{ "cells": [ { "cell_type": "markdown", "id": "5cbc8640", "metadata": {}, "source": [ "# **Enzyme Kinetics** : \n", "#### An _accurate numerical solution_ of an enzymatic reaction `E + S <-> ES -> E + P` - compared to the **Michaelis-Menten** model approximation and to the alternative **Morrison** model, using real-life kinetic parameters. \n", "\n", "#### Scenario with _small amount of Enzyme_, relative to the initial Substrate concentration.\n", "\n", "#### Unlike in experiment `enzyme_1_a`, we'll use data from a reaction that **VIOLATES the customary Michaelis-Menten assumption** that the rate constants satisfy `k1_reverse >> k2_forward`" ] }, { "cell_type": "markdown", "id": "604b150b-7812-4fd3-9403-69a06dd7e397", "metadata": {}, "source": [ "### THE REACTION: \n", "the enzyme `Aminopeptidase` with the substrate `Leu-Ala-DED`, \n", "and the initial concentration values choosen below, all satisfy the customary Michaelis-Menten assumptions that `[E] << [S]` BUT the reaction rate constants DON'T satisfy `k1_reverse >> k2_forward`\n", "\n", "For this reaction: k1_forward = 160 , k1_reverse = 0.089 , k2_forward = 0.58 \n", "\n", "Source of kinetic parameters: *page 16 of \"Analysis of Enzyme Reaction Kinetics, Vol. 1\", by F. Xavier Malcata, Wiley, 2023*" ] }, { "cell_type": "markdown", "id": "c123db4f-c802-47f0-a3d3-0b857314d8e5", "metadata": {}, "source": [ "### TAGS : \"uniform compartment\", \"chemistry\", \"numerical\", \"enzymes\"" ] }, { "cell_type": "code", "execution_count": 1, "id": "6e9d0902-6fc9-4692-ac39-0651d08902ca", "metadata": {}, "outputs": [], "source": [ "LAST_REVISED = \"Sep. 2, 2025\"\n", "LIFE123_VERSION = \"1.0.0rc6\" # Library version this experiment is based on" ] }, { "cell_type": "code", "execution_count": 2, "id": "1e0ae9a9-9d0c-4edf-a5f2-1c589419e6cf", "metadata": {}, "outputs": [], "source": [ "#import set_path # Using MyBinder? Uncomment this before running the next cell!" ] }, { "cell_type": "code", "execution_count": 3, "id": "a29db1c7", "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 pandas as pd\n", "\n", "from life123 import check_version, ChemData, UniformCompartment, ReactionEnzyme, PlotlyHelper" ] }, { "cell_type": "code", "execution_count": 4, "id": "af15ecf0-e083-4fef-b68e-abe794dcc86e", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "OK\n" ] } ], "source": [ "check_version(LIFE123_VERSION) # To check compatibility" ] }, { "cell_type": "code", "execution_count": null, "id": "3713fa4d-e9bb-4e33-8734-c1d4d23cf177", "metadata": {}, "outputs": [], "source": [] }, { "cell_type": "code", "execution_count": null, "id": "34d1cefc-f644-410a-9fe4-5204964742ac", "metadata": {}, "outputs": [], "source": [] }, { "cell_type": "markdown", "id": "91a729ee-ad2f-4771-bc84-49ecb9d6a851", "metadata": {}, "source": [ "# PART 1. Accurate numerical solution" ] }, { "cell_type": "code", "execution_count": 5, "id": "959cf00a-47b3-455b-9923-9ec8741cc863", "metadata": {}, "outputs": [], "source": [ "chem_data = ChemData(names=[\"P\", \"ES\"], plot_colors=[\"green\", \"red\"])" ] }, { "cell_type": "code", "execution_count": 6, "id": "90f9ec41-6e1b-4558-9400-47e35ece6efc", "metadata": {}, "outputs": [ { "data": { "text/html": [ "
| \n", " | name | \n", "label | \n", "plot_color | \n", "
|---|---|---|---|
| 0 | \n", "P | \n", "P | \n", "green | \n", "
| 1 | \n", "ES | \n", "ES | \n", "red | \n", "
| 2 | \n", "Aminopeptidase | \n", "E | \n", "violet | \n", "
| 3 | \n", "Leu-Ala-DED | \n", "S | \n", "darkturquoise | \n", "
| \n", " | SYSTEM TIME | \n", "P | \n", "ES | \n", "E | \n", "S | \n", "step | \n", "caption | \n", "
|---|---|---|---|---|---|---|---|
| 0 | \n", "0.000000 | \n", "0.000000e+00 | \n", "0.000000 | \n", "1.000000 | \n", "20.000000 | \n", "\n", " | Set concentration | \n", "
| 1 | \n", "0.000010 | \n", "0.000000e+00 | \n", "0.032000 | \n", "0.968000 | \n", "19.968000 | \n", "1 | \n", "1st reaction step | \n", "
| 2 | \n", "0.000021 | \n", "2.041600e-07 | \n", "0.066019 | \n", "0.933981 | \n", "19.933981 | \n", "2 | \n", "\n", " |
| 3 | \n", "0.000024 | \n", "3.305201e-07 | \n", "0.075849 | \n", "0.924151 | \n", "19.924151 | \n", "3 | \n", "\n", " |
| 4 | \n", "0.000025 | \n", "3.740726e-07 | \n", "0.078766 | \n", "0.921234 | \n", "19.921234 | \n", "4 | \n", "\n", " |
| ... | \n", "... | \n", "... | \n", "... | \n", "... | \n", "... | \n", "... | \n", "... | \n", "
| 169 | \n", "0.001437 | \n", "6.495498e-04 | \n", "0.988861 | \n", "0.011139 | \n", "19.010489 | \n", "169 | \n", "\n", " |
| 170 | \n", "0.001454 | \n", "6.594576e-04 | \n", "0.989435 | \n", "0.010565 | \n", "19.009906 | \n", "170 | \n", "\n", " |
| 171 | \n", "0.001472 | \n", "6.693712e-04 | \n", "0.989979 | \n", "0.010021 | \n", "19.009352 | \n", "171 | \n", "\n", " |
| 172 | \n", "0.001489 | \n", "6.792902e-04 | \n", "0.990494 | \n", "0.009506 | \n", "19.008827 | \n", "172 | \n", "\n", " |
| 173 | \n", "0.001506 | \n", "6.892144e-04 | \n", "0.990982 | \n", "0.009018 | \n", "19.008329 | \n", "173 | \n", "last reaction step | \n", "
174 rows × 7 columns
\n", "| \n", " | SYSTEM TIME | \n", "rxn0_rate_1 | \n", "rxn0_rate_2 | \n", "step | \n", "
|---|---|---|---|---|
| 0 | \n", "0.000000 | \n", "3200.000000 | \n", "0.000000 | \n", "0 | \n", "
| 1 | \n", "0.000010 | \n", "3092.640992 | \n", "0.018560 | \n", "1 | \n", "
| 2 | \n", "0.000021 | \n", "2978.868127 | \n", "0.038291 | \n", "2 | \n", "
| 3 | \n", "0.000024 | \n", "2946.061101 | \n", "0.043992 | \n", "3 | \n", "
| 4 | \n", "0.000025 | \n", "2936.333355 | \n", "0.045684 | \n", "4 | \n", "
| ... | \n", "... | \n", "... | \n", "... | \n", "... | \n", "
| 168 | \n", "0.001420 | \n", "35.636312 | \n", "0.573188 | \n", "168 | \n", "
| 169 | \n", "0.001437 | \n", "33.792720 | \n", "0.573539 | \n", "169 | \n", "
| 170 | \n", "0.001454 | \n", "32.046189 | \n", "0.573872 | \n", "170 | \n", "
| 171 | \n", "0.001472 | \n", "30.391600 | \n", "0.574188 | \n", "171 | \n", "
| 172 | \n", "0.001489 | \n", "28.824102 | \n", "0.574486 | \n", "172 | \n", "
173 rows × 4 columns
\n", "| \n", " | SYSTEM TIME | \n", "rxn0_rate_1 | \n", "rxn0_rate_2 | \n", "step | \n", "
|---|---|---|---|---|
| 0 | \n", "0.000000 | \n", "3200.000000 | \n", "0.000000 | \n", "0 | \n", "
| 1 | \n", "0.000010 | \n", "3092.640992 | \n", "0.018560 | \n", "1 | \n", "
| 2 | \n", "0.000021 | \n", "2978.868127 | \n", "0.038291 | \n", "2 | \n", "
| 3 | \n", "0.000024 | \n", "2946.061101 | \n", "0.043992 | \n", "3 | \n", "
| 4 | \n", "0.000025 | \n", "2936.333355 | \n", "0.045684 | \n", "4 | \n", "
| ... | \n", "... | \n", "... | \n", "... | \n", "... | \n", "
| 2162 | \n", "39.081690 | \n", "0.000009 | \n", "0.014678 | \n", "19889 | \n", "
| 2163 | \n", "39.195774 | \n", "0.000008 | \n", "0.013736 | \n", "19899 | \n", "
| 2164 | \n", "39.491678 | \n", "0.000007 | \n", "0.011552 | \n", "19909 | \n", "
| 2165 | \n", "39.836446 | \n", "0.000050 | \n", "0.009430 | \n", "19919 | \n", "
| 2166 | \n", "39.903868 | \n", "0.000005 | \n", "0.009068 | \n", "19929 | \n", "
2167 rows × 4 columns
\n", "| \n", " | SYSTEM TIME | \n", "P | \n", "ES | \n", "E | \n", "S | \n", "step | \n", "caption | \n", "P_rate | \n", "
|---|---|---|---|---|---|---|---|---|
| 0 | \n", "0.000000 | \n", "0.000000e+00 | \n", "0.000000 | \n", "1.000000 | \n", "20.000000 | \n", "\n", " | Set concentration | \n", "0.000000 | \n", "
| 1 | \n", "0.000010 | \n", "0.000000e+00 | \n", "0.032000 | \n", "0.968000 | \n", "19.968000 | \n", "1 | \n", "1st reaction step | \n", "0.018560 | \n", "
| 2 | \n", "0.000021 | \n", "2.041600e-07 | \n", "0.066019 | \n", "0.933981 | \n", "19.933981 | \n", "2 | \n", "\n", " | 0.038291 | \n", "
| 3 | \n", "0.000024 | \n", "3.305201e-07 | \n", "0.075849 | \n", "0.924151 | \n", "19.924151 | \n", "3 | \n", "\n", " | 0.043992 | \n", "
| 4 | \n", "0.000025 | \n", "3.740726e-07 | \n", "0.078766 | \n", "0.921234 | \n", "19.921234 | \n", "4 | \n", "\n", " | 0.045684 | \n", "
| ... | \n", "... | \n", "... | \n", "... | \n", "... | \n", "... | \n", "... | \n", "... | \n", "... | \n", "
| 2162 | \n", "39.081690 | \n", "1.997468e+01 | \n", "0.025307 | \n", "0.974693 | \n", "0.000014 | \n", "19889 | \n", "\n", " | 0.014678 | \n", "
| 2163 | \n", "39.195774 | \n", "1.997630e+01 | \n", "0.023682 | \n", "0.976318 | \n", "0.000014 | \n", "19899 | \n", "\n", " | 0.013736 | \n", "
| 2164 | \n", "39.491678 | \n", "1.998007e+01 | \n", "0.019917 | \n", "0.980083 | \n", "0.000011 | \n", "19909 | \n", "\n", " | 0.011552 | \n", "
| 2165 | \n", "39.836446 | \n", "1.998373e+01 | \n", "0.016258 | \n", "0.983742 | \n", "0.000010 | \n", "19919 | \n", "\n", " | 0.009430 | \n", "
| 2166 | \n", "39.903868 | \n", "1.998436e+01 | \n", "0.015634 | \n", "0.984366 | \n", "0.000009 | \n", "19929 | \n", "\n", " | 0.009068 | \n", "
2167 rows × 8 columns
\n", "| \n", " | SYSTEM TIME | \n", "P | \n", "ES | \n", "E | \n", "S | \n", "step | \n", "caption | \n", "P_rate | \n", "Michaelis_rate | \n", "
|---|---|---|---|---|---|---|---|---|---|
| 0 | \n", "0.000000 | \n", "0.000000e+00 | \n", "0.000000 | \n", "1.000000 | \n", "20.000000 | \n", "\n", " | Set concentration | \n", "0.000000 | \n", "0.579879 | \n", "
| 1 | \n", "0.000010 | \n", "0.000000e+00 | \n", "0.032000 | \n", "0.968000 | \n", "19.968000 | \n", "1 | \n", "1st reaction step | \n", "0.018560 | \n", "0.579879 | \n", "
| 2 | \n", "0.000021 | \n", "2.041600e-07 | \n", "0.066019 | \n", "0.933981 | \n", "19.933981 | \n", "2 | \n", "\n", " | 0.038291 | \n", "0.579878 | \n", "
| 3 | \n", "0.000024 | \n", "3.305201e-07 | \n", "0.075849 | \n", "0.924151 | \n", "19.924151 | \n", "3 | \n", "\n", " | 0.043992 | \n", "0.579878 | \n", "
| 4 | \n", "0.000025 | \n", "3.740726e-07 | \n", "0.078766 | \n", "0.921234 | \n", "19.921234 | \n", "4 | \n", "\n", " | 0.045684 | \n", "0.579878 | \n", "
| ... | \n", "... | \n", "... | \n", "... | \n", "... | \n", "... | \n", "... | \n", "... | \n", "... | \n", "... | \n", "
| 2162 | \n", "39.081690 | \n", "1.997468e+01 | \n", "0.025307 | \n", "0.974693 | \n", "0.000014 | \n", "19889 | \n", "\n", " | 0.014678 | \n", "0.002004 | \n", "
| 2163 | \n", "39.195774 | \n", "1.997630e+01 | \n", "0.023682 | \n", "0.976318 | \n", "0.000014 | \n", "19899 | \n", "\n", " | 0.013736 | \n", "0.001873 | \n", "
| 2164 | \n", "39.491678 | \n", "1.998007e+01 | \n", "0.019917 | \n", "0.980083 | \n", "0.000011 | \n", "19909 | \n", "\n", " | 0.011552 | \n", "0.001570 | \n", "
| 2165 | \n", "39.836446 | \n", "1.998373e+01 | \n", "0.016258 | \n", "0.983742 | \n", "0.000010 | \n", "19919 | \n", "\n", " | 0.009430 | \n", "0.001317 | \n", "
| 2166 | \n", "39.903868 | \n", "1.998436e+01 | \n", "0.015634 | \n", "0.984366 | \n", "0.000009 | \n", "19929 | \n", "\n", " | 0.009068 | \n", "0.001227 | \n", "
2167 rows × 9 columns
\n", "| \n", " | SYSTEM TIME | \n", "P | \n", "ES | \n", "E | \n", "S | \n", "step | \n", "caption | \n", "P_rate | \n", "Michaelis_rate | \n", "Morrison_rate | \n", "
|---|---|---|---|---|---|---|---|---|---|---|
| 0 | \n", "0.000000 | \n", "0.000000e+00 | \n", "0.000000 | \n", "1.000000 | \n", "20.000000 | \n", "\n", " | Set concentration | \n", "0.000000 | \n", "0.579879 | \n", "0.579872 | \n", "
| 1 | \n", "0.000010 | \n", "0.000000e+00 | \n", "0.032000 | \n", "0.968000 | \n", "19.968000 | \n", "1 | \n", "1st reaction step | \n", "0.018560 | \n", "0.579879 | \n", "0.579872 | \n", "
| 2 | \n", "0.000021 | \n", "2.041600e-07 | \n", "0.066019 | \n", "0.933981 | \n", "19.933981 | \n", "2 | \n", "\n", " | 0.038291 | \n", "0.579878 | \n", "0.579872 | \n", "
| 3 | \n", "0.000024 | \n", "3.305201e-07 | \n", "0.075849 | \n", "0.924151 | \n", "19.924151 | \n", "3 | \n", "\n", " | 0.043992 | \n", "0.579878 | \n", "0.579872 | \n", "
| 4 | \n", "0.000025 | \n", "3.740726e-07 | \n", "0.078766 | \n", "0.921234 | \n", "19.921234 | \n", "4 | \n", "\n", " | 0.045684 | \n", "0.579878 | \n", "0.579872 | \n", "
| ... | \n", "... | \n", "... | \n", "... | \n", "... | \n", "... | \n", "... | \n", "... | \n", "... | \n", "... | \n", "... | \n", "
| 2162 | \n", "39.081690 | \n", "1.997468e+01 | \n", "0.025307 | \n", "0.974693 | \n", "0.000014 | \n", "19889 | \n", "\n", " | 0.014678 | \n", "0.002004 | \n", "0.014624 | \n", "
| 2163 | \n", "39.195774 | \n", "1.997630e+01 | \n", "0.023682 | \n", "0.976318 | \n", "0.000014 | \n", "19899 | \n", "\n", " | 0.013736 | \n", "0.001873 | \n", "0.013685 | \n", "
| 2164 | \n", "39.491678 | \n", "1.998007e+01 | \n", "0.019917 | \n", "0.980083 | \n", "0.000011 | \n", "19909 | \n", "\n", " | 0.011552 | \n", "0.001570 | \n", "0.011510 | \n", "
| 2165 | \n", "39.836446 | \n", "1.998373e+01 | \n", "0.016258 | \n", "0.983742 | \n", "0.000010 | \n", "19919 | \n", "\n", " | 0.009430 | \n", "0.001317 | \n", "0.009395 | \n", "
| 2166 | \n", "39.903868 | \n", "1.998436e+01 | \n", "0.015634 | \n", "0.984366 | \n", "0.000009 | \n", "19929 | \n", "\n", " | 0.009068 | \n", "0.001227 | \n", "0.009034 | \n", "
2167 rows × 10 columns
\n", "