520000 | \n",
" aborted: excessive norm value(s) | \n",
" 3200.000000 | \n",
"
\n",
" \n",
" | 4 | \n",
" 0.000000 | \n",
" 0.002160 | \n",
" -6.912000 | \n",
" 6.912000 | \n",
" aborted: excessive norm value(s) | \n",
" 3200.000000 | \n",
"
\n",
" \n",
" | 5 | \n",
" 0.000000 | \n",
" 0.001296 | \n",
" -4.147200 | \n",
" 4.147200 | \n",
" aborted: excessive norm value(s) | \n",
" 3200.000000 | \n",
"
\n",
" \n",
" | 6 | \n",
" 0.000000 | \n",
" 0.000778 | \n",
" -2.488320 | \n",
" 2.488320 | \n",
" | \n",
" 3200.000000 | \n",
"
\n",
" \n",
" | 7 | \n",
" 0.000778 | \n",
" 0.000622 | \n",
" -1.879205 | \n",
" 1.879205 | \n",
" | \n",
" 3020.840960 | \n",
"
\n",
" \n",
" | 8 | \n",
" 0.001400 | \n",
" 0.000622 | \n",
" -1.795128 | \n",
" 1.795128 | \n",
" | \n",
" 2885.686820 | \n",
"
\n",
" \n",
" | 9 | \n",
" 0.002022 | \n",
" 0.000498 | \n",
" -1.371909 | \n",
" 1.371909 | \n",
" | \n",
" 2756.697133 | \n",
"
\n",
" \n",
" | 10 | \n",
" 0.002519 | \n",
" 0.000498 | \n",
" -1.322896 | \n",
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" | \n",
" 2658.211272 | \n",
"
\n",
" \n",
" | 11 | \n",
" 0.003017 | \n",
" 0.000498 | \n",
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" | \n",
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\n",
" \n",
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" | \n",
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\n",
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" | 13 | \n",
" 0.004012 | \n",
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" | \n",
" 2383.787110 | \n",
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\n",
" \n",
" | 14 | \n",
" 0.004510 | \n",
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" -1.144080 | \n",
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" | \n",
" 2298.901050 | \n",
"
\n",
" \n",
" | 15 | \n",
" 0.005008 | \n",
" 0.000498 | \n",
" -1.103376 | \n",
" 1.103376 | \n",
" | \n",
" 2217.109600 | \n",
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\n",
" \n",
" | 16 | \n",
" 0.005505 | \n",
" 0.000498 | \n",
" -1.064155 | \n",
" 1.064155 | \n",
" | \n",
" 2138.299488 | \n",
"
\n",
" \n",
" | 17 | \n",
" 0.006003 | \n",
" 0.000498 | \n",
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" | \n",
" 2062.361590 | \n",
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\n",
" \n",
" | 18 | \n",
" 0.006501 | \n",
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" | \n",
" 1989.190778 | \n",
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\n",
" \n",
" | 19 | \n",
" 0.006998 | \n",
" 0.000498 | \n",
" -0.954861 | \n",
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" | \n",
" 1918.685776 | \n",
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\n",
" \n",
" | 20 | \n",
" 0.007496 | \n",
" 0.000746 | \n",
" -1.381577 | \n",
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" | \n",
" 1850.749018 | \n",
"
\n",
" \n",
" | 21 | \n",
" 0.008243 | \n",
" 0.000746 | \n",
" -1.308275 | \n",
" 1.308275 | \n",
" | \n",
" 1752.555256 | \n",
"
\n",
" \n",
" | 22 | \n",
" 0.008989 | \n",
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" | \n",
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\n",
" \n",
" | 23 | \n",
" 0.009736 | \n",
" 0.000746 | \n",
" -1.173464 | \n",
" 1.173464 | \n",
" | \n",
" 1571.963289 | \n",
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\n",
" \n",
" | 24 | \n",
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" -1.667290 | \n",
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" | \n",
" 1488.992431 | \n",
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\n",
" \n",
" | 25 | \n",
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" 0.001120 | \n",
" -1.535536 | \n",
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" | \n",
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\n",
" \n",
" | 26 | \n",
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" 0.001120 | \n",
" -1.414563 | \n",
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" | \n",
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\n",
" \n",
" | 27 | \n",
" 0.013841 | \n",
" 0.001120 | \n",
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" | \n",
" 1164.090351 | \n",
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\n",
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" | \n",
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\n",
" \n",
" | 29 | \n",
" 0.016641 | \n",
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" -1.591470 | \n",
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" | \n",
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\n",
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" | 30 | \n",
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\n",
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" | \n",
" 740.761211 | \n",
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\n",
" \n",
" | 32 | \n",
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" 0.002519 | \n",
" -1.545670 | \n",
" 1.545670 | \n",
" | \n",
" 613.501217 | \n",
"
\n",
" \n",
" | 33 | \n",
" 0.025039 | \n",
" 0.002519 | \n",
" -1.283713 | \n",
" 1.283713 | \n",
" | \n",
" 509.526499 | \n",
"
\n",
" \n",
" | 34 | \n",
" 0.027558 | \n",
" 0.003779 | \n",
" -1.604351 | \n",
" 1.604351 | \n",
" | \n",
" 424.528524 | \n",
"
\n",
" \n",
" | 35 | \n",
" 0.031337 | \n",
" 0.003779 | \n",
" -1.210200 | \n",
" 1.210200 | \n",
" | \n",
" 320.231977 | \n",
"
\n",
" \n",
" | 36 | \n",
" 0.035116 | \n",
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" | \n",
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\n",
" \n",
" | 37 | \n",
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" 0.005669 | \n",
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" 1.070706 | \n",
" | \n",
" 188.880147 | \n",
"
\n",
" \n",
" | 38 | \n",
" 0.044564 | \n",
" 0.005669 | \n",
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" | \n",
" 127.992002 | \n",
"
\n",
" \n",
" | 39 | \n",
" 0.050233 | \n",
" 0.008503 | \n",
" -0.775920 | \n",
" 0.775920 | \n",
" | \n",
" 91.251951 | \n",
"
\n",
" \n",
" | 40 | \n",
" 0.058736 | \n",
" 0.006802 | \n",
" -0.390338 | \n",
" 0.390338 | \n",
" | \n",
" 57.382032 | \n",
"
\n",
" \n",
" | 41 | \n",
" 0.065538 | \n",
" 0.010204 | \n",
" -0.462090 | \n",
" 0.462090 | \n",
" | \n",
" 45.286669 | \n",
"
\n",
" \n",
" | 42 | \n",
" 0.075742 | \n",
" 0.008163 | \n",
" -0.278561 | \n",
" 0.278561 | \n",
" | \n",
" 34.125104 | \n",
"
\n",
" \n",
" | 43 | \n",
" 0.083905 | \n",
" 0.012244 | \n",
" -0.362328 | \n",
" 0.362328 | \n",
" | \n",
" 29.591334 | \n",
"
\n",
" \n",
" | 44 | \n",
" 0.096149 | \n",
" 0.009796 | \n",
" -0.239541 | \n",
" 0.239541 | \n",
" | \n",
" 24.454162 | \n",
"
\n",
" \n",
" | 45 | \n",
" 0.105945 | \n",
" 0.014693 | \n",
" -0.314953 | \n",
" 0.314953 | \n",
" | \n",
" 21.435190 | \n",
"
\n",
" \n",
" | 46 | \n",
" 0.120638 | \n",
" 0.014693 | \n",
" -0.257304 | \n",
" 0.257304 | \n",
" | \n",
" 17.511654 | \n",
"
\n",
" \n",
" | 47 | \n",
" 0.135332 | \n",
" 0.022040 | \n",
" -0.315733 | \n",
" 0.315733 | \n",
" | \n",
" 14.325526 | \n",
"
\n",
" \n",
" | 48 | \n",
" 0.157371 | \n",
" 0.022040 | \n",
" -0.229513 | \n",
" 0.229513 | \n",
" | \n",
" 10.413518 | \n",
"
\n",
" \n",
" | 49 | \n",
" 0.179411 | \n",
" 0.033060 | \n",
" -0.250266 | \n",
" 0.250266 | \n",
" | \n",
" 7.570088 | \n",
"
\n",
" \n",
" | 50 | \n",
" 0.212471 | \n",
" 0.033060 | \n",
" -0.147754 | \n",
" 0.147754 | \n",
" | \n",
" 4.469280 | \n",
"
\n",
" \n",
" | 51 | \n",
" 0.245531 | \n",
" 0.049590 | \n",
" -0.130860 | \n",
" 0.130860 | \n",
" | \n",
" 2.638855 | \n",
"
\n",
" \n",
" | 52 | \n",
" 0.295121 | \n",
" 0.074385 | \n",
" -0.075662 | \n",
" 0.075662 | \n",
" | \n",
" 1.017172 | \n",
"
\n",
" \n",
" | 53 | \n",
" 0.369506 | \n",
" 0.111577 | \n",
" -0.009068 | \n",
" 0.009068 | \n",
" | \n",
" 0.081275 | \n",
"
\n",
" \n",
"
\n",
"
"
],
"text/plain": [
" START_TIME time_step Delta A Delta B \\\n",
"0 0.000000 0.020000 NaN NaN \n",
"1 0.000000 0.010000 -32.000000 32.000000 \n",
"2 0.000000 0.006000 -19.200000 19.200000 \n",
"3 0.000000 0.003600 -11.520000 11.520000 \n",
"4 0.000000 0.002160 -6.912000 6.912000 \n",
"5 0.000000 0.001296 -4.147200 4.147200 \n",
"6 0.000000 0.000778 -2.488320 2.488320 \n",
"7 0.000778 0.000622 -1.879205 1.879205 \n",
"8 0.001400 0.000622 -1.795128 1.795128 \n",
"9 0.002022 0.000498 -1.371909 1.371909 \n",
"10 0.002519 0.000498 -1.322896 1.322896 \n",
"11 0.003017 0.000498 -1.275671 1.275671 \n",
"12 0.003515 0.000498 -1.230168 1.230168 \n",
"13 0.004012 0.000498 -1.186325 1.186325 \n",
"14 0.004510 0.000498 -1.144080 1.144080 \n",
"15 0.005008 0.000498 -1.103376 1.103376 \n",
"16 0.005505 0.000498 -1.064155 1.064155 \n",
"17 0.006003 0.000498 -1.026363 1.026363 \n",
"18 0.006501 0.000498 -0.989949 0.989949 \n",
"19 0.006998 0.000498 -0.954861 0.954861 \n",
"20 0.007496 0.000746 -1.381577 1.381577 \n",
"21 0.008243 0.000746 -1.308275 1.308275 \n",
"22 0.008989 0.000746 -1.238978 1.238978 \n",
"23 0.009736 0.000746 -1.173464 1.173464 \n",
"24 0.010482 0.001120 -1.667290 1.667290 \n",
"25 0.011602 0.001120 -1.535536 1.535536 \n",
"26 0.012722 0.001120 -1.414563 1.414563 \n",
"27 0.013841 0.001120 -1.303483 1.303483 \n",
"28 0.014961 0.001680 -1.802224 1.802224 \n",
"29 0.016641 0.001680 -1.591470 1.591470 \n",
"30 0.018320 0.001680 -1.406522 1.406522 \n",
"31 0.020000 0.002519 -1.866292 1.866292 \n",
"32 0.022519 0.002519 -1.545670 1.545670 \n",
"33 0.025039 0.002519 -1.283713 1.283713 \n",
"34 0.027558 0.003779 -1.604351 1.604351 \n",
"35 0.031337 0.003779 -1.210200 1.210200 \n",
"36 0.035116 0.003779 -0.923254 0.923254 \n",
"37 0.038896 0.005669 -1.070706 1.070706 \n",
"38 0.044564 0.005669 -0.725549 0.725549 \n",
"39 0.050233 0.008503 -0.775920 0.775920 \n",
"40 0.058736 0.006802 -0.390338 0.390338 \n",
"41 0.065538 0.010204 -0.462090 0.462090 \n",
"42 0.075742 0.008163 -0.278561 0.278561 \n",
"43 0.083905 0.012244 -0.362328 0.362328 \n",
"44 0.096149 0.009796 -0.239541 0.239541 \n",
"45 0.105945 0.014693 -0.314953 0.314953 \n",
"46 0.120638 0.014693 -0.257304 0.257304 \n",
"47 0.135332 0.022040 -0.315733 0.315733 \n",
"48 0.157371 0.022040 -0.229513 0.229513 \n",
"49 0.179411 0.033060 -0.250266 0.250266 \n",
"50 0.212471 0.033060 -0.147754 0.147754 \n",
"51 0.245531 0.049590 -0.130860 0.130860 \n",
"52 0.295121 0.074385 -0.075662 0.075662 \n",
"53 0.369506 0.111577 -0.009068 0.009068 \n",
"\n",
" caption rate \n",
"0 aborted: neg. conc. in `A` NaN \n",
"1 aborted: excessive norm value(s) 3200.000000 \n",
"2 aborted: excessive norm value(s) 3200.000000 \n",
"3 aborted: excessive norm value(s) 3200.000000 \n",
"4 aborted: excessive norm value(s) 3200.000000 \n",
"5 aborted: excessive norm value(s) 3200.000000 \n",
"6 3200.000000 \n",
"7 3020.840960 \n",
"8 2885.686820 \n",
"9 2756.697133 \n",
"10 2658.211272 \n",
"11 2563.317851 \n",
"12 2471.885360 \n",
"13 2383.787110 \n",
"14 2298.901050 \n",
"15 2217.109600 \n",
"16 2138.299488 \n",
"17 2062.361590 \n",
"18 1989.190778 \n",
"19 1918.685776 \n",
"20 1850.749018 \n",
"21 1752.555256 \n",
"22 1659.724818 \n",
"23 1571.963289 \n",
"24 1488.992431 \n",
"25 1371.327721 \n",
"26 1263.291238 \n",
"27 1164.090351 \n",
"28 1072.997749 \n",
"29 947.520208 \n",
"30 837.406853 \n",
"31 740.761211 \n",
"32 613.501217 \n",
"33 509.526499 \n",
"34 424.528524 \n",
"35 320.231977 \n",
"36 244.302890 \n",
"37 188.880147 \n",
"38 127.992002 \n",
"39 91.251951 \n",
"40 57.382032 \n",
"41 45.286669 \n",
"42 34.125104 \n",
"43 29.591334 \n",
"44 24.454162 \n",
"45 21.435190 \n",
"46 17.511654 \n",
"47 14.325526 \n",
"48 10.413518 \n",
"49 7.570088 \n",
"50 4.469280 \n",
"51 2.638855 \n",
"52 1.017172 \n",
"53 0.081275 "
]
},
"execution_count": 22,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# Concentration increments due to reaction 0 (A <-> B)\n",
"# Note that [C] is not affected\n",
"dynamics.diagnostics.get_diagnostic_rxn_data(rxn_index=0)"
]
},
{
"cell_type": "code",
"execution_count": 23,
"id": "71f62382-f07a-416d-876f-97832c1f79e5",
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Reaction: B <-> C\n"
]
},
{
"data": {
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\n",
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" | \n",
" START_TIME | \n",
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" Delta C | \n",
" caption | \n",
" rate | \n",
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" | \n",
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" | 47 | \n",
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" | \n",
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" | \n",
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" | \n",
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" | \n",
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\n",
" \n",
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\n",
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],
"text/plain": [
" START_TIME time_step Delta B Delta C \\\n",
"0 0.000000 0.010000 0.000000 0.000000 \n",
"1 0.000000 0.006000 0.000000 0.000000 \n",
"2 0.000000 0.003600 0.000000 0.000000 \n",
"3 0.000000 0.002160 0.000000 0.000000 \n",
"4 0.000000 0.001296 0.000000 0.000000 \n",
"5 0.000000 0.000778 0.000000 0.000000 \n",
"6 0.000778 0.000622 -0.018575 0.018575 \n",
"7 0.001400 0.000622 -0.032442 0.032442 \n",
"8 0.002022 0.000498 -0.036448 0.036448 \n",
"9 0.002519 0.000498 -0.044387 0.044387 \n",
"10 0.003017 0.000498 -0.051978 0.051978 \n",
"11 0.003515 0.000498 -0.059234 0.059234 \n",
"12 0.004012 0.000498 -0.066168 0.066168 \n",
"13 0.004510 0.000498 -0.072791 0.072791 \n",
"14 0.005008 0.000498 -0.079117 0.079117 \n",
"15 0.005505 0.000498 -0.085155 0.085155 \n",
"16 0.006003 0.000498 -0.090917 0.090917 \n",
"17 0.006501 0.000498 -0.096413 0.096413 \n",
"18 0.006998 0.000498 -0.101653 0.101653 \n",
"19 0.007496 0.000746 -0.159970 0.159970 \n",
"20 0.008243 0.000746 -0.170675 0.170675 \n",
"21 0.008989 0.000746 -0.180610 0.180610 \n",
"22 0.009736 0.000746 -0.189822 0.189822 \n",
"23 0.010482 0.001120 -0.297524 0.297524 \n",
"24 0.011602 0.001120 -0.315263 0.315263 \n",
"25 0.012722 0.001120 -0.330954 0.330954 \n",
"26 0.013841 0.001120 -0.344773 0.344773 \n",
"27 0.014961 0.001680 -0.535325 0.535325 \n",
"28 0.016641 0.001680 -0.559062 0.559062 \n",
"29 0.018320 0.001680 -0.577992 0.577992 \n",
"30 0.020000 0.002519 -0.889125 0.889125 \n",
"31 0.022519 0.002519 -0.914188 0.914188 \n",
"32 0.025039 0.002519 -0.928673 0.928673 \n",
"33 0.027558 0.003779 -1.402091 1.402091 \n",
"34 0.031337 0.003779 -1.400666 1.400666 \n",
"35 0.035116 0.003779 -1.381442 1.381442 \n",
"36 0.038896 0.005669 -2.025333 2.025333 \n",
"37 0.044564 0.005669 -1.937433 1.937433 \n",
"38 0.050233 0.008503 -2.749544 2.749544 \n",
"39 0.058736 0.006802 -2.001123 2.001123 \n",
"40 0.065538 0.010204 -2.763615 2.763615 \n",
"41 0.075742 0.008163 -1.940327 1.940327 \n",
"42 0.083905 0.012244 -2.618807 2.618807 \n",
"43 0.096149 0.009796 -1.778500 1.778500 \n",
"44 0.105945 0.014693 -2.344137 2.344137 \n",
"45 0.120638 0.014693 -1.917467 1.917467 \n",
"46 0.135332 0.022040 -2.352600 2.352600 \n",
"47 0.157371 0.022040 -1.710189 1.710189 \n",
"48 0.179411 0.033060 -1.864795 1.864795 \n",
"49 0.212471 0.033060 -1.100982 1.100982 \n",
"50 0.245531 0.049590 -0.975033 0.975033 \n",
"51 0.295121 0.074385 -0.563972 0.563972 \n",
"52 0.369506 0.111577 -0.066295 0.066295 \n",
"\n",
" caption rate \n",
"0 aborted: excessive norm value(s) NaN \n",
"1 aborted: excessive norm value(s) NaN \n",
"2 aborted: excessive norm value(s) NaN \n",
"3 aborted: excessive norm value(s) NaN \n",
"4 aborted: excessive norm value(s) NaN \n",
"5 NaN \n",
"6 29.859840 \n",
"7 52.150244 \n",
"8 73.237598 \n",
"9 89.190237 \n",
"10 104.443575 \n",
"11 119.023938 \n",
"12 132.956683 \n",
"13 146.266235 \n",
"14 158.976118 \n",
"15 171.108992 \n",
"16 182.686681 \n",
"17 193.730206 \n",
"18 204.259814 \n",
"19 214.295006 \n",
"20 228.634342 \n",
"21 241.944203 \n",
"22 254.283393 \n",
"23 265.707463 \n",
"24 281.549606 \n",
"25 295.562349 \n",
"26 307.903744 \n",
"27 318.718715 \n",
"28 332.850855 \n",
"29 344.121633 \n",
"30 352.908006 \n",
"31 362.855756 \n",
"32 368.605168 \n",
"33 371.008309 \n",
"34 370.631249 \n",
"35 365.544329 \n",
"36 357.283191 \n",
"37 341.777002 \n",
"38 323.359530 \n",
"39 294.176957 \n",
"40 270.845299 \n",
"41 237.699766 \n",
"42 213.877913 \n",
"43 181.562555 \n",
"44 159.538054 \n",
"45 130.499569 \n",
"46 106.742678 \n",
"47 77.595076 \n",
"48 56.406583 \n",
"49 33.302648 \n",
"50 19.661952 \n",
"51 7.581818 \n",
"52 0.594162 "
]
},
"execution_count": 23,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# Concentration increments due to reaction 1 (B <-> C) \n",
"# Also notice that the 0-th row from the A <-> B reaction isn't seen here (start time 0 and step 0.02), \n",
"# because that step was aborted early on, BEFORE even getting to THIS reaction\n",
"dynamics.diagnostics.get_diagnostic_rxn_data(rxn_index=1)"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "46fdcb5b-ed5f-43a9-9829-533039666d0c",
"metadata": {},
"outputs": [],
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"id": "de51ec27-1d0f-4bfb-83f0-b1c79b436d75",
"metadata": {
"tags": []
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"source": [
"# PART 3 : Re-run with very small constant steps, and compare with original run"
]
},
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"cell_type": "markdown",
"id": "ff9fffab-a1f5-4497-9d7b-c3db023d1bd2",
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"source": [
"We'll use **constant steps of size 0.0005** , which is 1/4 of the smallest steps (the \"substep\" size) previously used in the variable-step run"
]
},
{
"cell_type": "code",
"execution_count": 24,
"id": "93418d81-12fa-417d-9906-1dd24a40de48",
"metadata": {},
"outputs": [],
"source": [
"dynamics2 = UniformCompartment(chem_data=chem) # Re-use the same chemicals and reactions of the previous simulation"
]
},
{
"cell_type": "code",
"execution_count": 25,
"id": "ecf2029b-57a7-4db4-9b30-c854da91fb76",
"metadata": {
"tags": []
},
"outputs": [],
"source": [
"dynamics2.set_conc({\"A\": 50.}, snapshot=True)"
]
},
{
"cell_type": "code",
"execution_count": 26,
"id": "00777982-d633-4c12-8c92-2ece6bf00e5e",
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"800 total step(s) taken\n"
]
}
],
"source": [
"# Notice that we're using FIXED steps this time\n",
"dynamics2.single_compartment_react(initial_step=0.0005, duration=0.4,\n",
" variable_steps=False,\n",
" snapshots={\"initial_caption\": \"1st reaction step\",\n",
" \"final_caption\": \"last reaction step\"},\n",
" )"
]
},
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"execution_count": 27,
"id": "6bb61113-a081-4366-893d-f94894c44ec2",
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"plot_curves() WARNING: Excessive number of vertical lines (801) - only showing 1 every 6 lines\n"
]
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]
},
{
"cell_type": "code",
"execution_count": 28,
"id": "e884f562-a723-4208-88e5-ba0bf36e5e3a",
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"execution_count": 28,
"metadata": {},
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"cell_type": "code",
"execution_count": 29,
"id": "5b01f35a-17ac-4965-8a21-f300783c1e2c",
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"metadata": {},
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"cell_type": "code",
"execution_count": 30,
"id": "5dae6925-3f3b-49ca-81f9-3cb8f19fdd55",
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"\n",
"\n",
"
\n",
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" | \n",
" SYSTEM TIME | \n",
" A | \n",
" B | \n",
" C | \n",
" caption | \n",
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" | \n",
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\n",
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" ... | \n",
" ... | \n",
" ... | \n",
" ... | \n",
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" | 796 | \n",
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" SYSTEM TIME A B C caption\n",
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},
"execution_count": 31,
"metadata": {},
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],
"source": [
"df2 = dynamics2.get_history()\n",
"df2"
]
},
{
"cell_type": "markdown",
"id": "45d5883d-c2c6-40a7-80b4-8a5976b8dfcd",
"metadata": {},
"source": [
"## Notice that we now did 800 steps - vs. the 48 of the earlier variable-resolution run!"
]
},
{
"cell_type": "markdown",
"id": "0f590255-1555-43d5-9adb-ace15ac80aa4",
"metadata": {},
"source": [
"## Let's compare some entries with the coarser previous variable-time run"
]
},
{
"cell_type": "markdown",
"id": "bfef068e-d576-4c3c-bda8-b0b51c5f88da",
"metadata": {},
"source": [
"#### Let's compare the plots of [B] from the earlier (variable-step) run, and the latest (high-precision, fixed-step) one:"
]
},
{
"cell_type": "code",
"execution_count": 32,
"id": "b2917e15-1c3a-4597-bc9b-22f0a83b3e08",
"metadata": {},
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"fig2 = dynamics2.plot_history(chemicals='B', colors=['violet'], title=\"Fine fixed-step run\", \n",
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"source": [
"#### They overlap fairly well! The 800 fixed-timestep points vs. the 48 adaptable variable-timestep ones.\n",
"The adaptive algorithms avoided 752 extra steps of limited benefit..."
]
},
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