deletions | additions       diff --git a/section_Proofs_subsection_Bacterial_density__.tex b/section_Proofs_subsection_Bacterial_density__.tex  index 609be52..dc51545 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)  \end{equation}  \begin{equation}  \dot{N_R} = N \Gamma_0 \rho_B f = N f \Gamma_0 \left(\frac{bmc}{a}\right)  \end{equation]