# Parameters

## Counts

During the time course of invasion by Salmonella, a bacterium can be in 4 different generalized stages: swimming (looking to attach to a host) $$B$$, attached to a host cell $$B_a$$, invaded and vacuolar $$B_v$$, or invaded and cytosolic $$B_c$$. The counts of bacteria in these stages are constantly changing in time, but ignoring xenophagy and replication, the total number will remain constant:

$B_{\rm tot} = B(t) + B_a(t) + B_v(t) + B_c(t)$

We classify the relationship between host cells and bacteria in one of three ways: host cells with no bacteria $$H$$, with attached bacteria $$H_a$$, with invaded vacuolar bacteria $$H_v$$, and with invaded cytosolic bacteria $$H_c$$. A host cell may have a combination of attached, vacuolar and cytosolic bacteria at one time, so the total number of host cells is the the length of the union of these sets:

$H_{\rm tot} = \left\vert{ \left\{H(t) \right\} \cup \left\{H_a (t) \right\} \cup \left\{H_v (t) \right\} \cup \left\{H_c (t) \right\} }\right\vert$

We also define infected host cells $$H_x$$ as any with internalized bacteria (vacuolar or cytosolic):

$H_{x} = \left\vert{ \left\{H_v (t) \right\} \cup \left\{H_c (t) \right\} }\right\vert$

A well-studied feature of invasion by Salmonella is the formation of epithelial cell membrane ruffles, triggered by effectors secreted via the bacterium’s type III secretion system 1 (TTSS-1). We call the total number of ruffles $$R (t)$$, and the number of host cells with at least one ruffle $$H_r (t)$$.

An important and controllable quantity when performing invasion assays is the ratio of bacteria to host cells at inoculation. This is called the multiplicity of infection (MOI) and in terms of previously defined parameters, $$m = B_{\rm tot} / H_{\rm tot}$$. Another is the confluency, which is an estimate of the proportion of area covered by a monolayer of cells in a culture dish. In terms of average host cell area $$A$$ and side length of the square well $$L$$, we calculate confluency as $$c = H_{\rm tot} A / L^2$$. A similar, although time-dependent quantity, is the bacterial density per unit area $$\rho_B (t) = B(t) / L^2$$. More specifically, this is the density of bacteria have yet to attach to a host.

## Fractions

Since this is a mean field model, it is more useful to speak in terms of fractions or percentages, rather than discrete counts.

$$h (t) \equiv$$ fraction of host cells without attached and/or invaded bacteria $$= \frac{H}{H_{\rm tot}}$$

$$h_a (t) \equiv$$ fraction of host cells with attached bacteria $$= \frac{H_a}{H_{\rm tot}}$$

$$h_x (t) \equiv$$ fraction of host cells with invaded bacteria $$= \frac{H_x}{H_{\rm tot}}$$

$$h_v (t) \equiv$$ fraction of host cells with invaded vacuolar bacteria $$= \frac{H_v}{H_{\rm tot}}$$

$$h_c (t) \equiv$$ fraction of host cells with invaded cytosolic bacteria $$= \frac{H_c}{H_{\rm tot}}$$

$$h_r (t) \equiv$$ fraction of host cells with ruffles $$= \frac{H_r}{H_{\rm tot}}$$

$$b (t) \equiv$$ fraction of bacteria that are available for attachment $$= \frac{B}{B_{\rm tot}}$$

$$b_a (t) \equiv$$ fraction of bacteria that have attached to a host cell $$= \frac{B_a}{B_{\rm tot}}$$

$$b_x (t) \equiv$$ fraction of bacteria that have invaded a host cell $$= b_c + b_v$$

$$\quad b_v (t) \equiv$$ fraction of vacuolar bacteria $$= \frac{B_v}{B_{\rm tot}}$$

$$\quad b_c (t) \equiv$$ fraction of cytosolic bacteria $$= \frac{B_c}{B_{\rm tot}}$$

$$\tilde{r} (t) \equiv$$ ruffles per host cell $$= \frac{R}{H_r} = \frac{R}{h_r H_{\rm tot}}$$

$$\tilde{b}_a \equiv$$ bacteria attached per host cell $$= \frac{B_a}{H_a} = m \frac{b_a}{h_a}$$

$$\tilde{b}_x \equiv$$ bacteria invaded per host cell $$= \frac{B_x}{H_v} = m \frac{b_x}{h_v}$$

## Rates

To understand how these quantities evolve in time, we must define the rates which describe the various stochastic processes of bacterial invasion. A swimming bacterium can either attach to a host cell normally, or be recruited to a membrane ruffle.

$$\Gamma_a \equiv$$ primary attachment rate per bacterial density, per host cell

$$\Gamma_b \equiv$$ ruffle recruitment rate per bacterial density, per ruffle

Once attached, a bacterium can cause ruffling, invade the host cell and stay vacuolar, or invade the host cell and escape early into the cytosol.

$$\Gamma_r \equiv$$ ruffle formation rate per attached bacteria, per host cell

$$\Gamma_v \equiv$$ vacuolar invasion rate per attached bacteria, per host cell

$$\Gamma_c \equiv$$ cytosolic invasion rate per attached bacteria, per host cell

$$\quad \Gamma_x \equiv$$ combined invasion rate per attached bacteria, per host cell $$= \Gamma_v + \Gamma_c$$

There is evidence that suggests a physical limit to the number of ruffles per cell $$\tilde{r}_{\rm max}$$, and the number of invaded bacteria per cell $$\tilde{b}_{x, \rm max}$$, so we define the following effective rates.

$$\Gamma_r^{*} (t) \equiv$$ effective (limited) ruffle formation rate per attached bacteria $$= \Gamma_r (1 - \frac{\tilde{r} (t)}{\tilde{r}_{\mathrm{max}}})$$

$$\Gamma_x^{*} (t) \equiv$$ effective (limited) invasion rate per attached bacteria $$= \Gamma_x (1 - \frac{\tilde{b_x} (t)}{\tilde{b}_{x, \mathrm{max}}})$$

# Proofs

The bacterial density $$\rho_B (t)$$ is a quantity that will determine attachment rate, and is more helpful in terms of MOI, $$b (t)$$ and confluency.

$\rho_B (t) = \frac{B (t)}{L^2} = \frac{b B_{\rm tot }}{H_{\rm tot} A / c} = \frac{ b m c}{A} = bmc$

where in the last step, we have used the reduced units $$A = 1$$.

## Host cell dynamic equations

The change in the fraction of host cells with attached bacteria $$h_a$$ will only depend on the primary attachment rate $$\Gamma_a$$ and not secondary attachment (ruffle recruitment) rate $$\Gamma_b$$, because for bacteria to be recruited by ruffles, a bacterium has already attached to form that ruffle. This rate of change will be limited by the bacterial density $$\rho_B$$ and by the fraction of host cells without bacteria $$h$$.

$\dot{h}_a = \frac{\dot{H}_a}{H_{\rm tot}} = \frac{H}{H_{\rm tot}} \Gamma_a \rho_B = h \Gamma_a b m c$

The change in infectivity $$h_x$$ is controlled by invasion rates $$\Gamma_v$$ and $$\Gamma_c$$, and is limited by the fraction of uninfected cells and attached bacteria per cell $$\tilde{b}_a$$.

$\dot{h}_x = \frac{\dot{H_x}}{H_{\rm tot}} = \frac{H_a - H_x}{H_{\rm tot}} (\Gamma_v + \Gamma_c) \tilde{b}_a = (h_a - h_x) (\Gamma_v + \Gamma_c) \frac{b_a m}{h_a} = \left(1 - \frac{h_x}{h_a}\right) (\Gamma_v + \Gamma_c) b_a m$

Similar equations can be derived for host cells with vacuolar bacteria and host cells with cytosolic: $\dot{h}_v = \frac{\dot{H_v}}{H_{\rm tot}} = \frac{H_a - H_v}{H_{\rm tot}} \Gamma_v \tilde{b}_a = (h_a - h_v) \Gamma_v \frac{b_a m}{h_a} = \left(1 - \frac{h_v}{h_a}\right) \Gamma_v b_a m$ $\dot{h}_c = \frac{\dot{H_c}}{H_{\rm tot}} = \frac{H_a - H_c}{H_{\rm tot}} \Gamma_c \tilde{b}_a = (h_a - h_c) \Gamma_c \frac{b_a m}{h_a} = \left(1 - \frac{h_c}{h_a}\right) \Gamma_c b_a m$

Host cell ruffles form at rate $$\Gamma_r$$, depending on the number of attached bacteria.

$\dot{h}_r = \frac{\dot{H}_r}{H_{\rm tot}} = \frac{H_{a} - H_r}{H_{\rm tot}} \Gamma_r \tilde{b}_a = (h_a - h_r) \Gamma_r \frac{b_a m}{h_a} = \left(1 - \frac{h_r}{h_a} \right) \Gamma_r b_a m$

## Bacteria dynamic equations

A more complicated quantity is the fraction of bacteria which are attached to host cells, which has three means of change. The first is by regular primary attachment with rate $$\Gamma_a$$, the second is ruffle recruitment with rate $$\Gamma_b$$, and the third is a loss of attached bacteria as they invade with limited invasion rate $$\Gamma_x^*$$.

\begin{align} \dot{b}_a = \frac{\dot{B_a}}{B_{\mathrm{tot}}} &= \frac{H_{\rm tot} \Gamma_a \rho_B}{B_{\mathrm{tot}}} + \frac{R \Gamma_b \rho_B}{B_{\mathrm{tot}}} - \frac{B_a \Gamma_x^*}{B_{\mathrm{tot}}} \\ &= \Gamma_a b c + \tilde{r} h_r \Gamma_b b c - b_a \Gamma_x \left(1 - \frac{\tilde{b}_x}{\tilde{b}_{x, \rm max}}\right) \end{align}

Bacteria will either internalize and remain vacuolar at rate $$\Gamma_v^*$$ or escape early and become cytosolic at rate $$\Gamma_c^*$$: $\dot{b}_v = \frac{\dot{B_v}}{B_{\rm tot}} = \frac{B_a}{B_{\rm tot}} \Gamma_v^* = b_a \Gamma_v \left(1 - \frac{\tilde{b}_x}{\tilde{b}_{x, \rm max}}\right)$ $\dot{b}_c = \frac{\dot{B_c}}{B_{\rm tot}} = \frac{B_a}{B_{\rm tot}} \Gamma_c^* = b_a \Gamma_c \left(1 - \frac{\tilde{b}_x}{\tilde{b}_{x, \rm max}}\right)$

## Ruffle dynamic equation

Change in ruffle formation is another quantity whose rate $$\Gamma_r^*$$ depends on attached bacteria per cell $$\tilde{b}_a$$ and is limited by a maximum $$\tilde{r}_{\rm max}$$.

$\dot{\tilde{r}} = \frac{\dot{R}}{H_r} = \frac{B_a}{H_r} \Gamma_r^* = \Gamma_r \left(1 - \frac{\tilde{r}}{\tilde{r}_{\rm max}}\right) \frac{b_a m}{h_r} %\dot{\tilde{r}} = \frac{\dot{R}}{H_r} = \frac{B_a}{H_r} \Gamma_r^* \tilde{b}_a = \Gamma_r \left(1 - \frac{\tilde{r}}{\tilde{r}_{\rm max}}\right) \frac{b_a m}{h_a}$