Modelling the influence of the E-Cadherin - β-Catenin

pathway in cancer cell invasion: A multi-scale approach

Ignacio Ramis-Conde ∗, Dirk Drasdo†, Alexander R.A. Anderson‡, Mark A.J. Chaplain§

December 12, 2007

Abstract

In this paper, we show using a mathematical multi-scale model, how cell adhesionmay be regulated by interactions between E-cadherin and β-catenin and how the con-trol of cell adhesion may be related to cell migration, to the epithelial-mesenchymaltransition and to invasion in populations of eukaryotic cells. E-cadherin mediates cell-cell adhesion and plays a critical role in the formation and maintenance of junctionalcontacts between cells. Loss of E-cadherin mediated adhesion is a key feature of theepithelial-mesenchymal transition. β-catenin is an intracellular protein associated withthe actin cytoskeleton of a cell. E-cadherins bind to β-catenin to form a complex whichcan interact both with neighbouring cells to form bonds, and with the cytoskeleton ofthe cell. When cells detach from one another, β-catenin is released into the cyto-plasm, targeted for degradation and down-regulated. In this process there are multipleprotein-complexes involved which interact with β-catenin and E-cadherin. Within amathematical individual-based multi-scale model we are able to explain experimentallyobserved patterns solely by a variation of cell-cell adhesive interactions. Implicationsfor cell migration and cancer invasion are also discussed.

Keywords: Multi-scale modelling, Cell-adhesion, Agent-based models, Single-cell-based models, β-catenin, Cancer Invasion. 1

1 Introduction

Cancer is characterised by multiple mutations in a single cell leading to a loss of control incell replication accompanied by an uncontrolled growth of the total cell mass, eventually

1∗Corresponding author: ignacio.ramis conde@inria.fr, French National Institute for Research in Com-puter Science and Control (INRIA), Domaine de Voluceau-Rocquencourt,B.P. 105, 78153 Le Chesnay CedexFrance & Division of Mathematics, University of Dundee. †, dirk.drasdo@inria.fr, French National Insti-tute for Research in Computer Science and Control (INRIA), Domaine de Voluceau-Rocquencourt,B.P.105, 78153 Le Chesnay Cedex France & Interdisciplinary Centre for Bioinformatics (IZBI), Univer-sity of Leipzig, Germany.‡ chaplain@maths.dundee.ac.uk, Department of Mathematics The University ofDundee, Dundee DD1 4HN, §anderson@maths.dundee.ac.uk, Department of Mathematics, The Universityof Dundee, Dundee DD1 4HN

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leading to the formation of an in-situ solid tumour. After the tumour reaches a certainsize, genetic instability in the cancer cells may lead to further de-differentiation within themalignant cell mass. These secondary mutations are relevant in order for the tumour togain advantage over neighbouring cells and to invade further the local tissue and organs.In the transition from a normal cell to a malignant cell, the modification of intracellularpathways related to cell-cell adhesion and cell-matrix adhesion are important and determinethe compactness of the tumour surface and the invasiveness of the tumour [4]. Cadherins arecell-cell adhesion proteins which form part of multi-protein complexes at the cell membraneto bind neighbouring cells and determine the tissue architecture. The different types ofcadherins are named from the type of tissue where they originate from e.g. E-cadherinin epithelia, N-cadherin in the nervous system, etc. Of particular interest is E-cadherin,sometimes considered as a tumour suppressor protein due to its functionality in maintainingthe compactness of the epithelium. The role of E-cadherin in the malfunction of cell-celladhesion observed in colorectal cancer, and in the β-catenin degradation system aftermutations that affect the wnt-pathway, belong to the most studied examples [1, 2, 3].Over 80 percent of colorectal tumours show malfunctions in APC, a key protein in thewnt-pathway related also to intracellular interactions where E-cadherin plays a main role.These mutations are correlated with higher cancer invasion and therefore poorer prognosis.

Mathematical modelling of cell adhesion has been approached by different strategiesincluding continuum models [5, 6] (for a general review of mathematical continuum modelsof cancer see [7]), individual-based lattice-free models [8, 10, 11, 12] and lattice-basedmodels where each lattice site can at most be occupied by one single cell [13, 14, 15] (forgeneral reviews of individual-based models, see [16, 17, 18, 19]). Although all these differentapproaches have shown the importance of cell adhesion to keep tumour compactness andprevent invasion, there is still a wide field to explore linking the intracellular dynamicsof signalling pathways to the adhesion molecules at the cell surface and the extracellularconsequences in invasive tumours. In this paper we approach this problem using a multi-scale individual-based lattice-free model which accounts for the intracellular dynamics ofthe E-cadherin-β-catenin interactions and the physical forces on the cells.

2 The biological background

When a cell adheres to adjacent neighbours, the E-cadherin molecules are situated in aninter-membrane position, forming bonds with local neighbours at the intercellular space.The cytoplasmatic tail of the E-cadherin molecule binds to the proteins of the cateninfamily: p120-catenin, α-catenin and β-catenin. The α-catenin and β-catenin then form acomplex to link the actin filaments of the cytoskeleton and the E-cadherins. When bondsare released, caused by intra-cellular signalling or the effect of mechanical stress, the multi-protein complex is broken and the E-cadherins are internalised i.e. transported into thecytoplasm by the endocytosis apparatus within cadherin-coated vesicles. It is not well

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known yet if, after being endocytosed, E-cadherins are degraded or if they are kept by thevesicles for later recycling [20].

When the bond is broken, β-catenin is released in a phosphorylated state. In thisform it is ready to interact with other molecules and can be recognised and degraded inthe proteasome systems. Intra-cellular control of β-catenin concentration is important inpreserving the tissue architecture. Up-regulation of free β-catenin (also known as solubleβ-catenin) is related to cell migration and the epithelial-mesenchymal transition [22, 23],a process where a well-ordered and polarised layer of cells changes into an unstructuredconfiguration to facilitate collective cell migration. Sufficiently large concentrations of sol-uble β-catenin then move from the cytoplasm into the nucleus, where it interacts withtranscription factors which modify cell behaviour - for example, by promoting cell prolifer-ation. Although the precise mechanisms which relate the β-catenin nuclear translocationto cell migration are not yet very well known, it has been observed that invasive cellsshow a higher nuclear accumulation of soluble β-catenin. Gumbiner and collaborators [22]used matrigel chambers and different cell chimeras to show how up-regulation of solubleβ-catenin induces cell migration. They used cell clones with different malfunctions in thecytoplasmatic tail of E-cadherin and found that invasion was enhanced when β-catenin wasnot able to bind E-cadherin, despite cell-cell bonds being formed at the extra-cellular partwhere it was intact. Different biological mechanisms capable to trigger invasion after acti-vating the β-catenin pathway have been proposed [24][25][26]. In particular, Kemler et. al.[27] proposed that β-catenin is translocated into the nucleus above a certain concentrationthreshold leading to down-regulation of E-cadherin mediated adhesion, partly referring onearlier work by Huber et al. [25] and Jamora et al. [28].

In this paper we study the possible effects of a nuclear up-regulation of soluble β-catenin as a consequence of being released from the E-cadherin-β-catenin complex, andthe biological consequences of the existence of a β-catenin threshold which may down-regulate cell adhesion during the epithelial-mesenchymal transition. Instead of consideringthe pathways that involve β-catenin in every detail known we focus on a simplified β-catenin pathway which captures the key features of the cell adhesion process. The maininteractions of β-catenin in our model are with the E-cadherins at the cell membrane anda generic proteasome-related complex in the cytoplasm which accounts for the whole setof proteins involved in the process of β-catenin degradation. We include soluble β-cateninand the E-cadherin-β-catenin complex as the main variables of our model. Up-regulationof soluble β-catenin is assumed to interact with transcription factors in the nucleus, andinduce cell migration [26].

3 The β-catenin kinetics

We first present our model of the intra-cellular β-catenin dynamics and show the impor-tance of this regulation system on the E-cadherin system. This is then followed by our

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single-cell model and a subsection on how the intra-cellular dynamics couples to the cell-biophysical and cell-biological single-cell parameters.It is not completely known how E-cadherin is transported from the cytosol to the inter-membrane position to form bonds [29]. However, there is some evidence that cytoplasmaticE-cadherin translocates to the membrane after binding to β-catenin at the endoplasmic

reticulum [30]. For simplicity we assume that the complexes are formed in the cell mem-brane. This is translated into our model considering three possible different states of theE-cadherins: catenin free in the cytoplasm ([Ec]), catenin free at the cell membrane ([Em]),and the complex E-cadherin-β-catenin forming bonds at the cell membrane ([E/β]). Asone cell comes into contact with another cell, the cadherin in the cytoplasm moves tothe cell surface. Here we make the assumption that the amount of cadherin stimulatedto move is proportional to the contact area so that the surface density of β-catenin-E-cadherin receptors is constant. At the cell membrane, the cadherin binds to β-catenin andthe cell forms bonds with its neighbouring cells. On the other hand, if cell detachmentoccurs, then the β-catenin-E-cadherin complex is ruptured, the β-catenin becomes solubleβ-catenin and E-cadherin is sequestered into the cytoplasm by endocytosis. Further weassume that E-cadherin can be recruited to form bonds again. In this way we can considerthe total E-cadherin concentration (ET = 100nM) to be constant:

ET = [Ec] + [Em] + [E/β] (1)

The other reactions described above can be written as

[Ec] →{ contact} [Em], (2)

[β] + [Em]ν→ [E/β], (3)

[E/β] →{ detachment} [Ec] + [β]. (4)

Figure (1) summarises the intra-cellular interactions of the different forms of E-cadherin.(We also considered the backward reaction in equation (3) and found that it neither mod-ifies the β-catenin steady state concentrations within a cell, nor changes the multi-scalequalitative dynamics as we validated by fixed point analysis and by computer simulations).Following Heinrich and co-workers [31], we assume that the production of β-catenin oc-curs at a constant rate (km = 0.01nMmin−1). The degradation process takes place afterforming a complex with the proteasome. In the framework of our model, this proteasome

variable should be understood as a complex of proteins which after different biochemicalinteractions degrade soluble β-catenin i.e.

[β] + [P ]k+

⇋

k−[C]

k2→ [P ] + ω, (5)

where [P ] denotes the concentration of proteasome, [C] is the concentration of β-catenin-proteasome complex and ω is the final product of the degradation process. The important

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wnt-pathway is taken into account within this degradation process in a very simple way: ifwnt is activated, and therefore β-catenin cannot bind to actin and be marked for degrada-tion, degradation is at low levels which in our model is equivalent to k+ ≈ 0. If wnt is notactivated, β-catenin can be degraded and therefore down-regulated.

In a similar way to the E-cadherin concentration, we assume that the proteasome totalconcentration remains constant (PT = 0.33514nM), and therefore

PT = [C] + [P ]. (6)

Using mass conservation we obtain the following system of reaction equations from theabove chemical reactions for each individual cell i:

dt[Ec] = −ci(t)[Ec] + di(t)[E/β], (7)

dt[E/β] = ν(ET − [Ec] − [E/β])[β]

−di(t)[E/β], (8)

dt[β] = −ν(ET − [Ec] − [E/β])[β]

+di(t)[E/β] − k

+[β](PT − [C]) + k−[C] + km, (9)

dt[C] = k+[β](PT − [C]) − k−[C] − k2[C], (10)

where ν = 100min−1 is the rate at which the complex [E/β] is produced, k+ = 100min−1

and k− = 19min−1 are the β-catenin-proteasome association and dissociation rates, re-spectively. The functions ci(t) and di(t) measure the amount of cadherin stimulated toform bonds by physical contact with neighbouring cells. The function ci(t) is defined as

ci(t) =∑

new contacts

ac,j(t)ρc,

where ac,j(t) models the area of the newly formed cell-cell contact of cell i with its localneighbour j, ρc = 200min−1 is the rate at which E-cadherin concentration translocatesfrom cytoplasm to the inter-membrane position when induced by cell-cell contact stimuli.The function di(t) describes the equivalent effect if detachment occurs i.e.

di(t) =∑

new detachments

ad,j(t)ρd,

where ad,j(t) models the contact area lost by the detachment of cell i from cell j at timet. ρd = 200min−1 is the rate at which E-cadherin concentration translocates from theinter-membrane to the cytoplasm after cell-cell detachment.The functions a(t)c,j and a(t)d,j determine the area stimulated to interchange the cadherinfrom the cytosol to the membrane and from the membrane to the cytosol respectively. We

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define these functions as

ac,j(t) =

{ ∂

∂ta(t)j , if

∂

∂ta(t)j > 0,

0, otherwise,

and

ad,j(t) =

{

‖∂

∂ta(t)j‖, if

∂

∂ta(t)j < 0,

0, otherwise,

where a(t)j is the approximated area in contact with the cell j at time t, calculated by thespherical caps in contact (this approximation has also been used in Ref. [8]).Hence, both attachment and detachment of cells lead to an exchange of E-cadherin betweenthe membrane in the contact zone of the interacting cells. We assume that equations (8)and (9) determine the concentrations of β-catenin and β-catenin-E-cadherin only as longas the soluble β-catenin concentration is below a threshold cT . In the case [β] > cT , weconsider the soluble β-catenin in cytoplasm to be large enough and as a consequence it isfree to enter the nucleus and interact with transcription factors causing the cell to migrate.As a necessary step of migration, cell detachment occurs. To model the active detachmentprocess we assume that for [β] > cT ,

dt[E/β] = −(α + di(t))[E/β], (11)

dt[β] = (α + di(t))[E/β] −

k+[β](PT − [C]) + k−[C] + km. (12)

replace the original equations (8) and (9), respectively. α is the rate at which the complexis dissociated once the migration decision has been made.Since the molecular kinetics of β-catenin and its interaction varies between different cells,each cell within our model is considered as an individual entity which intracellular dynamicsare governed by the explained equations. Motivated by the observations that cells inisolation tend to aggregate [32], we assume that an invasive cell changes into an non-invasive state again if it comes into contact with other cells to which it can attach to. Inthis case equations (8) and (9) are recovered.The intra-cellular parameters values were taken, when possible, from Heinrich et al. [31].For those which were not found in literature, they were chosen in a similar range to theones they used. All the parameters values are given in Table 1.

4 The biophysical model of a single cell

We model each cell as an isotropic elastic object capable of migration and division andparameterise it by cell-kinetic, biophysical and cell-biological parameters that can be ex-perimentally measured. We now describe below the key features of this modelling approach.

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Cell-cell shape: We assume that an individual cell in isolation is spherical. We charac-terise the cell shape of a spherical cell by its radius R.

Cell division: In our model the cell cycle is sub-divided into two phases - the mitoticphase where a cell divides into two daughter cells and the interphase, the period of timebetween two mitotic events. We assume that during interphase the cell doubles its massand model the process of cell division by replacing two cells of size R by two daughter cellsof radius R/21/3 which then gradually grow during interphase to their original radius R.This radius value to determine the size of the new daughter cells was taken in accordingto the experiments performed by Galle et al. [8], where they reproduced realistic tumourgrowth curves using an individual force-based model of similar characteristics to ours.

Cell-cell interaction: With decreasing distance between the centres of two cells (e.g.upon compression), both their contact area and the number of adhesive contacts increase,resulting in an attractive interaction. On the other hand, if cells are spheroidal in isolation,a large contact area between them significantly stresses their cytoskeleton and membranes.Furthermore, experiments suggest that cells only have a small compressibility (the Poissonnumbers are close to 0.5, [33, 34]). In this instance, both the limited deformability and thelimited compressibility give rise to a repulsive interaction. We model the combination ofthe repulsive and attractive energy contributions by a modified Hertz-model [8, 10] wherethe potential Vij between two cells of radius Ri and Rj is given by

Vij = (Ri + Rj − dij)5/2 1

5Eij

√

RiRj

Ri + Rj︸ ︷︷ ︸

repulsive forces

+ ǫs︸︷︷︸

adhesive forces

. (13)

The first term of the equation models the repulsive interaction, the second term the adhesiveinteraction, and Eij is defined by

E−1ij =

3

4

(

1 − σ2i

Ei+

1 − σ2j

Ej

)

. (14)

Here, Ei, Ej are the elastic moduli of the cells i, j, σi, σj the Poisson ratios of the spheres.ǫs ≈ mAijWs, where Ws ≈ 25kBT (T : temperature, kB : Boltzmann constant) is theenergy of a single bond, Aij the contact area between cells i, j and m is the density ofsurface adhesion molecules in the contact zone, in our case the density of E-cadherin, [35].The interaction force results from deriving the potential function

F ij = −(∂Vij/∂dij)(d(dij)/dx, d(dij)/dy, d(dij)/dz). (15)

The modified Hertz-model approximates a cell as an elastic sphere and superimposes therepulsive force that emerge in case of a deformation or compression of the sphere with

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an attractive contribution due to cell-cell adhesion. In some cancer cell lines a hysteresiseffect has been observed i.e. the attachment and detachment of cells occur at differentdistances between the cell centres [21]. However, in [9] have been shown that the effectof the hysteresis is only a delay in the detachment, while the qualitative behaviour of thedetachment process does not depend on the existence of the hysteresis. The advantageof the modified Hertz model is that both the interaction energy and the force can berepresented as an analytical expression, while for models that represent the hysteresiseffect such as the Johnson-Kendall-Roberts (JKR)-model, the force has to be calculatednumerically from an implicit equation [11].

Cell movement: The direct use of equations of motion for the cells permits one toinclude more easily the limiting case of very small (or no) noise and is more intuitive. Inthis approach cells move under the influence of forces and a random contribution to thelocomotion which results from the local exploration of space. Moreover, in some of thescenarios explained below, a chemoattractant chemical generates a force-like term in theLangevin equation as suggested in [36]:

Γfis

vi︸ ︷︷ ︸

s-friction

+∑

j nn i

Γfij

(vi − vj

)

︸ ︷︷ ︸

cell-cell friction

=∑

i nn j

F ij

︸ ︷︷ ︸

forces

+ fi(t)

︸︷︷︸

noise

+ χ∇Q(t)︸ ︷︷ ︸

chemotaxis

. (16)

Inertia terms have been neglected due to the high friction of cells with their environment,and we also do not consider the existence of any memory term as in [8]. vi is the velocityof the cell i at time t, F ij is the force of cell j on cell i (previously calculated from theimplicit equations (13) and (14), and the sums are over the nearest neighbours in contactwith cell i. The s-friction term determines the friction with the substrate and the cell-cell

friction determines the friction with the nearest neighbours. The tensors Γfij

and Γfis

denote

cell-cell friction and cell-substrate friction, respectively. In our model cells are consideredto be spherical and to be surrounded by a homogeneous and isotropic environment, either agel or a homogeneous intra-cellular matrix, depending on the experimental situation underconsideration. Under these assumptions, the cell-substrate friction tensor takes the form

Γfis

= γI,

where I denotes the identity matrix and γ is the friction coefficient of the medium. Thecell-cell friction is described by the tensor [9]

Γfij

= γ(ij)‖ nijnij + γ

(ij)⊥

(I − nijnij

).

xi and xj are the position of the centre of mass of the cells and nij =xj−xi

|xj−xi|. nijnij here

denotes the dyadic product i.e. it is a 3 × 3-matrix. γ(ij)‖ and γ

(ij)⊥ are the parallel and

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perpendicular friction constants, respectively. The force term is the force that cell i exertson the other cells in contact with it. The noise term models the random component in thecell movement (the micro-motility) and is chosen to be uncorrelated as explained in [8, 9]i.e.

< f(t)f(t′) >≈ 2Γδ(t − t′)

and zero mean< f(t) >= 0.

Γ denotes the amplitude of the autocorrelated noise. Here we approximate Γ ≈ 2γ2DI. Dis the diffusion constant and characterises the free random movement of isolated cells inthe medium. Typically, D ≈ 10−12cm2/s.

The chemotaxis term is the chemotactic/haptotactic response towards a gradient ofmorphogen Q(t) and χ is the cell sensitivity to the chemical. This last term is onlyincluded in some of our simulations scenarios as is specified below.

5 Coupling of cell parameters to intra-cellular molecule con-

centrations

In our model the adhesion forces between cells are controlled by the density of E-cadherinin the cell membrane within the cell-cell contact zone. We could not find in the literaturespecific values for E-cadherin concentration per cell-cell bond. In their simulations Galleet al. [8], used an adhesion energy per unit of area of Wsm = 100µNm−1, so that thesurface receptor density is m = 100µNm−1W−1

s . We use this value as a maximum densityof the cadherin-β-catenin complex in the membrane and define the actual density by

=[E/β]

ETm.

In our simulations we non-dimensionalise all cadherin concentrations by ET , so that thenon-dimensional concentrations of free E-cadherin in the cytoplasm and in the membrane,and of E-cadherin within E-cadherin-β-catenin complexes are all in the interval [0, 1].

Figure (2) shows the resulting force function depending on the different ijm values.

By modifying the intra-cellular concentration of β-catenin the cells can control the con-centration of [Eerin/β]-complexes and thereby the strength of the intercellular adhesionforce.

The active decision of a cell to migrate can be triggered in different ways, all of theminvolving an upregulation of the soluble β-catenin which overcomes the critical thresholdcT . One case happens if the cytoplasmic concentration of β-catenin is up-regulated dueto a failure in the proteasome system. A further case happens if the detachment of localneighbours upregulates the soluble β-catenin concentration. In both cases β-catenin enters

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the nucleus and triggers cell migration. One way that this could cause rupture of the cell-cell contacts is by physical forces that a cell that starts to migrate exerts on the cadherinbonds in the cell-cell contact area to its neighbours. In our simulations we have chosenthe last term in equation (16) that represents chemotaxis so large that the cells at thetumour surface were not able to detach by breaking the cell-cell contacts but they need todown-regulate their adhesion molecules. However, detachment could also be triggered byan increase of the intrinsic random movement component of a cell represented by the noise

term which we do not consider here.

6 Results

In this section we present the results of computational simulations carried out on our modelgiven by equations (7)-(12) and (16). In the first set of simulations in Figures 3 and 4, wepresent results from a numerical simulation of the system of ordinary differential equations(7)-(12) governing the concentrations of various forms of E-cadherin, β-catenin and theproteasome in a single cell.

In order to illustrate the response of possible malfunctions in the intra-cellular controlon the β-catenin concentration, we study simulations for different attachment/detachmentscenarios. If the cell remains adhered to its neighbours, almost all of the β-catenin remainsbound to the E-cadherin complexes at the cell membrane. Alternatively, if a cell detaches,then the concentration of soluble β-catenin increases.

Figure (3) shows the concentration of the intracellular variables of a single cell whenattaches other cells, as can be seen from the figure, soluble β-catenin (yellow line) is rapidlysequestered from the cytoplasm by the cadherins to form the [E/β] complex (red line). Aslong as the contacts are maintained the soluble β-catenin concentration remains at a lowlevel. If some of the neighbouring cells detach then the concentration of E-cadherin formingbonds will be partially reduced.

Figure (4) shows concentrations of the intracellular variables for two different detach-ment scenarios. In the Figure on the left we assume that the cell loses all its bonds with theneighbours at t ≈ 0.4 which triggers a dramatic increase of the β-catenin concentration inthe cytoplasm. This soluble β-catenin enters the nucleus in excess of the threshold concen-tration necessary to initiate migration and promote cell movement via transcription. Onthe right figure, at time t ≈ 0.4, the cell has lost only 1/4 of its bonds with the neighboursand the soluble β-catenin concentration is insufficient to cause cell migration.We implement the intra-cellular dynamics model explained above in every single cell of theindividual-force-based model. The advantage of using this type of modelling approach isthat not only does it enable us to explicitly include the influence of intra-cellular pathways,but also provides a realistic approach to model the biophysical properties of individual cellswhich cannot be neglected when studying tissue organisation. We reproduce in silico dif-

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ferent scenarios of relevance in cancer growth and invasion and study the behaviour ofdetachment waves in epithelial layers and how it can produce the epithelial-mesenchymaltransition. We also study the β-catenin distribution in small tumours and how its up-regulation can induce invasion.

6.1 Detachment waves in epithelial layers

Figures (5) and (6) show the spatio-temporal dynamics in a hexagonal lattice of the cellsβ-catenin concentration, similar to the natural configuration of an epithelial layer. Wechoose here a tissue architecture where we have a layer of cells, with each cell in the layerbeing attached to each neighbouring cell, and initial values of E-cadherin and β-cateninat the steady state for the intra-cellular model. The intra-cellular concentration of β-catenin is denoted by the colour of the cell: Yellow denotes high concentration of solubleβ-catenin and red low. We note that as we have assumed that high concentrations of β-catenin induce cell-cell detachment, this choice of colours is equivalent to saying that reddenotes strong cell adhesion and yellow denotes weak cell adhesion. Within the cell layer,we insert a cell which has no regulation activity in the β-catenin pathway (we force itsintra-cellular dynamics to rise [β] > cT after a certain time and therefore detach from itsnearest neighbours). Figure (5) shows how this malfunction produces a wave caused bythe intra-cellular upregulation of β-catenin in the nearest neighbours. In the way this wavefront is moving, it can be observed how it induces a temporal cell-cell detachment withinthe whole epithelial layer. A particular feature of these waves that should be highlightedis the fact that they do not satisfy the principle of superposition.

6.2 The epithelial mesenchymal transition

Figure (7) shows how the three dimensional spatio-temporal dynamics influence cell mi-gration in a scenario where cells migrate away from a 2d epithelial layer to produce theepithelial-messenchymal transition. The so-called epithelial-mesenchymal transition is aprocess where a well-polarised layer of cells become diffuse and lose their initial structureand compactness. This transition occurs in a similar way at the tumour surface wheninvasion occurs - the mass of outer cells lose contact and invade further into the tissue.To model the epithelial-mesenchymal transition, we set as initial conditions a layer of cellswith fixed boundaries (i.e. unable to move). We include a constant force term as if it werea constant source of chemoattractant (term 5 in eq 9) that diffuses towards the tissue inthe form of the equation

∂Q(t)

∂t= DQ△Q(t),

Figure (7) shows how the cells migrate and the configuration of the epithelial layer is lost.When the proteasome system is down-regulated and a cell detaches it induces the same

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behaviour in neighbouring cells, creating a mass of migrating cells. This is a very similaroutcome to the epithelial-mesenchymal transition. The same migration mechanism can beobserved in a small tumour - cells detach from the outer rim and migrate towards a sourceof attractants, figure (10).

6.3 Tumour growth, aggregation and invasion

Figures (8) and (9) show the distribution of soluble catenin in a 2d cross section of a 3dtumour, and a 2d aggregation process of a culture of metastatic cells on a petri dish. Theprocess of tumour growth involves different cellular interactions which alter and remodelthe E-cadherin configuration at the cell surface with a subsequent impact on β-cateninconcentration. In a proliferative mass of cancer cells, cell detachment, via internal sig-nalling or physical forces, is a necessary process in order to release the stress and allow theproliferation.

Figure (8) shows a section of the tumour (radius ≈ 100µm) where we can observe thedifferent distribution of the intra-cellular catenin, the outer rim of the growing tumour hasa higher number of cells with nuclear β-catenin concentration, while in the centre of thetumour cells downregulate the intra-cellular β-catenin concentration when it is sequesteredby the E-cadherins forming bonds at the cell membrane. These simulations are in verygood agreement with the findings of Kirchner et al. [1].

Figure (9) shows the same patterns found when we study the aggregation process intwo dimensional layers. These findings show how our model can reproduce not only theepithelial-mesenchymal transition but also recover the compactness of the subsequent dis-tant metastatic clusters via aggregation and growth. The same travelling wave patterns asthe ones found in the previous scenarios can be observed within the tumour when allowingthe threshold value cT to be small enough. For threshold values high enough (cT > ET ) thewaves can be avoided but the β-catenin distribution observed by Kirchner et al. [1] remains.

In figure (10) we show results from the same growing tumour scenario stimulated by asource of chemoattractant. If cell detachment occurs, then cells will migrate towards thesource of attractants. The proteasome functionality has been down-regulated to the pointwhere migration occurs. It can be observed that the outer cells migrate and detach fromthe main tumour mass, the new cells at the outer rim lose part of their E-cadherin bonds,upregulate soluble β-catenin and enhance their migration and invasion. These findingssuggest how invasion can be a gradual process produced by subsequent layers of cells thatdetach the tumour surface.

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We performed a steady state and a sensitivity analysis of the intracellular reactionequations. Table 2 reports the result of the sensitivity analysis. Shown is the percentageof the soluble β-catenin in relation to the steady state obtained for the parameters ofTable 1 (e.g. ”100” means the corresponding value has not changed). The intracellulardynamics is very robust with regard to changes of most parameters. The largest sensitivityhas been found for variations of the β-catenin production rate (km) and in the proteasomedegradation rate (k2).The steady states of the β-catenin-proteasome complex and soluble β-catenin are

[Co] =km

k2(17)

and

[βo] =[Co](k

− + k2)

k+(Pt − [Co])=

km(k− + k2)

k+(Pt − kmk2), (18)

respectively. From (18), Pt > [Co] which is satisfied for the chosen parameters values. Inagreement with the results obtained from the sensitivity analysis (Table 2), the solubleβ-catenin concentration at the steady state is controlled by the degradation rate of theβ-catenin-proteasome complex, k2 if k2 is large.We studied the dependency of the multi-scale dynamics on the intracellular β-catenindegradation rate (k2). We simulated an invasion assay of similar characteristics to theones performed by experimentalists in Matrigel Chambers [22]. The initial conditions weretaken as in the scenario simulated in figure (10) where a growing tumour is stimulatedby a source of chemoattractant. We performed the migration assay for three differentdegradation rates: fast (k2 = 10 min−1), medium (k2 = 1 min−1) and no degradation(k2 = 0 min−1), and plot over time the number of cells which achieve to escape from theinitial tumour to a distance of 150 µm. Figure 11 shows the results of the migration assay.It can be seen that the intensity of degradation activity of the proteasome determines thecapacity of invasion of the malignant cells.

7 Discussion

In this chapter we have studied the intra-cellular and extra-cellular dynamics that wouldcause a possible soluble β-catenin upregulation via the release of E-cadherin bonds. Weused as framework a similar approach to the experiments performed by Kirchner and co-workers [1] and we have shown the same patterns of intra-cellular catenin distributionunder a growth process and an aggregation process. Kirchner and co-workers [1] lookingat the intra-cellular concentration of β-catenin postulated how tumour progression wasdriven by interactions with the tumour environment. In our findings we have shown howthis interaction may be mainly driven by the tumour cells themselves. We have shown howdownregulation of β-catenin can be mainly driven by cell-cell contacts and how this fact

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gives an invasive advantage to the cells that are positioned at the outer rim of the tumour.We have simulated the different main steps involved in an invasion process and shown howthe epithelial-mesenchymal transition can be achieved (migration) and reversed (aggrega-tion and growth) depending on the regulation of soluble β catenin by local contacts.More intriguing is how these simple dynamics can create waves of temporal cell-cell detach-ments. When considering the structure of a human tissue, we have to bear in mind how itis exposed to continuous mechanical stress and remodelling. Cell migration, apoptosis andcell mitosis are, probably, the three most important events that can alter the physical con-figuration of the layer. Takeichi [29] and co-workers have shown how E-cadherin expressionwas transient in mouse embryonic brain morphogenesis. They stained different zones of thebrain and neural tube and showed how E-cadherin expression followed particular patternsof expression where positive E-cadherin cells were isolated in different configurations fromnegative E-cadherin cells. In order to maintain the natural tissue configuration and thestress levels under a threshold that permits cell survival, a local mechanism of signallingand migration of neighbouring cells is necessary. This mechanism must allow cells to detachand attach looking for an optimal position such as is seen in an epithelial configuration,but must also be efficient enough to avoid unnecessary detachments and disorganisation ofthe tissue. We do not claim that our model of the internal dynamics of the cell is the exactmodel that occurs in reality but what we want to highlight is how a simple mechanismcan show transient catenin/cadherin (recall that in our model cadherin is surface expressedwhen forming bonds) expression and differentiate the tissue into two separate parts (cad-herin/catenin positive and cadherin/catenin negative). In our simulations we have shownhow a mechanism bi-regulated by cell-cell junctions and an independent degradation sys-tem can be sufficient to do this work. When a cell needs to migrate and releases the bondswith its neighbours, a “travelling wave” of detachment happens. This wave facilitates thecell layer to re-accommodate and release the stress caused by the new movement. Further-more, this model produces a self-regulation mechanism - when two waves collide both ofthem vanish. This helps to prevent tissue disorganisation caused by an excess of signalling.If many cells in the layer are sending detachment signals at the same time the whole layerwill become detached and chaotic, while if the detachment waves vanish when they collide,every single cell is releasing one time its bonds as if it were exist only a single wave.

References

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[2] Natke, I.S. 2004. The Adenomatous, Polyopsis Coli Protein: The achilles heel of thegut epithelium. Annu. Rev. Cell. Dev. Biol. 20: 337-366.

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[3] Hardy, R.G., S.J. Meltzer , J. A. Jankowsky. 2000. ABC of colorectal cancer, Molecularbasis of risk factors. Clinical Rev. 321: 886-889.

[4] Chaplain, M.A.J., L. Graziano, L. Preziosi. 2006. Mathematical modelling of the lossof tissue compression responsiveness and its role in solid tumour development. Math.Med. Biol. 23: 197-229.

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[6] Armstrong, N. J., K. J. Painter, J. A. Sherratt. 2006. A continuum approach tomodelling cell-cell adhesion. J. Theor. Biol. 243: 98-113.

[7] Araujo, R.P., D.L. McElwain. 2004. A history of the study of solid tumour growth:the contribution of mathematical models. Bull. Math. Biol. 66: 1039-1091.

[8] Galle, J., M. Loeffer, D. Drasdo. 2005. Modelling the effect of deregulated proliferationand apoptosis on the growth dynamics of epithelial cell populations in vitro. Biophys.88: 62-75.

[9] Drasdo, D., S. Hoehme, M. Block. 2007. On the role of physics in the growth andpattern formation of multi-cellular systems: What can we learn from individual-cellbased models? J. Stat. Phys. (in press; available online at J. Stat. Phys. homepage)

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[12] Ramis-Conde, I., M. A. J. Chaplain, A.R.A. Anderson. 2006. Mathematical modellingof cancer cell invasion of tissue. Math. and Comp. Modelling. To appear.

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[17] Alber, M.S., M.A. Kiskowski, J.A. Glazier, Y. Jiang. 2002. On Cellular AutomatonApproaches to Modeling Biological Cells. In: Mathematical Systems Theory in Biol-ogy, Communication, and Finance (eds. J. Rosenthal and D. S. Gilliam), 1-40, IMA142, Springer, New York.

[18] Drasdo, D. 2003. On selected individual-based approaches to the dynamics of multicel-lular systems. In: Multiscale modeling. (eds. W. Alt, M. Griebel, J. Lenz), Birkhauser,Basel.

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[23] Friedl, P. 2004. Prespecification and plasticity: Shifting mechanisms of cell migration.Curr. Opin. Cell Biol. 16(1), 14-23.

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[25] Huber, O., c. Bierkamp, R. Kemler. 1996. Cadherins and catenins in development.Curr. Opin. Cell Biol. 8: 685-691.

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[30] Chen Y-T., D. B. Stewart, W.J. Nelson. 1999. Coupling Assembly of the E-cadherin/βcomplex to efficient endoplasmic reticulum exit and basal-lateral membrane targetingof e-cadherin in polarized MDCK cells. JCB. 144: 687-699.

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Legends

Figure 1

The three states of E-cadherin considered in the model: free in the cytoplasm, just ar-rived to the cell membrane and forming bonds in a multiprotein complex which includesβ-catenin. When two cells become in contact, the cadherin travels to the membrane de-termined by the function c(t), it binds, between other molecules, to β-catenin and forms abond with the neighbour cell. When detachment occurs, the complex β-catenin-E-cadherinis broken at a rate governed by the function d(t).

Figure 2

The left plot shows the force function between two cells, variables are cell-cell distance (inµm) and adhesion energy per unit of area in contact (in µN/m). The right plot shows thevertical view of the same graph, where it can be better observed the adhesive interactionbetween cells depending on the E-cadherin concentration forming bonds. The orange zonedetermines the zone where adhesive forces act.

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Table 1: The tabular shows the parameter values used in the intracellular simulations.When possible the parameters were taken from literature. ⋆ Those intracellular parametersnot found in experimental data were assumed in order to maintain the β-catenin steadyvalue at 35nM as observed by Heinrich et al.

Parameter Definition value Source

ν E-cadherin-β-catenin binding rate 100 min−1 ⋆ estimatedk+ β-catenin proteasome normal binding rate 100 min−1 ⋆ estimatedk+ β-catenin proteasome downregulated binding rate 0 min−1 estimatedk− β-catenin-proteasome dissociation rate 19 min−1 ⋆ estimatedk2 β-catenin degradation rate in proteasome 0.03 min−1 ⋆ estimatedkm β-catenin production rate 0.01 nM/min−1 [31]Pt Proteasome total concentration 0.33514 nM ⋆ estimatedEt E-cadherin total concentration 100 nM estimatedρi E-cadherin surface translocation rate 200 min−1 ⋆ estimatedα E-cadherin-β catenin dissociation rate 2 min−1 estimatedcT β-catenin threshold value cT = 50 estimatedR0 Cell radius 5µm [8]E Young modulus 1 kPa [8]ν Poisson ratio 1/3 [8]γ‖ Parallel friction constant 0.5 ∗ 1011 Ns/m3 [8]

γ⊥ Perpendicular friction constant 0.5 ∗ 1011 Ns/m3 [8]Ws Adhesion energy 200 µN/m [8]γ Cell-medium friction constant 0.4 Ns/m3 [8]D Cell diffusion constant 10−12cm2/s [8]

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Figure 3

Plot showing the concentrations over time of the intracellular variables of a cell that at-taches to a group of cells at t ≈ 0.2 minutes. β-catenin is rapidly sequestered by thecadherins that travel to the cell surface to form bonds.

Figure 4

Plots showing the concentrations over time of the intracellular variables under two differentscenarios. On the left plot, a cell looses its contact with its neighbours at t ≈ 0.4 minutes.The β-catenin concentration increases dramatically and it enhances mechanisms whichpromote invasion. On the right plot only a few of the neighbours are detached, solubleβ-catenin is maintained under the threshold levels (cT = 0.5) that enhances migration.

Figure 5

Plots of the spatio-temporal dynamics of [β] in a layer of cells where a single cell withupregulated soluble catenin (yellow) is situated on a layer with defective proteasome system.As can be seen from the plots, it produces a wave of upregulated β catenin (yellow) causedby the induced decision of detachment in other cells. After the wave has passed, strongadhesion is recovered (red). Unit of time is measure in minutes.

Table 2: The table shows the results of the sensitivity analysis performed to study thevariation of soluble β-catenin. The values obtained from running simulations in differentorders of magnitude are given in percentages with respect to the β-catenin steady stateconcentration obtained from substituting the parameters of table 1 in equation (18). Itcan be observed that the intracellular dynamics are very robust with respect to most ofthe parameters.

Parameter ×10 ×102 ×10−1 ×10−2

ν 100 100 100 100k+ 99.974 100 101.847 100.200k− 100.237 101.234 99.974 99.972k2 55.056 0.005 104.502 104.952km 144.900 594.976 95.50 95.054ρi 100 100 100 100α 100 100 100 100

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Figure 6

Plots showing the spatio-temporal dynamics of a scenario where two cells with upregu-lated soluble β-catenin are situated on a layer with defective proteasome system (yellow),inducing two detachment waves. The detachment waves collide and vanish. This outcomeprevents the cell layer to become disorganized due to an excess of detachment signal. Unitof time is measure in minutes.

Figure 7

Plots showing how malfunctions in the proteasome system can alter the layer configurationproducing the epithelial-mesenchymal transition. In this figure, the cells migrate towardsa source of attractants escaping from the initial epithelial configuration. Migration canoccur only when the catenin levels are over a determined threshold (yellow). Unit of timeis measure in minutes.

Figure 8

Plot showing a transversal section of a tumour shows how the catenin spatial distributiondepends on the tumour geometry. Cells in orange, have less binding neighbours and thecatenin concentration is not attached to the cadherins and free to go into the nucleus. Cellsin the center of the tumour show how catenin is better downregulated by a larger numberof binding neighbours (red).

Figure 9

Plot showing a scenario where cells aggregate and grow in a two dimensional configurationin a petri dish where they show similar patterns of β-catenin distribution as those found byKirschner and co-workers [1]. Orange cells denote a higher β-catenin nuclear concentration.

Figure 10

Plot showing how a small tumour invades further tissue stimulated by a source of mor-phogen located at the r.h.s. of the tumour. Cells decide to detach gradually when theintracellular concentration of β-catenin is upregulated (Yellow). Unit of time is measurein minutes.

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Figure 11

Plot showing the simulation results of a cell invasion assay. The plot shows the numberof cells that achieve a migration distance of 150µm away from the principal tumour overtime. It can be observed that as the β-catenin degradation rate is decreased (k2 = 10, 1and 0 min−1), the malignant cells become more invasive.

Figure 1:

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Figure 3:

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Figure 5:

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Figure 6:

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Figure 8:

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Figure 10:

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Figure 11:

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