Trichome patterning in Arabidopsis serves as a model system for de novo pattern formation in plants.It is thought to typify the theoretical activator-inhibitor mechanism, although this hypothesis has never been challenged by a combined experimental and theoretical approach. By integrating the key genetic and molecular data of the trichome patterning system, we developed a new theoretical model that allows the direct testing of the effect of experimental interventions and in the prediction of patterning phenotypes. We show experimentally that the trichome inhibitor TRIPTYCHON is transcriptionally activated by the known positive regulators GLABRA1 and GLABRA3. Further, we demonstrate by particle bombardment of protein fusions with GFP that TRIPTYCHON and CAPRICE but not GLABRA1 and GLABRA3 can move between cells. Finally, theoretical considerations suggest promoter swapping and basal overexpression experiments by means of which we are able to discriminate three biologically meaningful variants of the trichome patterning model. Our study demonstrates that the mutual interplay between theory and experiment can reveal a new level of understanding of how biochemical mechanisms can drive biological patterning processes.

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Although plant development is highly reproducible, some stochasticity exists. This developmental stochasticity may be caused by noisy gene expression. Here we analyze the fluctuation of protein expression in Arabidopsis thaliana. Using the photoconvertible KikGR marker, we show that the protein expressions of individual cells fluctuate over time. A dual reporter system was used to study extrinsic and intrinsic noise of marker gene expression. We report that extrinsic noise is higher than intrinsic noise and that extrinsic noise in stomata is clearly lower in comparison to several other tissues/cell types. Finally, we show that cells are coupled with respect to stochastic protein expression in young leaves, hypocotyls and roots but not in mature leaves. Our data indicate that stochasticity of gene expression can vary between tissues/cell types and that it can be coupled in a non-cell-autonomous manner.

Birth-jump models are designed to describe population models for which growth and spatial spread cannot be decoupled. A birth-jump model is a nonlinear integro-differential equation. We present two different derivations of this equation, one based on a random walk approach and the other based on a two-compartmental reaction-diffusion model. In the case that the redistribution kernels are highly concentrated, we show that the integro-differential equation can be approximated by a reaction-diffusion equation, in which the proliferation rate contributes to both the diffusion term and the reaction term. We completely solve the corresponding critical domain size problem and the minimal wave speed problem. Birth-jump models can be applied in many areas in mathematical biology. We highlight an application of our results in the context of forest fire spread through spotting. We show that spotting increases the invasion speed of a forest fire front.

While pattern formation is studied in various areas of biology, little is known about the noise leading to variations between individual realizations of the pattern. One prominent example for de novo pattern formation in plants is the patterning of trichomes on Arabidopsis leaves, which involves genetic regulation and cell-to-cell communication. These processes are potentially variable due to, e.g., the abundance of cell components or environmental conditions. To elevate the understanding of regulatory processes underlying the pattern formation it is crucial to quantitatively analyze the variability in naturally occurring patterns. Here, we review recent approaches toward characterization of noise on trichome initiation. We present methods for the quantification of spatial patterns, which are the basis for data-driven mathematical modeling and enable the analysis of noise from different sources. Besides the insight gained on trichome formation, the examination of observed trichome patterns also shows that highly regulated biological processes can be substantially affected by variability.