  The key of tip growth in eukaryotes is the polarized distribution on plasma
membrane of a particle named ROP1. This distribution is the result of a
positive feedback loop, whose mechanism can be described by a Differential
Equation parametrized by two meaningful parameters kpf and knf . We introduce a
mechanistic Integro-Differential Equation (IDE) derived from a spatiotemporal
model of cell polarity and we show how this model can be fitted to real data,
i.e., ROP1 intensities measured on pollen tubes. At first, we provide an
existence and uniqueness result for the solution of our IDE model under certain
conditions. Interestingly, this analysis gives a tractable expression for the
likelihood, and our approach can be seen as the estimation of a constrained
nonlinear model. Moreover, we introduce a population variability by a
constrained nonlinear mixed model. We then propose a constrained Least Squares
method to fit the model for the single pollen tube case, and two methods,
constrained Methods of Moments and constrained Restricted Maximum Likelihood
(REML) to fit the model for the multiple pollen tubes case. The performances of
all three methods are studied through simulations and are used on an in-house
multiple pollen tubes dataset generated at UC Riverside.
