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\begin{document}
\title{Mean Field Model of Infection}
\author{Taylor Dunn}
\affil{Affiliation not available}
\author{Andrew Rutenberg}
\affil{Affiliation not available}
\date{\today}
\maketitle
\section*{Parameters}
\subsection{Counts}
$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
\subsection{Fractions}
$m \equiv $ multiplicity of infection (MOI) $ = \frac{N_B(t=0)}{N}$
$c \equiv $ confluency $ = \frac{N a}{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
$\quad \tilde{b}_R(t=0) = 1$
\subsection*{Rates}
$\Gamma_0 \equiv $ primary attachment rate per bacterial density
$\Gamma_1 \equiv $ ruffle recruitment rate per bacterial density
\section*{Proofs}
\subsection*{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 }}{H A / c} = \frac{(1-b) mc}{A}
\end{equation}
\subsection*{Rate of infectivity}
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}{A L^2}\right)
\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}
\subsection*{Rate of ruffle formation}
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 = N f \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}
\subsection*{Rate of ruffling}
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)
\end{equation*}
\begin{equation}
\dot{r} = (1 - r) f \Gamma_0 \left(\frac{bmc}{a}\right)
\end{equation}
\subsection*{Rate of bacteria capture}
Bacteria are captured through one of two ways: primary attachment to host cells, or secondary recruitment to host cell ruffles.
\begin{equation*}
\begin{align}
\dot{N}_B =& \ \dot{b} N_B (0) = \dot{b} m N \\
=& - N \Gamma_0 \rho_B - N_R \Gamma_1 \rho_B = -N \rho_B (\Gamma_0 + \tilde{r} \Gamma_1)
\end{align}
\end{equation*}
\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}
\section{Integration}
\subsection{Initial values}
$ 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$
\subsection{Universal values}
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.
\subsection{Experimental values}
These values are unique to experimental setup and conditions: $m$, $c$, $t_{\mathrm{max}}$, and $L$.
\subsection{Rescaling}
$t_{\mathrm{max}}' = t_{\mathrm{max}} \frac{c}{a} = t_{\mathrm{max}} k$
The rescaled form of the differential equations are
\begin{equation}
\label{eqn:diff}
\dot{x} = (1-x) \Gamma_0 b m \quad \dot{b} = -b (\Gamma_0 + \tilde{r} \Gamma_1) \quad \dot{r} = (1-r) f \Gamma_0 b m
\quad \dot{\tilde{r}} = f \Gamma_0 b m
\end{equation}
\section{Examples}
\subsection{Time dependence}
The following are examples of integrating the differential equations of \ref{eqn:diff} from 0 to $t_{\mathrm{max}}'$ using the fourth-order Runge Kutta method.\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/vary-tmax/vary-tmax}
\caption{{\textbf{Fig. 1} The effect of varying $t_{\mathrm{max}}'$ on the infectivity $x(t)$, the remaining bacteria $b(t)$ and the ruffling fraction $r(t)$. This integration was calculated for 100 MOI, $f = 0.7$, $\Gamma_0 = 0.02$, $\Gamma_1 = 0.1$ and $\Delta t = 10^{-2}$.%
}}
\end{center}
\end{figure}\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/vary-dt/vary-dt}
\caption{{\textbf{Fig. 2} The effect of varying $\Delta t$ on the infectivity $x(t)$, the remaining bacteria $b(t)$ and the ruffling fraction $r(t)$. This integration was calculated for 100 MOI, $f = 0.7$, $\Gamma_0 = 0.02$, $\Gamma_1 = 0.1$ and $t_{\mathrm{max}}' = 10.0$.%
}}
\end{center}
\end{figure}\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/vary-moi/vary-moi}
\caption{{\textbf{Fig. 3} The effect of varying MOI. This integration was calculated for $f = 0.7$, $\Gamma_0 = 0.02$, $\Gamma_1 = 0.1$, $t_{\mathrm{max}}' = 10.0$ and $\Delta t = 10^{-2}$. Due to ruffle recruitment, higher bacterial density (MOI) leads to faster uptake of available bacteria.%
}}
\end{center}
\end{figure}\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/vary-g0/vary-g0}
\caption{{\textbf{Fig. 4} The effect of varying the primary attachment rate $\Gamma_0$. This integration was calculated for 100 MOI, $f = 0.7$, $\Gamma_1 = 0.1$, $t_{\mathrm{max}}' = 10.0$ and $\Delta t = 10^{-2}$.%
}}
\end{center}
\end{figure}\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/vary-g1/vary-g1}
\caption{{\textbf{Fig. 5} The effect of varying the recruitment rate $\Gamma_1$. This integration was calculated for 100 MOI, $f = 0.7$, $\Gamma_0 = 0.02$, $t_{\mathrm{max}}' = 10.0$ and $\Delta t = 10^{-2}$. For low values of $\Gamma_1$ (i.e. little or no cooperative invasion), bacterial uptake is much slower and the majority of bacteria remain long after all the host cells have been infected. For high values, where recruitment is the dominant channel of uptake, the pool of available bacteria is quickly depleted leaving some host cells uninvaded.%
}}
\end{center}
\end{figure}
\section{Literature}
\subsection{Misselwitz 2012}
\cite{Misselwitz_2012} studied target-site selection and cooperative invasion of \textit{Salmonella} Typhimurium. The following is from Figure 9, and focuses on the efficiency of docking to ruffling cells compared to non-ruffling cells.\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/misselwitz-2012-fig9/misselwitz-2012-fig9}
\caption{{\textbf{Fig. 6} To analyze a preference for docking on ruffles, HeLa cells were infected with a 1:1 mixture of helper strain S. Tm$^{\rm SopE}$ (causes ruffles, carries deletions of the effectors \textit{sipA}, \textit{sopB} and \textit{sopE2}) and reporter strain S. Tm$^{\rm \Delta 4}$ (pGFP, does not trigger ruffles). Infection lasted for 6 minutes at 62 and 250 MOI. The bars summarize 170-220 cells/ruffles from two independent experiments. Materials and methods: HeLa cells were seeded in 96-well Microclear plates (full size, Grenier) at 6000 cells per well 24 hours prior to infection. After infection and fixation, extracellular \textit{S.} Typhimurium was stained using an anti-\textit{Salmonella} LPS antibody and a Cy5-labelled secondary antibody. Nuclei and actin were stained using DAPI and TRITC-phalloidin, respectively. Bound \textit{S.} Typhimurium bacteria were manually quantified using a 40x-objective.%
}}
\end{center}
\end{figure}
It is clear to see from the above figure that ruffling facilitates bacterial docking. In the context of our infection model, we therefore expect $\Gamma_1 > \Gamma_0$.
Figure 10 of \cite{Misselwitz_2012} had a number of interesting observations related to cooperative invasion.\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=1.00\columnwidth]{figures/misselwitz-2012-fig1/misselwitz-2012-fig1}
\caption{{\textbf{Fig. 7} HeLa cells were incubated with S. Tm$^{\rm SopE}$ (pGFP) for 9 minutes from 4 to 500 MOI. (A) explains the quantification strategies used for the below plots. (B) Left panel: fraction of cells carrying ruffles. Middle panel: quantification of "inside" and "outside" bacteria per individual ruffle. Right panel: invaded bacteria per cell, calculated by multiplying the number of intracellular bacteria per ruffle with the fraction of ruffling cells. The black line is an extrapolation from low MOI, assuming invasion efficiency is linear with MOI. Specifically, extrapolation was from 4 and 8 MOI, and only ruffles with 1 bacterium were considered to analyze invasion without "support" from other bacteria. Each data point summarizes 4x150 cells for the analysis of cellular ruffling and 4x25 cells for inside and outside bacteria from 2 independent experiments. Error bars: standard deviation. Materials and methods: as described in Fig. 6.%
}}
\end{center}
\end{figure}\selectlanguage{english}
\begin{table}
\caption{{Data extracted from Figure 10 of \cite{Misselwitz_2012} using the WebPlotDigitizer application.}}
\begin{tabular}{ c c c c c c}
MOI & Ruffling fraction & Bacteria per ruffle inside & Bacteria per ruffle outside & Bacteria invaded per cell (measured) & Bacteria invaded per cell (extrapolated) \\
4 & 0.01926 & 0.519 & 0.412 & 0.01 & 0.01 \\
8 & 0.04133 & 0.600 & 0.482 & 0.02 & 0.02 \\
16 & 0.08212 & 1.034 & 0.562 & 0.08 & 0.03 \\
31 & 0.15844 & 1.205 & 0.770 & 0.2 & 0.07\\
63 & 0.30108 & 1.928 & 1.161 & 0.61 & 0.13 \\
125 & 0.51314 & 2.928 & 1.823 & 1.51 & 0.22 \\
250 & 0.73332 & 5.080 & 3.271 & 3.77 & 0.49 \\
500 & 0.86570 & 7.702 & 5.628 & 6.80 & 1.02 \\
\end{tabular}
\end{table}
I question the legitimacy of 'bacteria invaded per cell (measured)' which was calculated as (ruffling fraction) * (bacteria per ruffle inside). There doesn't seem to be any mention of the number of ruffles a typical ruffling cell has, so the calculated number of bacteria invaded per cell is only accurate if the cell has one ruffle. Since this is likely an underestimate, the authors were maybe trying to just show the large difference between measured invasion and invasion extrapolated from low MOI.
\cite{Misselwitz_2012} performed an analogous experiment using automated image acquisition and analysis (CellProfiler) instead of manual quantification as above. Supplemental Figure S6 is shown below.\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.84\columnwidth]{figures/misselwitz-2012-figs6/misselwitz-2012-figs6}
\caption{{\textbf{Fig. 8} HeLa cells were infected for the indicated times at an MOI of 250 with a mixture of reporter and helper strains. The reporter strain (\textit{gfp}-labeled) was either \textit{S.} Tm$^{\rm \Delta 4}$ or \textit{S.} Tm$^{\rm -T1}$ (motile, but lacks a functional TTSS-1 apparatus required for efficient docking and triggering of invasion), and the helper strain (\textit{rfp}-labeled) was either incapable of triggering ruffles (\textit{S.} Tm$^{\rm \Delta 4}$ or \textit{S.} Tm$^{\rm -T1}$) or capable (\textit{S.} Tm$^{\rm SopE}$). (A-D) These panels show various combinations of reporter (green) and helper (red). Comparing panels A and B (no ruffling) to panels C and D, one can see the effects of cooperative invasion by the increased size of the yellow bars. (E, F) These panels show the sum of the green and yellow bars of panels A-D for the two reporter strains. Docking in this instance means both intracellular and extracellular, as they were indistinguishable. (G, H) A parallel experiment was performed where the reporter strains carried plasmid pM975 which specifically induces \textit{gfp} production intracellularly. (I, J) The fraction of cells exclusively harboring extracellular bacteria was calculated by subtracting the fraction of cells with docked bacteria (panels E and F) from the fraction of cells with intracellular bacteria (G, H). This calculation is a conservative estimation because cells can habor both docked and intracellular bacteria.%
}}
\end{center}
\end{figure}
There is no doubt that ruffling facilitates docking to the outside of the cell, but I don't believe that this shows ruffling stimulates cellular invasion, as the authors concluded. The difference in invasion fraction between ruffling (\textit{S.} TM$^{\rm SopE}$) and non-ruffling (\textit{S.} TM$^{\rm \Delta 4}$, \textit{S.} TM$^{\rm -T1}$) helpers shown in panels G and H may simply be due to the fact that ruffling cells have more bacteria per cell, and therefore are more likely to have at least one bacterium become intracellular. It is interesting, however, that in the case non-ruffling helper strains, close to zero percent of cells had invaded bacteria, despite $>$60\% (panel A, \textit{S.} TM$^{\rm \Delta 4}$) and ~20\% (panel B, \textit{S.} TM$^{\rm -T1}$) cells having docked bacteria. Yes, overall infection percentage will increase with ruffling but without comparing the number of extracellular versus intracellular bacteria, one cannot say whether cooperative invasion on ruffles increase the probability of invading on the single bacterium level, or just the probability of docking.
\subsection{Huang 1998}
\cite{Huang_1998} examined limitations on \textit{Salmonella} Typhi invasion of INT407 (HeLa) cells. Below is Figure 1 from their study.\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.98\columnwidth]{figures/huang-1998-fig1/huang-1998-fig1}
\caption{{\textbf{Fig. 8} HeLa cells were infected with \textit{S. Typhi} Ty2W (a spontaneous Vi capsule-negative isogenic derivative of wild-type Ty2, which consistently showed ~2-fold greater invasion ability) at MOI ranging from 0.4 to 4000. A 1 hour invasion period and a 1 hour gentamicin kill period were used prior to plate count enumeration of internalized bacteria. (A) Invasion efficiency was calculated as IE = (number of internalized bacteria at end of assay / starting inoculum) x 100. (B) Total internalized \textit{S. Typhi} per assay well is expressed as mean CFU/well (colony-forming unit) $\pm$ standard deviation. (C) The number of internalized bacteria per epithelial cell was calculated by dividing the total number of internalized bacteria by the 6 x 10$^5$ epithelial cells in each well. Materials and methods: all assays were conducted in duplicate and repeated on separate days at least three times. 24-well tissue culture plates (Sarstedt Inc.). Each well, as assessed by representative hemocytometer counts of sample wells, contained ~6 x 10$^5$ INT 407 cells in 1 ml of medium. Infected monolayers were incubated in 5\% CO$_2$ at 37$\degree$C. 1 ml of tissue culture medium containing gentamicin (100\mu g/ml) was used to kill remaining extracellular bacteria. Infected monolayers were washed and lysed with 1 ml of 0.1\% Triton X-100 for 10 minutes. After serial dilution in PBS, released intracellular bacteria were quantified by viable count after growth for 24 hours on L agar.%
}}
\end{center}
\end{figure}
Due to some doubts in accuracy of the viable bacteria counting method, fluorescence microscopy was also used to examine the number of infected cells (Table 1 of \cite{Huang_1998} below).\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=1.00\columnwidth]{figures/huang-1998-table1/huang-1998-table1}
\caption{{\textbf{Fig. 9} HeLa cells were infected with \textit{S. Typhi} Ty2W at ~40 MOI for various times. (Left) The plate count assay method followed the steps as described in Fig. 8, but with invasion time ranging from 0 to 60 minutes. (Right) Another set of time course-infected monolayers were examined using AO-CV staining (viable intracellular bacteria stain bright green, nonviable intracellular bacteria stain red/orange, and extracellular bacteria counterstain deep violet/black). The second column gives mean bacteria per infected cell, and in brackets the predicted number of bacteria for all cells after an hour of growth. This is included for comparison to the plate count assay bacteria/cell, which involves a 1 hour gentamicin kill period during which intracellular bacteria double according to control studies (data not shown). For example, the calculation for the 10 minute data would be (1.9 bacteria/cell) x (10\% of cells) x (2) \approx \ 0.4 bacteria/cell%
}}
\end{center}
\end{figure}
This part of the experiment was mainly used as a way of validating the method of the plate count assay data in Fig. 8, with the added unexpected observation that internalized bacteria existed in a limited number of clusters or foci of infection (2 $\pm$ 1 foci per infected INT407 cell at ~40 MOI). This may be a consequence of cooperative invasion and ruffles acting as a common point of entry for invading bacteria. Interestingly, foci per cell remained consistent at ~400 MOI (data not shown), suggesting that \textit{S. Typhi} entry is saturable and may occur at a limited number of possible entry points (ruffles) per cell.
In terms of our model parameters, invasion efficiency is the probability of internalization multiplied by the fraction of bacteria attached to host cells.
\begin{equation}
\mathrm{IE} = p (1 - b)
\end{equation}
\selectlanguage{english}
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