Search-and-capture model of cytoneme-mediated morphogen gradient formation

Morphogen protein gradients play an essential role in the spatial regulation of patterning during embryonic development. The most commonly accepted mechanism of protein gradient formation involves the diffusion and degradation of morphogens from a localized source. Recently, an alternative mechanism has been proposed, which is based on cell-to-cell transport via thin actin-rich cellular extensions known as cytonemes. Very little is currently known about the precise nature of the contacts between cytonemes and their target cells. Important unresolved issues include how cytoneme tips find their targets, how they are stabilized at their contact sites, and how vesicles are transferred to a receiving cell and subsequently internalized. It has been hypothesized that cytonemes find their targets via a random search process based on alternating periods of retraction and growth, perhaps mediated by some chemoattractant. This is an actin-based analog of the search-and-capture model of microtubules of the mitotic spindle searching for cytochrome binding sites (kinetochores) prior to separation of cytochrome pairs. In this paper we develop a search-and-capture model of cytoneme-based morphogenesis, in which nucleating cytonemes from a source cell dynamically grow and shrink along the surface of a one-dimensional array of target cells until making contact with one of the target cells. We analyze the first-passage-time problem for making contact and then use this to explore the formation of morphogen gradients under the mechanism proposed for Wnt in vertebrates. That is, we assume that morphogen is localized at the tip of a growing cytoneme, which is delivered as a "morphogen burst" to a target cell when the cytoneme makes temporary contact with a target cell before subsequently retracting. We show how multiple rounds of search-and-capture, morphogen delivery, cytoneme retraction, and nucleation events lead to the formation of a morphogen gradient. We proceed by formulating the morphogen bursting model as a queuing process, analogous to the study of translational bursting in gene networks. In order to analyze the expected times for cytoneme contact, we introduce an efficient method for solving first-passage-time problems in the presence of sticky boundaries, which exploits some classical concepts from probability theory, namely, stopping times and the strong Markov property. We end the paper by demonstrating how this method simplifies previous analyses of a well-studied problem in cell biology, namely, the search-and-capture model of microtubule-kinetochore attachment. Although the latter is completely unrelated to cytoneme-based morphogenesis from a biological perspective, it shares many of the same mathematical elements.