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Abstract As a universal second messenger, calcium (Ca2+) transmits specific cellular signals via a spatiotemporal signature generated from its extracellular source and internal stores. Our knowledge of the mechanisms underlying the generation of a Ca2+ signature is hampered by limited tools for simultaneously monitoring dynamic Ca2+ levels in multiple subcellular compartments. To overcome the limitation and to further improve spatiotemporal resolutions, we have assembled a molecular toolset (CamelliA lines) in Arabidopsis (Arabidopsis thaliana) that enables simultaneous and high-resolution monitoring of Ca2+ dynamics in multiple subcellular compartments through imaging different single-colored genetically encoded calcium indicators. We uncovered several Ca2+ signatures in three types of Arabidopsis cells in response to internal and external cues, including rapid oscillations of cytosolic Ca2+ and apical plasma membrane Ca2+ influx in fast-growing Arabidopsis pollen tubes, the spatiotemporal relationship of Ca2+ dynamics in four subcellular compartments of root epidermal cells challenged with salt, and a shockwave-like Ca2+ wave propagating in laser-wounded leaf epidermis. These observations serve as a testimony to the wide applicability of the CamelliA lines for elucidating the subcellular sources contributing to the Ca2+ signatures in plants. 

Partial differential equations are common models in biology for predicting and explaining complex behaviors. Nevertheless, deriving the equations and estimating the corresponding parameters remains challenging from data. In particular, the fine description of the interactions between species requires care for taking into account various regimes such as saturation effects. We apply a method based on neural networks to discover the underlying PDE systems, which involve fractional terms and may also contain integration terms based on observed data. Our proposed framework, called Frac-PDE-Net, adapts the PDE-Net 2.0 by adding layers that are designed to learn fractional and integration terms. The key technical challenge of this task is the identifiability issue. More precisely, one needs to identify the main terms and combine similar terms among a huge number of candidates in fractional form generated by the neural network scheme due to the division operation. In order to overcome this barrier, we set up certain assumptions according to realistic biological behavior. Additionally, we use an L2-norm based term selection criterion and the sparse regression to obtain a parsimonious model. It turns out that the method of Frac-PDE-Net is capable of recovering the main terms with accurate coefficients, allowing for effective long term prediction. We demonstrate the interest of the method on a biological PDE model proposed to study the pollen tube growth problem. 

Polar cell growth is a process that couples the establishment of cell polarity with growth and is extremely important in the growth, development, and reproduction of eukaryotic organisms, such as pollen tube growth during plant fertilization and neuronal axon growth in animals. Pollen tube growth requires dynamic but polarized distribution and activation of a signaling protein named ROP1 to the plasma membrane via three processes: positive feedback and negative feedback regulation of ROP1 activation and its lateral diffusion along the plasma membrane. In this paper, we introduce a mechanistic integro-differential equation (IDE) along with constrained semiparametric regression to quantitatively describe the interplay among these three processes that lead to the polar distribution of active ROP1 at a steady state. Moreover, we introduce a population variability by a constrained nonlinear mixed model. Our analysis of ROP1 activity distributions from multiple pollen tubes revealed that the equilibrium between the positive and negative feedbacks for pollen tubes with similar shapes are remarkably stable, permitting us to infer an inherent quantitative relationship between the positive and negative feedback loops that defines the tip growth of pollen tubes and the polarity of tip growth.