A spatially-averaged mathematical model of kidney branching morphogenesis

The developing murine kidney actively negotiates geometric packing conflicts to avoid defects

The physiological functions of several organs rely on branched epithelial tubule networks bearing specialized structures for secretion, gas exchange, or filtration. Little is known about conflicts in development between building enough tubules for adequate function and geometric constraints imposed by organ size. We show that the mouse embryonic kidney epithelium negotiates a physical packing conflict between increasing tubule tip numbers through branching and limited organ surface area. Through imaging of whole kidney explants, combined with computational and soft material modeling of tubule families, we identify six possible geometric packing phases, including two defective ones. Experiments in explants show that a radially oriented tension on tubule families is necessary and sufficient for them to switch to a vertical packing arrangement that increases surface tip density while avoiding defects. These results reveal developmental contingencies in response to physical limitations and create a framework for classifying congenital kidney defects.

Growth control of the kidney

The functional mass of kidney tissue in an adult is an important determinant of human health. Kidney formation during development is an essential determinant of the final nephron endowment of the adult organ, and no evidence has been reported that mice or humans are able to generate new nephrons after the developmental period. Mechanisms controlling organ growth after development are essential to establish the final adult organ size. The potential for organ growth is maintained in adult life and the size of one kidney may be significantly increased by loss of the contralateral kidney. The mouse has provided a model system for investigators to critically explore genetic, cell biological, and hormonal control of developmental and juvenile kidney growth. This article reviews three basic aspects of kidney size regulation: (1) Mechanisms that control nephron formation and how these are altered by the cessation of nephrogenesis at the end of the developmental period. (2) Applicability of the general model for growth hormone-insulin like growth factor control to kidney growth both pre- and postnatally. (3) Commonalities between mechanisms of juvenile kidney growth and the compensatory growth that is stimulated in adult life by reduction of kidney mass. Understanding the mechanisms that determine set-points for cell numbers and size in the kidney may inform ongoing efforts to generate kidney tissue from stem cells.

Modelling Cellular Interactions and Dynamics During Kidney Morphogenesis

Kidney disease and renal disorders account for a significant proportion of health complications in mid-late adulthood worldwide. Many renal deficiencies are due to improper formation of the kidneys before birth, which are caused by disorders in the developmental process that arise from genetic and/or environmental factors. Mathematical modelling can help build on experimental knowledge to increase our understanding of the complexities of kidney organogenesis. In this paper, we present a discrete cell-based model of kidney development. Specifically, we model the tip of the developing ureteric tree to investigate the behaviours of cap mesenchyme cells which are required to sustain ureteric tip growth. We find that spatial regulation of the differentiation of cap mesenchyme cells through cellular signalling is sufficient to ensure robust ureteric tip development. Additionally, we find that increased adhesion interactions between cap mesenchyme cells and the ureteric tip surface can lead to a more stable tip-cap unit. Our analysis of the various processes on this scale highlights essential components for healthy kidney growth and provides insight into mechanisms to be studied further in order to replicate the process in vitro.

Clearly imaging and quantifying the kidney in 3D

For decades, measurements of kidney microanatomy using 2-dimensional sections has provided us with a detailed knowledge of kidney morphology under physiological and pathological conditions. However, the rapid development of tissue clearing methods in recent years, in combination with the development of novel 3-dimensional imaging modalities have provided new insights into kidney structure and function. This review article describes a range of novel insights into kidney development and disease obtained recently using these new methodological approaches. For example, in the developing kidney these approaches have provided new understandings of ureteric branching morphogenesis, nephron progenitor cell proliferation and commitment, interactions between ureteric tip cells and nephron progenitor cells, and the establishment of nephron segmentation. In whole adult mouse kidneys, tissue clearing combined with light sheet microscopy can image and quantify the total number of glomeruli, a major breakthrough in the field. Similar approaches have provided new insights into the structure of the renal vasculature and innervation, tubulointerstitial remodeling, podocyte loss and hypertrophy, cyst formation, the evolution of cellular crescents, and the structure of the glomerular filtration barrier. Many more advances in the understanding of kidney biology and pathology can be expected as additional clearing and imaging techniques are developed and adopted by more investigators.

Multi-scale simulation of early kidney branching morphogenesis

An important feature of the branch morphogenesis during kidney development is the termination of the tips on the outer surface of a kidney. This feature requires the avoidance of the intersection between the tips and existing ducts inside the kidney. Here, we started from a continuous model and implemented the coarse grained rules into a fast and discrete simulations. The ligand-receptor-based Turing mechanism suggests a repulsion that decreases exponentially with distance between interacting branches, preventing the intersection between neighboring branches. We considered this repulsive effect in numerical simulations and successfully reproduce the key features of the experimentally observed branch morphology for an E15.5 kidney. We examine the similarity of several geometrical parameters between the simulation results and experimental observations. The good agreement between the simulations and experiments suggests that the concentration decay caused by the absorption of glial cell line derived neurotrophic factor might be the key factor to affect the geometry in early kidney development.

Simple Rules Determine Distinct Patterns of Branching Morphogenesis

Many metazoan organs are comprised of branching trees of epithelial tubes; how patterning occurs in these trees is a fundamental problem of development. Commonly, branch tips fill the volume of the organ approximately uniformly, e.g., in mammalian lung, airway branch tips are dispersed roughly uniformly throughout the volume of the lung. In contrast, in the developing metanephric kidney, the tips of the ureteric bud tree are located close to the outer surface of the kidney rather than filling the kidney. Here, we describe a simple alteration in the branching rules that accounts for the difference between the kidney pattern that leads to tips near the organ surface versus previously known patterns that lead to the branch tips being dispersed throughout the organ. We further use a simple toy model to deduce from first principles how this rule change accounts for the differences in the two types of trees.

Mathematical and computational models for bone tissue engineering in bioreactor systems

Research into cellular engineered bone grafts offers a promising solution to problems associated with the currently used auto- and allografts. Bioreactor systems can facilitate the development of functional cellular bone grafts by augmenting mass transport through media convection and shear flow-induced mechanical stimulation. Developing successful and reproducible protocols for growing bone tissue in vitro is dependent on tuning the bioreactor operating conditions to the specific cell type and graft design. This process, largely reliant on a trial-and-error approach, is challenging, time-consuming and expensive. Modelling can streamline the process by providing further insight into the effect of the bioreactor environment on the cell culture, and by identifying a beneficial range of operational settings to stimulate tissue production. Models can explore the impact of changing flow speeds, scaffold properties, and nutrient and growth factor concentrations. Aiming to act as an introductory reference for bone tissue engineers looking to direct their experimental work, this article presents a comprehensive framework of mathematical models on various aspects of bioreactor bone cultures and overviews modelling case studies from literature.

Mathematical Approaches of Branching Morphogenesis

Many organs require a high surface to volume ratio to properly function. Lungs and kidneys, for example, achieve this by creating highly branched tubular structures during a developmental process called branching morphogenesis. The genes that control lung and kidney branching share a similar network structure that is based on ligand-receptor reciprocal signalling interactions between the epithelium and the surrounding mesenchyme. Nevertheless, the temporal and spatial development of the branched epithelial trees differs, resulting in organs of distinct shape and size. In the embryonic lung, branching morphogenesis highly depends on FGF10 signalling, whereas GDNF is the driving morphogen in the kidney. Knockout of Fgf10 and Gdnf leads to lung and kidney agenesis, respectively. However, FGF10 plays a significant role during kidney branching and both the FGF10 and GDNF pathway converge on the transcription factors ETV4/5. Although the involved signalling proteins have been defined, the underlying mechanism that controls lung and kidney branching morphogenesis is still elusive. A wide range of modelling approaches exists that differ not only in the mathematical framework (e.g., stochastic or deterministic) but also in the spatial scale (e.g., cell or tissue level). Due to advancing imaging techniques, image-based modelling approaches have proven to be a valuable method for investigating the control of branching events with respect to organ-specific properties. Here, we review several mathematical models on lung and kidney branching morphogenesis and suggest that a ligand-receptor-based Turing model represents a potential candidate for a general but also adaptive mechanism to control branching morphogenesis during development.

Kidney Nephron Determination

The nephron is a multifunctional filtration device equipped with an array of sophisticated sensors. For appropriate physiological function in the human and mouse, nephrons must be stereotypically arrayed in large numbers, and this essential structural property that defines the kidney is determined during its fetal development. This review explores the process of nephron determination in the fetal kidney, providing an overview of the foundational literature in the field as well as exploring new developments in this dynamic research area. Mechanisms that ensure that a large number of nephrons can be formed from a small initial number of progenitor cells are central to this process, and the question of how the nephron progenitor cell population balances epithelial differentiation with renewal in the progenitor state is a subject of particular interest. Key growth factor signaling pathways and transcription factor networks are discussed.

Bayesian inference of agent-based models: a tool for studying kidney branching morphogenesis

The adult mammalian kidney has a complex, highly-branched collecting duct epithelium that arises as a ureteric bud sidebranch from an epithelial tube known as the nephric duct. Subsequent branching of the ureteric bud to form the collecting duct tree is regulated by subcellular interactions between the epithelium and a population of mesenchymal cells that surround the tips of outgrowing branches. The mesenchymal cells produce glial cell-line derived neurotrophic factor (GDNF), that binds with RET receptors on the surface of the epithelial cells to stimulate several subcellular pathways in the epithelium. Such interactions are known to be a prerequisite for normal branching development, although competing theories exist for their role in morphogenesis. Here we introduce the first agent-based model of ex vivo kidney uretic branching. Through comparison with experimental data, we show that growth factor-regulated growth mechanisms can explain early epithelial cell branching, but only if epithelial cell division depends in a switch-like way on the local growth factor concentration; cell division occurring only if the driving growth factor level exceeds a threshold. We also show how a recently-developed method, "Approximate Approximate Bayesian Computation", can be used to infer key model parameters, and reveal the dependency between the parameters controlling a growth factor-dependent growth switch. These results are consistent with a requirement for signals controlling proliferation and chemotaxis, both of which are previously identified roles for GDNF.

A Unifying Theory of Branching Morphogenesis

The morphogenesis of branched organs remains a subject of abiding interest. Although much is known about the underlying signaling pathways, it remains unclear how macroscopic features of branched organs, including their size, network topology, and spatial patterning, are encoded. Here, we show that, in mouse mammary gland, kidney, and human prostate, these features can be explained quantitatively within a single unifying framework of branching and annihilating random walks. Based on quantitative analyses of large-scale organ reconstructions and proliferation kinetics measurements, we propose that morphogenesis follows from the proliferative activity of equipotent tips that stochastically branch and randomly explore their environment but compete neutrally for space, becoming proliferatively inactive when in proximity with neighboring ducts. These results show that complex branched epithelial structures develop as a self-organized process, reliant upon a strikingly simple but generic rule, without recourse to a rigid and deterministic sequence of genetically programmed events.

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Branching morphogenesis follows conserved statistical rules in multiple organs

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Ductal tips grow and branch as default state and stop dividing in high-density regions

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Model reproduces quantitatively organ properties in a parameter-free manner

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Shows that complex organ formation proceeds in a stochastic, self-organized manner

Growth Factor Regulation in the Nephrogenic Zone of the Developing Kidney

New nephrons are induced by the interaction between mesenchymal progenitor cells and collecting duct tips, both of which are located at the outer edge of the kidney. This leading edge of active nephron induction is known as the nephrogenic zone. Cell populations found within this zone include collecting duct tips, cap mesenchyme cells, pretubular aggregates, nephrogenic zone interstitium, hemoendothelial progenitor cells, and macrophages. The close association of these dynamic progenitor cell compartments enables the intricate and synchronized patterning of the epithelial and the vascular components of the nephron. Understanding signaling interactions between the distinct progenitor cells of the nephrogenic zone are essential to determining the basis for new nephron formation, an important goal in regenerative medicine. A variety of technologies have been applied to define essential signaling pathways, including organ culture, mouse genetics, and primary cell culture. This chapter provides an overview of essential signaling pathways and discusses how these may be integrated.

Genetic Syndromes Affecting Kidney Development

Renal anomalies are common birth defects that may manifest as a wide spectrum of anomalies from hydronephrosis (dilation of the renal pelvis and calyces) to renal aplasia (complete absence of the kidney(s)). Aneuploidies and mosaicisms are the most common syndromes associated with CAKUT. Syndromes with single gene and renal developmental defects are less common but have facilitated insight into the mechanism of renal and other organ development. Analysis of underlying genetic mutations with transgenic and mutant mice has also led to advances in our understanding of mechanisms of renal development.

A Geometrically-Constrained Mathematical Model of Mammary Gland Ductal Elongation Reveals Novel Cellular Dynamics within the Terminal End Bud

Mathematics is often used to model biological systems. In mammary gland development, mathematical modeling has been limited to acinar and branching morphogenesis and breast cancer, without reference to normal duct formation. We present a model of ductal elongation that exploits the geometrically-constrained shape of the terminal end bud (TEB), the growing tip of the duct, and incorporates morphometrics, region-specific proliferation and apoptosis rates. Iterative model refinement and behavior analysis, compared with biological data, indicated that the traditional metric of nipple to the ductal front distance, or percent fat pad filled to evaluate ductal elongation rate can be misleading, as it disregards branching events that can reduce its magnitude. Further, model driven investigations of the fates of specific TEB cell types confirmed migration of cap cells into the body cell layer, but showed their subsequent preferential elimination by apoptosis, thus minimizing their contribution to the luminal lineage and the mature duct.

Our paper describes a mathematical model of mammary ductal elongation during pubertal development. We make several conclusions that will be of interest to scientists studying mammary gland biology, epithelial tube formation, and branching morphogenesis. First, our model indicates that a common measurement of developmental outgrowth (‘percent fat pad filled’) underestimates the total growth and leads to mischaracterization of mutant phenotypes. Second, we show that cap cells, a population enriched with putative mammary stem cells, do not contribute to the luminal lineage as previously hypothesized. Further, we find that a high percentage of proliferation in these cells is not used productively to actually form the mammary duct. We believe our model has future application to other branching organs and also for the modeling of disease states in the breast.

The Life Cycle of the Nephron Progenitor

Although we know that mesenchymal progenitors give rise to nephrons in the kidney, how they balance self-renewal versus differentiation is still unclear. In this issue of Developmental Cell, Chen et al. (2015) show that nephron progenitors age, but not necessarily irreversibly: old progenitors can be "rejuvenated" by a young crowd.