# 6. Coupling Constraint Derivations¶

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.

## 6.1. Parameters¶

The parameters for the mRNA coupling coefficient derivations are listed below:

$\begin{split}\mathrm{ P = total \ cellular \ protein \ mass \ fraction \ (\frac{g_{aa}}{gDW_{cell}}) \\ R = total \ cellular \ RNA \ mass \ fraction \ (\frac{g_{nt}}{gDW_{cell}}) \\ \mu = specific \ growth \ rate \ (\frac{1}{hr}) \\ f_{rRNA} = mass \ fraction \ of \ RNA \ that \ is \ rRNA (\frac{g_{nt}}{g_{nt_{total}}}) \\ f_{tRNA} = mass \ fraction \ of \ RNA \ that \ is \ tRNA (\frac{g_{nt}}{g_{nt_{total}}}) \\ f_{mRNA} = mass \ fraction \ of \ RNA \ that \ is \ mRNA (\frac{g_{nt}}{g_{nt_{total}}}) \\ m_{aa}= molecular \ weight \ of \ average \ amino \ acid (\frac{g_{aa}}{mol_{aa}}) \\ m_{nt} = molecular \ weight \ of \ average \ mRNA \ nucleotide (\frac{g_{nt}}{mol_{nt}}) \\ m_{tRNA} = molecular \ weight \ of \ average \ tRNA (\frac{g_{tRNA}}{mol_{tRNA}}) \\ m_{rr} = mass \ of \ rRNA \ per \ ribosome (\frac{g_{nt}}{mol_{ribosome}}) \\ k_{deg}^{mRNA} = first-order \ mRNA \ degradation \ constant (\frac{1}{hr} ) }\end{split}$

Along with an experical relationship between measured ratio of RNA (R) to Protein (P)

$\mathrm{\frac{R}{P} = \frac{\mu}{\kappa_{\tau}} + r_0 = \frac{\mu + \kappa_{\tau} \cdot r_0}{\kappa_{\tau}}}$

For E. coli grown at $$37^o C$$, (Scott et al., 2010) emperically found $$r_0 = 0.087$$ and $$\kappa_{\tau} =4.5 \frac{1}{hr}$$.

## 6.2. Derivation of mRNA coupling coefficients¶

To derive the mRNA dilution and degradation coupling coefficients, we assume that these processes are coupled together as follows.

$\begin{split}\mathrm{v_{dilution_{nt_{mRNA}}} = \alpha_1 \cdot v_{degradation_{nt_{mRNA}}}} \\ \mathrm{v_{degradation_{nt_{mRNA}}} = \alpha_2 \cdot v_{translation_{aa_{protein}}}}\end{split}$

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.

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:

$\mathrm{v_{dilution} = \mu \cdot [nt_{mRNA}]}$

similarly the degradation rate can be found using the first order rate constant of mRNA degradation

$\mathrm{v_{degradation} = k_{deg}^{mRNA} \cdot [nt_{mRNA}]}$

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:

$\mathrm{v_{translation} = \frac{\mu \cdot P}{m_{aa}}}$

The concentration of mRNA nucleotides in units of ($$\frac{mol_{nt}}{gDW_{cell}}$$) can be defined as:

$\begin{split}\mathrm{[nt_{mRNA}] = \frac{R \cdot f_{mRNA}}{m_{nt}}}\\\end{split}$

### 6.2.1. Solving for mRNA coupling coefficients¶

Solving for each of these coupling terms gives:

$\begin{split}\boxed{\mathrm{\alpha_1 = \frac{v_{dilution}}{v_{degradation}} = \frac{\mu \cdot [nt_{mRNA}]}{k_{deg}^{mRNA} \cdot [nt_{mRNA}]} = \frac{\mu}{k_{deg}^{mRNA}}}}\\ \mathrm{\alpha_2 = \frac{v_{degradation}}{v_{translation}} = \frac{k_{deg}^{mRNA} \cdot [nt_{mRNA}]}{\frac{\mu \cdot P}{m_{aa}}}}\\\end{split}$

substituting for [mRNA] gives:

\begin{align}\begin{aligned} \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}}\\simplifying:\end{aligned}\end{align}
$\mathrm{\alpha_2 = \frac{k_{deg}^{mRNA}}{\mu} \cdot \frac{R}{P} \cdot \frac{f_{mRNA} \cdot m_{aa}}{m_{nt}}}$

substitution for $$\frac{R}{P}$$ gives:

$\mathrm{\boxed{\alpha_2 = \frac{k_{deg}^{mRNA}}{\mu} \cdot \frac{\mu + \kappa_{\tau} \cdot r_0}{\kappa_{\tau}} \cdot \frac{f_{mRNA} \cdot m_{aa}}{m_{nt}}}}$

Simplifying the above relationship, the coupling of dilution to translation is represented by:

\begin{align}\begin{aligned} \mathrm{v_{dilution} = \alpha_1 \cdot \alpha_2 \cdot v_{translation}}\\where:\end{aligned}\end{align}
$\boxed{\mathrm{\alpha_1 \cdot \alpha_2 = \frac{\mu + \kappa_{\tau} \cdot r_0}{\kappa_{\tau}} \cdot \frac{f_{mRNA} \cdot m_{aa}}{m_{nt}}}}$

Therefore $$\mathrm{\frac{\mu}{k_{mRNA}} = \alpha_1 \cdot \alpha_2}$$ and:

$\mathrm{k_{mRNA} = \frac{\mu}{\alpha_1 \cdot \alpha_2} = \frac{\mu \cdot \kappa_{\tau}}{\mu + \kappa_{\tau} \cdot r_0} \cdot \frac{m_{nt}}{f_{mRNA} \cdot m_{aa}}}$

### 6.2.2. Units of mRNA coupling¶

Based on the $$[\mathrm{nt_{mRNA}}]$$ expression above, the units will be:

$\mathrm{[nt_{mRNA}] = \frac{R \cdot f_{mRNA}}{m_{nt}} \xrightarrow{units} \frac{(\frac{g_{nt_{total}}}{gDW_{cell}}) \cdot (\frac{g_{nt}}{g_{nt_{total}}})}{(\frac{g_{nt}}{mol_{nt}})}= (\frac{mol_{nt}}{gDW_{cell}})}$

therefore $$\mathrm{v_{degradation}}$$ will be :

$\mathrm{v_{degradation} = k_{deg}^{mRNA} \cdot [nt_{mRNA}] \xrightarrow{units} (\frac{1}{hr}) \cdot (\frac{mol_{nt}}{gDW_{cell}}) = (\frac{mol_{nt}}{gDW_{cell} \cdot hr})}$

and for $$\mathrm{v_{translation}}$$:

$\mathrm{v_{translation} = \frac{\mu \cdot P}{m_{aa}} \xrightarrow{units} \frac{(\frac{1}{hr}) \cdot (\frac{g_{aa}}{gDW_{cell}})}{(\frac{g_{aa}}{mol_{aa}})} = (\frac{mol_{aa}}{gDW_{cell} \cdot hr})}$

and for $$\mathrm{v_{dilution}}$$:

$\mathrm{v_{dilution} = \mu \cdot [nt_{mRNA}] \xrightarrow{units} (\frac{1}{hr}) \cdot (\frac{mol_{nt}}{gDW_{cell}}) = (\frac{mol_{nt}}{gDW_{cell} \cdot hr})}$

### 6.2.3. Applying mRNA coupling to translation¶

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:

$\mathrm{v_{dilution_i} = \alpha_1 \cdot \alpha_2 \cdot \frac{len_{peptide_i}}{len_{mRNA_i}} \cdot v_{translation_i}}$

the length terms are required due to the fact that $$\mathrm{v_{dilution_i}\ and \ v_{translation_i}}$$ will have units of $$\mathrm{\frac{mol_{mRNA_i}}{gDW \cdot hr} \ and \ \frac{mol_{protein_i}}{gDW \cdot hr}}$$, respectively.

Since:

$\mathrm{\alpha_1 \cdot \alpha_2 = \frac{v_{dilution}}{v_{translation}} \xrightarrow{units} \frac{mol_{nt}}{mol_{aa}}}$

therefore:

$\mathrm{\alpha_1 \cdot \alpha_2 \cdot \frac{len_{peptide_i}}{len_{mRNA_i}} \xrightarrow{units} (\frac{mol_{nt}}{mol_{aa}}) \cdot (\frac{\frac{mol_{aa}}{mol_{peptide_i}}}{\frac{mol_{nt}}{mol_{mRNA_i}}})= \frac{mol_{mRNA_i}}{mol_{protein_i}}}$

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 ($$\mathrm{ \frac{1}{3} \frac{mol_{aa} \cdot mol_{mRNA_i}}{mol_{protein_i} \cdot mol_{nt}}})$$

therefore the final coupling of dilution to translation will be:

$\boxed{\mathrm{v_{dilution_i} = \alpha_1 \cdot \alpha_2 \cdot \frac{1}{3} \cdot v_{translation_i}}}$

$\boxed{\mathrm{v_{degradation_i} = \alpha_2 \cdot \frac{1}{3} \cdot v_{translation_i}}}$

### 6.2.4. Plugging ribosome coupling into a ME-model reaction¶

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:

$\mathrm{x \cdot charged\_tRNAs + (\frac{1}{3} \cdot \alpha_1 \cdot \alpha_2 + \frac{1}{3} \cdot \alpha_2) \cdot mRNA_i + y \cdot ribosome \xrightarrow{v_{translation_i}} protein_i + \frac{1}{3} \cdot \alpha_2 \cdot nucleotides}$

with the coupling coefficients substituted:

$\begin{split}\mathrm{x \cdot charged\_tRNAs + (\frac{1}{3} \cdot \frac{\mu + \kappa_{\tau} \cdot r_0}{\kappa_{\tau}} \cdot \frac{f_{mRNA} \cdot m_{aa}}{m_{nt}} + \\ \frac{1}{3} \cdot \frac{k_{deg}^{mRNA}}{\mu} \cdot \frac{\mu + \kappa_{\tau} \cdot r_0}{\kappa_{\tau}} \cdot \frac{f_{mRNA} \cdot m_{aa}}{m_{nt}}) \cdot mRNA_i + y \cdot ribosome \\ \xrightarrow{v_{translation_i}} protein_i + (\frac{1}{3} \cdot \frac{k_{deg}^{mRNA}}{\mu} \cdot \frac{\mu + \kappa_{\tau} \cdot r_0}{\kappa_{\tau}} \cdot \frac{f_{mRNA} \cdot m_{aa}}{m_{nt}}) \cdot nucleotides}\end{split}$

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.

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 $$\frac{1}{3}$$.

## 6.3. Derivation of ribosome coupling coefficients¶

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

$\begin{split}\mathrm{v_{dilution_{ribosome}} = \alpha_3 \cdot v_{translation_{aa_{protein}}}} \\\end{split}$

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.

The translation of protein is defined as above in the mRNA coupling derivations:

$\mathrm{v_{translation} = \frac{\mu \cdot P}{m_{aa}}}$

and:

$\begin{split}\mathrm{v_{dilution} = \mu \cdot [ribosome]} \\\end{split}$

The concentration of ribosome in units of ($$\mathrm{\frac{mol_{ribosome}}{gDW_{cell}}}$$) :

$\mathrm{[ribosome] = \frac{R \cdot f_{rRNA}}{m_{rr}}}$

plugging in this expression for [ribosome] and solving for $$\alpha_3$$ gives :

$\mathrm{\alpha_3 = \frac{\frac{R \cdot f_{rRNA}}{m_{rr}} \cdot \mu}{\frac{\mu \cdot P}{m_{aa}}} = \frac{R}{P} \cdot \frac{f_{rRNA} \cdot m_{aa}}{m_{rr}}}$

plugging in the above empirical expression for $$\frac{R}{P}$$:

$\boxed{\mathrm{\alpha_3 = \frac{\mu + \kappa_{\tau} \cdot r_0}{\kappa_{\tau}} \cdot \frac{f_{rRNA} \cdot m_{aa}}{m_{rr}}}}$

### 6.3.1. Units of ribosome coupling¶

$\begin{split}\mathrm{v_{dilution} = \mu \cdot [ribosome] \xrightarrow{units} (\frac{1}{hr}) \cdot \frac{(\frac{g_{nt_{total}}}{gDW}) \cdot (\frac{g_{nt_{ribosome}}}{g_{nt_{total}}})}{(\frac{g_{nt_{ribosome}}}{mol_{ribosome}})} = (\frac{mol_{ribosome}}{gDW_{cell} \cdot hr})\\ v_{translation} = \frac{\mu \cdot P}{m_{aa}} \xrightarrow{units} = \frac{(\frac{1}{hr}) \cdot (\frac{g_{aa}}{gDW_{cell}})}{(\frac{g_{aa}}{mol_{aa}})} = (\frac{mol_{aa}}{gDW_{cell} \cdot hr}) }\end{split}$

### 6.3.3. Applying ribosome coupling to translation¶

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:

$\mathrm{v_{dilution_i} = \alpha_3 \cdot len_{peptide_i} \cdot v_{translation_i}}$

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 of $$\mathrm{\frac{mol_{ribosome}}{gDW_{cell} \cdot hr} \ and \ \frac{mol_{protein_i}}{gDW_{cell} \cdot hr}}$$, respectively.

Since:

$\mathrm{\alpha_3 = \frac{v_{dilution}}{v_{translation}} \xrightarrow{units} \frac{mol_{ribosome}}{mol_{aa}}}$

therefore:

$\mathrm{(\alpha_3) \cdot (len_{peptide_i}) \xrightarrow{units} (\frac{mol_{ribosome}}{mol_{aa}}) \cdot (\frac{mol_{aa}}{mol_{peptide_i}})= \frac{mol_{ribosome}}{mol_{protein_i}}}$

therefore plugging this into the final coupling of dilution to translation will be:

$\boxed{\mathrm{v_{dilution_i} = \alpha_3 \cdot len_{protein_i} \cdot v_{translation_i}}}$

confirming units:

$\mathrm{v_{dilution_i} = \alpha_3 \cdot len_{protein_i} \cdot v_{translation_i} \xrightarrow{units} (\frac{mol_{ribosome}}{mol_{protein_i}}) \cdot (\frac{mol_{protein_i}}{gDW_{cell}}) = \frac{mol_{ribosome}}{gDW_{cell}}}$

### 6.3.3. Applying ribosome coupling to translation¶

When further imposing ribosome dilution coupling in the ME-model, a translation reaction will look similar to following:

$\mathrm{x \cdot charged\_tRNAs + (\frac{1}{3} \cdot \alpha_1 \cdot \alpha_2 + \frac{1}{3} \cdot \alpha_2) \cdot mRNA_i + len_{protein_i} \cdot \alpha_3 \cdot ribosome \xrightarrow{v_{translation_i}} protein_i + \frac{1}{3} \cdot \alpha_2 \cdot nucleotides}$

with the coupling coefficients substituted:

$\begin{split}\mathrm{x \cdot charged\_tRNAs + (\frac{1}{3} \cdot \frac{\mu + \kappa_{\tau} \cdot r_0}{\kappa_{\tau}} \cdot \frac{f_{mRNA} \cdot m_{aa}}{m_{nt}} + \\ \frac{1}{3} \cdot \frac{k_{deg}^{mRNA}}{\mu} \cdot \frac{\mu + \kappa_{\tau} \cdot r_0}{\kappa_{\tau}} \cdot \frac{f_{mRNA} \cdot m_{aa}}{m_{nt}}) \cdot mRNA_i + (len_{protein} \cdot \frac{\mu + \kappa_{\tau} \cdot r_0}{\kappa_{\tau}} \cdot \frac{f_{rRNA} \cdot m_{aa}}{m_{rr}}) \cdot ribosome \xrightarrow{v_{translation_i}} \\ protein_i + (\frac{1}{3} \cdot \frac{k_{deg}^{mRNA}}{\mu} \cdot \frac{\mu + \kappa_{\tau} \cdot r_0}{\kappa_{\tau}} \cdot \frac{f_{mRNA} \cdot m_{aa}}{m_{nt}}) \cdot nucleotides}\end{split}$

where x represents the coupling coefficient for the tRNAs.