deletions | additions       diff --git a/section_Proofs_subsection_Bacterial_density__.tex b/section_Proofs_subsection_Bacterial_density__.tex  index 42d63d3..609be52 100644  --- a/section_Proofs_subsection_Bacterial_density__.tex  +++ b/section_Proofs_subsection_Bacterial_density__.tex 
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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} = \Gamma_0 f \left(\frac{bmc}{a}\right)