deletions | additions       diff --git a/section_Proofs_subsection_Bacterial_density__.tex b/section_Proofs_subsection_Bacterial_density__.tex  index 086e756..8bb4b83 100644  --- a/section_Proofs_subsection_Bacterial_density__.tex  +++ b/section_Proofs_subsection_Bacterial_density__.tex 
...
\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*}