Fitting multiple nls functions with dplyr

I wish to fit multiple nls functions using a group_by from the dplyr package but I am unsure how I can pass multiple starting values. Let's take a simpler example (see ?nls for the inspiration).

DNase1 <- subset(DNase, Run == 1)
modelDNase1 <- DNase1 %>% 
  do(model = nls(density ~ 1/(1 + exp((xmid - log(conc))/scal)),
             data = .,
             start = list(xmid = 0, scal = 1),
             algorithm = "plinear"))

So here I am fitting a single model. But what if I want to extend this so I am fitting the following:

DNase$Run <- factor(DNase$Run)
modelDNase <- DNase %>%
  group_by(Run) %>% 
  do(model = nls(density ~ 1/(1 + exp((xmid - log(conc))/scal)),
             data = .,
             start = list(xmid = 0, scal = 1),
             algorithm = "plinear"))

How would I pass on multiple start parameters? Would the purrr package be of any use?