Constrained Nonlinear and Mixed Effects Differential Equation Models for   Dynamic Cell Polarity Signaling
  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. 
Zhen Xiao, Nicolas Brunel, Zhenbiao Yang, Xinping Cui
stat.ME