Stochastic clonal expansion of "superstars" enhances the reserve capacity of enteric nervous system precursor cells.

We quantified cell population increase in the quail embryo enteric nervous system (ENS) from E2.5 (about 1500 cells) to E12 (about 8 million cells). We then probed ENS proliferative capacity by grafting to the chorio-allantoic membrane large (600 cells) and small (40 cells) populations of enteric neural crest (ENC) cells with aneural gut. This demonstrated that ENC cells show an extremely high capacity to regulate their proliferation while forming the ENS. Previous mathematical models and clonal label experiments revealed that a few dominant ENS "superstar" cell clones emerge but most clones are small. The model implied that "superstars" arise stochastically, but the same outcome could arise if "superstars" were pre-determined. We investigated these two modes mathematically and by grafting experiments with large and small numbers of ENCs, each including one EGFP-labelled ENC cell. The stochastic model predicts that the frequency of "superstar" detection increases as the ENC population decreases, the pre-determined model does not. Experimentally, as predicted by the stochastic model, the frequency of "superstar" detection increased with small ENC cell number. We conclude that ENS "superstar" clones achieve this status stochastically. Clonal dominance implies that clonal diversity is greatly reduced and in this case, somatic mutations may affect the phenotype. We suggest that somatic mutations coupled with loss of clonal diversity may contribute to variable penetrance and expressivity in individuals with genetically identical ENS pathologies.

A Multicellular Model of Intestinal Crypt Buckling and Fission.

Crypt fission is an in vivo tissue deformation process that is involved in both intestinal homeostasis and colorectal tumourigenesis. Despite its importance, the mechanics underlying crypt fission are currently poorly understood. Recent experimental development of organoids, organ-like buds cultured from crypt stem cells in vitro, has shown promise in shedding light on crypt fission. Drawing inspiration from observations of organoid growth and fission in vivo, we develop a computational model of a deformable epithelial tissue layer. Results from in silico experiments show the stiffness of cells and the proportions of cell subpopulations affect the nature of deformation in the epithelial layer. In particular, we find that increasing the proportion of stiffer cells in the layer increases the likelihood of crypt fission occurring. This is in agreement with and helps explain recent experimental work.

Differential Clonal Expansion in an Invading Cell Population: Clonal Advantage or Dumb Luck?

In neoplastic cell growth, clones and subclones are variable both in size and mutational spectrum. The largest of these clones are believed to represent those cells with mutations that make them the most "fit," in a Darwinian sense, for expansion in their microenvironment. Thus, the degree of quantitative clonal expansion is regarded as being determined by innate qualitative differences between the cells that originate each clone. Here, using a combination of mathematical modelling and clonal labelling experiments applied to the developmental model system of the forming enteric nervous system, we describe how cells which are qualitatively identical may consistently produce clones of dramatically different sizes: most clones are very small while a few clones we term "superstars" contribute most of the cells to the final population. The basis of this is minor stochastic variations ("luck") in the timing and direction of movement and proliferation of individual cells, which builds a local advantage for daughter cells that is cumulative. This has potentially important consequences. In cancers, especially before strongly selective cytotoxic therapy, the assumption that the largest clones must be the cells with deterministic proliferative ability may not always hold true. In development, the gradual loss of clonal diversity as "superstars" take over the population may erode the resilience of the system to somatic mutations, which may have occurred early in clonal growth.

Is cell migration or proliferation dominant in the formation of linear arrays of oligodendrocytes?

Oligodendrocytes are the myelin-producing cells of the central nervous system that are responsible for electrically insulating axons to speed the propagation of electrical impulses. A striking feature of oligodendrocyte development within white matter is that the cell bodies of many oligodendrocyte progenitor cells become organised into discrete linear arrays of three or more cells before they differentiate into myelin-producing oligodendrocytes. These linear arrays align parallel to the direction of the axons within white matter tracts and are believed to play an important role in the co-ordination of myelination. Guided by experimental data on the abundance and composition of linear arrays in the corpus callosum of the postnatal mouse brain, we construct discrete and continuous models of linear array generation to specifically investigate the relative influence of cell migration, proliferation, differentiation and death of oligodendroglia upon the genesis of linear arrays during early postnatal development. We demonstrate that only models that incorporate significant cell migration can replicate all of the experimental observations on number of arrays, number of cells in arrays and total cell count of oligodendroglia within a given area of the corpus callosum. These models are also necessary to accurately reflect experimental data on the abundance of linear arrays composed of oligodendrocytes that derive from progenitors of different clonal origins.

Incomplete penetrance: The role of stochasticity in developmental cell colonization.

Cell colonization during embryonic development involves cells migrating and proliferating over growing tissues. Unsuccessful colonization, resulting from genetic causes, can result in various birth defects. However not all individuals with the same mutation show the disease. This is termed incomplete penetrance, and it even extends to discordancy in monozygotic (identical) twins. A one-dimensional agent-based model of cell migration and proliferation within a growing tissue is presented, where the position of every cell is recorded at any time. We develop a new model that approximates this agent-based process - rather than requiring the precise configuration of cells within the tissue, the new model records the total number of cells, the position of the most advanced cell, and then invokes an approximation for how the cells are distributed. The probability mass function (PMF) for the most advanced cell is obtained for both the agent-based model and its approximation. The two PMFs compare extremely well, but using the approximation is computationally faster. Success or failure of colonization is probabilistic. For example for sufficiently high proliferation rate the colonization is assured. However, if the proliferation rate is sufficiently low, there will be a lower, say 50%, chance of success. These results provide insights into the puzzle of incomplete penetrance of a disease phenotype, especially in monozygotic twins. Indeed, stochastic cell behavior (amplified by disease-causing mutations) within the colonization process may play a key role in incomplete penetrance, rather than differences in genes, their expression or environmental conditions.

Proliferation and cell cycle dynamics in the developing stellate ganglion.

Cell proliferation during nervous system development is poorly understood outside the mouse neocortex. We measured cell cycle dynamics in the embryonic mouse sympathetic stellate ganglion, where neuroblasts continue to proliferate following neuronal differentiation. At embryonic day (E) 9.5, when neural crest-derived cells were migrating and coalescing into the ganglion primordium, all cells were cycling, cell cycle length was only 10.6 h, and S-phase comprised over 65% of the cell cycle; these values are similar to those previously reported for embryonic stem cells. At E10.5, Sox10(+) cells lengthened their cell cycle to 38 h and reduced the length of S-phase. As cells started to express the neuronal markers Tuj1 and tyrosine hydroxylase (TH) at E10.5, they exited the cell cycle. At E11.5, when >80% of cells in the ganglion were Tuj1(+)/TH(+) neuroblasts, all cells were again cycling. Neuroblast cell cycle length did not change significantly after E11.5, and 98% of Sox10(-)/TH(+) cells had exited the cell cycle by E18.5. The cell cycle length of Sox10(+)/TH(-) cells increased during late embryonic development, and ∼25% were still cycling at E18.5. Loss of Ret increased neuroblast cell cycle length at E16.5 and decreased the number of neuroblasts at E18.5. A mathematical model generated from our data successfully predicted the relative change in proportions of neuroblasts and non-neuroblasts in wild-type mice. Our results show that, like other neurons, sympathetic neuron differentiation is associated with exit from the cell cycle; sympathetic neurons are unusual in that they then re-enter the cell cycle before later permanently exiting.

Simple rules for a "simple" nervous system? Molecular and biomathematical approaches to enteric nervous system formation and malformation.

We review morphogenesis of the enteric nervous system from migratory neural crest cells, and defects of this process such as Hirschsprung disease, centering on cell motility and assembly, and cell adhesion and extracellular matrix molecules, along with cell proliferation and growth factors. We then review continuum and agent-based (cellular automata) models with rules of cell movement and logistical proliferation. Both movement and proliferation at the individual cell level are modeled with stochastic components from which stereotyped outcomes emerge at the population level. These models reproduced the wave-like colonization of the intestine by enteric neural crest cells, and several new properties emerged, such as colonization by frontal expansion, which were later confirmed biologically. These models predict a surprising level of clonal heterogeneity both in terms of number and distribution of daughter cells. Biologically, migrating cells form stable chains made up of unstable cells, but this is not seen in the initial model. We outline additional rules for cell differentiation into neurons, axon extension, cell-axon and cell-cell adhesions, chemotaxis and repulsion which can reproduce chain migration. After the migration stage, the cells re-arrange as a network of ganglia. Changes in cell adhesion molecules parallel this, and we describe additional rules based on Steinberg's Differential Adhesion Hypothesis, reflecting changing levels of adhesion in neural crest cells and neurons. This was able to reproduce enteric ganglionation in a model. Mouse mutants with disturbances of enteric nervous system morphogenesis are discussed, and these suggest future refinement of the models. The modeling suggests a relatively simple set of cell behavioral rules could account for complex patterns of morphogenesis. The model has allowed the proposal that Hirschsprung disease is mostly an enteric neural crest cell proliferation defect, not a defect of cell migration. In addition, the model suggests an explanations for zonal and skip segment variants of Hirschsprung disease, and also gives a novel stochastic explanation for the observed discordancy of Hirschsprung disease in identical twins.

Spatial and temporal dynamics of cell generations within an invasion wave: a link to cell lineage tracing.

Mathematical models of a cell invasion wave have included both continuum partial differential equation (PDE) approaches and discrete agent-based cellular automata (CA) approaches. Here we are interested in modelling the spatial and temporal dynamics of the number of divisions (generation number) that cells have undergone by any time point within an invasion wave. In the CA framework this is performed from agent lineage tracings, while in the PDE approach a multi-species generalized Fisher equation is derived for the cell density within each generation. Both paradigms exhibit qualitatively similar cell generation densities that are spatially organized, with agents of low generation number rapidly attaining a steady state (with average generation number increasing linearly with distance) behind the moving wave and with evolving high generation number at the wavefront. This regularity in the generation spatial distributions is in contrast to the highly stochastic nature of the underlying lineage dynamics of the population. In addition, we construct a method for determining the lineage tracings of all agents without labelling and tracking the agents, but through either a knowledge of the spatial distribution of the generations or the number of agents in each generation. This involves determining generation-dependent proliferation probabilities and using these to define a generation-dependent Galton-Watson (GDGW) process. Monte-Carlo simulations of the GDGW process are used to determine the individual lineage tracings. The lineages of the GDGW process are analyzed using Lorenz curves and found to be similar to outcomes generated by direct lineage tracing in CA realizations. This analysis provides the basis for a potentially useful technique for deducing cell lineage data when imaging every cell is not feasible.

Spatial analysis of multi-species exclusion processes: application to neural crest cell migration in the embryonic gut.

Hindbrain (vagal) neural crest cells become relatively uniformly distributed along the embryonic intestine during the rostral to caudal colonization wave which forms the enteric nervous system (ENS). When vagal neural crest cells are labeled before migration in avian embryos by in ovo electroporation, the distribution of labeled neural crest cells in the ENS varies vastly. In some cases, the labeled neural crest cells appear evenly distributed and interspersed with unlabeled neural crest cells along the entire intestine. However, in most specimens, labeled cells occur in relatively discrete patches of varying position, area, and cell number. To determine reasons for these differences, we use a discrete cellular automata (CA) model incorporating the underlying cellular processes of neural crest cell movement and proliferation on a growing domain, representing the elongation of the intestine during development. We use multi-species CA agents corresponding to labeled and unlabeled neural crest cells. The spatial distributions of the CA agents are quantified in terms of an index. This investigation suggests that (i) the percentage of the initial neural crest cell population that is labeled and (ii) the ratio of cell proliferation to motility are the two key parameters producing the extreme differences in spatial distributions observed in avian embryos.

Neural crest regionalisation for enteric nervous system formation: implications for Hirschsprung's disease and stem cell therapy.

Midbrain, hindbrain and vagal neural crest (NC) produced abundant enteric nervous system (ENS) in co-grafted aneural hindgut and midgut, using chick-quail chorio-allantoic membrane grafts, forming complete myenteric and submucosal plexuses. This ability dropped suddenly in cervical and thoracic NC levels, furnishing an incomplete ENS in one or both plexuses. Typically, one plexus was favoured over the other. This deficiency was not caused by lower initial trunk NC number, yet overloading the initial number decreased the deficiency. No qualitative difference in neuronal and glial differentiation between cranial and trunk levels was observed. All levels formed HuC/D+ve, NOS+ve, ChAT+ve, and TH-ve enteric neurons with SoxE+ve, GFAP+ve, and BFABP+ve glial cells. We mathematically modelled a proliferative difference between NC populations, with a plexus preference hierarchy, in the context of intestinal growth. High proliferation achieved an outcome similar to cranial NC, while low proliferation described the trunk NC outcome of incomplete primary plexus and even more deficient secondary plexus. We conclude that cranial NC, relative to trunk NC, has a positionally-determined proliferation advantage favouring ENS formation. This has important implications for proposed NC stem cell therapy for Hirschsprung's disease, since such cells may need to be optimised for positional identity.

Building stable chains with motile agents: Insights into the morphology of enteric neural crest cell migration.

A defining characteristic of the normal development of the enteric nervous system (ENS) is the existence of an enteric neural crest (ENC) cell colonization wave, where the ENC cells form stable chains often associated with axons and near the vascular network. However, within this evolving neural network, the individual ENC cell elements constantly move, change direction and appear to act independently of neighbors. Three possible hypotheses are investigated. The simplest of these postulates that the ENS follows the vascular network as a template. We present evidence which does not support this hypothesis. Two viable alternatives are either that (i) the axons muster the ENC cells, providing the pattern for the chain migration or (ii) ENC cells form chains and the axons follow these paths. These two hypotheses are explored by developing a stochastic cellular automata model, where ENC agents follow simple rules, which reflect the underlying biology of movement, proliferation and differentiation. By simulating ENC precursors and the associated neurons and axons, two models with different fundamental mechanisms are developed. From local rules, a mesoscale network pattern with lacunae emerges, which can be analyzed quantitatively. Simulation and analysis establishes the parameters that affect the morphology of the resulting network. This investigation into the axon/ENC and ENC/ENC interplay suggests possible explanations for observations in mouse and avian embryos in normal and abnormal ENS development, as well as further experimentation.

On the role of differential adhesion in gangliogenesis in the enteric nervous system.

A defining characteristic of the normal development of the enteric nervous system (ENS) is the existence of mesoscale patterned entities called ganglia. Ganglia are clusters of neurons with associated enteric neural crest (ENC) cells, which form in the simultaneously growing gut wall. At first the precursor ENC cells proliferate and gradually differentiate to produce the enteric neurons; these neurons form clusters with ENC scattered around and later lying on the periphery of neuronal clusters. By immunolabelling neural cell-cell adhesion molecules, we infer that the adhesive capacity of neurons is greater than that of ENC cells. Using a discrete mathematical model, we test the hypothesis that local rules governing differential adhesion of neuronal agents and ENC agents will produce clusters which emulate ganglia. The clusters are relatively stable, relatively uniform and small in size, of fairly uniform spacing, with a balance between the number of neuronal and ENC agents. These features are attained in both fixed and growing domains, reproducing respectively organotypic in vitro and in vivo observations. Various threshold criteria governing ENC agent proliferation and differentiation and neuronal agent inhibition of differentiation are important for sustaining these characteristics. This investigation suggests possible explanations for observations in normal and abnormal ENS development.

When are cellular oscillators sufficient for sequential segmentation?

Sequential segmentation during embryogenesis involves the generation of a repeated pattern along the embryo, which is concurrently undergoing axial elongation by cell division. Most mathematical models of sequential segmentation involve inherent cellular oscillators, acting as a segmentation clock. The cellular oscillation is assumed to be governed by the cell's physiological age or by its interaction with an external morphogen gradient. Here, we address the issue of when cellular oscillators alone are sufficient for predicting segmentation, and when a morphogen gradient is required. The key to resolving this issue lies in how cells determine positional information in the model--this is directly related to the distribution of cell divisions responsible for axial elongation. Mathematical models demonstrate that if axial elongation occurs through cell divisions restricted to the posterior end of the unsegmented region, a cell can obtain its positional information from its physiological age, and therefore cellular oscillators will suffice. Alternatively, if axial elongation occurs through cell divisions distributed throughout the unsegmented region, then positional information can be obtained through another mechanism, such as a morphogen gradient. Two alternative ways to establish a morphogen gradient in tissue with distributed cell divisions are presented--one with diffusion and the other without diffusion. Our model produces segment polarity and a distribution of segment size from the anterior-to-posterior ends, as observed in some systems. Furthermore, the model predicts segment deletions when there is an interruption in cell division, just as seen in heat shock experiments, as well as the growth and final shrinkage of the presomitic mesoderm during somitogenesis.

Growth patterns in Onychophora (velvet worms): lack of a localised posterior proliferation zone.

BACKGROUND: During embryonic development of segmented animals, body segments are thought to arise from the so-called "posterior growth zone" and the occurrence of this "zone" has been used to support the homology of segmentation between arthropods, annelids, and vertebrates. However, the term "posterior growth zone" is used ambiguously in the literature, mostly referring to a region of increased proliferation at the posterior end of the embryo. To determine whether such a localised posterior proliferation zone is an ancestral feature of Panarthropoda (Onychophora + Tardigrada + Arthropoda), we examined cell division patterns in embryos of Onychophora. RESULTS: Using in vivo incorporation of the DNA replication marker BrdU (5-bromo-2'-deoxyuridine) and anti-phospho-histone H3 immunolabelling, we found that a localised posterior region of proliferating cells does not occur at any developmental stage in onychophoran embryos. This contrasts with a localised pattern of cell divisions at the posterior end of annelid embryos, which we used as a positive control. Based on our data, we present a mathematical model, which challenges the paradigm that a localised posterior proliferation zone is necessary for segment patterning in short germ developing arthropods. CONCLUSIONS: Our findings suggest that a posterior proliferation zone was absent in the last common ancestor of Onychophora and Arthropoda. By comparing our data from Onychophora with those from annelids, arthropods, and chordates, we suggest that the occurrence of a "posterior growth zone" currently cannot be used to support the homology of segmentation between these three animal groups.

Exclusion processes on a growing domain.

A discrete model provides a useful framework for experimentalists to understand the interactions between growing tissues and other biological mechanisms. A cellular automata (CA) model with domain growth, cell motility and cell proliferation, based on cellular exclusion processes, is developed here. Average densities can be defined from the CA model and a continuum representation can be determined. The domain growth mechanism in the CA model gives rise to a Fokker-Planck equation in the corresponding continuum model, with a diffusive and a convective term. Deterministic continuum models derived from conservation laws do not include this diffusive term. The new diffusive term arises because of the stochasticity inherited from the CA mechanism for domain growth. We extend the models to multiple species and investigate the influence of the flux terms arising from the exclusion processes. The averaged CA agent densities are well approximated by the solution of nonlinear advection-diffusion equations, provided that the relative size of the proliferation processes to the diffusion processes is sufficiently small. This dual approach provides an understanding of the microscopic and macroscopic scales in a developmental process.

Cell lineage tree models of neurogenesis.

The production of neurons to form the mammalian cortex, known as embryonic cortical neurogenesis, is a complex developmental process. Insight into the process of cell division during neurogenesis is provided by murine cortical cell lineage trees, recorded through experimental observation. Recurring patterns within cell lineage trees may be indicative of predetermined cell behaviour. The application of mathematical modelling to this process requires careful consideration and identification of the key features to be incorporated into the model. A biologically plausible stochastic model of evolution of cell lineage trees is developed, based on the most important known features of neurogenesis. Tractable means of measuring lineage tree shape are discussed. Symmetry is identified as a significant feature of shape and is measured using Colless's Index of Imbalance. Distributions of tree size and imbalance for large tree sizes are computed and results compared to experimental data. Several refinements to the model are investigated, when the cell division probabilities are weighted according to cell generation. Two models involving generation-dependent cell division probabilities produce imbalance distributions which are the most consistent with the available experimental results. The results indicate that a stochastic cell division mechanism is a plausible basis of mammalian neurogenesis.

A traveling wave model for invasion by precursor and differentiated cells.

We develop and investigate a continuum model for invasion of a domain by cells that migrate, proliferate and differentiate. The model is applicable to neural crest cell invasion in the developing enteric nervous system, but is presented in general terms and is of broader applicability. Two cell populations are identified and modeled explicitly; a population of precursor cells that migrate and proliferate, and a population of differentiated cells derived from the precursors which have impaired migration and proliferation. The equation describing the precursor cells is based on Fisher's equation with the addition of a carrying-capacity limited differentiation term. Two variations of the proliferation term are considered and compared. For most parameter values, the model admits a traveling wave solution for each population, both traveling at the same speed. The traveling wave solutions are investigated using perturbation analysis, phase plane methods, and numerical techniques. Analytical and numerical results suggest the existence of two wavespeed selection regimes. Regions of the parameter space are characterized according to existence, shape, and speed of traveling wave solutions. Our observations may be used in conjunction with experimental results to identify key parameters determining the invasion speed for a particular biological system. Furthermore, our results may assist experimentalists in identifying the resource that is limiting proliferation of precursor cells.

Distinguishing between directed and undirected cell motility within an invading cell population.

Cell invasion is the basis of several fundamental biological systems including developmental morphogenesis and disease progression. Invasion processes involve combined cell motility and proliferation. Standard experimental approaches to characterize invasion systems focus on measuring population-level wavespeed data. However, continuum models which incorporate either directed or undirected motility both give rise to traveling wave solutions with a well-defined wavespeed in terms of the motility parameters. Therefore, such population-level models and experimental data cannot be used to determine whether the motility is directed or undirected. This is a major impediment limiting our ability to interpret experimental observations of cell invasion. We demonstrate how to overcome this difficulty using individual-level data and discrete models. This approach can be used to interpret and design time-lapse imaging data to determine whether the cell motility is directed or undirected. Making a distinction between directed and undirected motility has profound implications regarding our ability to design strategies to manage development and disease associated with cell invasion.

Pathlines in exclusion processes.

Trajectory data from observations of a random-walk process are often used to characterize macroscopic transport coefficients and to make inferences about motility mechanisms. Continuum equations describing the average moments of the position of an agent in an exclusion process are derived and validated with simulation data. Unlike standard noninteracting random walks, the moment equations for the exclusion process explicitly represent the interaction of agents since they depend on the averaged macroscopic agent density. Key issues associated with the validity of the continuum equations and interpretation of experimental data are discussed.

Addressing Interdisciplinary Difficulties in Developmental Biology/Mathematical Collaborations: A Neural Crest Example.

Mathematical modeling can allow insight into the biological processes that can be difficult to access by conventional biological means alone. Such projects are becoming increasingly attractive with the appearance of faster and more powerful quantitative techniques in both biological data acquisition and data storage, manipulation, and presentation. However, as is frequent in interdisciplinary research, the main hurdles are not within the mindset and techniques of each discipline but are usually encountered in attempting to meld the different disciplines together. Based upon our experience in applying mathematical methods to investigate how neural crest cells interact to form the enteric nervous system, we present our views on how to pursue biomathematical modeling projects, what difficulties to expect, and how to overcome, or at least survive, these hurdles. The main advice being: persevere.