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

Transfer of polarity information via diffusion of Wnt ligands in C. elegans embryos

Different signaling mechanisms concur to ensure robust tissue patterning and cell fate instruction during animal development. Most of these mechanisms rely on signaling proteins that are produced, transported, and detected. The spatiotemporal dynamics of signaling molecules are largely unknown, yet they determine signal activity's spatial range and time frame. Here, we use the Caenorhabditis elegans embryo to study how Wnt ligands, an evolutionarily conserved family of signaling proteins, dynamically organize to establish cell polarity in a developing tissue. We identify how Wnt ligands, produced in the posterior half of the embryos, spread extracellularly to transmit information to distant target cells in the anterior half. With quantitative live imaging and fluorescence correlation spectroscopy, we show that Wnt ligands diffuse through the embryo over a timescale shorter than the cell cycle, in the intercellular space, and outside the tissue below the eggshell. We extracted diffusion coefficients of Wnt ligands and their receptor Frizzled and characterized their co-localization. Integrating our different measurements and observations in a simple computational framework, we show how fast diffusion in the embryo can polarize individual cells through a time integration of the arrival of the ligands at the target cells. The polarity established at the tissue level by a posterior Wnt source can be transferred to the cellular level. Our results support a diffusion-based long-range Wnt signaling, which is consistent with the dynamics of developing processes.

Close encounters of the sticky kind: Brownian motion at absorbing boundaries

Encounter-based models of diffusion provide a probabilistic framework for analyzing the effects of a partially absorbing reactive surface, in which the probability of absorption depends upon the amount of surface-particle contact time. In this paper we develop a class of encounter-based models that deal with absorption at sticky boundaries. Sticky boundaries occur in a diverse range of applications, including cell biology, colloidal physics, finance, and human crowd dynamics. They also naturally arise in active matter, where confined active particles tend to spontaneously accumulate at boundaries even in the absence of any particle-particle interactions. We begin by constructing a one-dimensional encounter-based model of sticky Brownian motion (BM), which is based on the zero-range limit of nonsticky BM with a short-range attractive potential well near the origin. In this limit, the boundary-contact time is given by the amount of time (occupation time) that the particle spends at the origin. We calculate the joint probability density or propagator for the particle position and the occupation time, and then identify an absorption event as the first time that the occupation time crosses a randomly generated threshold. We illustrate the theory by considering diffusion in a finite interval with a partially absorbing sticky boundary at one end. We show how various quantities, such as the mean first passage time (MFPT) for single-particle absorption and the relaxation to steady state at the multiparticle level, depend on moments of the random threshold distribution. Finally, we determine how sticky BM can be obtained by taking a particular diffusion limit of a sticky run-and-tumble particle (RTP).

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.

Morphogen-directed cell fate boundaries: slow passage through bifurcation and the role of folded saddles

One of the fundamental mechanisms in embryogenesis is the process by which cells differentiate and create tissues and structures important for functioning as a multicellular organism. Morphogenesis involves diffusive process of chemical signalling involving morphogens that pre-pattern the tissue. These morphogens influence cell fate through a highly nonlinear process of transcriptional signalling. In this paper, we consider this multiscale process in an idealised model for a growing domain. We focus on intracellular processes that lead to robust differentiation into two cell lineages through interaction of a single morphogen species with a cell fate variable that undergoes a bifurcation from monostability to bistability. In particular, we investigate conditions that result in successful and robust pattern formation into two well-separated domains, as well as conditions where this fails and produces a pinned boundary wave where only one part of the domain grows. We show that successful and unsuccessful patterning scenarios can be characterised in terms of presence or absence of a folded saddle singularity for a system with two slow variables and one fast variable; this models the interaction of slow morphogen diffusion, slow parameter drift through bifurcation and fast transcription dynamics. We illustrate how this approach can successfully model acquisition of three cell fates to produce three-domain "French flag" patterning, as well as for a more realistic model of the cell fate dynamics in terms of two mutually inhibiting transcription factors.

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.

A Reduced 1D Stochastic Model of Bleb-driven Cell Migration

Blebs are pressure-driven protrusions that have been observed in cells undergoing apoptosis, cytokinesis, or migration, including tumour cells that use blebs to escape their organs of origin. Here, we present a minimal 1D model of bleb-driven cell motion that combines a simple mechanical model with turnover kinetics of the actin cortex and adhesions between the membrane and the cortex. The deterministic version of this model is used to study the properties of individual blebbing events. We further introduce stochastic turnover of the adhesions, which allows for spontaneous initiation of repeated blebbing events, thus leading to sustained cell travel. We explore how the main parameters of the system control the properties of the blebbing events and the speed of cell travel. Finally, we derive a further simplification by deriving a Langevin approximation to this stochastic model.

Regulatory mechanisms of cytoneme-based morphogen transport

During development and tissue homeostasis, cells must communicate with their neighbors to ensure coordinated responses to instructional cues. Cues such as morphogens and growth factors signal at both short and long ranges in temporal- and tissue-specific manners to guide cell fate determination, provide positional information, and to activate growth and survival responses. The precise mechanisms by which such signals traverse the extracellular environment to ensure reliable delivery to their intended cellular targets are not yet clear. One model for how this occurs suggests that specialized filopodia called cytonemes extend between signal-producing and -receiving cells to function as membrane-bound highways along which information flows. A growing body of evidence supports a crucial role for cytonemes in cell-to-cell communication. Despite this, the molecular mechanisms by which cytonemes are initiated, how they grow, and how they deliver specific signals are only starting to be revealed. Herein, we discuss recent advances toward improved understanding of cytoneme biology. We discuss similarities and differences between cytonemes and other types of cellular extensions, summarize what is known about how they originate, and discuss molecular mechanisms by which their activity may be controlled in development and tissue homeostasis. We conclude by highlighting important open questions regarding cytoneme biology, and comment on how a clear understanding of their function may provide opportunities for treating or preventing disease.

Seven challenges in the multiscale modeling of multicellular tissues

The growth and dynamics of multicellular tissues involve tightly regulated and coordinated morphogenetic cell behaviors, such as shape changes, movement, and division, which are governed by subcellular machinery and involve coupling through short- and long-range signals. A key challenge in the fields of developmental biology, tissue engineering and regenerative medicine is to understand how relationships between scales produce emergent tissue-scale behaviors. Recent advances in molecular biology, live-imaging and ex vivo techniques have revolutionized our ability to study these processes experimentally. To fully leverage these techniques and obtain a more comprehensive understanding of the causal relationships underlying tissue dynamics, computational modeling approaches are increasingly spanning multiple spatial and temporal scales, and are coupling cell shape, growth, mechanics, and signaling. Yet such models remain challenging: modeling at each scale requires different areas of technical skills, while integration across scales necessitates the solution to novel mathematical and computational problems. This review aims to summarize recent progress in multiscale modeling of multicellular tissues and to highlight ongoing challenges associated with the construction, implementation, interrogation, and validation of such models. This article is categorized under: Reproductive System Diseases > Computational Models Metabolic Diseases > Computational Models Cancer > Computational Models.

Ultrastructural changes associated to the neuroendocrine transdifferentiation of the lung adenocarcinoma cell line A549

The neuroendocrine transdifferentiation has been found in many cancer cell types, such as prostate, lung and gastrointestinal cells and is accompanied by a lower patient life expectancy. The transdifferentiation process has been induced in vitro by the exposure to different stimuli in human lung adenocarcinoma. The aim of this work was to identify the morphological characteristics of the neuroendocrine phenotype in a human lung cancer cell line, induced by two cAMP elevating agents (IBMX and FSK). Our results showed two phenotypes, one produced by IBMX with higher volume, cell size and increased number of secondary projections, and the other produced by FSK with higher area, roughness of the membrane, cell neurite percentage, number of outgrowths per cell and increased number of primary projections. In conclusion, we describe some morphological and ultrastructural characteristics of the neuroendocrine phenotype in A549 human lung cancer cell line promoted by IBMX and FSK to contribute to the understanding of the autocrine or paracrine signaling within the tumor microenvironment.

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.

First-passage processes and the target-based accumulation of resources

A random search for one or more targets in a bounded domain occurs widely in nature, with examples ranging from animal foraging to the transport of vesicles within cells. Most theoretical studies take a searcher-centric viewpoint, focusing on the first passage time (FTP) problem to find a target. This single search-and-capture event then triggers a downstream process or provides the searcher with some resource such as food. In this paper we take a target-centric viewpoint by considering the accumulation of resources in one or more targets due to multiple rounds of search-and-capture events combined with resource degradation; whenever a searcher finds a target it delivers a resource packet to the target, after which it escapes and returns to its initial position. The searcher is then resupplied with cargo and a new search process is initiated after a random delay. It has previously been shown how queuing theory can be used to derive general expressions for the steady-state mean and variance of the resulting resource distributions. Here we apply the theory to some classical FPT problems involving diffusion in simple geometries with absorbing boundaries, including concentric spheres, wedge domains, and branching networks. In each case, we determine how the resulting Fano factor depends on the degradation rate, the delay distribution, and various geometric parameters. We thus establish that the Fano factor can deviate significantly from Poisson statistics and exhibits a nontrivial dependence on model parameters, including nonmonotonicity and crossover behavior. This indicates the nontrivial nature of the higher-order statistics of resource accumulation.

Asymptotic analysis of extended two-dimensional narrow capture problems

In this paper, we extend our recent work on two-dimensional diffusive search-and-capture processes with multiple small targets (narrow capture problems) by considering an asymptotic expansion of the Laplace transformed probability flux into each target. The latter determines the distribution of arrival or capture times into an individual target, conditioned on the set of events that result in capture by that target. A characteristic feature of strongly localized perturbations in two dimensions is that matched asymptotics generates a series expansion in ν  = −1/ln ϵ rather than ϵ , 0 <  ϵ  ≪ 1, where ϵ specifies the size of each target relative to the size of the search domain. Moreover, it is possible to sum over all logarithmic terms non-perturbatively. We exploit this fact to show how a Taylor expansion in the Laplace variable s for fixed ν provides an efficient method for obtaining corresponding asymptotic expansions of the splitting probabilities and moments of the conditional first-passage-time densities. We then use our asymptotic analysis to derive new results for two major extensions of the classical narrow capture problem: optimal search strategies under stochastic resetting and the accumulation of target resources under multiple rounds of search-and-capture.

Occupation time of a run-and-tumble particle with resetting

We study the positive occupation time of a run-and-tumble particle (RTP) subject to stochastic resetting. Under the resetting protocol, the position of the particle is reset to the origin at a random sequence of times generated by a Poisson process with rate r. The velocity state is reset to ±v with fixed probabilities ρ_{1} and ρ_{-1}=1-ρ_{1}, where v is the speed. We exploit the fact that the moment-generating functions with and without resetting are related by a renewal equation, and the latter generating function can be calculated by solving a corresponding Feynman-Kac equation. This allows us to numerically locate in Laplace space the largest real pole of the moment-generating function with resetting, and thus derive a large deviation principle (LDP) for the occupation time probability density using the Gartner-Ellis theorem. We explore how the LDP depends on the switching rate α of the velocity state, the resetting rate r, and the probability ρ_{1}. First, we show that the corresponding LDP for a Brownian particle with resetting is recovered in the fast switching limit α→∞. We then consider the case of a finite switching rate. In particular, we investigate how a directional bias in the resetting protocol (ρ_{1}≠0.5) skews the LDP rate function so that its minimum is shifted away from the expected fractional occupation time of one-half. The degree of shift increases with r and decreases with α.

Target competition for resources under multiple search-and-capture events with stochastic resetting

We develop a general framework for analysing the distribution of resources in a population of targets under multiple independent search-and-capture events. Each event involves a single particle executing a stochastic search that resets to a fixed location x r at a random sequence of times. Whenever the particle is captured by a target, it delivers a packet of resources and then returns to x r , where it is reloaded with cargo and a new round of search and capture begins. Using renewal theory, we determine the mean number of resources in each target as a function of the splitting probabilities and unconditional mean first passage times of the corresponding search process without resetting. We then use asymptotic PDE methods to determine the effects of resetting on the distribution of resources generated by diffusive search in a bounded two-dimensional domain with N small interior targets. We show that slow resetting increases the total number of resources M tot across all targets provided that ∑ j = 1 N G ( x r , x j ) < 0 , where G is the Neumann Green's function and x j is the location of the j-th target. This implies that M tot can be optimized by varying r. We also show that the k-th target has a competitive advantage if ∑ j = 1 N G ( x r , x j ) > N G ( x r , x k ) .

Search processes with stochastic resetting and multiple targets

Search processes with stochastic resetting provide a general theoretical framework for understanding a wide range of naturally occurring phenomena. Most current models focus on the first-passage-time problem of finding a single target in a given search domain. Here we use a renewal method to derive general expressions for the splitting probabilities and conditional mean first passage times (MFPTs) in the case of multiple targets. Our analysis also incorporates the effects of delays arising from finite return times and refractory periods. Carrying out a small-r expansion, where r is the mean resetting rate, we obtain general conditions for when resetting increases the splitting probability or reduces the conditional MFPT to a particular target. This also depends on whether π_{tot}=1 or π_{tot}<1, where π_{tot} is the probability that the particle is eventually absorbed by one of the targets in the absence of resetting. We illustrate the theory by considering two distinct examples. The first consists of an actin-rich cell filament (cytoneme) searching along a one-dimensional array of target cells, a problem for which the splitting probabilities and MFPTs can be calculated explicitly. In particular, we highlight how the resetting rate plays an important role in shaping the distribution of splitting probabilities along the array. The second example involves a search process in a three-dimensional bounded domain containing a set of N small interior targets. We use matched asymptotics and Green's functions to determine the behavior of the splitting probabilities and MFPTs in the small-r regime. In particular, we show that the splitting probabilities and MFPTs depend on the "shape capacitance" of the targets.

Queueing theory of search processes with stochastic resetting

We use queueing theory to develop a general framework for analyzing search processes with stochastic resetting, under the additional assumption that following absorption by a target, the particle (searcher) delivers a packet of resources to the target and the search process restarts at the reset point x_{r}. This leads to a sequence of search-and-capture events, whereby resources accumulate in the target under the combined effects of resource supply and degradation. Combining the theory of G/M/∞ queues with a renewal method for analyzing resetting processes, we derive general expressions for the mean and variance of the number of resource packets within the target at steady state. These expressions apply to both exponential and nonexponential resetting protocols and take into account delays arising from various factors such as finite return times, refractory periods, and delays due to the loading or unloading of resources. In the case of exponential resetting, we show how the resource statistics can be expressed in terms of the MFPTs T_{r}(x_{r}) and T_{r+γ}(x_{r}), where r is the resetting rate and γ is the degradation rate. This allows us to derive various general results concerning the dependence of the mean and variance on the parameters r,γ. Our results are illustrated using several specific examples. Finally, we show how fluctuations can be reduced either by allowing the delivery of multiple packets that degrade independently or by having multiple independent searchers.

Modeling of Wnt-mediated tissue patterning in vertebrate embryogenesis

During embryogenesis, morphogens form a concentration gradient in responsive tissue, which is then translated into a spatial cellular pattern. The mechanisms by which morphogens spread through a tissue to establish such a morphogenetic field remain elusive. Here, we investigate by mutually complementary simulations and in vivo experiments how Wnt morphogen transport by cytonemes differs from typically assumed diffusion-based transport for patterning of highly dynamic tissue such as the neural plate in zebrafish. Stochasticity strongly influences fate acquisition at the single cell level and results in fluctuating boundaries between pattern regions. Stable patterning can be achieved by sorting through concentration dependent cell migration and apoptosis, independent of the morphogen transport mechanism. We show that Wnt transport by cytonemes achieves distinct Wnt thresholds for the brain primordia earlier compared with diffusion-based transport. We conclude that a cytoneme-mediated morphogen transport together with directed cell sorting is a potentially favored mechanism to establish morphogen gradients in rapidly expanding developmental systems.

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.