Team:INSA-UPS France/Model/Complete Model
iGEM UPS-INSA Toulouse 2017
Complete model
\textit{Vibrio cholerae}, \textit{Vibrio harveyi} and \textit{Pichia pastoris} growth depends on their growth rate and their concentration. To account for the lag phase, we assumed no growth during a first period.
\subsection{\textit{Vibrio cholerae}}
\begin{equation*}
V_{growth,Vc} = \mu_{\textit{Vc}}\cdot[\textit{Vc}]_W
\end{equation*}
\[
\left\{
\begin{array}{r c l}
t < t_{l,Vc} \ ;\ \mu_{\textit{Vc}}=0\text{ }h^{-1} \\
t \geq t_{l,Vc}\ ;\ \mu_{\textit{Vc}}=\mu_{MAX,\textit{Vc}} \\
\end{array}
\right.
\].
\begin{itemize}
\item $[\textit{Vc}]_W$: \textit{Vibrio cholerae} concentration in water (cell/L)
\item $\mu_{\textit{Vc}}$: \textit{Vibrio cholerae} growth rate ($s^{-1}$)
\item $t_{l,Vc}$: \textit{Vibrio cholerae} lag time (s)
\end{itemize}
\subsection{\textit{Vibrio harveyi}}
\begin{equation*}
V_{growth,Vh} = \mu_{\textit{Vh}}\cdot[\textit{Vh}]_D
\end{equation*}
\[
\left\{
\begin{array}{r c l}
t < t_{l,Vh} \ ;\ \mu_{\textit{Vh}}=0\text{ }h^{-1} \\
t \geq t_{l,Vh} ;\ \mu_{\textit{Vh}}=\mu_{MAX,\textit{Vh}} \\
\end{array}
\right.
\].
\begin{itemize}
\item $[\textit{Vh}]_D$: \textit{Vibrio harveyi} concentration in the device (cell/L)
\item $\mu_{\textit{Vh}}$: \textit{Vibrio harveyi} growth rate ($s^{-1}$)
\item $t_{l,Vh}$: \textit{Vibrio harveyi} lag time (s)
\end{itemize}
\subsection{\textit{Pichia pastoris}}
\begin{equation*}
V_{growth,Pp} = \mu_{\textit{Pp}}\cdot[\textit{Pp}]_D
\end{equation*}
\[
\left\{
\begin{array}{r c l}
t < t_{l,Pp} \ ;\ \mu_{\textit{Pp}}=0\text{ }h^{-1} \\
t \geq t_{l,Pp}\ ;\ \mu_{\textit{Pp}}=\mu_{MAX,\textit{Pp}} \\
\end{array}
\right.
\].
\begin{itemize}
\item $[\textit{Pp}]_D$: \textit{Pichia pastoris} concentration in the device (cell/L)
\item $\mu_{\textit{Pp}}$: \textit{Pichia pastoris} growth rate ($s^{-1}$)
\item $t_{l,Pp}$: \textit{Pichia pastoris} lag time (s)
\end{itemize}
\section{Death}
\textit{Vibrio cholerae}, \textit{Vibrio harveyi} and \textit{Pichia pastoris} death depends on their death rate with the antimicrobial peptides (AMP) and their concentration. This process is modeled by a Michaelian model ruled by the cells IC50 with respect to AMP and on the antimicrobial peptides concentration in the compartment.
\subsection{\textit{Vibrio cholerae}}
\begin{equation*}
V_{death,\textit{Vc}}=\frac{[AMP]_W}{[AMP]_W+IC50_{\textit{Vc}}}\cdot k_{kill,\textit{Vc}} \cdot[\textit{Vc}]_W
\end{equation*}
\begin{itemize}
\item $[AMP]_W$: antimicrobial peptides concentration in water (mol/L)
\item $ k_{kill,\textit{Vc}} $: \textit{Vibrio cholerae} death rate with peptides ($s^{-1}$)
\item$IC50_{\textit{Vc}}$: IC50 value of the antimicrobial peptides, for \textit{Vibrio cholerae} (mol/L)
\end{itemize}
\subsection{\textit{Vibrio harveyi}}
\begin{equation*}
V_{death,\textit{Vh}}=\frac{[AMP]_D}{[AMP]_D+IC50_{\textit{Vh}}}\cdot k_{kill,\textit{Vh}} \cdot[\textit{Vh}]_D
\end{equation*}
\begin{itemize}
\item $[AMP]_D$: antimicrobial peptides concentration in the device (mol/L)
\item $k_{kill,\textit{Vh}}$: \textit{Vibrio harveyi} death rate with peptides ($s^{-1}$)
\item$IC50_{\textit{Vh}}$: IC50 value of the antimicrobial peptides, for \textit{Vibrio harveyi} (mol/L)
\end{itemize}
\subsection{\textit{Pichia pastoris}}
\begin{equation*}
V_{death,\textit{Pp}} = \frac{[AMP]_D}{[AMP]_D+IC50_{\textit{Pp}}}\cdot k_{kill,\textit{Pp}} \cdot[\textit{Pp}]_D
\end{equation*}
\begin{itemize}
\item $k_{kill,\textit{Pp}}$: \textit{Pichia pastoris} death rate with peptides ($s^{-1}$)
\item$IC50_{\textit{Pp}}$: IC50 value of the antimicrobial peptides, for \textit{Pichia pastoris} (mol/L)
\end{itemize}
\section{CAI-1 transfer}
Initial CAI-1 concentration is high in the presence of \textit{Vibrio cholerae}, thus there is no need to calculate the kinetics of CAI-1 production. CAI-1, initially in water, can freely diffuse through the membrane. A passive diffusion model is used for this process.
\begin{equation*}
V_{diff,CAI\text{-}1,W\to D} = K.([CAI\text{-}1]_W - [CAI\text{-}1]_D)
\end{equation*}
\medbreak
\begin{itemize}
\item $[CAI\text{-1}]_W$: CAI-1 concentration in water (mol/L)
\item $[CAI\text{-1}]_D$: CAI-1 concentration in the device (mol/L)
\item K: transfer coefficient through the membrane ($s^{-1}$)
\end{itemize}
\section{CqsS - CAI-1 complexation}
\subsection{CqsS* concentration}
The CqsS* concentration is obtained approximating the number of protein per cell, using the \textit{Vibrio harveyi} concentration and the Avogadro number.
\begin{equation*}
[\text{CqsS*}]_D =(\text{Number of CqsS*}/cell).\frac{[Vh]_D}{N_A}
\end{equation*}
\begin{itemize}
\item $[\text{CqsS*}]_D$: concentration of the CqsS* protein in the device (mol/L)
\item Number of CqsS*/cell: number of CqsS* proteins per \textit{Vibrio harveyi} cell
\item $N_A$: Avogadro number ($mol^{-1}$)
\end{itemize}
\subsection{Complexation}
%The complexation is defined as :
%\\~
\begin{center}
\ce{CAI\text{-1} + CqsS\text{*} <=> CqsS\text{*-}CAI\text{-1}}
\end{center}
Assuming kinetics of CAI-1-CqsS* complexation is fast compared to the rest of the system, we assumed that the free and complexed forms are at equilibrum.
\[
V_{complexation} = V_{dissociation}
\]
\[
k_1.[CAI\text{-}1]_D.[\text{CqsS*}]_D = k_2.[CqsS\text{*-}CAI\text{-}1]_D
\]
\begin{equation*}
[CqsS\text{*-}CAI\text{-}1]_D=\frac{[CAI\text{-1}]_D.[\text{CqsS*}]_D}{K_{eq,CqsS\text{*-CAI-1}}}
\end{equation*}
\[
\text{With } K_{eq,CqsS\text{-CAI-1}} = k_2/k_1
\]
\begin{itemize}
\item $k_{1}$: velocity constant of dissociation ($L.mol^{-1}.s^{-1}$)
\item $k_{2}$: velocity constant of complexation ($s^{-1}$)
\item $K_{eq,CqsS\text{*-CAI-1}}$: equilibrum constant of the CqsS*-CAI-1 complexation (mol/L)
\end{itemize}
\section{ALS production}
The production of the ALS enzyme from the ALS gene is divided into two steps: transcription and translation.
\medbreak
\ce{{ALS_{DNA,inactivated}} -> ALS_{DNA} -> ALS_{RNA} -> ALS_{protein}}
\subsection{Activation}
The ALS gene is activated. This is modeled using a Michaelian formalism depending on it activator (CqsS*-CAI-1 complex) concentration. The promoter strength is also taken into account.
\begin{equation*}
ALS_{DNA/cell} = ALS_{DNA,0/cell}.\frac{[CqsS\text{*-}CAI\text{-}1]_D}{K_{a,CqsS\text{*-CAI-1}}+[CqsS\text{*-}CAI\text{-}1]_D}.k_{p,ALS}
\end{equation*}
\begin{itemize}
\item $ALS_{DNA,0/cell}$: total number of ALS DNA per cell
\item $ALS_{DNA/cell}$: number of activated ALS DNA per cell
\item $K_{a,CqsS\text{*-CAI-1}}$: activation constant of the CqsS*-CAI-1 complex (mol/L)
\item $k_{p,ALS}$: ALS promoter influence
\end{itemize}
\subsection{Transcription}
The ALS gene is transcribed into mRNA. The ALS transcription depends on the transcription velocity of the strain and the length of the ALS gene. The transcription velocity is expressed in number of mRNA per time unit. The Avogadro number is used to deduce the molar concentration of mRNA in a theoretical compartment representing the space occupied by all the \textit{Vibrio harveyi} cells.
%\begin{equation*}
%\frac{dALS_{ARN}}{dt} = ALS_{DNA}.\frac{V_{transcription,Vh}}{\text{DNA size}}
%\end{equation*}
\begin{equation*}
V_{transcription,ALS/cell} = \frac{ALS_{DNA/cell}.V_{transcription,Vh}.(\text{RNA polymerase/gene})}{\text{DNA length}}
\end{equation*}
\begin{equation*}
V_{transcription,ALS} = V_{transcription,ALS/cell}.(\text{Number of cells})
\end{equation*}
\begin{equation*}
\text{Number of cells} = [\textit{Vh}].\mathcal{V}_{D}
\end{equation*}
\begin{itemize}
\item $ALS_{DNA}$: number of ALS gene per cell
\item $V_{transcription,Vh}$: \textit{Vibrio harveyi} transcription rate (nucleotides/s)
\item RNA polymerase/gene: number of RNA polymerase per gene
\item DNA length (ALS): number of nucleotides on the ALS gene
\end{itemize}
\begin{equation*}
[ALS_{RNA}]_{\textit{Vh}} = \frac{ALS_{RNA}}{N_A.\mathcal{V}_{Vh}}
\end{equation*}
\begin{equation*}
\mathcal{V}_{Vh} = \mathcal{V}_{intracell}.[\textit{Vh}].\mathcal{V}_{D}
\end{equation*}
\begin{itemize}
\item $[ALS_{RNA}]_{Vh}$: ALS mRNA concentration ($mol/L$)
\item $ALS_{RNA}$: number of ALS mRNA considering the entire system
\item $\mathcal{V}_{intracell}$: volume of a bacterial cell (L)
\item $\mathcal{V}_{Vh}$: volume occupied by all the \textit{Vibrio harveyi} cells
\end{itemize}
\subsection{Translation}
The ALS mRNA is translated into protein. The ALS translation depends on the translation velocity of the strain, the mRNA length and the quantity of mRNA. The translation velocity is expressed in number of protein per time unit. The Avogadro number is used to deduce the molecular concentration of ALS enzyme in a theoretical compartment representing the space occupied by all the \textit{Vibrio harveyi} cells.
\begin{equation*}
V_{translation,ALS/cell} = \frac{ALS_{RNA}.V_{translation,Vh}.(\text{Ribosomes/RNA})}{\text{RNA length}}
\end{equation*}
\begin{equation*}
V_{translation,ALS} = V_{translation,ALS/cell}.(\text{Number of cells})
\end{equation*}
\begin{itemize}
\item $V_{translation,Vh}$: \textit{Vibrio harveyi} translation rate (nucleotides/s)
\item Ribosomes/RNA: number of ribosomes per mRNA
\item RNA length (ALS): number of nucleotides on the ALS mRNA
\end{itemize}
\begin{equation*}
[ALS]_{Vh}= \frac{ALS_{protein}}{N_A.\mathcal{V}_{Vh}}
\end{equation*}
\begin{itemize}
\item $[ALS]_{Vh}$: ALS enzyme concentration ($mol/L$)
\item $ALS_{protein}$: number of ALS proteins considering the entire system ($mol/L$)
\end{itemize}
\subsection{Degradation}
Some of the ALS enzymes and mRNA are degraded. A degradation constant is used to model the degradation velocity.
\begin{equation*}
V_{degradation,ALS} = K_{deg,ALS}.ALS_{protein}
\end{equation*}
\begin{itemize}
\item $K_{deg,ALS}$: ALS degradation constant ($s^{-1}$)
\end{itemize}
\begin{equation*}
V_{degradation,ALS RNA} = K_{deg,ALS RNA}.ALS_{RNA}
\end{equation*}
\begin{itemize}
\item $K_{deg,ALS RNA}$: ALS mRNA degradation constant ($s^{-1}$)
\end{itemize}
%\subsection{General equation}
%Considering one \textit{Vibrio harveyi} cell :
%\begin{equation*}
%\frac{d^2ALS_{protein}}{dt^2} = \frac{dALS_{RNA}}{dt}.\frac{V_{traduction,Vh}}{\text{RNA size}} - K_{deg,ALS}.\frac{dALS_{protein}}{dt}
%\end{equation*}
%\begin{equation*}
%\frac{d^2ALS_{protein}}{dt^2} = ALS_{DNA}.\frac{V_{transcription,Vh}}{\text{DNA size}}.\frac{V_{traduction,Vh}}{\text{RNA size}} - K_{deg}.\frac{dALS_{protein}}{dt}
%\end{equation*}
%Considering all the cells :
%\begin{equation*}
%\frac{d^2[ALS_{protein}]}{dt^2} = ALS_{DNA}.\frac{V_{transcription,Vh}}{\text{DNA size}}.\frac{V_{traduction,Vh}}{\text{RNA size}}.\frac{[Vh]}{N_A} - K_{deg}.\frac{d[ALS_{protein}]}{dt}.
%\end{equation*}
%\begin{itemize}
%\item $[ALS_{protein}]$ : ALS protein concentration (mol/L)
%\end{itemize}
\section{Diacetyl production}
%\begin{equation*}
% \frac{d[dac]_B}{dt} = k_{cat,ALS}.[ALS_{protein}].\frac{[S]}{K_M+[S]} + K.([dac]_A-[dac]_B)
%\end{equation*}
Diacetyl is produced by \textit{Vibrio harveyi} through the reaction catalyzed by ALS and is modeled assuming a Michaelis-Menten kinetics. The theoretical volume of all the \textit{Vibrio harveyi} cells and the device volume are taken into account to determine the diacetyle concentration in the device.
\begin{equation*}
V_{prod,dac} = k_{cat,ALS}.[ALS]_{Vh}.\frac{[S]_D}{K_M+[S]_D}.\frac{\mathcal{V}_{Vh}}{\mathcal{V}_D}
\end{equation*}
\begin{itemize}
\item $k_{cat,ALS}$: catalytic constant of the ALS enzyme ($s^{-1}$)
\item $[S]_D$: initial substrate concentration (mol/L)
\item $K_{M,ALS}$: Michaelis constant of the ALS enzyme (mol/L)
\end{itemize}
\section{Diacetyl transfer}
Diacetyl produced in compartment B can freely diffuse through the membrane. This process is taken into account through a passive diffusion model.
\begin{equation*}
V_{diff,dac,W \to D} = K.([dac]_W-[dac]_D)
\end{equation*}
\begin{itemize}
\item $[dac]_D$: diacetyl concentration in the device (mol/L)
\item $[dac]_W$: diacetyl concentration in water (mol/L)
\end{itemize}
\section{Diacetyl - Odr10 complexation}
\subsection{Odr10 concentration}
The Odr10 concentration is obtained approximating the number of protein per cell, using the \textit{Pichia pastoris} concentration and the Avogadro number.
\begin{equation*}
[Odr10]_D = (\text{Number of Odr10}/cell).\frac{[Pp]_D}{N_A}
\end{equation*}
\begin{itemize}
\item $[Odr10]_D$: concentration of the Odr10 protein in the device (mol/L)
\item Number of Odr10/cell: approximative number of Odr10 proteins per \textit{Pichia pastoris} cell
\end{itemize}
\subsection{Complexation}
%The complexation is defined by :
%\\~
\begin{center}
\ce{dac + \text{Odr10} <=> \text{Odr10-dac}}
\end{center}
The kinetics of complexation is described assuming a first order system. The Odr10-dac complex is assumed to be at the equilibrum.
\[
V_{complexation} = V_{dissociation}
\]
\[
k_1.[dac]_D.[Odr10]_D = k_2.[\text{Odr10-dac}]_D
\]
\begin{equation*}
[Odr10\text{-dac}]_D=\frac{[dac]_D.[Odr10]_D}{K_{eq,Odr10\text{-dac}}}
\end{equation*}
\[
\text{With } K_{eq} = k_2/k_1
\]
\begin{itemize}
\item $k_{1}$: velocity constant of dissociation ($L.mol^{-1}.s^{-1}$)
\item $k_{2}$: velocity constant of complexation ($s^{-1}$)
\item $K_{eq,Odr10\text{-dac}}$: equilibrum constant of the Odr10-dac complexation (mol/L)
\end{itemize}
\section{Antimicrobial peptide (AMP) production}
The production of the antimicrobial peptides (AMP) from the AMP gene is divided into two steps: transcription and translation.
\medbreak
%The ALS gene is activated. This is modeled using a Michaelian formalism depending on it activator (CqsS*-CAI-1 complex) concentration. The promoter strength is also taken into account. The ALS gene is transcribed into mRNA. The ALS transcription depends on the transcription velocity of the strain and the length of the ALS gene. The strain concentration and the Avogadro number are used to express the velocity as a molar concentration per time unit.
%The ALS RNA is translated into protein. The ALS translation depends on the translation velocity of the strain and the RNA size of the translated mRNA.
\ce{AMP_{DNA,inactivated} -> AMP_{DNA} -> AMP_{RNA} -> AMP_{peptide}}
\subsection{Activation}
The AMP gene is activated. This is modeled using a Michaelian formalism depending on the Odr10-dac complex concentration. The promoter strength is also taken into account.
\begin{equation*}
AMP_{DNA/cell} = AMP_{DNA,0/cell}.\frac{[\text{Odr10-}dac]_D}{K_{a,\text{Odr10-dac}}+[\text{Odr10-}dac]_D}.k_{p,AMP}
\end{equation*}
\begin{itemize}
\item $AMP_{DNA,0}$: total number of AMP DNA per cell
\item $AMP_{DNA}$: number of activated AMP DNA per cell
\item $K_{a,\text{Odr10-dac}}$: activation constant of the Odr10-dac complex (mol/L)
\item $k_{p,AMP}$: AMP promoter influence
\end{itemize}
\subsection{Transcription}
The AMP DNA is transcribed into mRNA. The AMP transcription depends on the transcription velocity of the strain and the length of the AMP gene. The transcription velocity is expressed in number of mRNA per time unit. The Avogadro number is used to deduce the molar concentration of mRNA in a theoretical compartment representing the space occupied by all the \textit{Pichia pastoris} cells.
\begin{equation*}
V_{transcription,AMP/cell} = \frac{AMP_{DNA/cell}.V_{transcription,Pp}.(\text{RNA polymerase/gene})}{\text{DNA length}}
\end{equation*}
\begin{equation*}
V_{transcription,AMP} = V_{transcription,AMP/cell}.(\text{Number of cells})
\end{equation*}
\begin{equation}
\text{Number of cells} = [Pp]_D.\mathcal{V}_D
\end{equation}
\begin{itemize}
\item $V_{transcription,Pp}$ : \textit{Pichia pastoris} transcription rate (nucleotides/s)
\item RNA polymerase/gene: number of RNA polymerase per gene
\item DNA length (AMP): number of nucleotides on the AMP gene
\end{itemize}
\begin{equation*}
[AMP_{RNA}]_{Vh}=\frac{AMP_{RNA}}{N_A.\mathcal{V}_{Pp}}
\end{equation*}
\begin{equation*}
\mathcal{V}_{Pp}=\mathcal{V}_{intracell}.[Pp]_D.\mathcal{V}_D
\end{equation*}
\begin{itemize}
\item $[AMP_{RNA}]_{Vh}$: AMP mRNA concentration (mol/L)
\item $AMP_{RNA}$: number of AMP mRNA considering the entire system
\item $\mathcal{V}_{Pp}$: volume occupied by all the \textit{Pichia pastoris} cells
\item $\mathcal{V}_{intracell}$: volume of a yeast cell (L)
\end{itemize}
\subsection{Translation}
The AMP RNA is translated into protein. The AMP translation depends on the translation velocity of the strain, the mRNA length and the quantity of mRNA. The translation velocity is expressed in number of peptides per time unit. The Avogadro number and the theoretical volume of all the \textit{Pichia pastoris} cells are used to deduce the molecular concentration of peptides in cells.
\begin{equation*}
V_{translation,AMP/cell} = \frac{AMP_{RNA}.V_{translation,Pp}.(\text{Ribosomes/RNA})}{\text{RNA length}}
\end{equation*}
\begin{equation*}
V_{translation,AMP} = V_{translation,AMP/cell}.(\text{Number of cells})
\end{equation*}
\begin{itemize}
\item $V_{translation,Pp}$: \textit{Pichia pastoris} translation rate (nucleotides/s)
\item Ribosomes/RNA: number of ribosomes per mRNA
\item RNA length (AMP): number of nucleotides on the AMP mRNA
\end{itemize}
\begin{equation*}
[AMP]_{Pp} = \frac{AMP_{peptide}}{N_A.\mathcal{V}_{Pp}}
\end{equation*}
\begin{itemize}
\item $AMP_{peptide}$: number of AMP in the system
\item $[AMP]_D$: AMP peptides concentration in the device (mol/L)
\end{itemize}
\subsection{Diffusion through membrane}
The diffusion of peptides through the membrane is considered using a passive diffusion model.
\begin{equation*}
V_{diffusion,AMP,Pp \to D} = D.([AMP]_{Pp}.\mathcal{V}_{Pp} - [AMP]_D.\mathcal{V}_{D})/\mathcal{V}_{D}
\end{equation*}
\begin{itemize}
\item D: diffusion coefficient through the cellular membrane ($s^{-1}$)
\end{itemize}
\subsection{Degradation}
Some of the AMP and mRNA are degraded. A degradation constant is used to model the degradation velocity.
\begin{equation*}
V_{degradation,AMP} =K_{deg,AMP}.[AMP]_D
\end{equation*}
\begin{itemize}
\item $K_{deg,AMP}$: AMP degradation constant ($s^{-1}$)
\end{itemize}
\begin{equation*}
V_{degradation,AMP RNA} = K_{deg,AMP RNA}.AMP_{RNA}
\end{equation*}
\begin{itemize}
\item $K_{deg,AMP RNA}$: AMP RNA degradation constant ($s^{-1}$)
\item $[AMP_{RNA}]$: AMP RNA concentration ($mol/L$)
\end{itemize}
\subsection{AMP transfer}
The antimicrobial peptides (AMP), initially in the device can freely diffuse through the membrane. A passive diffusion model is used for this process.
\begin{eqnarray*}
V_{diff,AMP,W \to D} = K([AMP_{peptide}]_W - [AMP_{peptide}]_D)
\end{eqnarray*}
\begin{itemize}
\item $[AMP]_W$: AMP protein concentration in water (mol/L)
\end{itemize}
\newpage
\section*{System of ODEs}
The complete set of ODEs is detailed here :
\begin{equation}
\frac{d[\textit{Vc}]_W}{dt} = V_{growth,Vc} - V_{death,Vc}
\end{equation}
\begin{equation}
\frac{d[\textit{Vh}]_D}{dt} = V_{growth,Vh} - V_{death,Vh}
\end{equation}
\begin{equation}
\frac{d[\textit{Pp}]_D}{dt} = V_{growth,Pp} - V_{death,Pp}
\end{equation}
\begin{equation}
\frac{d[CAI\text{-}1]_D}{dt} = \frac{V_{diff,CAI\text{-}1,W\to D}}{\mathcal{V}_D}
\end{equation}
\begin{equation}
\frac{d[CAI\text{-}1]_W}{dt} = -V_{diff,CAI\text{-}1,W\to D}
\end{equation}
\begin{equation}
\frac{dALS_{RNA}}{dt} = V_{transcription,ALS} - V_{degradation,ALS RNA}
\end{equation}
\begin{equation}
\frac{dALS_{enzyme}}{dt} = V_{translation,ALS} - V_{degradation,ALSenzyme}
\end{equation}
\begin{equation}
\frac{d[dac]_D}{dt}=V_{prod,dac}+\frac{V_{diff,dac,W \to D}}{\mathcal{V}_D}
\end{equation}
\begin{equation}
\frac{d[dac]_W}{dt}=- V_{diff,dac,W \to D}
\end{equation}
\begin{equation}
\frac{dAMP_{RNA}}{dt}=V_{transcription,AMP} - V_{degradation,AMP RNA}
\end{equation}
\begin{equation}
\frac{dAMP_{peptide,Pp}}{dt}=V_{translation,AMP}
\end{equation}
\begin{equation}
\frac{d[AMP]_D}{dt} = V_{diff,AMP,Pp\to D} - V_{degradation,AMP} + \frac{V_{diff,AMP,W \to D}}{\mathcal{V}_D}
\end{equation}
\begin{equation}
\frac{d[AMP]_W}{dt} = -V_{diff,AMP,W\to D}
\end{equation}
\newpage
\section*{Model parameters and initial concentration}
\subsection*{Microorganisms properties}
\begin{itemize}
\item $\mu_{MAX,\textit{Vc}}$: \textit{Vibrio cholerae} maximum growth rate ($s^{-1}$)
\item $\mu_{MAX,\textit{Vh}}$: \textit{Vibrio harveyi} maximum growth rate ($s^{-1}$)
\item $\mu_{MAX,\textit{Pp}}$: \textit{Pichia pastoris} maximum growth rate ($s^{-1}$)
\item $t_{l,Vc}$: \textit{Vibrio cholerae} lag time (s)
\item $t_{l,Vh}$: \textit{Vibrio harveyi} lag time (s)
\item $t_{l,Pp}$: \textit{Pichia pastoris} lag time (s)
\item$IC50_{\textit{Vh}}$: IC50 value of the antimicrobial peptides, for \textit{Vibrio harveyi} (mol/L)
\item$IC50_{\textit{Vc}}$: IC50 value of the antimicrobial peptides, for \textit{Vibrio cholerae} (mol/L)
\item$IC50_{\textit{Pp}}$: IC50 value of the antimicrobial peptides, for \textit{Pichia pastoris} (mol/L)
\item $k_{kill,Vc}$: \textit{Vibrio cholerae} death rate with peptides ($s^{-1}$)
\item $k_{kill,Vh}$: \textit{Vibrio harveyi} death rate with peptides ($s^{-1}$)
\item $k_{kill,Pp}$: \textit{Pichia pastoris} death rate with peptides ($s^{-1}$)
\item $\mathcal{V}_{intracell}$: volume of a cell (respectively of a bacteria and a yeast) (L)
\end{itemize}
\subsection*{Technical parameters}
\begin{itemize}
\item K: transfer coefficient through the membrane $(s^{-1})$
\item $\mathcal{V}_B$: device volume (L), considering a water volume of 1L
\end{itemize}
\subsection*{Molecular and genetic properties}
\begin{itemize}
\item $ALS_{DNA,0/cell}$: total number of ALS DNA \textit{Vibrio harveyi} per cell
\item $AMP_{DNA,0/cell}$: total number of AMP DNA per cell
\item $V_{transcription,Vh}$: \textit{Vibrio harveyi} transcription rate (nucleotides/s)
\item $V_{translation,Vh}$: \textit{Vibrio harveyi} translation rate (nucleotides/s)
\item $k_{p,ALS}$: ALS promoter influence
\item DNA length (ALS): number of nucleotides on the ALS gene
\item Number of CqsS*/cell: approximative number of CqsS* proteins per \textit{Vibrio harveyi} cell
\item RNA length (ALS): number of translated nucleotides on the ALS mRNA
\item Number of Odr10/cell: approximative number of Odr10 proteins per \textit{Pichia pastoris} cell
\item $k_{p,AMP}$: AMP promoter influence
\item $V_{transcription,Pp}$: \textit{Pichia pastoris} transcription rate (nucleotides/h)
\item $V_{translation,Pp}$: \textit{Pichia pastoris} translation rate (nucleotides/h)
\item DNA length (AMP): number of nucleotides on the AMP gene
\item RNA length (AMP): number of nucleotides on the AMP mRNA
\item RNA polymerase/gene: number of RNA polymerase per gene, respectively for a bacteria and a yeast
\item Ribosomes/RNA: number of ribosomes per mRNA, respectively for a bacteria and a yeast
\end{itemize}
\subsection*{Biochemical properties}
\begin{itemize}
\item $K_{eq,CqsS\text{*-CAI-1}}$: equilibrum constant of the CqsS*-CAI-1 complexation (mol/L)
\item $K_{a,CqsS\text{*-CAI-1}}$: activation constant of the CqsS*-CAI-1 complex (mol/L)
\item $K_{deg,ALS}$: ALS degradation constant ($s^{-1}$)
\item $K_{deg,ALS RNA}$: ALS RNA degradation constant ($s^{-1}$)
\item $K_{eq,Odr10\text{-dac}}$: equilibrum constant of the Odr10-dac complexation (mol/L)
\item $k_{cat,ALS}$: catalytic constant of the ALS enzyme ($s^{-1}$)
\item $K_{M,ALS}$: Michaelis constant of the ALS enzyme (mol/L)
\item $K_{a,\text{Odr10-dac}}$: activation constant of the Odr10-dac complex (mol/L)
\item $K_{deg,AMP}$: AMP degradation constant ($s^{-1}$)
\item $K_{deg,AMP RNA}$: AMP RNA degradation constant ($s^{-1}$)
\item D: diffusion coefficient through the cellular membrane ($s^{-1}$)
\end{itemize}
\subsection*{Initial concentration}
\begin{itemize}
\item $[\textit{Vc}]_{0,W}$: \textit{Vibrio cholerae} initial concentration (cell/L)
\item $[\textit{Vh}]_{0,D}$: \textit{Vibrio harveyi} initial concentration (cell/L)
\item $[\textit{Pp}]_{0,D}$: \textit{Pichia pastoris} initial concentration (cell/L)
\item $[S]_0$: initial substrate concentration (mol/L)
\item $[\text{CAI-1}]_{W,0}$: CAI-1 initial concentration in water (mol/L)
\end{itemize}
\end{document}
Equations
