{"title":"mRNA expression","code":"# /// script\n# requires-python = \">=3.12\"\n# dependencies = [\"marimo>=0.25.1\", \"numpy\", \"matplotlib\"]\n# ///\n\nimport marimo\n\n__generated_with = \"0.25.1\"\napp = marimo.App(width=\"medium\")\n\n\n@app.cell(hide_code=True)\ndef _(mo):\n mo.md(\n r\"\"\"\n # How long does an mRNA dose make protein?\n\n An mRNA therapeutic or vaccine delivers instructions, not the protein\n itself. Once the mRNA reaches the cytoplasm, ribosomes translate it\n until it is degraded, and the protein it encodes is degraded in turn.\n The simplest model that describes single-cell transfection data has two\n first-order steps (Leonhardt et al. 2014):\n\n $$\\frac{dm}{dt} = -\\delta\\, m, \\qquad \\frac{dp}{dt} = k\\, m - \\beta\\, p$$\n\n with mRNA $m$, protein $p$, translation rate $k$, and degradation rates\n $\\delta = \\ln 2 / t_{1/2}^{\\text{mRNA}}$ and\n $\\beta = \\ln 2 / t_{1/2}^{\\text{protein}}$. Starting from $m_0$ at\n $t = 0$ the protein level is\n\n $$p(t) = \\frac{k\\, m_0}{\\delta - \\beta}\\left(e^{-\\beta t} - e^{-\\delta t}\\right).$$\n \"\"\"\n )\n return\n\n\n@app.cell\ndef _():\n import marimo as mo\n import numpy as np\n import matplotlib.pyplot as plt\n return mo, np, plt\n\n\n@app.cell\ndef _(mo):\n mrna_half_life = mo.ui.slider(0.5, 48, step=0.5, value=10, label=\"mRNA half-life (h)\")\n protein_half_life = mo.ui.slider(1, 96, step=1, value=26, label=\"Protein half-life (h)\")\n mo.vstack([mrna_half_life, protein_half_life])\n return mrna_half_life, protein_half_life\n\n\n@app.cell\ndef _(np):\n def protein(t, mrna_half, protein_half, k=1.0, m0=1.0):\n \"\"\"Protein level after one dose at t = 0, in units of k * m0 * hours.\"\"\"\n delta, beta = np.log(2) / mrna_half, np.log(2) / protein_half\n t = np.asarray(t, dtype=float)\n if np.isclose(delta, beta):\n level = k * m0 * t * np.exp(-beta * t)\n else:\n level = k * m0 / (delta - beta) * (np.exp(-beta * t) - np.exp(-delta * t))\n return np.where(t >= 0, level, 0.0)\n\n def peak_time(mrna_half, protein_half):\n delta, beta = np.log(2) / mrna_half, np.log(2) / protein_half\n if np.isclose(delta, beta):\n return 1 / beta\n return np.log(delta / beta) / (delta - beta)\n\n return peak_time, protein\n\n\n@app.cell\ndef _(mo, mrna_half_life, np, peak_time, plt, protein, protein_half_life):\n hours = np.linspace(0, 168, 800)\n level = protein(hours, mrna_half_life.value, protein_half_life.value)\n mrna = np.exp(-np.log(2) / mrna_half_life.value * hours)\n t_peak = peak_time(mrna_half_life.value, protein_half_life.value)\n exposure = mrna_half_life.value * protein_half_life.value / np.log(2) ** 2\n\n fig, ax = plt.subplots(figsize=(7, 3.6))\n ax.plot(hours / 24, level / level.max(), color=\"#0969da\", lw=2, label=\"protein\")\n ax.plot(hours / 24, mrna, color=\"#9a6700\", lw=1.5, ls=\"--\", label=\"mRNA\")\n ax.axvline(t_peak / 24, color=\"#57606a\", lw=0.8)\n ax.set(xlabel=\"days after the dose\", ylabel=\"fraction of peak\", xlim=(0, 7), ylim=(0, 1.05))\n ax.legend(frameon=False)\n fig.tight_layout()\n\n mo.vstack(\n [\n fig,\n mo.md(\n f\"Protein peaks **{t_peak:.1f} hours** after the dose. Total protein made \"\n f\"over the whole course is $k m_0 / (\\\\delta\\\\beta)$ = **{exposure:,.0f}** \"\n \"in units of $k m_0$ hours squared.\"\n ),\n ]\n )\n return\n\n\n@app.cell(hide_code=True)\ndef _(mo):\n mo.md(\n r\"\"\"\n ## Why mRNA stability matters\n\n The area under the protein curve is $k m_0 / (\\delta \\beta)$: it is\n proportional to the mRNA half-life times the protein half-life. Doubling\n how long the mRNA survives doubles the total protein made from the same\n dose, without changing the dose. Much of the engineering of therapeutic\n mRNA (modified nucleosides such as N1-methylpseudouridine, optimised\n untranslated regions, the cap structure) works on $k$ and $\\delta$.\n The curves below keep the protein half-life you chose and vary only the\n mRNA half-life.\n \"\"\"\n )\n return\n\n\n@app.cell\ndef _(np, plt, protein, protein_half_life):\n days = np.linspace(0, 10, 800)\n fig2, ax2 = plt.subplots(figsize=(7, 3.4))\n for _half, _shade in zip((2, 6, 12, 24), (\"#d0d7de\", \"#8c959f\", \"#57606a\", \"#0969da\")):\n ax2.plot(\n days,\n protein(days * 24, _half, protein_half_life.value),\n color=_shade,\n lw=2,\n label=f\"mRNA half-life {_half} h\",\n )\n ax2.set(xlabel=\"days after the dose\", ylabel=\"protein (k m₀ h)\")\n ax2.legend(frameon=False)\n fig2.tight_layout()\n fig2\n return\n\n\n@app.cell(hide_code=True)\ndef _(mo):\n mo.md(\n r\"\"\"\n ## Repeated doses\n\n The model is linear, so repeated doses add up. The protein level after\n a schedule is the sum of the single-dose curves, each shifted to its\n dose time. Choose an interval and a number of doses to see whether\n expression stays above a threshold between doses.\n \"\"\"\n )\n return\n\n\n@app.cell\ndef _(mo):\n interval = mo.ui.slider(6, 96, step=6, value=24, label=\"Dose every (h)\")\n doses = mo.ui.slider(1, 10, step=1, value=5, label=\"Number of doses\")\n threshold = mo.ui.slider(0.05, 1.0, step=0.05, value=0.5, label=\"Threshold (fraction of one-dose peak)\")\n mo.vstack([mo.hstack([interval, doses], justify=\"start\"), threshold])\n return doses, interval, threshold\n\n\n@app.cell\ndef _(\n doses,\n interval,\n mo,\n mrna_half_life,\n np,\n plt,\n protein,\n protein_half_life,\n threshold,\n):\n span = interval.value * doses.value + 96\n t = np.linspace(0, span, 1600)\n single_peak = protein(np.linspace(0, 400, 4000), mrna_half_life.value, protein_half_life.value).max()\n total = sum(\n protein(t - n * interval.value, mrna_half_life.value, protein_half_life.value)\n for n in range(doses.value)\n ) / single_peak\n covered = t[(t > 0) & (t < interval.value * doses.value)]\n above = np.mean(np.interp(covered, t, total) >= threshold.value) if covered.size else 0.0\n\n fig3, ax3 = plt.subplots(figsize=(7, 3.4))\n ax3.plot(t / 24, total, color=\"#0969da\", lw=2)\n ax3.axhline(threshold.value, color=\"#cf222e\", lw=0.8, ls=\"--\")\n for _n in range(doses.value):\n ax3.axvline(_n * interval.value / 24, color=\"#d0d7de\", lw=0.8)\n ax3.set(xlabel=\"days\", ylabel=\"protein / one-dose peak\")\n fig3.tight_layout()\n mo.vstack(\n [\n fig3,\n mo.md(\n f\"During dosing, expression is above the threshold **{above:.0%}** of the time. \"\n f\"The highest level reached is **{total.max():.2f}** times a single dose's peak.\"\n ),\n ]\n )\n return\n\n\n@app.cell(hide_code=True)\ndef _(mo):\n mo.md(\n r\"\"\"\n ---\n **Model and limits.** One compartment, first-order kinetics, all mRNA\n released at $t = 0$. Real delivery is spread over hours, cells differ\n widely in all three rates, and the immune response to a vaccine is not\n modelled. Levels are relative; set $k$ and $m_0$ from your own data to\n get absolute amounts. The default protein half-life, 26 hours, is the\n value reported for EGFP in mammalian cells (Corish and Tyler-Smith 1999).\n **References.** A. Leonhardt et al., *Nanomedicine: NBM* 10 (2014) 679;\n P. Corish and C. Tyler-Smith, *Protein Eng.* 12 (1999) 1035;\n K. Karikó et al., *Mol. Ther.* 16 (2008) 1833.\n Fork it to fit the model to your own time-course data.\n \"\"\"\n )\n return\n\n\nif __name__ == \"__main__\":\n app.run()\n","revision":"2026-10-13T02:22:13.172Z","appConfig":{"width":"medium"},"firstPaint":{"notebook":{"cells":[{"code":"mo.md(\n r\"\"\"\n # How long does an mRNA dose make protein?\n\n An mRNA therapeutic or vaccine delivers instructions, not the protein\n itself. Once the mRNA reaches the cytoplasm, ribosomes translate it\n until it is degraded, and the protein it encodes is degraded in turn.\n The simplest model that describes single-cell transfection data has two\n first-order steps (Leonhardt et al. 2014):\n\n $$\\frac{dm}{dt} = -\\delta\\, m, \\qquad \\frac{dp}{dt} = k\\, m - \\beta\\, p$$\n\n with mRNA $m$, protein $p$, translation rate $k$, and degradation rates\n $\\delta = \\ln 2 / t_{1/2}^{\\text{mRNA}}$ and\n $\\beta = \\ln 2 / t_{1/2}^{\\text{protein}}$. Starting from $m_0$ at\n $t = 0$ the protein level is\n\n $$p(t) = \\frac{k\\, m_0}{\\delta - \\beta}\\left(e^{-\\beta t} - e^{-\\delta t}\\right).$$\n \"\"\"\n)","code_hash":"a8bb96a4b9e7161047b17c1c90418d3c","config":{"column":null,"disabled":false,"expand_output":false,"hide_code":true},"id":"Hbol","name":"_"},{"code":"import marimo as mo\nimport numpy as np\nimport matplotlib.pyplot as plt","code_hash":"ef0032c4370d821ec8e83d0954968f66","config":{"column":null,"disabled":false,"expand_output":false,"hide_code":false},"id":"MJUe","name":"_"},{"code":"mrna_half_life = mo.ui.slider(0.5, 48, step=0.5, value=10, label=\"mRNA half-life (h)\")\nprotein_half_life = mo.ui.slider(1, 96, step=1, value=26, label=\"Protein half-life (h)\")\nmo.vstack([mrna_half_life, protein_half_life])","code_hash":"b711204af78d2545cdcb4b6b999bc17e","config":{"column":null,"disabled":false,"expand_output":false,"hide_code":false},"id":"vblA","name":"_"},{"code":"def protein(t, mrna_half, protein_half, k=1.0, m0=1.0):\n \"\"\"Protein level after one dose at t = 0, in units of k * m0 * hours.\"\"\"\n delta, beta = np.log(2) / mrna_half, np.log(2) / protein_half\n t = np.asarray(t, dtype=float)\n if np.isclose(delta, beta):\n level = k * m0 * t * np.exp(-beta * t)\n else:\n level = k * m0 / (delta - beta) * (np.exp(-beta * t) - np.exp(-delta * t))\n return np.where(t >= 0, level, 0.0)\n\ndef peak_time(mrna_half, protein_half):\n delta, beta = np.log(2) / mrna_half, np.log(2) / protein_half\n if np.isclose(delta, beta):\n return 1 / beta\n return np.log(delta / beta) / (delta - beta)","code_hash":"1b9b9ffb75d516e7aec712bf77685213","config":{"column":null,"disabled":false,"expand_output":false,"hide_code":false},"id":"bkHC","name":"_"},{"code":"hours = np.linspace(0, 168, 800)\nlevel = protein(hours, mrna_half_life.value, protein_half_life.value)\nmrna = np.exp(-np.log(2) / mrna_half_life.value * hours)\nt_peak = peak_time(mrna_half_life.value, protein_half_life.value)\nexposure = mrna_half_life.value * protein_half_life.value / np.log(2) ** 2\n\nfig, ax = plt.subplots(figsize=(7, 3.6))\nax.plot(hours / 24, level / level.max(), color=\"#0969da\", lw=2, label=\"protein\")\nax.plot(hours / 24, mrna, color=\"#9a6700\", lw=1.5, ls=\"--\", label=\"mRNA\")\nax.axvline(t_peak / 24, color=\"#57606a\", lw=0.8)\nax.set(xlabel=\"days after the dose\", ylabel=\"fraction of peak\", xlim=(0, 7), ylim=(0, 1.05))\nax.legend(frameon=False)\nfig.tight_layout()\n\nmo.vstack(\n [\n fig,\n mo.md(\n f\"Protein peaks **{t_peak:.1f} hours** after the dose. Total protein made \"\n f\"over the whole course is $k m_0 / (\\\\delta\\\\beta)$ = **{exposure:,.0f}** \"\n \"in units of $k m_0$ hours squared.\"\n ),\n ]\n)","code_hash":"284614b01f0fbe2a68e001676cef3477","config":{"column":null,"disabled":false,"expand_output":false,"hide_code":false},"id":"lEQa","name":"_"},{"code":"mo.md(\n r\"\"\"\n ## Why mRNA stability matters\n\n The area under the protein curve is $k m_0 / (\\delta \\beta)$: it is\n proportional to the mRNA half-life times the protein half-life. Doubling\n how long the mRNA survives doubles the total protein made from the same\n dose, without changing the dose. Much of the engineering of therapeutic\n mRNA (modified nucleosides such as N1-methylpseudouridine, optimised\n untranslated regions, the cap structure) works on $k$ and $\\delta$.\n The curves below keep the protein half-life you chose and vary only the\n mRNA half-life.\n \"\"\"\n)","code_hash":"76c7e6d54473f13ec075334f0d7f1334","config":{"column":null,"disabled":false,"expand_output":false,"hide_code":true},"id":"PKri","name":"_"},{"code":"days = np.linspace(0, 10, 800)\nfig2, ax2 = plt.subplots(figsize=(7, 3.4))\nfor _half, _shade in zip((2, 6, 12, 24), (\"#d0d7de\", \"#8c959f\", \"#57606a\", \"#0969da\")):\n ax2.plot(\n days,\n protein(days * 24, _half, protein_half_life.value),\n color=_shade,\n lw=2,\n label=f\"mRNA half-life {_half} h\",\n )\nax2.set(xlabel=\"days after the dose\", ylabel=\"protein (k m₀ h)\")\nax2.legend(frameon=False)\nfig2.tight_layout()\nfig2","code_hash":"9ad1781ead99e3d21bcd4fa88018c0f2","config":{"column":null,"disabled":false,"expand_output":false,"hide_code":false},"id":"Xref","name":"_"},{"code":"mo.md(\n r\"\"\"\n ## Repeated doses\n\n The model is linear, so repeated doses add up. The protein level after\n a schedule is the sum of the single-dose curves, each shifted to its\n dose time. Choose an interval and a number of doses to see whether\n expression stays above a threshold between doses.\n \"\"\"\n)","code_hash":"713a376c2a78285cd27821f91f04a2a8","config":{"column":null,"disabled":false,"expand_output":false,"hide_code":true},"id":"SFPL","name":"_"},{"code":"interval = mo.ui.slider(6, 96, step=6, value=24, label=\"Dose every (h)\")\ndoses = mo.ui.slider(1, 10, step=1, value=5, label=\"Number of doses\")\nthreshold = mo.ui.slider(0.05, 1.0, step=0.05, value=0.5, label=\"Threshold (fraction of one-dose peak)\")\nmo.vstack([mo.hstack([interval, doses], justify=\"start\"), threshold])","code_hash":"7fbe9b45be88401e800ae3703fc7eb35","config":{"column":null,"disabled":false,"expand_output":false,"hide_code":false},"id":"BYtC","name":"_"},{"code":"span = interval.value * doses.value + 96\nt = np.linspace(0, span, 1600)\nsingle_peak = protein(np.linspace(0, 400, 4000), mrna_half_life.value, protein_half_life.value).max()\ntotal = sum(\n protein(t - n * interval.value, mrna_half_life.value, protein_half_life.value)\n for n in range(doses.value)\n) / single_peak\ncovered = t[(t > 0) & (t < interval.value * doses.value)]\nabove = np.mean(np.interp(covered, t, total) >= threshold.value) if covered.size else 0.0\n\nfig3, ax3 = plt.subplots(figsize=(7, 3.4))\nax3.plot(t / 24, total, color=\"#0969da\", lw=2)\nax3.axhline(threshold.value, color=\"#cf222e\", lw=0.8, ls=\"--\")\nfor _n in range(doses.value):\n ax3.axvline(_n * interval.value / 24, color=\"#d0d7de\", lw=0.8)\nax3.set(xlabel=\"days\", ylabel=\"protein / one-dose peak\")\nfig3.tight_layout()\nmo.vstack(\n [\n fig3,\n mo.md(\n f\"During dosing, expression is above the threshold **{above:.0%}** of the time. \"\n f\"The highest level reached is **{total.max():.2f}** times a single dose's peak.\"\n ),\n ]\n)","code_hash":"ce69e665af88198c912622eb14ff62d6","config":{"column":null,"disabled":false,"expand_output":false,"hide_code":false},"id":"RGSE","name":"_"},{"code":"mo.md(\n r\"\"\"\n ---\n **Model and limits.** One compartment, first-order kinetics, all mRNA\n released at $t = 0$. Real delivery is spread over hours, cells differ\n widely in all three rates, and the immune response to a vaccine is not\n modelled. Levels are relative; set $k$ and $m_0$ from your own data to\n get absolute amounts. The default protein half-life, 26 hours, is the\n value reported for EGFP in mammalian cells (Corish and Tyler-Smith 1999).\n **References.** A. Leonhardt et al., *Nanomedicine: NBM* 10 (2014) 679;\n P. Corish and C. Tyler-Smith, *Protein Eng.* 12 (1999) 1035;\n K. Karikó et al., *Mol. Ther.* 16 (2008) 1833.\n Fork it to fit the model to your own time-course data.\n \"\"\"\n)","code_hash":"efe2ee33c64ab49f344dae1860c364c4","config":{"column":null,"disabled":false,"expand_output":false,"hide_code":true},"id":"Kclp","name":"_"}],"metadata":{"marimo_version":"0.25.1"},"version":"1"},"session":{"cells":[{"code_hash":"a8bb96a4b9e7161047b17c1c90418d3c","console":[],"id":"Hbol","outputs":[{"data":{"text/markdown":"How long does an mRNA dose make protein?
\nAn mRNA therapeutic or vaccine delivers instructions, not the protein\nitself. Once the mRNA reaches the cytoplasm, ribosomes translate it\nuntil it is degraded, and the protein it encodes is degraded in turn.\nThe simplest model that describes single-cell transfection data has two\nfirst-order steps (Leonhardt et al. 2014):\n||[\\frac{dm}{dt} = -\\delta\\, m, \\qquad \\frac{dp}{dt} = k\\, m - \\beta\\, p||]with mRNA ||(m||), protein ||(p||), translation rate ||(k||), and degradation rates\n||(\\delta = \\ln 2 / t_{1/2}^{\\text{mRNA}}||) and\n||(\\beta = \\ln 2 / t_{1/2}^{\\text{protein}}||). Starting from ||(m_0||) at\n||(t = 0||) the protein level is\n||[p(t) = \\frac{k\\, m_0}{\\delta - \\beta}\\left(e^{-\\beta t} - e^{-\\delta t}\\right).||]"},"type":"data"}]},{"code_hash":"ef0032c4370d821ec8e83d0954968f66","console":[],"id":"MJUe","outputs":[{"data":{"text/plain":""},"type":"data"}]},{"code_hash":"b711204af78d2545cdcb4b6b999bc17e","console":[],"id":"vblA","outputs":[{"data":{"text/html":"
"},"type":"data"}]},{"code_hash":"1b9b9ffb75d516e7aec712bf77685213","console":[],"id":"bkHC","outputs":[{"data":{"text/plain":""},"type":"data"}]},{"code_hash":"284614b01f0fbe2a68e001676cef3477","console":[],"id":"lEQa","outputs":[{"data":{"text/html":"Protein peaks 22.4 hours after the dose. Total protein made over the whole course is ||(k m_0 / (\\delta\\beta)||) = 541 in units of ||(k m_0||) hours squared.
"},"type":"data"}]},{"code_hash":"76c7e6d54473f13ec075334f0d7f1334","console":[],"id":"PKri","outputs":[{"data":{"text/markdown":"Why mRNA stability matters
\nThe area under the protein curve is ||(k m_0 / (\\delta \\beta)||): it is\nproportional to the mRNA half-life times the protein half-life. Doubling\nhow long the mRNA survives doubles the total protein made from the same\ndose, without changing the dose. Much of the engineering of therapeutic\nmRNA (modified nucleosides such as N1-methylpseudouridine, optimised\nuntranslated regions, the cap structure) works on ||(k||) and ||(\\delta||).\nThe curves below keep the protein half-life you chose and vary only the\nmRNA half-life."},"type":"data"}]},{"code_hash":"9ad1781ead99e3d21bcd4fa88018c0f2","console":[],"id":"Xref","outputs":[{"data":{"application/vnd.marimo+mimebundle":"{\"image/png\":\"https://pppqewd2askr4srw3c4biffsvi.notebooks.sciencehubusercontent.com/_outputs/eyJuYW1lc3BhY2VJZCI6Im5zLXdicjZya2tkOWZqMG1neGUiLCJwcm9qZWN0SWQiOiJwcm9qLXJ6dGEwZnlkZ3FxOGtkNTEiLCJub3RlYm9va0lkIjoibmItYWQ5bmMyODdzMXQ0MGFheiIsImhhc2giOiI1MTQ1YmE4MWFhOTQ1MWI0MzFlMzBlNWY3NTU4M2JhMiJ9.0.YO0sQEtMT41-bA67UwwGsixZlmeE6ghvuYI84O_BHIs\",\"__metadata__\":{\"image/png\":{\"width\":690,\"height\":330}}}"},"type":"data"}]},{"code_hash":"713a376c2a78285cd27821f91f04a2a8","console":[],"id":"SFPL","outputs":[{"data":{"text/markdown":"Repeated doses
\nThe model is linear, so repeated doses add up. The protein level after\na schedule is the sum of the single-dose curves, each shifted to its\ndose time. Choose an interval and a number of doses to see whether\nexpression stays above a threshold between doses."},"type":"data"}]},{"code_hash":"7fbe9b45be88401e800ae3703fc7eb35","console":[],"id":"BYtC","outputs":[{"data":{"text/html":""},"type":"data"}]},{"code_hash":"ce69e665af88198c912622eb14ff62d6","console":[],"id":"RGSE","outputs":[{"data":{"text/html":"During dosing, expression is above the threshold 96% of the time. The highest level reached is 2.77 times a single dose's peak.
"},"type":"data"}]},{"code_hash":"efe2ee33c64ab49f344dae1860c364c4","console":[],"id":"Kclp","outputs":[{"data":{"text/markdown":"
\nModel and limits. One compartment, first-order kinetics, all mRNA\nreleased at ||(t = 0||). Real delivery is spread over hours, cells differ\nwidely in all three rates, and the immune response to a vaccine is not\nmodelled. Levels are relative; set ||(k||) and ||(m_0||) from your own data to\nget absolute amounts. The default protein half-life, 26 hours, is the\nvalue reported for EGFP in mammalian cells (Corish and Tyler-Smith 1999).\nReferences. A. Leonhardt et al., Nanomedicine: NBM 10 (2014) 679;\nP. Corish and C. Tyler-Smith, Protein Eng. 12 (1999) 1035;\nK. Karikó et al., Mol. Ther. 16 (2008) 1833.\nFork it to fit the model to your own time-course data."},"type":"data"}]}],"metadata":{"marimo_version":"0.25.1","script_metadata_hash":null},"version":"1"},"appConfig":{"sql_output":"auto","width":"medium"},"layout":null,"renderedOn":"browser","formulas":{"||[\\frac{dm}{dt} = -\\delta\\, m, \\qquad \\frac{dp}{dt} = k\\, m - \\beta\\, p||]":"dtdm=−δm,dtdp=km−βp","||(m||)":"m","||(p||)":"p","||(k||)":"k","||(\\delta = \\ln 2 / t_{1/2}^{\\text{mRNA}}||)":"δ=ln2/t1/2mRNA","||(\\beta = \\ln 2 / t_{1/2}^{\\text{protein}}||)":"β=ln2/t1/2protein","||(m_0||)":"m0","||(t = 0||)":"t=0","||[p(t) = \\frac{k\\, m_0}{\\delta - \\beta}\\left(e^{-\\beta t} - e^{-\\delta t}\\right).||]":"p(t)=δ−βkm0(e−βt−e−δt).","||(k m_0 / (\\delta\\beta)||)":"km0/(δβ)","||(k m_0||)":"km0","||(k m_0 / (\\delta \\beta)||)":"km0/(δβ)","||(\\delta||)":"δ"}},"discussions":{"cells":{},"newHref":"/sciencehub-team/mrna-expression/discussions/new?notebook=mrna-expression&version=ver_01M4SH2J4KJZJE7G8JZQK7HWBH","listHref":"/sciencehub-team/mrna-expression/discussions?notebook=mrna-expression&version=ver_01M4SH2J4KJZJE7G8JZQK7HWBH"}}