Liver Zonation

E. Kolbe, S. Aleithe, C. Rennert, L. Spormann, F. Ott, D. Meierhofer, R. Gajowski, C. Stöpel, S. Hoehme, M. Kücken, L. Brusch, M. Seifert, W. von Schoenfels, C. Schafmayer, M. Brosch, U. Hofmann, G. Damm, D. Seehofer, J. Hampe, R. Gebhardt, M. Matz-Soja (Authors)

M. Kücken, M. Seifert, L. Brusch (Contributors)

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How does the architectural heterogeneity of the liver with its array of peri-central zones lead to the metabolic zonation across the liver lobule?

Introduction

Liver micro-architecture is set by an array of central veins interspersed with portal triads. Hepatocytes between these landmarks show a position-dependent or zonated metabolism, both in mice and human. Mutual interactions of Wnt and Hh signaling pathways self-organize gradients of ligand concentration that instruct hepatocyte zonation, see Fig. 1. The peri-central (pc) tissue acts as a source of Wnt ligands which registers the ligand gradients with the array of central veins. These spatio-temporal processes are modeled using coupled partial differential equations.

Figure 1. Sketch of signaling gradients (blue) that instruct zonated metabolism (red) in register with architectural landmarks ([CC BY 4.0](https://creativecommons.org/licenses/by/4.0/): [**Kolbe _et al._**](#reference)). — **Figure 1.** Sketch of signaling gradients (blue) that instruct zonated metabolism (red) in register with architectural landmarks (CC BY 4.0: Kolbe *et al.*).

Description

The model considers a 2D cross-section of 7 liver lobules. The positions of 7 central veins (red dots in Video 1) in a hexagonal arrangement are given by an analytical expression. Five partial differential equations for w=Wnt ligands (green), b=Intracellular Wnt signal (orange), i=IHH (purple), h=SHH (red) and g=GLI3 (brown) determine the time course (video) of dynamic liver zonation, starting from an artificial uniform distribution.

The original model Liver_zonation_model_wnt_hh.xml had used the partial differential equations as described above. But to visualize the two intracellular signals together with cell turnover of hepatocytes, that model has here been extended by a CPM in Liver_zonation_model_Wnt_Hh_main.xml.

Results

The final state of this spatio-temporal simulation reproduces the control condition in Fig. 5B of the publication Kolbe et al., 2019, see Fig. 2.

Figure 2. Final state of the model, with five variables as given above each panel, showing opposing signaling gradients for Wnt versus SHH. Reproduced from the published [Fig. 5B](https://www.sciencedirect.com/science/article/pii/S2211124719316043?via%3Dihub#fig5) ([CC BY 4.0](https://creativecommons.org/licenses/by/4.0/): [**Kolbe _et al._**](#reference)) — **Figure 2.** Final state of the model, with five variables as given above each panel, showing opposing signaling gradients for Wnt versus SHH. Reproduced from the published Fig. 5B (CC BY 4.0: Kolbe *et al.*)

The full spatio-temporal dynamics are shown in the videos Video 1, for the original model Liver_zonation_model_wnt_hh.xml, and Video 2, for the extended model Liver_zonation_model_Wnt_Hh_main.xml.

Video 1. Spatio-temporal dynamics for the original model Liver_zonation_model_wnt_hh.xml.

Video 2. Spatio-temporal dynamics for the extended model Liver_zonation_model_Wnt_Hh_main.xml.

Reference

This model is the original used in the publication, up to technical updates:

E. Kolbe, S. Aleithe, C. Rennert, L. Spormann, F. Ott, D. Meierhofer, R. Gajowski, C. Stöpel, S. Hoehme, M. Kücken, L. Brusch, M. Seifert, W. von Schoenfels, C. Schafmayer, M. Brosch, U. Hofmann, G. Damm, D. Seehofer, J. Hampe, R. Gebhardt, M. Matz-Soja: Mutual Zonated Interactions of Wnt and Hh Signaling Are Orchestrating the Metabolism of the Adult Liver in Mice and Human. Cell Rep. 29: 4553–4567, 2019.

Model

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Download: Liver_zonation_model_Wnt_Hh_main.xml

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<?xml version='1.0' encoding='UTF-8'?>
<MorpheusModel version="4">
    <Description>
        <Details>Model ID:		https://identifiers.org/morpheus/M6749
  Software:       	Morpheus (open-source). Download from https://morpheus.gitlab.io
  File type:		Main model
Full title:		Liver Zonation
Authors:		E. Kolbe, S. Aleithe, C. Rennert, L. Spormann, F. Ott, D. Meierhofer, R. Gajowski, C. Stöpel, S. Hoehme, M. Kücken, L. Brusch, M. Seifert, W. von Schoenfels, C. Schafmayer, M. Brosch, U. Hofmann, G. Damm, D. Seehofer, J. Hampe, R. Gebhardt, M. Matz-Soja
Submitters:		M. Kücken, M. Seifert, L. Brusch
Curators:		D. Jahn
Date:		24.12.2019
Units:		[time] = N/A
		[space] = N/A
Reference:		This model is an extension of the original used in the publication:
		E. Kolbe, S. Aleithe, C. Rennert, L. Spormann, F. Ott, D. Meierhofer, R. Gajowski, C. Stöpel, S. Hoehme, M. Kücken, L. Brusch, M. Seifert, W. von Schoenfels, C. Schafmayer, M. Brosch, U. Hofmann, G. Damm, D. Seehofer, J. Hampe, R. Gebhardt, M. Matz-Soja. Mutual Zonated Interactions of Wnt and Hh Signaling Are Orchestrating the Metabolism of the Adult Liver in Mice and Human. Cell Reports 29, 4553, 2019.
		https://doi.org/10.1016/j.celrep.2019.11.104
Comment:		To visualize the intracellular signals together with cell turnover, the published model has here been extended by a CPM.
</Details>
        <Title>M6749 Liver Zonation (main)</Title>
    </Description>
    <Space>
        <Lattice class="square">
            <Neighborhood>
                <Order>2</Order>
            </Neighborhood>
            <Size symbol="size" value="256,256, 0"/>
            <NodeLength value="0.1"/>
            <BoundaryConditions>
                <Condition type="periodic" boundary="x"/>
                <Condition type="periodic" boundary="-x"/>
                <Condition type="periodic" boundary="y"/>
                <Condition type="periodic" boundary="-y"/>
            </BoundaryConditions>
        </Lattice>
        <SpaceSymbol symbol="s"/>
    </Space>
    <Time>
        <StartTime value="0"/>
        <StopTime value="100"/>
        <TimeSymbol symbol="t"/>
    </Time>
    <Global>
        <Field symbol="cvein" name="peri-central (pc) Wnt source" value="if((s.x-128)*(s.x-128)+(s.y-128)*(s.y-128)&lt;=25, 1, 0) + if((s.x-28)*(s.x-28)+(s.y-128)*(s.y-128)&lt;=25, 1, 0) + if((s.x-228)*(s.x-228)+(s.y-128)*(s.y-128)&lt;=25, 1, 0) + if((s.x-78)*(s.x-78)+(s.y-215)*(s.y-215)&lt;=25, 1, 0) + if((s.x-78)*(s.x-78)+(s.y-41)*(s.y-41)&lt;=25, 1, 0) + if((s.x-178)*(s.x-178)+(s.y-215)*(s.y-215)&lt;=25, 1, 0) + if((s.x-178)*(s.x-178)+(s.y-41)*(s.y-41)&lt;=25, 1, 0)"/>
        <Field symbol="w" name="Wnt ligands (w)" value="0.0">
            <Diffusion rate="1.0"/>
        </Field>
        <Field symbol="b" name="Intracellular Wnt signal (b)" value="0.01">
            <Diffusion rate="0.001"/>
        </Field>
        <Field symbol="i" name="IHH (i)" value="0.001"/>
        <Field symbol="h" name="SHH (h)" value="0.01">
            <Diffusion rate="0.5"/>
        </Field>
        <Field symbol="g" name="GLI3 (g)" value="0.01">
            <Diffusion rate="1.0"/>
        </Field>
        <Constant symbol="a" value="0.2"/>
        <Constant symbol="c" value="1.0"/>
        <Constant symbol="k2" value="12.0"/>
        <Constant symbol="k1" value="0.1"/>
        <Constant symbol="k3" value="0.2"/>
        <Constant symbol="k4" value="12.0"/>
        <Constant symbol="k5" value="1.0"/>
        <Constant symbol="k6" value="0.5"/>
        <Constant symbol="k7" value="2.0"/>
        <Constant symbol="k8" value="1.0"/>
        <System time-step="0.1" solver="Runge-Kutta [fixed, O(4)]">
            <DiffEqn symbol-ref="w">
                <Expression>0.001-k1*w + cvein</Expression>
            </DiffEqn>
            <DiffEqn symbol-ref="b">
                <Expression>b*(w-a)*(1/(1+k2*h)-b) </Expression>
            </DiffEqn>
            <DiffEqn symbol-ref="h">
                <Expression>k3*c-k4*w*h-h</Expression>
            </DiffEqn>
            <DiffEqn symbol-ref="i">
                <Expression>b*c*c-k5*i</Expression>
            </DiffEqn>
            <DiffEqn symbol-ref="g">
                <Expression>k6*i+k7*h-k8*g</Expression>
            </DiffEqn>
        </System>
    </Global>
    <Analysis>
        <Gnuplotter log-commands="false" time-step="1" decorate="true">
            <Terminal name="png"/>
            <Plot>
                <Field symbol-ref="cvein"/>
            </Plot>
            <Plot>
                <Field symbol-ref="w" max="0.3" min="0">
                    <ColorMap>
                        <Color value="1" color="green"/>
                        <Color value="0" color="white"/>
                    </ColorMap>
                </Field>
            </Plot>
            <Plot>
                <Cells flooding="true" max="0.3" min="0" value="bCatenin">
                    <ColorMap>
                        <Color value="0.3" color="orange"/>
                        <Color value="0" color="white"/>
                    </ColorMap>
                </Cells>
            </Plot>
            <Plot>
                <Field symbol-ref="i" max="0.3" min="0">
                    <ColorMap>
                        <Color value="0.3" color="violet"/>
                        <Color value="0" color="white"/>
                    </ColorMap>
                </Field>
            </Plot>
            <Plot>
                <Field symbol-ref="h" max="0.2" min="0">
                    <ColorMap>
                        <Color value="0.2" color="red"/>
                        <Color value="0" color="white"/>
                    </ColorMap>
                </Field>
            </Plot>
            <Plot>
                <Cells flooding="true" max="0.3" min="0" value="Gli3">
                    <ColorMap>
                        <Color value="0.3" color="brown4"/>
                        <Color value="0" color="white"/>
                    </ColorMap>
                </Cells>
            </Plot>
        </Gnuplotter>
        <ModelGraph format="svg" reduced="false" include-tags="#untagged"/>
    </Analysis>
    <CellTypes>
        <CellType class="biological" name="hepatocyte">
            <VolumeConstraint target="100" strength="1"/>
            <SurfaceConstraint target="1" strength="1" mode="aspherity"/>
            <CellDeath>
                <Condition>rand_uni(0,1) &lt; 0.001</Condition>
            </CellDeath>
            <CellDivision division-plane="major">
                <Condition>rand_uni(0,1) &lt; 0.001</Condition>
            </CellDivision>
            <Property symbol="bCatenin" name="Intracellular Wnt signal (b)" value="0.0"/>
            <Property symbol="Gli3" name="GLI3 (g)" value="0.0"/>
            <Mapper time-step="1.0" name="bCatenin">
                <Input value="b"/>
                <Output symbol-ref="bCatenin" mapping="average"/>
            </Mapper>
            <Mapper time-step="1.0" name="Gli3">
                <Input value="g"/>
                <Output symbol-ref="Gli3" mapping="average"/>
            </Mapper>
        </CellType>
    </CellTypes>
    <CellPopulations>
        <Population type="hepatocyte" size="10">
            <!--    <Disabled>
        <InitHexLattice randomness="1"/>
    </Disabled>
-->
            <InitRectangle random-offset="1" number-of-cells="1000" mode="random">
                <Dimensions origin="0.0, 0.0, 0.0" size="size.x, size.y, 1.0"/>
            </InitRectangle>
            <InitVoronoi/>
        </Population>
    </CellPopulations>
    <CPM>
        <Interaction/>
        <ShapeSurface scaling="norm">
            <Neighborhood>
                <Order>2</Order>
            </Neighborhood>
        </ShapeSurface>
        <MonteCarloSampler stepper="edgelist">
            <MCSDuration value="1"/>
            <MetropolisKinetics temperature="1"/>
            <Neighborhood>
                <Order>2</Order>
            </Neighborhood>
        </MonteCarloSampler>
    </CPM>
</MorpheusModel>

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Last updated on Fri, 05 Sep 2025