# Endy:Measuring PoPS on a plate reader

*Back to plate reader.*

## Introduction

This is a naive attempt to infer PoPS from plate reader measurements of fluorescence. This analysis depends on many assumptions, only a small number of which are explicitly stated at the moment. The document and model are at a draft stage so please correct typos, math/unit errors.

## Measuring protein numbers

See Standardized GFP quantification for a methodology to relate relative fluorescence measurements from the plate reader back to copies of a fluorescent reporter per well. Previous work, here and here has shown that absorbance measurements on the plate reader can be related to colony forming units (cfu). Assuming these calibrations have been done, we can use the protein copy measurements as an input to a simple model relating protein production rate to PoPS/cell.

## Protein production model

Without derivation, here is a simple, widely used model of protein production within a single cell. The model consists of two differential equations, one governing the rate of accumulation of reporter mRNA per cell and the other governing reporter protein accumulation per cell.

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \dot{[M]} = k_{M}.[D]-\gamma_M[M]\qquad(1) }**

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \dot{[P]} = k_{P}.[M]-\gamma_P[P]\qquad(2) }**

Where,

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \emph [M]}**
= Reporter mRNA concentration per cell, [mRNA]

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \emph [P]}**
= Reporter protein concentration per cell, [Protein]

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \emph [D]}**
= Reporter DNA molecule concentration per cell, [DNA]

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \emph k_{M}}**
= mRNA production rate (PoPS), [mRNA]/[DNA]/s

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \emph k_{P}}**
= Protein production rate (RiPS), [Protein]/[mRNA]/s

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \gamma_M}**
= mRNA degradation rate, 1/s

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \gamma_P}**
= Protein degradation rate. Note that for a stable protein, this becomes equal to dilution rate due to growth, 1/s

*Anyone want to fix the gammas?*

Since mRNA degradation rates are fast (~1min) and since mRNA production rates are also fast, we can assume that reporter mRNA levels are in pseudo steady state. Hence we can solve Eqn. 1 to yield -

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle PoPS,\ k_{M} = \frac{\gamma_M[M]}{[D]}\qquad(3)}**

We can rewrite Eqn. 2 in terms of **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \emph [M]}**
to yield -

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle [M] = \frac{1}{k_{P}}\left (\dot{[P]}+\gamma_P[P] \right )\qquad(4)}**

(3) and (4) can be combined to yield a relation between PoPS and protein levels and rates of production.

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle PoPS,\ k_{M} = \frac{\gamma_M}{k_{P}[D]}\left (\dot{[P]}+\gamma_P[P] \right)\qquad(5)}**

## Using the model

Jennifer Braff and Caitlin Conboy have measured mRNA degradation rates for GFP. In addition, they have measured RiPS for GFP from BBa_B0032 among other RBSs. As mentioned above we can approximate the protein degradation rate by the dilution rate due to growth. Given these parameters, the model can be used to crudely estimate PoPS from protein copy measurements on the plate reader. For this to be in any way accurate, the same parts and conditions should be used as were used by Jen and Caitlin.