Mean Field Model of Infection

\(N\equiv\) number of host cells

\(N_{I}\equiv\) number of infected cells

\(N_{B}\equiv\) number of bacteria

\(N_{R}\equiv\) number of ruffles

\(N_{r}\equiv\) number or ruffling cells (\(\geq 1\) ruffles)

\(t_{\mathrm{max}}\equiv\) total incubation time

\(m\equiv\) multiplicity of infection (MOI) \(=\frac{N_{B}(t=0)}{N}\)

\(c\equiv\) confluency \(=\frac{Na}{L^{2}}\)

\(\quad a\equiv\) mean cellular area

\(\quad L\equiv\) side length of square well

\(x\equiv\) fraction of host cells infected \(=\frac{N_{I}}{N}\)

\(b\equiv\) fraction of bacteria remaining (i.e. not landed on a host) \(=\frac{N_{B}}{N_{B}(0)}\)

\(f\equiv\) fraction of attached bacteria that form ruffles

\(r\equiv\) fraction of host cells with ruffling (\(\geq\) 1 ruffle)

\(\tilde{r}\equiv\) ruffles per cell \(=\frac{N_{R}}{N}\)

\(\tilde{b}_{R}\equiv\) bacteria per ruffle

\(\quad\tilde{b}_{R}(t=0)=1\)

\(\Gamma_{0}\equiv\) primary attachment rate per bacterial density

\(\Gamma_{1}\equiv\) ruffle recruitment rate per bacterial density

The bacterial density \(\rho_{B}\) is the number of bacteria (available for attachment) per unit area, but is more helpful in terms of MOI and \(b\).

\begin{equation}
\rho_{B}=\frac{B_{u}}{L^{2}}=\frac{(1-b)B_{\rm tot}}{HA/c}=\frac{(1-b)mc}{A}\\
\end{equation}

The rate of change of the number of host cells with bacteria (i.e. \(\geq\) 1 bacteria have attached) depends on the number of remaining cells without bacteria attached, the primary attachment rate and the bacterial density.

\begin{equation}
\dot{H}_{a}=\dot{a}H=(H-H_{a})\Gamma_{0}\rho_{B}=(H-H_{a})\Gamma_{0}\left(\frac{bmc}{AL^{2}}\right)\nonumber \\
\end{equation}

In our model, we assume limited invasion (i.e. we impose a maximum number of internalized bacteria per cell). The rate of change of infected cells

\begin{equation}
\dot{H}_{x}=\dot{x}H=(H-H_{x})\\
\end{equation}

\begin{equation}
\dot{x}=(1-x)\Gamma_{0}\left(\frac{bmc}{a}\right)\\
\end{equation}

The rate of change of the number of total ruffles over all cells depends on the number of total host cells, the primary attachment rate, the bacterial density and the probability of ruffle formation.

\begin{equation}
\dot{N}_{R}=N\Gamma_{0}\rho_{B}f=Nf\Gamma_{0}\left(\frac{bmc}{a}\right)\\
\end{equation}

\begin{equation}
\dot{\tilde{r}}=\frac{\dot{N}_{R}}{N}=f\Gamma_{0}\left(\frac{bmc}{a}\right)\\
\end{equation}

The rate of change of the number or ruffling cells (i.e. \(\geq\) 1 ruffle) is given by the number of cells without ruffles, the primary attachment rate, the bacterial density and the probability of ruffle formation.

\begin{equation}
\dot{N}_{r}=\dot{r}N=(N-N_{r})\Gamma_{0}\rho_{B}f=(N-N_{r})f\Gamma_{0}\left(\frac{bmc}{a}\right)\nonumber \\
\end{equation}

\begin{equation}
\dot{r}=(1-r)f\Gamma_{0}\left(\frac{bmc}{a}\right)\\
\end{equation}

Bacteria are captured through one of two ways: primary attachment to host cells, or secondary recruitment to host cell ruffles.

\begin{eqnarray}
\dot{N}_{B}= & \ \dot{b}N_{B}(0)=\dot{b}mN\nonumber \\
= & -N\Gamma_{0}\rho_{B}-N_{R}\Gamma_{1}\rho_{B}=-N\rho_{B}(\Gamma_{0}+\tilde{r}\Gamma_{1})\nonumber \\
\end{eqnarray}

\begin{equation}
\dot{b}=-\frac{bc}{a}(\Gamma_{0}+\tilde{r}\Gamma_{1})\\
\end{equation}

\begin{equation}
\dot{\tilde{b}}=\frac{-\dot{N}_{B}}{N}=-\dot{b}m\\
\end{equation}

\(x(0)=0\quad b(0)=1\quad\tilde{b}(0)=0\quad r(0)=0\quad\tilde{r}(0)=0\quad\tilde{b}_{R}(0)=1\)

These values are presumably consistent between experiments: \(\Gamma_{0}\), \(\Gamma_{1}\) and \(f\). For experiments with the same host cells (primarily HeLa), mean cellular area \(a\) should also be consistent.

These values are unique to experimental setup and conditions: \(m\), \(c\), \(t_{\mathrm{max}}\), and \(L\).

\(t_{\mathrm{max}}^{\prime}=t_{\mathrm{max}}\frac{c}{a}=t_{\mathrm{max}}k\)

The rescaled form of the differential equations are

\begin{equation}
\label{eqn:diff}\dot{x}=(1-x)\Gamma_{0}bm\quad\dot{b}=-b(\Gamma_{0}+\tilde{r}\Gamma_{1})\quad\dot{r}=(1-r)f\Gamma_{0}bm\quad\dot{\tilde{r}}=f\Gamma_{0}bm\\
\end{equation}

Taylor Dunn10 months ago · Public’Infected’ in this case means 1 or more bacteria attached, but not necessarily invaded. There may be a more appropriate word for this.