From 99c2fff8d3cb5128e2cda98c6345b2a67b6339a5 Mon Sep 17 00:00:00 2001 From: coltonlloyd Date: Sat, 24 Mar 2018 12:01:13 -0700 Subject: [PATCH] Update coupling constraint derivations in docs --- docs/MEModelFundamentals.ipynb | 38 ++--- docs/coupling_constraint_derivation.ipynb | 195 +++++++++++++++------- 2 files changed, 148 insertions(+), 85 deletions(-) diff --git a/docs/MEModelFundamentals.ipynb b/docs/MEModelFundamentals.ipynb index 82cdbb7..6c04be0 100644 --- a/docs/MEModelFundamentals.ipynb +++ b/docs/MEModelFundamentals.ipynb @@ -35,13 +35,12 @@ ] }, { - "cell_type": "raw", + "cell_type": "markdown", "metadata": { - "raw_mimetype": "text/restructuredtext" + "raw_mimetype": "text/markdown" }, "source": [ - ".. math::\n", - " \\frac{\\mu}{k_{eff}}" + "$$\\frac{\\mu}{k_{eff}}$$" ] }, { @@ -95,15 +94,16 @@ ] }, { - "cell_type": "raw", + "cell_type": "markdown", "metadata": { "raw_mimetype": "text/restructuredtext" }, "source": [ - ".. math::\n", - " \\textbf{Enzyme Priming: } \\mathrm{enzyme} \\xrightarrow{v0} \\mathrm{enzyme\\_primed + } \\alpha \\cdot \\mathrm{coupling} \\\\\n", - " \\textbf{Enzymatic Reaction: } \\mathrm{a + enzyme\\_primed} \\xrightarrow{v1} \\textrm{enzyme + b}\\\\\n", - " \\textbf{Dilution Coupling Reaction: } \\textrm{enzyme + coupling} \\xrightarrow{v2} \\emptyset \\\\" + "$$\n", + "\\textbf{Enzyme Priming: } \\mathrm{enzyme} \\xrightarrow{v0} \\mathrm{enzyme\\_primed + } \\alpha \\cdot \\mathrm{coupling} \\\\\n", + "\\textbf{Enzymatic Reaction: } \\mathrm{a + enzyme\\_primed} \\xrightarrow{v1} \\textrm{enzyme + b}\\\\\n", + "\\textbf{Dilution Coupling Reaction: } \\textrm{enzyme + coupling} \\xrightarrow{v2} \\emptyset \n", + "$$" ] }, { @@ -117,13 +117,12 @@ ] }, { - "cell_type": "raw", + "cell_type": "markdown", "metadata": { "raw_mimetype": "text/restructuredtext" }, "source": [ - ".. math::\n", - " \\textrm{a} + \\alpha \\cdot \\textrm{enzyme} \\xrightarrow{v1} \\textrm{ b}" + "$$ \\textrm{a} + \\alpha \\cdot \\textrm{enzyme} \\xrightarrow{v1} \\textrm{ b}$$ " ] }, { @@ -174,13 +173,12 @@ ] }, { - "cell_type": "raw", + "cell_type": "markdown", "metadata": { "raw_mimetype": "text/restructuredtext" }, "source": [ - ".. math::\n", - " v_i \\Rightarrow \\frac{mmol}{gDW \\cdot hr}" + "$$v_i \\Rightarrow \\frac{mmol}{gDW \\cdot hr}$$" ] }, { @@ -191,13 +189,12 @@ ] }, { - "cell_type": "raw", + "cell_type": "markdown", "metadata": { "raw_mimetype": "text/restructuredtext" }, "source": [ - ".. math::\n", - " molecular\\_weight \\Rightarrow kDA[\\frac{g}{mmol}]" + "$$molecular\\_weight \\Rightarrow kDA[\\frac{g}{mmol}]$$" ] }, { @@ -208,13 +205,12 @@ ] }, { - "cell_type": "raw", + "cell_type": "markdown", "metadata": { "raw_mimetype": "text/restructuredtext" }, "source": [ - ".. math::\n", - " molecular\\_weight \\cdot v_i \\Rightarrow hr^{-1}[\\mu]" + "$$molecular\\_weight \\cdot v_i \\Rightarrow hr^{-1}[\\mu]$$" ] } ], diff --git a/docs/coupling_constraint_derivation.ipynb b/docs/coupling_constraint_derivation.ipynb index 9daae09..9b41e02 100644 --- a/docs/coupling_constraint_derivation.ipynb +++ b/docs/coupling_constraint_derivation.ipynb @@ -6,22 +6,22 @@ "source": [ "# Coupling Constraint Derivations\n", "\n", - "This section will show how coupling coefficients for two macromolecules (**mRNA** and **ribosome**) are derived in detail. The remaining macromolecule coupling derivations follow a similar approach and logic. For remaining derivations reference O'Brien et al, 2013.\n", + "This section will show in detail how coupling coefficients for two macromolecules (**mRNA** and **ribosome**) are derived. The remaining macromolecule coupling derivations follow a similar approach and logic, therefore they are omitted here. For remaining derivations, reference [O'Brien et al, 2013](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3817402/).\n", "\n", "## Parameters\n", "The parameters for the mRNA coupling coefficient derivations are listed below:\n", "$$\n", "\\mathrm{\n", - "P = total \\ cellular \\ protein \\ mass \\ fraction \\ (\\frac{gDW_{aa}}{gDW}) \\\\\n", - "R = total \\ cellular \\ RNA \\ mass \\ fraction \\ (\\frac{gDW_{nt}}{gDW}) \\\\\n", + "P = total \\ cellular \\ protein \\ mass \\ fraction \\ (\\frac{g_{aa}}{gDW_{cell}}) \\\\\n", + "R = total \\ cellular \\ RNA \\ mass \\ fraction \\ (\\frac{g_{nt}}{gDW_{cell}}) \\\\\n", "\\mu = specific \\ growth \\ rate \\ (\\frac{1}{hr}) \\\\\n", - "f_{rRNA} = mass \\ fraction \\ of \\ RNA \\ that \\ is \\ rRNA (\\frac{gDW_{nt}}{gDW_{nt_{total}}}) \\\\\n", - "f_{tRNA} = mass \\ fraction \\ of \\ RNA \\ that \\ is \\ tRNA (\\frac{gDW_{nt}}{gDW_{nt_{total}}}) \\\\\n", - "f_{mRNA} = mass \\ fraction \\ of \\ RNA \\ that \\ is \\ mRNA (\\frac{gDW_{nt}}{gDW_{nt_{total}}}) \\\\\n", - "m_{aa}= molecular \\ weight \\ of \\ average \\ amino \\ acid (\\frac{gDW_{aa}}{mmol_{aa}}) \\\\\n", - "m_{nt} = molecular \\ weight \\ of \\ average \\ mRNA \\ nucleotide (\\frac{gDW_{nt}}{mmol_{nt}}) \\\\\n", - "m_{tRNA} = molecular \\ weight \\ of \\ average \\ tRNA (\\frac{gDW_{tRNA}}{mmol_{tRNA}}) \\\\\n", - "m_{rr} = mass \\ of \\ rRNA \\ per \\ ribosome (gDW_{nt}) \\\\\n", + "f_{rRNA} = mass \\ fraction \\ of \\ RNA \\ that \\ is \\ rRNA (\\frac{g_{nt}}{g_{nt_{total}}}) \\\\\n", + "f_{tRNA} = mass \\ fraction \\ of \\ RNA \\ that \\ is \\ tRNA (\\frac{g_{nt}}{g_{nt_{total}}}) \\\\\n", + "f_{mRNA} = mass \\ fraction \\ of \\ RNA \\ that \\ is \\ mRNA (\\frac{g_{nt}}{g_{nt_{total}}}) \\\\\n", + "m_{aa}= molecular \\ weight \\ of \\ average \\ amino \\ acid (\\frac{g_{aa}}{mol_{aa}}) \\\\\n", + "m_{nt} = molecular \\ weight \\ of \\ average \\ mRNA \\ nucleotide (\\frac{g_{nt}}{mol_{nt}}) \\\\\n", + "m_{tRNA} = molecular \\ weight \\ of \\ average \\ tRNA (\\frac{g_{tRNA}}{mol_{tRNA}}) \\\\\n", + "m_{rr} = mass \\ of \\ rRNA \\ per \\ ribosome (\\frac{g_{nt}}{mol_{ribosome}}) \\\\\n", "k_{deg}^{mRNA} = first-order \\ mRNA \\ degradation \\ constant (\\frac{1}{hr} )\n", "}\n", "$$\n", @@ -41,38 +41,38 @@ "source": [ "----\n", "## Derivation of mRNA coupling coefficients\n", - "To derive mRNA dilution and degradation coupling coefficients, we assume that these processes are coupled together as follows. \n", + "To derive the mRNA dilution and degradation coupling coefficients, we assume that these processes are coupled together as follows. \n", "\n", "$$\n", - "\\mathrm{v_{dilution_{mRNA}} = \\alpha_1 \\cdot v_{degradation_{mRNA}}} \\\\\n", - "\\mathrm{v_{degradation_{mRNA}} = \\alpha_2 \\cdot v_{translation_{mRNA}}}\n", + "\\mathrm{v_{dilution_{nt_{mRNA}}} = \\alpha_1 \\cdot v_{degradation_{nt_{mRNA}}}} \\\\\n", + "\\mathrm{v_{degradation_{nt_{mRNA}}} = \\alpha_2 \\cdot v_{translation_{aa_{protein}}}}\n", "$$\n", "\n", "\n", - "where $\\alpha_1 $ and $\\alpha_2$ represent the coupling of degradation to dilution and translation to degredation, respectively.\n", + "where $\\alpha_1$ and $\\alpha_2$ represent the coupling of degradation to dilution and translation to degredation, respectively. For the remainder of the mRNA coupling derivation we will abbreviate these reaction rates as $\\mathrm{v_{dilution}, v_{degradation} and \\ v_{translation}}$ for simplicity.\n", "\n", - "To find these coupling values, we will need to find $\\mathrm{v_{dilution_{mRNA}}, v_{degradation_{mRNA}} and \\ v_{translation_{mRNA}}}$. The dilution of mRNA as it is passed on to daughter cells is related to the concentration of mRNA and the growth rate as follows:\n", + "To find these coupling values, we will need to find $\\mathrm{v_{dilution}, v_{degradation} and \\ v_{translation}}$. The dilution of mRNA nucleotides as it is passed on to daughter cells is related to the concentration of mRNA nucleotides and the growth rate as follows:\n", "\n", "$$\n", - "\\mathrm{v_{dilution_{mRNA}} = \\mu \\cdot [mRNA]}\n", + "\\mathrm{v_{dilution} = \\mu \\cdot [nt_{mRNA}]}\n", "$$\n", "\n", - "similarly the degradation rate will be found using the first order rate constant of mRNA degradation\n", + "similarly the degradation rate can be found using the first order rate constant of mRNA degradation\n", "\n", "$$\n", - "\\mathrm{v_{degradation_{mRNA}} = k_{deg}^{mRNA} \\cdot [mRNA]}\n", + "\\mathrm{v_{degradation} = k_{deg}^{mRNA} \\cdot [nt_{mRNA}]}\n", "$$\n", "\n", - "the rate of translation / protein synthesis rate in ($\\frac{mol_{nt}}{hr}$) can be found using the following. The 3 term accounts for the 3 nucleotides per amino acid during translation. **Note:** There is a minor typo in the O'brien et al., 2013 coupling coefficient derivations causing the 3 term to be on the numerator of the $\\alpha_2$ expresion when it should be in the denominator. The 3 is also missing from the $k_{mRNA}$ expression. \n", + "the rate of translation / protein synthesis rate in ($\\frac{mol_{aa}}{hr}$) can be found using the following. This represents the rate which amino acid are incorporated into protein:\n", "\n", "$$\n", - "\\mathrm{v_{translation_{mRNA}} = \\frac{3 \\cdot \\mu \\cdot P}{m_{aa}}}\n", + "\\mathrm{v_{translation} = \\frac{\\mu \\cdot P}{m_{aa}}}\n", "$$\n", "\n", - "The concentration of mRNA nucleotides in units of ($\\frac{mol_{nt}}{gDW}$):\n", + "The concentration of mRNA nucleotides in units of ($\\frac{mol_{nt}}{gDW_{cell}}$) can be defined as:\n", "\n", "$$\n", - "\\mathrm{[mRNA] = \\frac{R \\cdot f_{mRNA}}{m_{nt}}}\\\\\n", + "\\mathrm{[nt_{mRNA}] = \\frac{R \\cdot f_{mRNA}}{m_{nt}}}\\\\\n", "$$\n" ] }, @@ -80,36 +80,41 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "### Solving for coupling terms\n", + "### Solving for mRNA coupling coefficients\n", "\n", "Solving for each of these coupling terms gives:\n", "\n", "$$\n", - "\\boxed{\\mathrm{\\alpha_1 = \\frac{v_{dilution_{mRNA}}}{v_{degradation_{mRNA}}} = \\frac{\\mu \\cdot [mRNA]}{k_{deg}^{mRNA} \\cdot [mRNA]} = \\frac{\\mu}{k_{deg}^{mRNA}}}}\\\\\n", - "\\mathrm{\\alpha_2 = \\frac{v_{degradation_{mRNA}}}{v_{translation_{mRNA}}} = \\frac{k_{deg}^{mRNA} \\cdot [mRNA]}{\\frac{3 \\cdot \\mu \\cdot P}{m_{aa}}}}\\\\\n", + "\\boxed{\\mathrm{\\alpha_1 = \\frac{v_{dilution}}{v_{degradation}} = \\frac{\\mu \\cdot [nt_{mRNA}]}{k_{deg}^{mRNA} \\cdot [nt_{mRNA}]} = \\frac{\\mu}{k_{deg}^{mRNA}}}}\\\\\n", + "\\mathrm{\\alpha_2 = \\frac{v_{degradation}}{v_{translation}} = \\frac{k_{deg}^{mRNA} \\cdot [nt_{mRNA}]}{\\frac{\\mu \\cdot P}{m_{aa}}}}\\\\\n", "$$\n", "\n", - "substituting for [mRNA] gives\n", + "substituting for [mRNA] gives:\n", "$$\n", - "\\mathrm{\\alpha_2 = \\frac{k_{deg}^{mRNA} \\cdot \\frac{R \\cdot f_{mRNA}}{m_{nt}}}{\\frac{3 \\cdot \\mu \\cdot P}{m_{aa}}} = \\frac{k_{deg}^{mRNA} \\cdot R \\cdot f_{mRNA} \\cdot m_{aa}}{m_{nt} \\cdot 3 \\cdot \\mu \\cdot P} \\\\\n", - "\\alpha_2 = \\frac{1}{3} \\cdot \\frac{k_{deg}^{mRNA}}{\\mu} \\cdot \\frac{R}{P} \\cdot \\frac{f_{mRNA} \\cdot m_{aa}}{m_{nt}}}\n", + "\\mathrm{\\alpha_2 = \\frac{k_{deg}^{mRNA} \\cdot \\frac{R \\cdot f_{mRNA}}{m_{nt}}}{\\frac{\\mu \\cdot P}{m_{aa}}} = \\frac{k_{deg}^{mRNA} \\cdot R \\cdot f_{mRNA} \\cdot m_{aa}}{m_{nt} \\cdot \\mu \\cdot P}}\n", + "$$\n", + "simplifying:\n", + "$$\n", + "\\mathrm{\\alpha_2 = \\frac{k_{deg}^{mRNA}}{\\mu} \\cdot \\frac{R}{P} \\cdot \\frac{f_{mRNA} \\cdot m_{aa}}{m_{nt}}}\n", "$$\n", "\n", "substitution for $\\frac{R}{P}$ gives:\n", "$$\n", - "\\mathrm{\\boxed{\\alpha_2 = \\frac{1}{3} \\cdot \\frac{k_{deg}^{mRNA}}{\\mu} \\cdot \\frac{\\mu + \\kappa_{\\tau} \\cdot r_0}{\\kappa_{\\tau}} \\cdot \\frac{f_{mRNA} \\cdot m_{aa}}{m_{nt}}}}\n", + "\\mathrm{\\boxed{\\alpha_2 = \\frac{k_{deg}^{mRNA}}{\\mu} \\cdot \\frac{\\mu + \\kappa_{\\tau} \\cdot r_0}{\\kappa_{\\tau}} \\cdot \\frac{f_{mRNA} \\cdot m_{aa}}{m_{nt}}}}\n", "$$\n", "\n", - "and coupling of dilution to translation give:\n", + "Simplifying the above relationship, the coupling of dilution to translation is represented by:\n", + "$$\n", + "\\mathrm{v_{dilution} = \\alpha_1 \\cdot \\alpha_2 \\cdot v_{translation}}\n", "$$\n", - "\\boxed{\\mathrm{\\alpha_1 \\cdot \\alpha_2 = \\frac{1}{3}\n", - "\\cdot \\frac{\\mu + \\kappa_{\\tau} \\cdot r_0}{\\kappa_{\\tau}} \\cdot \\frac{f_{mRNA} \\cdot m_{aa}}{m_{nt}}}}\n", + "where:\n", + "$$\n", + "\\boxed{\\mathrm{\\alpha_1 \\cdot \\alpha_2 = \\frac{\\mu + \\kappa_{\\tau} \\cdot r_0}{\\kappa_{\\tau}} \\cdot \\frac{f_{mRNA} \\cdot m_{aa}}{m_{nt}}}}\n", "$$\n", "\n", - "therefore $\\mathrm{\\frac{\\mu}{k_{mRNA}} = \\alpha_1 \\cdot \\alpha_2}$ and:\n", + "Therefore $\\mathrm{\\frac{\\mu}{k_{mRNA}} = \\alpha_1 \\cdot \\alpha_2}$ and:\n", "$$\n", - "\\mathrm{k_{mRNA} = \\frac{\\mu}{\\alpha_1 \\cdot \\alpha_2} = 3\n", - "\\cdot \\frac{\\mu \\cdot \\kappa_{\\tau}}{\\mu + \\kappa_{\\tau} \\cdot r_0} \\cdot \\frac{m_{nt}}{f_{mRNA} \\cdot m_{aa}}}\n", + "\\mathrm{k_{mRNA} = \\frac{\\mu}{\\alpha_1 \\cdot \\alpha_2} = \\frac{\\mu \\cdot \\kappa_{\\tau}}{\\mu + \\kappa_{\\tau} \\cdot r_0} \\cdot \\frac{m_{nt}}{f_{mRNA} \\cdot m_{aa}}}\n", "$$" ] }, @@ -117,22 +122,27 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "### Confirming units\n", - "Based on the $[mRNA]$ expression above, the units will be:\n", + "### Units of mRNA coupling\n", + "Based on the $[\\mathrm{nt_{mRNA}}]$ expression above, the units will be:\n", "$$\n", - "\\mathrm{[mRNA] \\xrightarrow{units} \\frac{(\\frac{gDW_{nt_{total}}}{gDW}) \\cdot (\\frac{gDW_{nt}}{gDW_{nt_{total}}})}{(\\frac{gDW_{nt}}{mol_{nt}})}:= (\\frac{mol_{nt}}{gDW})}\n", + "\\mathrm{[nt_{mRNA}] = \\frac{R \\cdot f_{mRNA}}{m_{nt}} \\xrightarrow{units} \\frac{(\\frac{g_{nt_{total}}}{gDW_{cell}}) \\cdot (\\frac{g_{nt}}{g_{nt_{total}}})}{(\\frac{g_{nt}}{mol_{nt}})}= (\\frac{mol_{nt}}{gDW_{cell}})}\n", "$$\n", "\n", - "therefore $\\mathrm{v_{degradation_{mRNA}}}$ will be :\n", + "therefore $\\mathrm{v_{degradation}}$ will be :\n", "\n", "$$\n", - "\\mathrm{v_{degradation_{mRNA}} = [mRNA] \\cdot k_{deg}^{mRNA} \\xrightarrow{units} (\\frac{mol_{nt}}{gDW}) \\cdot (\\frac{1}{hr}) := (\\frac{mol_{nt}}{gDW \\cdot hr})}\n", + "\\mathrm{v_{degradation} = k_{deg}^{mRNA} \\cdot [nt_{mRNA}] \\xrightarrow{units} (\\frac{1}{hr}) \\cdot (\\frac{mol_{nt}}{gDW_{cell}}) = (\\frac{mol_{nt}}{gDW_{cell} \\cdot hr})}\n", "$$\n", "\n", - "and the denominator $\\mathrm{v_{translation_{mRNA}}}$:\n", + "and for $\\mathrm{v_{translation}}$:\n", + "\n", + "$$\n", + "\\mathrm{v_{translation} = \\frac{\\mu \\cdot P}{m_{aa}} \\xrightarrow{units} \\frac{(\\frac{1}{hr}) \\cdot (\\frac{g_{aa}}{gDW_{cell}})}{(\\frac{g_{aa}}{mol_{aa}})} = (\\frac{mol_{aa}}{gDW_{cell} \\cdot hr})}\n", + "$$\n", "\n", + "and for $\\mathrm{v_{dilution}}$:\n", "$$\n", - "\\mathrm{v_{translation_{mRNA}} = \\frac{3 \\cdot \\mu \\cdot P}{m_{aa}} \\xrightarrow{units} \\frac{(\\frac{3 \\cdot mol_{nt}}{mol_{aa}}) \\cdot (\\frac{1}{hr}) \\cdot (\\frac{gDW_{aa}}{gDW})}{(\\frac{gDW_{aa}}{mol_{aa}})} := (\\frac{mol_{nt}}{gDW \\cdot hr})}\n", + "\\mathrm{v_{dilution} = \\mu \\cdot [nt_{mRNA}] \\xrightarrow{units} (\\frac{1}{hr}) \\cdot (\\frac{mol_{nt}}{gDW_{cell}}) = (\\frac{mol_{nt}}{gDW_{cell} \\cdot hr})}\n", "$$" ] }, @@ -141,20 +151,51 @@ "metadata": {}, "source": [ "### Applying mRNA coupling to translation\n", + "Note that the units for each reaction detailed in the above derivations describe the overall coupling of translation, dilution, and degradation cell-wide. For individual proteins and ME-model translation reactions, we will have:\n", + "\n", + "$$\\mathrm{v_{dilution_i} = \\alpha_1 \\cdot \\alpha_2 \\cdot \\frac{len_{peptide_i}}{len_{mRNA_i}} \\cdot v_{translation_i}}$$\n", + "\n", + "the length terms are required due to the fact that $\\mathrm{v_{dilution_i}\\ and \\ v_{translation_i}}$ will have units\n", + "of $\\mathrm{\\frac{mol_{mRNA_i}}{gDW \\cdot hr} \\ and \\ \\frac{mol_{protein_i}}{gDW \\cdot hr}}$, respectively.\n", + "\n", + "Since:\n", + "\n", + "$$\\mathrm{\\alpha_1 \\cdot \\alpha_2 = \\frac{v_{dilution}}{v_{translation}} \\xrightarrow{units} \\frac{mol_{nt}}{mol_{aa}}}$$\n", + "\n", + "therefore:\n", + "$$\\mathrm{\\alpha_1 \\cdot \\alpha_2 \\cdot \\frac{len_{peptide_i}}{len_{mRNA_i}} \\xrightarrow{units} (\\frac{mol_{nt}}{mol_{aa}}) \\cdot (\\frac{\\frac{mol_{aa}}{mol_{peptide_i}}}{\\frac{mol_{nt}}{mol_{mRNA_i}}})= \\frac{mol_{mRNA_i}}{mol_{protein_i}}}$$\n", + "\n", + "however the length of a peptide will always be 1/3 the length of the mRNA that encodes it (3 nucleotides in a codon) therefore we can replace ($\\mathrm{\\frac{len_{peptide_i}}{len_{mRNA_i}}}$) with ($\\mathrm{ \\frac{1}{3} \\frac{mol_{aa} \\cdot mol_{mRNA_i}}{mol_{protein_i} \\cdot mol_{nt}}})$\n", + "\n", + "therefore the final coupling of dilution to translation will be:\n", + "$$\\boxed{\\mathrm{v_{dilution_i} = \\alpha_1 \\cdot \\alpha_2 \\cdot \\frac{1}{3} \\cdot v_{translation_i}}}$$\n", + "\n", + "and similarly for degradation coupling:\n", + "$$\\boxed{\\mathrm{v_{degradation_i} = \\alpha_2 \\cdot \\frac{1}{3} \\cdot v_{translation_i}}}$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "### Plugging ribosome coupling into a ME-model reaction\n", + "\n", "The coupling of mRNA synthesis to translation will require considering the sum of the mRNA dilution and degration. When imposed in the ME-model, a translation reaction will look similar to following:\n", "\n", "$$\n", - "\\mathrm{x \\cdot charged\\_tRNAs + (\\alpha_1 \\cdot \\alpha_2 + \\alpha_2) \\cdot mRNA_i + y \\cdot ribosome \\longrightarrow protein_i + \\alpha_2 \\cdot nucleotides}\n", + "\\mathrm{x \\cdot charged\\_tRNAs + (\\frac{1}{3} \\cdot \\alpha_1 \\cdot \\alpha_2 + \\frac{1}{3} \\cdot \\alpha_2) \\cdot mRNA_i + y \\cdot ribosome \\xrightarrow{v_{translation_i}} protein_i + \\frac{1}{3} \\cdot \\alpha_2 \\cdot nucleotides}\n", "$$\n", "\n", "with the coupling coefficients substituted:\n", "\n", "$$\n", - "\\mathrm{x \\cdot charged\\_tRNAs + (\\frac{1}{3} \\cdot \\frac{\\mu + \\kappa_{\\tau} \\cdot r_0}{\\kappa_{\\tau}} \\cdot \\frac{f_{mRNA} \\cdot m_{aa}}{m_{nt}} + \\\\ \\frac{1}{3} \\cdot \\frac{k_{deg}^{mRNA}}{\\mu} \\cdot \\frac{\\mu + \\kappa_{\\tau} \\cdot r_0}{\\kappa_{\\tau}} \\cdot \\frac{f_{mRNA} \\cdot m_{aa}}{m_{nt}}) \\cdot mRNA_i + y \\cdot ribosome \\\\ \\longrightarrow protein_i + (\\frac{1}{3} \\cdot \\frac{k_{deg}^{mRNA}}{\\mu} \\cdot \\frac{\\mu + \\kappa_{\\tau} \\cdot r_0}{\\kappa_{\\tau}} \\cdot \\frac{f_{mRNA} \\cdot m_{aa}}{m_{nt}}) \\cdot nucleotides}\n", + "\\mathrm{x \\cdot charged\\_tRNAs + (\\frac{1}{3} \\cdot \\frac{\\mu + \\kappa_{\\tau} \\cdot r_0}{\\kappa_{\\tau}} \\cdot \\frac{f_{mRNA} \\cdot m_{aa}}{m_{nt}} + \\\\ \\frac{1}{3} \\cdot \\frac{k_{deg}^{mRNA}}{\\mu} \\cdot \\frac{\\mu + \\kappa_{\\tau} \\cdot r_0}{\\kappa_{\\tau}} \\cdot \\frac{f_{mRNA} \\cdot m_{aa}}{m_{nt}}) \\cdot mRNA_i + y \\cdot ribosome \\\\ \\xrightarrow{v_{translation_i}} protein_i + (\\frac{1}{3} \\cdot \\frac{k_{deg}^{mRNA}}{\\mu} \\cdot \\frac{\\mu + \\kappa_{\\tau} \\cdot r_0}{\\kappa_{\\tau}} \\cdot \\frac{f_{mRNA} \\cdot m_{aa}}{m_{nt}}) \\cdot nucleotides}\n", "$$\n", "\n", "\n", - "where x and y represnts the coupling coefficient for the tRNAs and ribosome (the ribosome coupling is derived below). The reaction will produce nucleotides with a coefficient of $\\alpha_2$ since these are the product of mRNA degradation." + "where x and y represents the coupling coefficient for the tRNAs and ribosome (the ribosome coupling is derived below). The reaction will produce nucleotides with a coefficient of $\\frac{1}{3} \\cdot \\alpha_2$ since these are the product of mRNA degradation.\n", + "\n", + "**Note:** There is a minor typo in the O'brien et al., 2013 coupling coefficient derivations where the $\\alpha_1$ and $\\alpha_2$ expresions are multiplied by 3 instead of $\\frac{1}{3}$." ] }, { @@ -165,33 +206,33 @@ "source": [ "----\n", "## Derivation of ribosome coupling coefficients\n", - "Like above, we will derive the coupling between translation and ribosome dilution to daughter cells during cell divition. Unlike mRNA, ribosomes and rRNA are stable and we assume they are degraded at a neglible rate.\n", - "\n", + "Like above, we will derive the coupling between translation and ribosome dilution to daughter cells during cell division. Unlike mRNA, ribosomes and rRNA are stable and we assume they are degraded at a neglible rate\n", "$$\n", - "\\mathrm{v_{dilution_{ribosome}} = \\alpha_3 \\cdot v_{translation_{protein}}}\n", + "\\mathrm{v_{dilution_{ribosome}} = \\alpha_3 \\cdot v_{translation_{aa_{protein}}}} \\\\\n", "$$\n", "\n", - "where :\n", + "As for the mRNA coupling derivation above, $\\alpha_3$ represent the coupling of translation to ribosome dilution. For the remainder of the ribosome coupling derivation, we will abbreviate these reaction rates as $\\mathrm{v_{dilution} and \\ v_{translation}}$ for simplicity.\n", + "\n", + "The translation of protein is defined as above in the mRNA coupling derivations:\n", "$$\n", - "\\mathrm{v_{dilution_{ribosome}} = \\mu \\cdot [ribosome]} \\\\\n", + "\\mathrm{v_{translation} = \\frac{\\mu \\cdot P}{m_{aa}}}\n", "$$\n", "\n", - "unlike the mRNA derivation, the ribosome coupling requires a consideration of the number of amino acids incorporated into the protein:\n", - "\n", + "and:\n", "$$\n", - "\\mathrm{v_{translation_{protein}} = \\frac{\\mu \\cdot P}{m_{aa}} \\cdot \\frac{1}{len_{protein}}}\n", + "\\mathrm{v_{dilution} = \\mu \\cdot [ribosome]} \\\\\n", "$$\n", "\n", - "the concentration of ribosome in units of ($\\mathrm{\\frac{mol_{ribosome}}{gDW}}$) :\n", + "The concentration of ribosome in units of ($\\mathrm{\\frac{mol_{ribosome}}{gDW_{cell}}}$) :\n", "\n", "$$\n", "\\mathrm{[ribosome] = \\frac{R \\cdot f_{rRNA}}{m_{rr}}}\n", "$$\n", "\n", - "pluggin in this expression for [ribosome] and solving for $\\alpha_3$ gives :\n", + "plugging in this expression for [ribosome] and solving for $\\alpha_3$ gives :\n", "\n", "$$\n", - "\\mathrm{\\alpha_3 = \\frac{\\frac{R \\cdot f_{rRNA}}{m_{rr}} \\cdot \\mu}{\\frac{\\mu \\cdot P}{m_{aa}} \\cdot \\frac{1}{len_{protein}}} = \\frac{R}{P} \\cdot \\frac{f_{rRNA} \\cdot m_{aa}}{m_{rr}} \\cdot len_{protein}}\n", + "\\mathrm{\\alpha_3 = \\frac{\\frac{R \\cdot f_{rRNA}}{m_{rr}} \\cdot \\mu}{\\frac{\\mu \\cdot P}{m_{aa}}} = \\frac{R}{P} \\cdot \\frac{f_{rRNA} \\cdot m_{aa}}{m_{rr}}}\n", "$$" ] }, @@ -199,9 +240,9 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "pluggin in the above empirical expression for $\\frac{R}{P}$:\n", + "plugging in the above empirical expression for $\\frac{R}{P}$:\n", "$$\n", - "\\boxed{\\mathrm{\\alpha_3 = \\frac{\\mu + \\kappa_{\\tau} \\cdot r_0}{\\kappa_{\\tau}} \\cdot \\frac{f_{rRNA} \\cdot m_{aa}}{m_{rr}} \\cdot len_{protein}}}\n", + "\\boxed{\\mathrm{\\alpha_3 = \\frac{\\mu + \\kappa_{\\tau} \\cdot r_0}{\\kappa_{\\tau}} \\cdot \\frac{f_{rRNA} \\cdot m_{aa}}{m_{rr}}}}\n", "$$" ] }, @@ -209,14 +250,40 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "### Confirming Units\n", + "### Units of ribosome coupling\n", "$$\n", - "\\mathrm{v_{dilution_{ribosome}} = \\mu \\cdot [ribosome] \\xrightarrow{units} (\\frac{1}{hr}) \\cdot \\frac{(\\frac{gDW_{nt_{total}}}{gDW}) \\cdot (\\frac{gDW_{nt_{ribosome}}}{gDW_{nt_{total}}})}{(\\frac{gDW_{nt_{ribosome}}}{mol_{ribosome}})} = (\\frac{mol_{ribosome}}{gDW \\cdot hr})\\\\\n", - "v_{translation_{protein}} = \\frac{\\mu \\cdot P}{m_{aa}} \\cdot \\frac{1}{len_{protein}} \\xrightarrow{units} \\frac{(\\frac{1}{hr}) \\cdot (\\frac{gDW_{aa}}{gDW})}{(\\frac{gDW_{aa}}{mol_{aa}})} \\cdot \\frac{1}{(\\frac{mol_{aa}}{mol_{protein}})} = (\\frac{mol_{protein}}{gDW \\cdot hr})\n", + "\\mathrm{v_{dilution} = \\mu \\cdot [ribosome] \\xrightarrow{units} (\\frac{1}{hr}) \\cdot \\frac{(\\frac{g_{nt_{total}}}{gDW}) \\cdot (\\frac{g_{nt_{ribosome}}}{g_{nt_{total}}})}{(\\frac{g_{nt_{ribosome}}}{mol_{ribosome}})} = (\\frac{mol_{ribosome}}{gDW_{cell} \\cdot hr})\\\\\n", + "v_{translation} = \\frac{\\mu \\cdot P}{m_{aa}} \\xrightarrow{units} = \\frac{(\\frac{1}{hr}) \\cdot (\\frac{g_{aa}}{gDW_{cell}})}{(\\frac{g_{aa}}{mol_{aa}})} = (\\frac{mol_{aa}}{gDW_{cell} \\cdot hr})\n", "}\n", "$$" ] }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "### Applying ribosome coupling to translation\n", + "Note that the units for each reaction detailed in the above derivations describe the overall coupling of translation to ribosome dilution on a cell-wide level. For individual proteins, we will have:\n", + "\n", + "$$\\mathrm{v_{dilution_i} = \\alpha_3 \\cdot len_{peptide_i} \\cdot v_{translation_i}}$$\n", + "\n", + "The length term is required due to the fact that in the ME-model $\\mathrm{v_{dilution_i}\\ and \\ v_{translation_i}}$ will have units\n", + "of $\\mathrm{\\frac{mol_{ribosome}}{gDW_{cell} \\cdot hr} \\ and \\ \\frac{mol_{protein_i}}{gDW_{cell} \\cdot hr}}$, respectively.\n", + "\n", + "Since:\n", + "\n", + "$$\\mathrm{\\alpha_3 = \\frac{v_{dilution}}{v_{translation}} \\xrightarrow{units} \\frac{mol_{ribosome}}{mol_{aa}}}$$\n", + "\n", + "therefore:\n", + "$$\\mathrm{(\\alpha_3) \\cdot (len_{peptide_i}) \\xrightarrow{units} (\\frac{mol_{ribosome}}{mol_{aa}}) \\cdot (\\frac{mol_{aa}}{mol_{peptide_i}})= \\frac{mol_{ribosome}}{mol_{protein_i}}}$$\n", + "\n", + "therefore plugging this into the final coupling of dilution to translation will be:\n", + "$$\\boxed{\\mathrm{v_{dilution_i} = \\alpha_3 \\cdot len_{protein_i} \\cdot v_{translation_i}}}$$\n", + "\n", + "confirming units:\n", + "$$\\mathrm{v_{dilution_i} = \\alpha_3 \\cdot len_{protein_i} \\cdot v_{translation_i} \\xrightarrow{units} (\\frac{mol_{ribosome}}{mol_{protein_i}}) \\cdot (\\frac{mol_{protein_i}}{gDW_{cell}}) = \\frac{mol_{ribosome}}{gDW_{cell}}}$$\n" + ] + }, { "cell_type": "markdown", "metadata": {}, @@ -225,18 +292,18 @@ "When further imposing ribosome dilution coupling in the ME-model, a translation reaction will look similar to following:\n", "\n", "$$\n", - "\\mathrm{x \\cdot charged\\_tRNAs + (\\alpha_1 \\cdot \\alpha_2 + \\alpha_2) \\cdot mRNA_i + \\alpha_3 \\cdot ribosome \\longrightarrow protein_i + \\alpha_2 \\cdot nucleotides} \\\\\n", + "\\mathrm{x \\cdot charged\\_tRNAs + (\\frac{1}{3} \\cdot \\alpha_1 \\cdot \\alpha_2 + \\frac{1}{3} \\cdot \\alpha_2) \\cdot mRNA_i + len_{protein_i} \\cdot \\alpha_3 \\cdot ribosome \\xrightarrow{v_{translation_i}} protein_i + \\frac{1}{3} \\cdot \\alpha_2 \\cdot nucleotides}\n", "$$\n", "\n", "with the coupling coefficients substituted:\n", "\n", "$$\n", - "\\mathrm{x \\cdot charged\\_tRNAs + (\\frac{1}{3} \\cdot \\frac{\\mu + \\kappa_{\\tau} \\cdot r_0}{\\kappa_{\\tau}} \\cdot \\frac{f_{mRNA} \\cdot m_{aa}}{m_{nt}} + \\\\ \\frac{1}{3} \\cdot \\frac{k_{deg}^{mRNA}}{\\mu} \\cdot \\frac{\\mu + \\kappa_{\\tau} \\cdot r_0}{\\kappa_{\\tau}} \\cdot \\frac{f_{mRNA} \\cdot m_{aa}}{m_{nt}}) \\cdot mRNA_i + (\\frac{\\mu + \\kappa_{\\tau} \\cdot r_0}{\\kappa_{\\tau}} \\cdot \\frac{f_{rRNA} \\cdot m_{aa}}{m_{rr}} \\cdot len_{protein}) \\cdot ribosome \\longrightarrow \\\\ protein_i + (\\frac{1}{3} \\cdot \\frac{k_{deg}^{mRNA}}{\\mu} \n", + "\\mathrm{x \\cdot charged\\_tRNAs + (\\frac{1}{3} \\cdot \\frac{\\mu + \\kappa_{\\tau} \\cdot r_0}{\\kappa_{\\tau}} \\cdot \\frac{f_{mRNA} \\cdot m_{aa}}{m_{nt}} + \\\\ \\frac{1}{3} \\cdot \\frac{k_{deg}^{mRNA}}{\\mu} \\cdot \\frac{\\mu + \\kappa_{\\tau} \\cdot r_0}{\\kappa_{\\tau}} \\cdot \\frac{f_{mRNA} \\cdot m_{aa}}{m_{nt}}) \\cdot mRNA_i + (len_{protein} \\cdot \\frac{\\mu + \\kappa_{\\tau} \\cdot r_0}{\\kappa_{\\tau}} \\cdot \\frac{f_{rRNA} \\cdot m_{aa}}{m_{rr}}) \\cdot ribosome \\xrightarrow{v_{translation_i}} \\\\ protein_i + (\\frac{1}{3} \\cdot \\frac{k_{deg}^{mRNA}}{\\mu} \n", "\\cdot \\frac{\\mu + \\kappa_{\\tau} \\cdot r_0}{\\kappa_{\\tau}} \\cdot \\frac{f_{mRNA} \\cdot m_{aa}}{m_{nt}}) \\cdot nucleotides}\n", "$$\n", "\n", "\n", - "where x represnts the coupling coefficient for the tRNAs." + "where x represents the coupling coefficient for the tRNAs." ] } ],