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Branch Mode Selection during Early Lung Development

Denis Menshykau, Conradin Kraemer, Dagmar Iber

arXiv:1203.1497v1q-bio.TO

TL;DR

Lung branching follows a stereotyped sequence, but the regulatory process that generates its branch patterns is not fully understood. This paper models a core FGF10-SHH-Ptc signaling module with reaction-diffusion equations. The model yields patterns consistent with wildtype and mutant observations and associates different growth speeds with different branching modes.

  • Problem

    Although lung branching uses a stereotyped sequence of modes, an integrated account of the regulatory network that generates the patterns is lacking.

    Source: Introduction · p. 2

  • Method

    The authors develop a reaction-diffusion model of the core lung-branching signaling module, focusing on FGF10, SHH, and Ptc.

    Source: Introduction · pp. 2–3

  • Results

    The model produces a Schnakenberg-type Turing pattern consistent with wildtype and mutant observations, including increased FGF10-pattern spacing in mutants with reduced FGF10 expression.

    Source: Discussion · pp. 8–9

  • Takeaways & Limitations

    In simulations, high growth speeds yield FGF10 patterns characteristic of lateral branching, while low speeds yield patterns characteristic of bifurcations.

    Source: Discussion · p. 9

    Further advances require three-dimensional models, additional signaling factors including BMPs, and experimental data on branching dynamics, tissue responses, and growth speed relative to FGF10.

    Source: Discussion · p. 10

Abstract

from arXiv · show

Many organs of higher organisms, such as the vascular system, lung, kidney, pancreas, liver and glands, are heavily branched structures. The branching process during lung development has been studied in great detail and is remarkably stereotyped. The branched tree is generated by the sequential, non-random use of three geometrically simple modes of branching (domain branching, planar and orthogonal bifurcation). While many regulatory components and local interactions have been defined an integrated understanding of the regulatory network that controls the branching process is lacking. We have developed a deterministic, spatio-temporal differential-equation based model of the core signaling network that governs lung branching morphogenesis. The model focuses on the two key signaling factors that have been identified in experiments, fibroblast growth factor (FGF10) and sonic hedgehog (SHH) as well as the SHH receptor patched (Ptc). We show that the reported biochemical interactions give rise to a Schnakenberg-type Turing patterning mechanisms that allows us to reproduce experimental observations in wildtype and mutant mice. The kinetic parameters as well as the domain shape are based on experimental data where available. The developed model is robust to small absolute and large relative changes in the parameter values. At the same time there is a strong regulatory potential in that the switching between branching modes can be achieved by targeted changes in the parameter values. We note that the sequence of different branching events may also be the result of different growth speeds: fast growth triggers lateral branching while slow growth favours bifurcations in our model. We conclude that the FGF10-SHH-Ptc1 module is sufficient to generate pattern that correspond to the observed branching modes

Introduction

The paper develops a reaction-diffusion model of lung branching to address how a stereotyped sequence of branch modes can emerge from a core signaling network. The model reproduces reported wildtype and mutant patterns and proposes growth speed as another influence on branch-mode selection.

Introduction

  • Lung branching uses three stereotyped, non-random modes, motivating a search for a regulatory process that can generate their sequence without encoding every branch point separately.The modes are domain branching, planar bifurcation, and orthogonal bifurcation.Source: Introduction · pp. 1–2
  • A reaction-diffusion Turing-type model reproduces available experimental data in wild-type and mutant mice and mechanistically explains different lung branching types; lung-bud growth rate can determine branching mode.Source: Introduction · p. 2
  • The model focuses on FGF10, SHH, and Ptc: FGF10 stimulates SHH, while SHH-Ptc binding represses FGF10; BMP4 modulation is excluded.The authors focus on the core patterning mechanism and note that BMP4 cannot directly feed back onto Fgf10 expression because its main receptor is expressed only in the epithelium.Source: Figure 1 · p. 2, Introduction · pp. 2–3
  • A quasi-steady-state approximation for the SHH-Ptc complex yields a receptor-production relation, and the model reduces to the classical Schnakenberg system under specified parameter conditions.The reduction applies when F is much smaller than KS, v equals 3dC, and dS equals zero.Source: Introduction · pp. 3–4
  • The model uses a 2D Cartesian lung-tip cross-section, with FGF10 and SHH diffusing in tissue and cavities while membrane-bound Ptc diffuses much more slowly.A mean-field approximation with a small, non-zero Ptc diffusion coefficient is presented as computationally efficient and qualitatively consistent with cell-resolved simulations.Source: Introduction · p. 3, Computational Domain Geometry,… · p. 4
  • The PDEs are solved by finite elements in COMSOL 4.1, while nondimensionalization and parameter-variation analyses assess model sensitivity.The model removes five independent parameters through nondimensionalization; local stability tests vary parameters individually, and spatial variability is modeled with normally distributed parameter changes.Source: Numerical Solution of PDEs · p. 4, Modelling Domain Growth · p. 4, Robustness to Parameter Variability · p. 4

Results

The model produces FGF10 patterns associated with lateral branching and bifurcation, with patterning supported as a diffusion-driven Turing mechanism. Its patterns are robust to several parameter, geometry, and boundary changes, while parameter values and growth speed can alter the branching mode.

Morphogen Distribution in the Steady-State

  • FGF10 patterns from uniform initial conditions correspond to lateral branching when tip-centered and bifurcation when concentrated at the sides; distributions qualitatively match experimental observations.The model cannot reproduce domain branching or distinguish planar from orthogonal bifurcations in its two-dimensional form.Source: Morphogen Distribution in the… · p. 5, Figure 2 · p. 5

Robustness of the Observed Pattern

  • Most relative parameters can vary by 20%–30% without changing the pattern, while nearly all except receptor degradation dP can switch lateral branching to bifurcation.The analysis identifies the pattern as robust to noise yet sensitive to regulation.Source: Robustness of the Observed Pattern · p. 6, Figure 3 · p. 6
  • Similar FGF10 patterns persist after domain deformations and with zero-flux boundaries; linear stability analysis identifies the latter patterns as diffusion-driven and Turing-type.The tested changes include larger epithelial or mesenchymal bud radii and a truncated stalk.Source: Robustness of the Observed Pattern · p. 6, Figure 4 · p. 6

Branching Mode Selection on a Growing Domain

  • With growth restricted to the lung tip, FGF10 develops high-concentration regions near the tip at vg~0:08, corresponding to lateral branching; at fourfold lower growth speed, vg~0:02, high-concentration regions appear only at the sides, corresponding to bifurcation.Source: Branching Mode Selection on a Growing… · p. 6, Figure 5 · p. 7
  • On growing domains, domain branching remains stable when random parameter standard deviations do not exceed 0.2–0.3 of their reference values.This test evaluates parameter variation drawn from Gaussian distributions.Source: Branching Mode Selection on a Growing… · p. 7

Mutant Phenotypes

  • A 25% reduction in FGF10 expression increases spacing between branching points, and this pattern persists across sensible growth rates.The simulations also show fewer branching points than in wild type.Source: Branching Mode Selection on a Growing… · p. 7, Figure 6 · p. 8
  • At constant FGF concentrations, SHH–Ptc patterning uncouples from the FGF equation and still produces a pattern consistent with the classical Schnakenberg Turing model.This limit agrees with observations of branching in isolated endoderm cultures exposed to FGF-loaded beads.Source: Branching Mode Selection on a Growing… · p. 7
  • Changing SHH or Ptc regulation alters modeled signaling patterns: reduced SHH delays Ptc production, while increased Ptc production rates lower maximum SHH concentration.The model also reproduces full knockouts, although these results are described as trivial because mutant lungs are severely truncated or absent.Source: Branching Mode Selection on a Growing… · pp. 7–8, Figure 6 · p. 8

Discussion

The FGF10–SHH–Ptc module produces a Schnakenberg-type Turing pattern consistent with observed wild-type and mutant expression patterns. In the model, parameter values and growth speed can shift FGF10 patterns associated with different branching modes.

Discussion

  • The experimentally described interactions generate a Schnakenberg-type Turing mechanism that produces FGF10 patterns from homogeneous initial conditions.The model focuses on core signaling interactions and produces patterns during in silico lung-bud development.Source: Discussion · p. 8
  • The model reproduces observed wild-type and mutant expression patterns and predicts increased FGF10-spot spacing in mutants with reduced FGF10 expression.The authors describe the spacing prediction as non-trivial and not explained by earlier models [36,42–44].Source: Discussion · pp. 8–9
  • The model uses relative parameter values because many absolute values are unknown, requiring FGF10 and SHH to diffuse about 100 times faster than Ptc.The authors report parameter values within physiological ranges established in other model systems; assuming quasi-steady-state SHH–Ptc complex formation does not alter patterns when binding and unbinding constants are of order 1.Source: Discussion · p. 9
  • The model’s 2D slice cannot distinguish planar from orthogonal bifurcations, and further work requires 3D models coupling local growth to FGF10 and adding signaling factors, especially BMPs.The authors also call for experimental data quantifying branching dynamics, epithelial and mesenchymal responses, and growth speed relative to FGF10 concentration.Source: Discussion · pp. 9–10
  • Parameter changes can shift FGF10 patterns toward bifurcation or lateral branching: increased diffusion or smaller domains favor bifurcation, while higher production or lower decay favor lateral branching.The model proposes that BMP-related changes could affect these rates, but further modeling and experiments are needed to clarify this possibility.Source: Discussion · p. 9
  • Fast growth produces FGF10 patterns characteristic of lateral branching, whereas slow growth produces patterns characteristic of bifurcation.The modeled growth rates are about 14 mm h^-1 for lateral branching and 3.6 mm h^-1 for bifurcation; the former yields two new branches per day separated by about 150–200 mm [17,24].Source: Discussion · p. 9
  • With rapid tip growth, new FGF10 spots form behind the tip, while more uniform growth also produces spots in proximal regions.The authors suggest both growth modes and patterning dynamics occur in developing lungs and may produce different branching sequences; the profile has highest FGF10 at the tip and lowest at a newly formed branch point behind it.Source: Discussion · pp. 9–10, Figure 7 · p. 10

Supporting Information

The supporting figures probe how FGF10 patterns behave under alternative domain assumptions, discretizations, parameter choices, and growth modes. They also show patterns for different branching modes and model configurations.

Supporting Information

  • On the alternative model’s domain, elongation occurs when the tip is far from an impermeable boundary, while planar bifurcation occurs near it; an infinite domain yields elongation.The figure describes a model with SHH fixed at the tip and SHH-promoted FGF production.Source: Figure S1 · p. 10
  • Schnakenberg-model patterns are shown for fast- and slow-diffusing components on both a continuous 2D plane and an array of spheres.Source: Figure S2
  • Cell-based patterns preserve the main continuous-model features at the mesenchyme–epithelium border when Ptc diffusion at cell edges is zero, though other regions are distorted.When edge diffusion is significantly lower than surface diffusion, the patterns are the same as in the continuous model.Source: Supporting Information · p. 10
  • Steady-state FGF10 distributions are presented for reaction-order values n = 1 and n = 3 and for three SHH–Ptc complex stoichiometries.The stoichiometries shown are SP3, S2P2, and SP1:2.Source: Figure S5 · p. 10, Figure S4
  • FGF10 distributions at about t = 90 are compared across mesh maximum element sizes of 0.4, 0.2, 0.1, and 0.05.Source: Figure S6
  • Steady-state FGF10, SHH, and Ptc patterns are shown for lateral-branching and bifurcation examples, including a second parameter setting with rS = 1300.Source: Supporting Information · p. 11
  • Three-dimensional steady-state FGF10 distributions are shown for lateral branching and bifurcation modes.Source: Figure S8
  • Growing-domain simulations compare uniform growth with local growth at the tip, and test parameter variability using normally distributed deviations of 0.1, 0.2, and 0.3 of the Table 1 values.Source: Figure S9, Figure S10
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