Beta Cell Turnover in Pancreatic Islet
Persistent Identifier
Use this permanent link to cite or share this Morpheus model:
How is the right islet size established during zebrafish pancreas development?
Introduction
Beta cells in the developing pancreas of zebrafish divide rapidly between 5 and 30 days post fertilization (dpf) and additional beta cells are recruited to the islet. Yet their overall number by 30 dpf remains well under control with the help of cell death.
Model Description
This Morpheus model combines four models of beta cell turnover in the pancreatic islet. One pair of models are ordinary differential equations for (the continuum approximation of) the beta cell number, the other pair are two-dimensional agent-based models (in z-layers ‘0’ for wildtype and ‘2’ for the mutant, separated by a fixed layer ‘1’) that explore stochastic effects of the combined cell behaviors and visualize the spatial density of cell death events. Of each pair, one model corresponds to the wildtype with all processes active and the other to the mutant.
This model is also featured at the BioModels repository from where it is linked, using RDF metadata, to further biological databases.
Results
The following movie shows the agent-based simulation:
Cell numbers from the simulation above are shown in the following graph and compared to ODE simulations with the same rate constants:
Reference
This model is the original used in the publication, up to technical updates:
M. N. Akhtar, A. Hnatiuk, L. Delgadillo-Silva, S. Geravandi, K. Sameith, S. Reinhardt, K. Bernhardt, S. P. Singh, K. Maedler, L. Brusch, N. Ninov: Developmental beta-cell death orchestrates the islet’s inflammatory milieu by regulating immune system crosstalk. The EMBO Journal, 2025.
Model
M2986-pancreas-islet_model.xml
XML Preview
<?xml version='1.0' encoding='UTF-8'?>
<MorpheusModel version="4">
<Description>
<Title>M2986 Pancreas Turnover</Title>
<Details>Model ID: https://identifiers.org/morpheus/M2986
Full title: Beta Cell Turnover in Pancreatic Islet
Date: 20.09.2024
Authors: M. N. Akhtar, L. Brusch, N. Ninov
BioModels: https://www.ebi.ac.uk/biomodels/MODEL2503030004
Species: https://identifiers.org/taxonomy:7955
Tissue: https://identifiers.org/UBERON:0000006
Cells: https://identifiers.org/CL:0000169, https://identifiers.org/CL:0000445
Units: [time] = 1 d
[space] = 1 µm
Software: Morpheus (open source). Download from: https://morpheus.gitlab.io
Reference: This Morpheus model was developed and analyzed in the original publication:
M. N. Akhtar, A. Hnatiuk, L. Delgadillo-Silva, S. Geravandi, K. Sameith, S. Reinhardt, K. Bernhardt, S. P. Singh, K. Maedler, L. Brusch, N. Ninov (2025). Developmental beta-cell death orchestrates the islet’s inflammatory milieu by regulating immune system crosstalk. The EMBO Journal 44 (4), 1131 - 1153, https://doi.org/10.1038/s44318-024-00332-w
Submitters: L. Brusch
Curators: D. Jahn
</Details>
</Description>
<Space>
<Lattice class="cubic">
<Neighborhood>
<Order>optimal</Order>
</Neighborhood>
<Size symbol="size" value="200, 200, 3"/>
<BoundaryConditions>
<Condition type="noflux" boundary="x"/>
<Condition type="noflux" boundary="-x"/>
<Condition type="noflux" boundary="y"/>
<Condition type="noflux" boundary="-y"/>
<Condition type="noflux" boundary="z"/>
<Condition type="noflux" boundary="-z"/>
</BoundaryConditions>
</Lattice>
<SpaceSymbol symbol="space"/>
</Space>
<Time>
<StartTime value="0"/>
<StopTime value="30"/>
<TimeSymbol symbol="time" name="Time (dpf)"/>
</Time>
<Analysis>
<ModelGraph include-tags="#untagged" format="dot" reduced="false"/>
<Gnuplotter time-step="0.1" name="Plot WT tissue">
<Terminal name="png"/>
<Plot title="WT cells" slice="0">
<Cells max="5" min="0" value="cell.type">
<ColorMap adaptive-range="false">
<Color color="white" value="0"/>
<Color color="green" value="1"/>
<Color color="red" value="2"/>
<Color color="green" value="3"/>
<Color color="red" value="4"/>
<Color color="blue" value="5"/>
</ColorMap>
</Cells>
</Plot>
</Gnuplotter>
<Gnuplotter time-step="0.1" name="Plot p35 mutant tissue">
<Terminal name="png"/>
<Plot title="p35 cells" slice="2">
<Cells max="5" min="0" value="cell.type">
<ColorMap adaptive-range="false">
<Color color="white" value="0"/>
<Color color="green" value="1"/>
<Color color="red" value="2"/>
<Color color="green" value="3"/>
<Color color="red" value="4"/>
<Color color="blue" value="5"/>
</ColorMap>
</Cells>
</Plot>
</Gnuplotter>
<Logger time-step="0.1" name="Plot time courses">
<Input/>
<Output>
<TextOutput file-format="csv" separator="comma"/>
</Output>
<Plots>
<Plot title="Cell number" time-step="-1">
<Style style="lines"/>
<Terminal terminal="png"/>
<X-axis>
<Symbol symbol-ref="time"/>
</X-axis>
<Y-axis minimum="0" maximum="350">
<Symbol symbol-ref="N_WT"/>
<Symbol symbol-ref="N_p35"/>
<Symbol symbol-ref="ode.Nsum_WT"/>
<Symbol symbol-ref="ode.Nsum_p35"/>
</Y-axis>
</Plot>
</Plots>
<Restriction condition="time>time.start-0.1"/>
</Logger>
</Analysis>
<CellTypes>
<CellType name="Medium" class="medium">
<Annotation>Surrounding tissue with fluid-like properties, hence no interactions with CellTypes.</Annotation>
</CellType>
<CellType name="Cell_WT_p" class="biological">
<Annotation>Cycling beta cell in WT condition (https://identifiers.org/CL:0000169)</Annotation>
<VolumeConstraint target="V_target" strength="Lam_V"/>
<SurfaceConstraint target="1" strength="Lam_S" mode="aspherity"/>
<CellDivision division-plane="random">
<Condition>time >= time.birth+Tc</Condition>
<Triggers>
<Rule symbol-ref="Tc">
<Expression>exp(rand_norm(Tc.ln_mean,Tc.ln_std))</Expression>
</Rule>
<Rule symbol-ref="time.birth">
<Expression>time</Expression>
</Rule>
</Triggers>
</CellDivision>
<Property symbol="Tc" value="Tc.mean"/>
<Property symbol="time.birth" value="rand_uni(time.start-Tc,time.start)"/>
<Property symbol="time.inact" value="time.start"/>
<!-- <Disabled>
<CellDeath name="apoptosis">
<Condition>(time>=time.onset) and (time<time.onset+time.duration) and rand_uni(0,1)<(d*mcs_duration)</Condition>
</CellDeath>
</Disabled>
-->
<ChangeCellType time-step="mcs_duration" newCellType="Cell_WT_q" name="inactivation">
<Condition>rand_uni(0,1)<(k.inact*mcs_duration) and time>time.start</Condition>
<Triggers>
<Rule symbol-ref="time.inact">
<Expression>time</Expression>
</Rule>
</Triggers>
</ChangeCellType>
<AddCell name="neogenesis" overwrite="true">
<Count>(rand_uni(0,1)<(v*celltype.Cell_WT_p.size/N_WT*mcs_duration)) and (time>time.start)</Count>
<Distribution>space.z==0 and ((space.x-size.x/2)^2+(space.y-size.y/2)^2)<20^2</Distribution>
<Triggers>
<Rule symbol-ref="time.birth">
<Expression>time</Expression>
</Rule>
</Triggers>
</AddCell>
<ChangeCellType time-step="mcs_duration" newCellType="Cell_WT_apoptosis" name="apoptosis">
<Condition>(time>=time.onset) and (time<time.onset+time.duration) and rand_uni(0,1)<(d*mcs_duration)</Condition>
<Triggers>
<Rule symbol-ref="time.birth">
<Expression>time</Expression>
</Rule>
</Triggers>
</ChangeCellType>
</CellType>
<CellType name="Cell_WT_q" class="biological">
<Annotation>Quiescent beta cell in WT condition (https://identifiers.org/CL:0000169)</Annotation>
<VolumeConstraint target="V_target" strength="Lam_V"/>
<SurfaceConstraint target="1" strength="Lam_S" mode="aspherity"/>
<Property symbol="Tc" value="Tc.mean"/>
<Property symbol="time.birth" value="rand_uni(time.start-Tc,time.start)"/>
<Property symbol="time.inact" value="time.start"/>
<!-- <Disabled>
<CellDeath name="apoptosis">
<Condition>(time>=time.onset) and (time<time.onset+time.duration) and rand_uni(0,1)<(d*mcs_duration)</Condition>
</CellDeath>
</Disabled>
-->
<ChangeCellType time-step="mcs_duration" newCellType="Cell_WT_p" name="activation">
<Condition>rand_uni(0,1)<(k.act*mcs_duration) and time>time.start</Condition>
<Triggers>
<Rule symbol-ref="time.birth">
<Expression>time-(time.inact-time.birth)</Expression>
</Rule>
</Triggers>
</ChangeCellType>
<AddCell name="neogenesis" overwrite="true">
<Count>(rand_uni(0,1)<(v*celltype.Cell_WT_q.size/N_WT*mcs_duration)) and (time>time.start)</Count>
<Distribution>space.z==0 and ((space.x-size.x/2)^2+(space.y-size.y/2)^2)<20^2</Distribution>
<Triggers>
<Rule symbol-ref="time.birth">
<Expression>time</Expression>
</Rule>
</Triggers>
</AddCell>
<ChangeCellType time-step="mcs_duration" newCellType="Cell_WT_apoptosis" name="apoptosis">
<Condition>(time>=time.onset) and (time<time.onset+time.duration) and rand_uni(0,1)<(d*mcs_duration)</Condition>
<Triggers>
<Rule symbol-ref="time.birth">
<Expression>time</Expression>
</Rule>
</Triggers>
</ChangeCellType>
</CellType>
<CellType name="Cell_p35_p" class="biological">
<Annotation>Cycling beta cell in p35 mutant condition (https://identifiers.org/CL:0000169)</Annotation>
<VolumeConstraint target="V_target" strength="Lam_V"/>
<SurfaceConstraint target="1" strength="Lam_S" mode="aspherity"/>
<CellDivision division-plane="random">
<Condition>time >= time.birth+Tc</Condition>
<Triggers>
<Rule symbol-ref="Tc">
<Expression>exp(rand_norm(Tc.ln_mean,Tc.ln_std))</Expression>
</Rule>
<Rule symbol-ref="time.birth">
<Expression>time</Expression>
</Rule>
</Triggers>
</CellDivision>
<Property symbol="Tc" value="Tc.mean"/>
<Property symbol="time.birth" value="rand_uni(time.start-Tc,time.start)"/>
<Property symbol="time.inact" value="time.start"/>
<ChangeCellType time-step="mcs_duration" newCellType="Cell_p35_q" name="inactivation">
<Condition>rand_uni(0,1)<(k.inact*mcs_duration) and time>time.start</Condition>
<Triggers>
<Rule symbol-ref="time.inact">
<Expression>time</Expression>
</Rule>
</Triggers>
</ChangeCellType>
<AddCell name="neogenesis" overwrite="true">
<Count>(rand_uni(0,1)<(v*celltype.Cell_p35_p.size/N_p35*mcs_duration)) and (time>time.start)</Count>
<Distribution>space.z==2 and ((space.x-size.x/2)^2+(space.y-size.y/2)^2)<20^2</Distribution>
<Triggers>
<Rule symbol-ref="time.birth">
<Expression>time</Expression>
</Rule>
</Triggers>
</AddCell>
</CellType>
<CellType name="Cell_p35_q" class="biological">
<Annotation>Quiescent beta cell in p35 mutant condition (https://identifiers.org/CL:0000169)</Annotation>
<VolumeConstraint target="V_target" strength="Lam_V"/>
<SurfaceConstraint target="1" strength="Lam_S" mode="aspherity"/>
<Property symbol="Tc" value="Tc.mean"/>
<Property symbol="time.birth" value="rand_uni(time.start-Tc,time.start)"/>
<Property symbol="time.inact" value="time.start"/>
<ChangeCellType time-step="mcs_duration" newCellType="Cell_p35_p" name="activation">
<Condition>rand_uni(0,1)<(k.act*mcs_duration) and time>time.start</Condition>
<Triggers>
<Rule symbol-ref="time.birth">
<Expression>time-(time.inact-time.birth)</Expression>
</Rule>
</Triggers>
</ChangeCellType>
<AddCell name="neogenesis" overwrite="true">
<Count>(rand_uni(0,1)<(v*celltype.Cell_p35_q.size/N_p35*mcs_duration)) and (time>time.start)</Count>
<Distribution>space.z==2 and ((space.x-size.x/2)^2+(space.y-size.y/2)^2)<20^2</Distribution>
<Triggers>
<Rule symbol-ref="time.birth">
<Expression>time</Expression>
</Rule>
</Triggers>
</AddCell>
</CellType>
<CellType name="Separator" class="biological">
<Annotation>Static dummy object to separate two conditions in z-layers above and below.</Annotation>
<FreezeMotion>
<Condition>1</Condition>
</FreezeMotion>
</CellType>
<CellType name="Cell_WT_apoptosis" class="biological">
<Annotation>Apoptotic beta cell (https://identifiers.org/CL:0000445)</Annotation>
<VolumeConstraint target="V_target" strength="Lam_V"/>
<SurfaceConstraint target="1" strength="Lam_S" mode="aspherity"/>
<Property symbol="time.birth" value="0"/>
<CellDeath name="apoptosis">
<Condition>time>=time.birth+time.apoptotic</Condition>
</CellDeath>
</CellType>
</CellTypes>
<CPM>
<Interaction/>
<ShapeSurface scaling="norm">
<Neighborhood>
<Order>optimal</Order>
</Neighborhood>
</ShapeSurface>
<MonteCarloSampler stepper="edgelist">
<MCSDuration value="0.01"/>
<MetropolisKinetics temperature="1"/>
<Neighborhood>
<Order>optimal</Order>
</Neighborhood>
</MonteCarloSampler>
</CPM>
<CellPopulations>
<Population type="Separator" name="Separator" size="1">
<InitCellObjects mode="distance">
<Arrangement repetitions="1, 1, 1" displacements="1, 1, 1">
<Box origin="0.0, 0.0, 1.0" size="size.x, size.y, 1.0"/>
</Arrangement>
</InitCellObjects>
</Population>
<Population type="Cell_WT_p" name="3 proliferative WT" size="0">
<InitCircle mode="regular" number-of-cells="3">
<Dimensions center="size.x/2, size.y/2, 0.0" radius="20.0"/>
</InitCircle>
</Population>
<Population type="Cell_WT_q" name="41 quiescent WT" size="0">
<InitCircle mode="regular" number-of-cells="41">
<Dimensions center="size.x/2, size.y/2, 0.0" radius="20.0"/>
</InitCircle>
</Population>
<Population type="Cell_p35_p" name="3 proliferative p35" size="0">
<InitCircle mode="regular" number-of-cells="3">
<Dimensions center="size.x/2, size.y/2, 2.0" radius="20.0"/>
</InitCircle>
</Population>
<Population type="Cell_p35_q" name="41 quiescent p35" size="0">
<InitCircle mode="regular" number-of-cells="41">
<Dimensions center="size.x/2, size.y/2, 2.0" radius="20.0"/>
</InitCircle>
</Population>
</CellPopulations>
<Global>
<Constant symbol="Tc.mean" name="Mean cell cycle duration" value="1.05">
<Annotation>Mean cell cycle duration.</Annotation>
</Constant>
<Constant symbol="Tc.std" value="0.05"/>
<Constant symbol="Tc.ln_mean" value="ln(Tc.mean^2/sqrt(Tc.mean^2+Tc.std^2))"/>
<Constant symbol="Tc.ln_std" value="sqrt(ln(1+Tc.std^2/Tc.mean^2))"/>
<Constant symbol="v" name="neogenesis" value="3.3">
<Annotation>Recruitment flux per day of de-novo beta cells into the islet.</Annotation>
</Constant>
<Constant symbol="d" name="apoptosis" value="0.03">
<Annotation>Apoptotic death rate per day for beta cells.</Annotation>
</Constant>
<Constant symbol="time.onset" value="16.3">
<Annotation>Onset time of apoptosis for beta cells.</Annotation>
</Constant>
<Constant symbol="time.duration" value="20">
<Annotation>Duration in days of apoptosis phase.</Annotation>
</Constant>
<Constant symbol="k.act" value="0.03">
<Annotation>Rate of cell cycle activation per day (quiescence -> cycling).</Annotation>
</Constant>
<Constant symbol="k.inact" value="1.0">
<Annotation>Rate of cell cycle inactivation per day (cycling -> quiescence).</Annotation>
</Constant>
<Constant symbol="time.start" value="5.0">
<Annotation>Cell dynamics, plotting and exp. data start from time.start=5dpf. The CPM model uses the burn-in phase from 0 to 5dpf to equilibrate cell shapes.</Annotation>
</Constant>
<Constant symbol="time.apoptotic" value="1.0">
<Annotation>Duration in days when apoptotic cells remain observable before clearance by immune cells.</Annotation>
</Constant>
<Variable symbol="ode.Np_WT" value="3.0"/>
<Variable symbol="ode.Nq_WT" value="41.0"/>
<Variable symbol="ode.Np_p35" value="3.0"/>
<Variable symbol="ode.Nq_p35" value="41.0"/>
<Function symbol="ode.Nsum_WT">
<Expression>ode.Np_WT+ode.Nq_WT</Expression>
</Function>
<Function symbol="ode.Nsum_p35">
<Expression>ode.Np_p35+ode.Nq_p35</Expression>
</Function>
<System time-step="0.001" name="ODE models for WT and p35 mutant" solver="Dormand-Prince [adaptive, O(5)]">
<DiffEqn symbol-ref="ode.Np_WT">
<Expression>(time>=5.0)*(v*ode.Np_WT/ode.Nsum_WT + k.act*ode.Nq_WT - k.inact*ode.Np_WT + ln(2)/Tc.mean*ode.Np_WT - d*(time>=time.onset)*(time<(time.onset+time.duration))*ode.Np_WT)</Expression>
</DiffEqn>
<DiffEqn symbol-ref="ode.Nq_WT">
<Expression>(time>=5.0)*(v*ode.Nq_WT/ode.Nsum_WT - k.act*ode.Nq_WT + k.inact*ode.Np_WT - d*(time>=time.onset)*(time<(time.onset+time.duration))*ode.Nq_WT)</Expression>
</DiffEqn>
<DiffEqn symbol-ref="ode.Np_p35">
<Expression>(time>=5.0)*(v*ode.Np_p35/ode.Nsum_p35 + k.act*ode.Nq_p35 - k.inact*ode.Np_p35 + ln(2)/Tc.mean*ode.Np_p35)</Expression>
</DiffEqn>
<DiffEqn symbol-ref="ode.Nq_p35">
<Expression>(time>=5.0)*(v*ode.Nq_p35/ode.Nsum_p35 - k.act*ode.Nq_p35 + k.inact*ode.Np_p35)</Expression>
</DiffEqn>
</System>
<Constant symbol="mcs_duration" value="0.01"/>
<Constant symbol="V_target" value="100">
<Annotation>Target cell size in number of lattice nodes.</Annotation>
</Constant>
<Constant symbol="Lam_V" name="Volume strength" value="1"/>
<Constant symbol="Lam_S" name="Surface strangth" value="1"/>
<Function symbol="N_WT">
<Expression>celltype.Cell_WT_p.size+celltype.Cell_WT_q.size</Expression>
</Function>
<Function symbol="N_p35">
<Expression>celltype.Cell_p35_p.size+celltype.Cell_p35_q.size</Expression>
</Function>
</Global>
</MorpheusModel>
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