DOI:10.1016/j.jtbi.2025.112354 | Morpheus

Patterned Tissue Growth Regulation along Turing Patterns from Gierer-Meinhardt Model

Tue, 09 Jun 2026 12:53:26 +0000

Can morphodynamic feedback generate complex body shapes like that of a long and thin flatworm?

Introduction

During flatworm development and regeneration, morphogens like diffusible Wnt ligands self-organize concentration gradients that contribute to tissue morphogenesis, differentiation and the establishment of the body axes. Kaity et al. have considered planarian regeneration and studied a center-based model coupled to Turing patterns to explore tissue morphogenesis through patterned cell proliferation and cell death with morphodynamic feedback onto the Turing pattern from a Gierer-Meinhardt model. The two CPM-based models here extend spherical tissue models and qualitatively reproduce the published results.

Description

The spatial domain is a 2D square lattice and the initial condition are just two cells. Here, cells are represented with individual shape (occupying between A00/2 $=20$ and A00*2 $=80$ lattice nodes per cell) and follow CPM dynamics (weak cell-cell adhesion) as opposed to spherical cells with dynamic radii in the reference paper. The radius growth kinetics of the CPM model has been calibrated to that of the center-based model of the reference paper.

The Gierer-Meinhardt pattern formation model, in the reference paper and the models here, is not the classical coupled PDE system but a cell-cell interaction version of it. Cells here exchange morphogens m1 and m2 with their neighbors through fluxes proportional to the concentration difference that is measured by the model’s NeighborhoodReporters. Radius growth (or shrinkage) of each cell depends on neighbor packing and this may differ between the circular cells in the reference paper and the more flexible cells here. We have therefore inserted a global correction factor to the neighbor count that is varied between 1.2 and 1.5 in the simulations below. All other parameter values are taken as published.

Results

The reference paper in its Fig. 12 A,B compares tissue morphogenesis for different parameter values $\lambda$, which affects the wavelength of activator stripes in the Turing pattern. We correspondingly studied two closely related models (see the second XML file patterned_growth_labyrinth.xml in the list of files at the bottom of the page).

We first set $\lambda=0.075$ as in Fig. 12B and the neighborhood correction factor to 1.5 and observed the emergence of filamentous tissue shapes, see Fig. 1 and Movie 1 below.

Fig. 1. (A) Results of CPM model in Morpheus at matching time points as indicated. (B) Reference results from center-based model. Cells are color coded by local activator (m2) with values spanning cyan for m2 $=1$ to dark-blue for m2 $=0$. (CC BY-NC 4.0: Kaity et al., Fig. 12B)

Video 1. Simulation video of patterned_growth_filament_main.xml

We then reduce $\lambda=0.05$ as in Fig. 12A and the neighborhood correction factor to 1.2 and observe the emergence of compact tissue shapes with labyrinth patterns, see Fig. 2 and Movie 2 below.

Fig. 2. (A) Results of CPM model in Morpheus at matching time points as indicated. (B) Reference results from center-based model. Cells are color coded by local activator (m2) with values spanning cyan for m2 $=1$ to dark blue for m2 $=0$. (CC BY-NC 4.0: Kaity et al., Fig. 12A)

Video 2. Simulation video of patterned_growth_labyrinth.xml

Reference

This model reproduces a published result, originally obtained with a different simulator:

B. Kaity, D. Lobo: Emergent stable tissue shapes from the regulatory feedback between morphogens and cell growth. J. Theor. Biol. 620: 112354, 2026.

Model

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Download: patterned_growth_filament_main.xml

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<?xml version='1.0' encoding='UTF-8'?>
<MorpheusModel version="4">
<Description>
<Details>Model ID: https://identifiers.org/morpheus/M5933
File type: Main model
Software: Morpheus (open source). Download from: https://morpheus.gitlab.io
Full title: Patterned Tissue Growth Regulation along Turing Patterns from Gierer-Meinhardt Model
Authors: B. Kaity, D. Lobo
Submitter: Y. Korotkyi, G. Börner
Curator: J. Bürger Michaelis
Date: 07.05.2026
Reference: This model reproduces a published result, originally obtained with a different simulator:
B. Kaity, D. Lobo: Emergent stable tissue shapes from the regulatory feedback between morphogens and cell growth. J. Theor. Biol. 620: 112354, 2026.
https://doi.org/10.1016/j.jtbi.2025.112354</Details>
<Title>M5933 Patterned Growth Regulation (Filament Shape)</Title>
</Description>
<Space>
<Lattice class="square">
<Neighborhood>
<Order>2</Order>
</Neighborhood>
<Size symbol="size" value="600, 600, 0"/>
<BoundaryConditions>
<Condition type="constant" boundary="x"/>
<Condition type="constant" boundary="-x"/>
<Condition type="constant" boundary="y"/>
<Condition type="constant" boundary="-y"/>
</BoundaryConditions>
</Lattice>
<SpaceSymbol symbol="space"/>
</Space>
<Time>
<StartTime value="0"/>
<StopTime value="10000"/>
<TimeSymbol symbol="time"/>
</Time>
<Analysis>
<ModelGraph include-tags="#untagged" format="dot" reduced="false"/>
<Gnuplotter time-step="20">
<Terminal name="png" size="1600, 1600, 0"/>
<Plot title="Tissue (radius)">
<Cells value="r"/>
</Plot>
<Plot title="Morphogen m1">
<Cells per-frame-range="true" value="m2">
<ColorMap adaptive-range="true">
<Color color="blue" value="0"/>
<Color color="cyan" value="0.1"/>
</ColorMap>
</Cells>
</Plot>
<!-- <Disabled>
<Plot title="Activator m2">
<Cells value="m2"/>
</Plot>
</Disabled>
-->
<Plot title="Inhibitor m3">
<Cells value="m3"/>
</Plot>
<Plot title="Growth rate g">
<Cells value="g"/>
</Plot>
</Gnuplotter>
<Gnuplotter time-step="10">
<Terminal name="png" size="1600, 1600, 0"/>
<!-- <Disabled>
<Plot title="Neighbor count">
<Cells min="0" value="neighbor_count"/>
</Plot>
</Disabled>
-->
<Plot title="Activator m2">
<Cells max="1" min="0" value="m2">
<ColorMap adaptive-range="false">
<Color color="dark-blue" value="0"/>
<Color color="cyan" value="1"/>
</ColorMap>
</Cells>
</Plot>
<!-- <Disabled>
<Plot title="Inhibitor m3">
<Cells value="m3"/>
</Plot>
</Disabled>
-->
<!-- <Disabled>
<Plot title="Growth rate g">
<Cells value="g"/>
</Plot>
</Disabled>
-->
</Gnuplotter>
<Logger time-step="10">
<Input/>
<Output>
<TextOutput file-format="csv"/>
</Output>
<Plots>
<Plot time-step="1000">
<Style style="points"/>
<Terminal terminal="png"/>
<X-axis>
<Symbol symbol-ref="time"/>
</X-axis>
<Y-axis>
<Symbol symbol-ref="celltype.cell.size"/>
</Y-axis>
</Plot>
</Plots>
</Logger>
</Analysis>
<CellTypes>
<CellType name="cell" class="biological">
<VolumeConstraint target="A0" strength="Lam_V"/>
<SurfaceConstraint target="1" strength="Lam_S" mode="aspherity"/>
<!-- Cell radius (state variable, integrated by ODE) -->
<Property symbol="r" name="Cell radius" value="sqrt(A00/pi)"/>
<Function symbol="A0" name="Target area">
<Expression>pi*r^2</Expression>
</Function>
<!-- Morphogen concentrations (one scalar per cell, like the paper) -->
<Property symbol="m2" name="Activator" value="0.5"/>
<Property symbol="m3" name="Inhibitor" value="0.5"/>
<!-- Reaction terms (Eq. 16, Gierer-Meinhardt) -->
<Property symbol="reaction_m2" value="0.0"/>
<Property symbol="reaction_m3" value="0.0"/>
<Equation symbol-ref="reaction_m2">
<Expression>Lambda*(q2 + gamma*m2^2/((m3+epsilon)*(1 + delta*m2^2)) - lambda2*m2 - g*m2)</Expression>
</Equation>
<Equation symbol-ref="reaction_m3">
<Expression>Lambda*(q3 + gamma*m2^2 - lambda3*m3 - g*m3)</Expression>
</Equation>
<!-- Cell-to-cell diffusion (Eq. 9, paper-faithful)
alpha_eff = alpha_paper * (A00/pi) is the unit-converted diffusion constant.
All distances are in pixels; d_eq and d_max are also in pixels. -->
<Property symbol="D_m2" value="0.0"/>
<NeighborhoodReporter time-step="0.1">
<Input scaling="cell" value="if(cell.type == celltype.cell.id&#xa; and sqrt((cell.center.x-local.cell.center.x)^2 + (cell.center.y-local.cell.center.y)^2) &lt; d_max,&#xa; (alpha2_eff/(pi*local.r^2))&#xa; * (d_max - sqrt((cell.center.x-local.cell.center.x)^2 + (cell.center.y-local.cell.center.y)^2)) / (d_max - d_eq)&#xa; * (m2 - local.m2),&#xa; 0)"/>
<Output symbol-ref="D_m2" mapping="sum"/>
</NeighborhoodReporter>
<Property symbol="D_m3" value="0.0"/>
<NeighborhoodReporter time-step="0.1">
<Input scaling="cell" value="if(cell.type == celltype.cell.id&#xa; and sqrt((cell.center.x-local.cell.center.x)^2 + (cell.center.y-local.cell.center.y)^2) &lt; d_max,&#xa; (alpha3_eff/(pi*local.r^2))&#xa; * (d_max - sqrt((cell.center.x-local.cell.center.x)^2 + (cell.center.y-local.cell.center.y)^2)) / (d_max - d_eq)&#xa; * (m3 - local.m3),&#xa; 0)"/>
<Output symbol-ref="D_m3" mapping="sum"/>
</NeighborhoodReporter>
<!-- Combined reaction-diffusion ODE system -->
<System time-step="0.001" solver="Euler [fixed, O(1)]">
<DiffEqn symbol-ref="m2">
<Expression>reaction_m2 + D_m2</Expression>
</DiffEqn>
<DiffEqn symbol-ref="m3">
<Expression>reaction_m3 + D_m3</Expression>
</DiffEqn>
</System>
<!-- Growth rate (Eq. 10): activator m2 acts as the growth morphogen -->
<Function symbol="g">
<Expression>((g_max-g_min)*m2^h/(s^h+m2^h)+g_min)*(c^h/(c^h+(neighbor_count*1.5)^h))</Expression>
</Function>
<!-- Cell radius growth (Eq. 11): dr/dt = g*r/2 -->
<System time-step="1.0" name="Cell growth" solver="Euler [fixed, O(1)]">
<DiffEqn symbol-ref="r">
<Expression>g*r/2</Expression>
</DiffEqn>
</System>
<!-- Mitosis when cell volume reaches 2*A00 -->
<CellDivision division-plane="random">
<Condition>cell.volume >= 2*A00</Condition>
<Triggers>
<Rule symbol-ref="r">
<Expression>sqrt(cell.volume/pi)</Expression>
</Rule>
</Triggers>
</CellDivision>
<!-- Apoptosis when cell shrinks below half target -->
<CellDeath>
<Condition>cell.volume &lt;= A00/2</Condition>
</CellDeath>
<!-- Neighbor count for crowding inhibition -->
<Property symbol="neighbor_count" value="0.0"/>
<NeighborhoodReporter time-step="0.1">
<Input scaling="cell" value="if(cell.type == celltype.cell.id&#xa; and sqrt((cell.center.x-local.cell.center.x)^2 + (cell.center.y-local.cell.center.y)^2) &lt; d_max,&#xa; 1,&#xa; 0)"/>
<Output symbol-ref="neighbor_count" mapping="sum"/>
</NeighborhoodReporter>
</CellType>
<CellType name="Medium" class="medium">
<Constant symbol="m2" value="0.0"/>
<Constant symbol="m3" value="0.0"/>
<Constant symbol="g" value="0.0"/>
<Constant symbol="r" value="0.0"/>
</CellType>
</CellTypes>
<CPM>
<Interaction>
<Contact type2="cell" type1="cell" value="-0.05"/>
</Interaction>
<ShapeSurface scaling="norm">
<Neighborhood>
<Order>optimal</Order>
</Neighborhood>
</ShapeSurface>
<MonteCarloSampler stepper="edgelist">
<MCSDuration value="0.1"/>
<MetropolisKinetics temperature="0.01"/>
<Neighborhood>
<Order>optimal</Order>
</Neighborhood>
</MonteCarloSampler>
</CPM>
<CellPopulations>
<Population type="cell" name="Initial cell" size="0">
<InitCellObjects mode="distance">
<Arrangement repetitions="2, 1, 0" displacements="2*sqrt(A00/pi), 2*sqrt(A00/pi), 0">
<Sphere center="size.x/2, size.y/2, 0" radius="sqrt(A00/pi)"/>
</Arrangement>
</InitCellObjects>
</Population>
</CellPopulations>
<Global>
<!-- ============== GEOMETRY ============== -->
<Constant symbol="A00" value="40"/>
<!-- target cell area in pixels -->
<Constant symbol="Lam_V" name="Volume strength" value="0.1"/>
<Constant symbol="Lam_S" name="Surface strength" value="0.1"/>
<!-- Interaction distances (in pixels)
Paper convention: d_eq = 2*r, d_max = 1.5*d_eq = 3*r -->
<Constant symbol="d_eq" value="2*sqrt(A00/pi)"/>
<Constant symbol="d_max" value="3*sqrt(A00/pi)"/>
<!-- ============== UNIT CONVERSION ============== -->
<!-- Paper's alpha is in [cell_radius²/time]; convert to [pixel²/time]
by multiplying by (pixels per cell_radius)² = A00/pi = r² -->
<Constant symbol="alpha2_paper" value="0.1"/>
<Constant symbol="alpha3_paper" value="2.0"/>
<Constant symbol="alpha2_eff" value="alpha2_paper * A00/pi"/>
<Constant symbol="alpha3_eff" value="alpha3_paper * A00/pi"/>
<!-- ============== TURING (GIERER-MEINHARDT) ============== -->
<Constant symbol="q2" value="0"/>
<!-- basal activator production -->
<Constant symbol="q3" value="0"/>
<!-- basal inhibitor production -->
<Constant symbol="lambda2" value="1.2"/>
<!-- activator decay -->
<Constant symbol="lambda3" value="1.0"/>
<!-- inhibitor decay -->
<Constant symbol="gamma" value="1.0"/>
<!-- coupling -->
<Constant symbol="delta" value="0.4"/>
<!-- activator saturation -->
<Constant symbol="Lambda" value="0.075"/>
<!-- scaling factor (Fig. 12B = filament) -->
<Constant symbol="epsilon" value="1e-6"/>
<!-- divide-by-zero guard -->
<!-- ============== GROWTH ============== -->
<Constant symbol="g_min" value="-0.01"/>
<Constant symbol="g_max" value="0.01"/>
<Constant symbol="s" value="0.5*pi/A00"/>
<!-- Fig. 12 threshold -->
<Constant symbol="c" value="6"/>
<!-- crowding threshold -->
<Constant symbol="h" value="8"/>
<!-- Hill coefficient -->
</Global>
</MorpheusModel>

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Steady State Growth Regulation

Tue, 09 Jun 2026 12:53:26 +0000

How to stop tissue growth at a specific size?

Introduction

The growth and regeneration of organisms into complex shapes remains largly an open question. This model page presents the reproduction of a theoretical study of Kaity et al. where cell growth is mediated by morphogens. The original model is a lattice free, center based model where cells are modeled as 2D spheres. Cell growth/shrinkage is a function of the intracellular morphogen concentration and the neighbor count. When cells are growing to twice their initial size, they divide; when they shrink to half their original size, they go into apoptosis. This growth mechanism depends on a growth agent, whose concentration is governed by a diffusion equation. Diffusion is approximated as a rate equation, that is, a concentration balance between cells and their neighbors. In combination, this leads to growth alongside the patterns emerging from the diffusion of the morphogen.

On this model page, we present the equilibrium radial growth by two different models. The first model has a single producer cell at the center, which produces morphogen at a constant rate. The morphogen then diffuses to the edges of the cell sphere. The second model has no dedicated producer cell, instead the cells contain a Schnakenberg reaction diffusion system. The parameters are calibrated such that the Turing pattern resulting from the reaction diffusion system is a spot of high activator concentration. When the simulation is run long enough, this leads to a similar equilibrium growth.

Description

Constant Production Equilibrium Growth

In contrast to the original publication, our model is lattice based. Thus, we define a global cell size and adjust our parameters according to this. The model contains two cell types: producer cell and normal growth cells. In producer cells, the growth mechanism is shut off (that is, the $r$ parameter is a constant) and it contains a production term ($q$). The normal cells function as described in the introduction of Kaity et al., and their production term $q$ is 0.

Schnakenberg Equilibrium Growth

Instead of relying on producer cells, the Schnakenberg growth model implements a reaction system with an activator and an inhibitor morphogen, resulting in the emergence of a Turing pattern. In this case, the pattern is a simple sphere of high activator concentration. The activator functions as the input for the growth mechanism.

Results

Constant Production Equilibrium Growth

We initialize with a production cell in the center and a normal cell next to it. Due to the constant production of morphogen, the normal cell starts to divide and the tissue grows until the equilibrium state is reached. In Figure 1, we present a comparison to the original results.

Fig. 1. Top layer: our simulation results. Bottom layer: Original Results. Single organizer source cell (red) secreting a growth morphogen (blue), resulting in a steady-state circular shape. Time points of the original correspond to MCS in our simulation (CC BY-NC 4.0: Kaity et al., Fig. 2A)

Video 1. Simulation of the single producer cell model in Morpheus (simple_equilibrium_rate_main.xml)

When we compare the resulting cell count from the original publication to ours, we observe the saturation cell count to be similar, see Figure 2.

Fig. 2. Cell count over time in the single producer cell equilibrium model. The Figure shows our results as dots and the original results digitzed and visualized as a red line. (CC BY-NC 4.0: Kaity et al., Fig. 3A)

Schnakenberg Equilibrium Growth

We initialize a single cell in the center with a moderate concentration of both activator and inhibitor. In order to have a single spot emerge, we increase the growth speed in the beginning – otherwise, the initial randomness of the cell divisions would lead to the emergence of multiple spots. Once the tissue is large enough to incorporate the Turing wavelength of a single spot, we decrease the growth speed. This leads to the formation of a stable spherical tissue, as shown in Fig. 3.

Fig. 3. Top layer: Our simulation results. Bottom layer: Original results. Shows a spot Turing pattern emerging from a Schnakenberg system. Time points of the original correspond to MCS in our simulation. (CC BY-NC 4.0: Kaity et al., Fig. 9)

When cutting away part from the Schnakenberg pattern, the tissue regenerates into its original spherical shape, see Fig. 4. We obtain the same results with our Morpheus reproduction with the difference of our regeneration being roughly an order of magnitude faster than in the original publication. This is because in the Morpheus simulation, the neighbors are evaluated differently, leading to faster growth at the edge.

Fig. 4. Top layer: Our simulation results. Bottom layer: Original results. Shows the regeneration of the spherical equilibrium state after cutting away parts of the tissue (CC BY-NC 4.0: Kaity et al., Fig. 9)

In Video 2, we show both the emergence and regeneration of the Schnakenberg pattern.

Video 2. Simulation of the stable Schnakenberg spot pattern emergence and regeneration (simple_turing.xml). At a runtime of $t = 10000$ (approximately the midpoint of the video), almost half of the cells on the right-hand side of the tissue are artificially deleted, after which regeneration of the pattern is observed.