Bidirectional transport model of morphogen gradient formation via cytonemes

Optimality of intercellular signaling: Direct transport versus diffusion

Intercellular signaling has an important role in organism development, but not all communication occurs using the same mechanism. Here, we analyze the energy efficiency of intercellular signaling by two canonical mechanisms: Diffusion of signaling molecules and direct transport mediated by signaling cellular protrusions. We show that efficient contact formation for direct transport can be established by an optimal rate of projecting protrusions, which depends on the availability of information about the location of the target cell. The optimal projection rate also depends on how signaling molecules are transported along the protrusion, in particular the ratio of the energy cost for contact formation and molecule synthesis. Also, we compare the efficiency of the two signaling mechanisms, under various model parameters. We find that direct transport is favored over diffusion when transporting a large amount of signaling molecules. There is a critical number of signaling molecules at which the efficiencies of the two mechanisms are the same. The critical number is small when the distance between cells is far, which helps explain why protrusion-based mechanisms are observed in long-range cellular communications.

Zebrafish airinemes optimize their shape between ballistic and diffusive search

In addition to diffusive signals, cells in tissue also communicate via long, thin cellular protrusions, such as airinemes in zebrafish. Before establishing communication, cellular protrusions must find their target cell. Here we demonstrate that the shapes of airinemes in zebrafish are consistent with a finite persistent random walk model. The probability of contacting the target cell is maximized for a balance between ballistic search (straight) and diffusive search (highly curved, random). We find that the curvature of airinemes in zebrafish, extracted from live cell microscopy, is approximately the same value as the optimum in the simple persistent random walk model. We also explore the ability of the target cell to infer direction of the airineme's source, finding that there is a theoretical trade-off between search optimality and directional information. This provides a framework to characterize the shape, and performance objectives, of non-canonical cellular protrusions in general.

Insights from chemical systems into Turing-type morphogenesis

In 1952, Alan Turing proposed a theory showing how morphogenesis could occur from a simple two morphogen reaction-diffusion system [Turing, A. M. (1952) Phil. Trans. R. Soc. Lond. A 237, 37-72. (doi:10.1098/rstb.1952.0012)]. While the model is simple, it has found diverse applications in fields such as biology, ecology, behavioural science, mathematics and chemistry. Chemistry in particular has made significant contributions to the study of Turing-type morphogenesis, providing multiple reproducible experimental methods to both predict and study new behaviours and dynamics generated in reaction-diffusion systems. In this review, we highlight the historical role chemistry has played in the study of the Turing mechanism, summarize the numerous insights chemical systems have yielded into both the dynamics and the morphological behaviour of Turing patterns, and suggest future directions for chemical studies into Turing-type morphogenesis. This article is part of the theme issue 'Recent progress and open frontiers in Turing's theory of morphogenesis'.

Improving the understanding of cytoneme-mediated morphogen gradients by in silico modeling

Morphogen gradients are crucial for the development of organisms. The biochemical properties of many morphogens prevent their extracellular free diffusion, indicating the need of an active mechanism for transport. The involvement of filopodial structures (cytonemes) has been proposed for morphogen signaling. Here, we describe an in silico model based on the main general features of cytoneme-meditated gradient formation and its implementation into Cytomorph, an open software tool. We have tested the spatial and temporal adaptability of our model quantifying Hedgehog (Hh) gradient formation in two Drosophila tissues. Cytomorph is able to reproduce the gradient and explain the different scaling between the two epithelia. After experimental validation, we studied the predicted impact of a range of features such as length, size, density, dynamics and contact behavior of cytonemes on Hh morphogen distribution. Our results illustrate Cytomorph as an adaptive tool to test different morphogen gradients and to generate hypotheses that are difficult to study experimentally.

Graded distribution of signaling molecules (morphogens) is crucial for the development of organisms. Signaling membrane protrusions, called Cytonemes, have been experimentally demonstrated to be involved in morphogen transport and reception. Here, we have developed an in silico model for gradient formation based on key features of cytoneme mediated signaling. We have also implemented the model into an open software tool we named Cytomorph, and validated it by comparing its simulations with experimental data obtained from Hedgehog morphogen distribution. Finally, we have generated in silico predictions for the impact of different cytoneme features such as length, size, density, dynamics and contact behavior. Our results show that Cytomorph is an adaptive tool that can facilitate the study of other cytoneme-dependent morphogen gradients, besides being able to generate hypotheses about aspects that remain elusive to experimental approaches.

A Random Walk Approach to Transport in Tissues and Complex Media: From Microscale Descriptions to Macroscale Models

The biological processes necessary for the development and continued survival of any organism are often strongly influenced by the transport properties of various biologically active species. The transport phenomena involved vary over multiple temporal and spatial scales, from organism-level behaviors such as the search for food, to systemic processes such as the transport of oxygen from the lungs to distant organs, down to microscopic phenomena such as the stochastic movement of proteins in a cell. Each of these processes is influenced by many interrelated factors. Identifying which factors are the most important, and how they interact to determine the overall result is a problem of great importance and interest. Experimental observations are often fit to relatively simple models, but in reality the observations are the output of complicated functions of the physicochemical, topological, and geometrical properties of a given system. Herein we use multistate continuous-time random walks and generalized master equations to model transport processes involving spatial jumps, immobilization at defined sites, and stochastic internal state changes. The underlying spatial models, which are framed as graphs, may have different classes of nodes, and walkers may have internal states that are governed by a Markov process. A general form of the solutions, using Fourier-Laplace transforms and asymptotic analysis, is developed for several spatially infinite regular lattices in one and two spatial dimensions, and the theory is developed for the analysis of transport and internal state changes on general graphs. The goal in each case is to shed light on how experimentally observable macroscale transport coefficients can be explained in terms of microscale properties of the underlying processes. This work is motivated by problems arising in transport in biological tissues, but the results are applicable to a broad class of problems that arise in other applications.

Influence of survival, promotion, and growth on pattern formation in zebrafish skin

The coloring of zebrafish skin is often used as a model system to study biological pattern formation. However, the small number and lack of movement of chromatophores defies traditional Turing-type pattern generating mechanisms. Recent models invoke discrete short-range competition and long-range promotion between different pigment cells as an alternative to a reaction-diffusion scheme. In this work, we propose a lattice-based “Survival model,” which is inspired by recent experimental findings on the nature of long-range chromatophore interactions. The Survival model produces stationary patterns with diffuse stripes and undergoes a Turing instability. We also examine the effect that domain growth, ubiquitous in biological systems, has on the patterns in both the Survival model and an earlier “Promotion” model. In both cases, domain growth alone is capable of orienting Turing patterns above a threshold wavelength and can reorient the stripes in ablated cells, though the wavelength for which the patterns orient is much larger for the Survival model. While the Survival model is a simplified representation of the multifaceted interactions between pigment cells, it reveals complex organizational behavior and may help to guide future studies.

Generation of extracellular morphogen gradients: the case for diffusion

Cells within developing tissues rely on morphogens to assess positional information. Passive diffusion is the most parsimonious transport model for long-range morphogen gradient formation but does not, on its own, readily explain scaling, robustness and planar transport. Here, we argue that diffusion is sufficient to ensure robust morphogen gradient formation in a variety of tissues if the interactions between morphogens and their extracellular binders are considered. A current challenge is to assess how the affinity for extracellular binders, as well as other biophysical and cell biological parameters, determines gradient dynamics and shape in a diffusion-based transport system. Technological advances in genome editing, tissue engineering, live imaging and in vivo biophysics are now facilitating measurement of these parameters, paving the way for mathematical modelling and a quantitative understanding of morphogen gradient formation and modulation.

Modelling the motion of organelles in an elongated cell via the coordination of heterogeneous drift-diffusion and long-range transport

 Cellular distribution of organelles in living cells is achieved via a variety of transport mechanisms, including directed motion, mediated by molecular motors along microtubules (MTs), and diffusion which is predominantly heterogeneous in space. In this paper, we introduce a model for particle transport in elongated cells that couples poleward drift, long-range bidirectional transport and diffusion with spatial heterogeneity in a three-dimensional space. Using stochastic simulations and analysis of a related population model, we find parameter regions where the three-dimensional model can be reduced to a coupled one-dimensional model or even a one-dimensional scalar model. We explore the efficiency with which individual model components can overcome drift towards one of the cell poles to reach an approximately even distribution. In particular, we find that if lateral movement is well mixed, then increasing the binding ability of particles to MTs is an efficient way to overcome a poleward drift, whereas if lateral motion is not well mixed, then increasing the axial diffusivity away from MTs becomes an efficient way to overcome the poleward drift. Our three-dimensional model provides a new tool that will help to understand the mechanisms by which eukaryotic cells organize their organelles in an elongated cell, and in particular when the one-dimensional models are applicable.

Diffusion vs. direct transport in the precision of morphogen readout

Morphogen profiles allow cells to determine their position within a developing organism, but not all morphogen profiles form by the same mechanism. Here, we derive fundamental limits to the precision of morphogen concentration sensing for two canonical mechanisms: the diffusion of morphogen through extracellular space and the direct transport of morphogen from source cell to target cell, for example, via cytonemes. We find that direct transport establishes a morphogen profile without adding noise in the process. Despite this advantage, we find that for sufficiently large values of profile length, the diffusion mechanism is many times more precise due to a higher refresh rate of morphogen molecules. We predict a profile lengthscale below which direct transport is more precise, and above which diffusion is more precise. This prediction is supported by data from a wide variety of morphogens in developing Drosophila and zebrafish.

Impulsive signaling model of cytoneme-based morphogen gradient formation

Morphogen protein gradients play a vital role in regulating spatial pattern formation during development. The most commonly accepted mechanism of protein gradient formation involves the diffusion and degradation of morphogens from a localized source. However, there is growing experimental evidence for a direct cell-to-cell signaling mechanism via thin actin-rich cellular extensions known as cytonemes. Recent modeling studies of cytoneme-based morphogenesis in invertebrates ignore the discrete nature of vesicular transport along cytonemes, focusing on deterministic continuum models. In this paper, we develop an impulsive signaling model of morphogen gradient formation in invertebrates, which takes into account the discrete and stochastic nature of vesicular transport along cytonemes. We begin by solving a first passage time problem with sticky boundaries to determine the expected time to deliver a vesicle to a target cell, assuming that there is a 'nucleation' time for injecting the vesicle into the cytoneme. We then use queuing theory to analyze the impulsive model of morphogen gradient formation in the case of multiple cytonemes and multiple targets. In particular, we determine the steady-state mean and variance of the morphogen distribution across a one-dimensional array of target cells. The mean distribution recovers the spatially decaying morphogen gradient of previous deterministic models. However, the burst-like nature of morphogen transport can lead to Fano factors greater than unity across the array of cells, resulting in significant fluctuations at more distant target sites.

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.