deletions | additions       diff --git a/untitled.tex b/untitled.tex  index 25a04c2..890ed22 100644  --- a/untitled.tex  +++ b/untitled.tex 
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$\quad b_x \equiv $ fraction of bacteria that have invaded a host cell $ = \frac{B_x}{B_{\rm tot}}$  %$\tilde{r} $\tilde{r}  \equiv $ ratio of ruffles to host cells with ruffles $ = \frac{R}{H_r}$ %$\tilde{b} $\tilde{b}  \equiv $ ratio of attached invaded  bacteria to infected  host cellswith bacteria  $ = \frac{B_a}{H_a}$ \frac{B_x}{H_x}$  \subsection{Rates} 
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There is evidence that suggests a physical limit to the number of ruffles per cell, and the number of invaded bacteria per cell, so we define the following effective rates.  $\Gamma_r $\Gamma_r^{*}  \equiv $ effective (limited) ruffle formation rate per attached bacteria $ = \Gamma_r (1 - \frac{\tilde{r}}{\tilde{r}_{\mathrm{max}}})$ $\Gamma_x $\Gamma_x^{*}  \equiv $ effective (limited) invasion rate per attached bacteria $ = \Gamma_x (1 - \frac{\tilde{b}}{\tilde{b}_{\mathrm{max}}})$ $\Gamma_x \equiv $ invasion rate per attached bacteria