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A Computer Model for Reshaping of Cells in EpitheliaDue to In-plane Deformation and AnnealingG. WAYNE BRODLAND a & JIM H. VELDHUIS aa Department of Civil Engineering , University of Waterloo , Waterloo , CanadaPublished online: 04 Jun 2013.

To cite this article: G. WAYNE BRODLAND & JIM H. VELDHUIS (2003) A Computer Model for Reshaping of Cells in Epithelia Dueto In-plane Deformation and Annealing, Computer Methods in Biomechanics and Biomedical Engineering, 6:2, 89-98, DOI:10.1080/1025584031000078934

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Computer Methods in Biomechanics and Biomedical Engineering, 2003 VoL. 6 (2), pp. 89-98

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A Computer Model for Reshaping of Cells in Epithelia Due to In-plane Deformation and Annealing

G. WAYNE BRODLAND* and JIM H. VELDHUIS

Department of Civil Engineering, University of Waterloo, Waterloo, ON, Canada N2L JG J

(Received 3 July 2002; In final form 22 November 2002)

Although cell reshaping is fundamental to the mechanics of epithelia, technical barriers have prevented the methods of mechanics from being used to investigate it. These barriers have recently been overcome by the cell-based finite element formulation of Chen and Brodland. Here, parameters to describe the fabric of an epithelium in terms of cell shape and orientation and cell edge density are defined. Then, rectangular "patches" of model epithelia having various initial fabric parameters are generated and are either allowed to anneal or are subjected to one of several patterns of in-plane deformation. The simulations show that cell reshaping lags the deformation history, that it is allayed by cell rearrangement and that it causes the epithelium as a whole to exhibit viscoelastic mechanical properties. Equations to describe changes in cell shape due to annealing and in-plane deformation are presented.

Keywords: Computer model; Epithelia; Cell shape; Cell rearrangement; Cell mechanics; Finite element method

INTRODUCTION

During embryo morphogenesis, wound healing and other important biological processes, structured monolayers of cells called epithelia (Fig.l) undergo dramatic in-plane deformations that include stretching, narrowing and shear [I -5]. In some cases, the cells in these epithelia undergo significant reshaping, while in others they do not because deformation of the epithelium is accommodated almost entirely by cell rearrangement.

Although previous analyses and computer simulations have shown that the stresses in an epithelium depend strongly on the shapes of its cells (6], a fundamental understanding of cell reshaping, and equations to describe it, have been conspicuously lacking. This is because the successive geometric and topological changes that occur in the cells of the epithelium rule out purely analytical methods and substantially complicate numerical approaches. The purpose of this paper is to use the cell­based computer formulation of Chen and Brodland (7] to investigate cell reshaping and to devise equations that can be used to describe it.

Because the stresses in an epithelium depend so strongly on cell shape, equations to predict cell reshaping are a critical step towards the development of cell-based constitutive equations. Constitutive equations of this kind

are needed so that accurate computer simulations can be carried out of the morphogenetic movements that epithelia undergo as they reshape to form organs and other critical structures. Thus, they are critical to understanding the mechanics of normal morphogenesis and to identification of the mechanical causes of malformation-type birth defects.

To model the mechanics of the cell-cell interactions of interest here, it is necessary that the model cells have realistic contact angles so that the critical force interactions that occur at their triple junctions can be modelled properly [8]. This requires that the model cells be space filling. It is also necessary to have suitable material models for the cytoplasm, and to update the forces generated by the cytoplasm based on the current geometry of each cell. Many current families of numerical models [9-13] satisfy some of these requirements, but only finite element formulations like that of Chen and Brodland [7] satisfy them all. In their formulation, forces generated by cytoskeletal components, the cell membrane and cell-cell adhesions are resolved into an equivalent surface tension 'Y (Fig. 2(a)). This approach is similar to that used in many other computer simulations of cells [9-121. However, unlike other approaches, the formu­lation of Chen and Brodland models the cytoplasm, its associated filamentous networks and the basement

*Corresponding author. Tel.: + 1-519-888-4567. Ext. 6211. Fax: + 1-519-888-6197. E-mail: brodland @uwaterloo.ca

ISSN Hl25-5842 priniiiSSN 1476-8259 online © 2003 Taylor & Francis Ltd DO!: 10.108011025584031000078934

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90 G. W. BRODLAND AN D J.H. VELDHUIS

Cell Membrane -

Nucleus

Control Blochemlcals

Basement Membrane

Mlcrofllament Bundle

Cell Adhesion Molecules

Extracellular Matrix

FIGURE I A schematic of a piece o f ep ithelium. Component s o f mechanica l importance are shown.

me mbrane by an e ffective viscosity 1-L· The mechani ca l e ffect of the cytoplas m is recalcul ated at every time step based o n the current geometry o f each ce ll. Multiple finite e le me nts are used to model each cell (Fig. 2(b)), and cell s can rearrange within the plane. The result is a powerful and general-purpose " first-order" cell mode l.

Previous studies [ 141 based o n the formul ati on o f Chen and Brodl and have shown that when a mode l epithe lium is stretched , the cell s beg in to elongate in the directi on o f stretch. When the cell s have reached a certain aspec t rati o, it becomes energeti call y pre ferable for cell s to rearrange to accommodate ongoing stretching o f the epithelium rather than for individua l cell s to stretch furth er. The aspec t rati o at which thi s happens depends on the rati o o f y to /-L . on the cell edge density, and on the strain rate e. In those studi es, cell reshaping was unidirecti onal and genera l pu rpose fabri c parameters were not required .

Microfllaments

Cell Adhesion Molecules (CAMs)

(a)

(b)

Cell Membrane

Microtubules

Cell Cytoplasm

FIGURE 2 The ce ll and lini tc e lement mode ls. (a) A typica l ce ll shown in plan view. (b) The corresponding ll nitc .: lcmcnt modc l. See Ref. 171 for deta il s.

That formul ati on has also proved useful for investigating the effects of mitosis on mode l epithe li a r 15].

The purpose of thi s study is to use the model of Chen a nd Brodl and to investi ga te ce ll reshapin g under conditi ons that are more general and to dev ise equations to describe it. The first step in thi s process is to de fine paramete rs th at desc ribe the ce llul ar fabri c o f an epithe lium in terms o f cell shape and ori entatio n, and cell edge density. Nex t, a seri es o f computer simulati ons are used to investi gate mode l epithe li a that have various initi a l fabric parameters and that subseque ntl y undergo ex tension, narrowing and shear. The changes to cell orientati o n and aspect rati o over time are then plotted and used to deve lop equations to desc ribe the m. The simul ati ons show that cell reshaping lags the strain hi story of the epithe lium, that it is a ll ayed by cell rearrangement and that these e ffects cause the epithe lium as a who le to exhibit mechanica l properti es similar to those o f a viscoelasti c liquid .

FORMULATION

Definition of the Fabric Parameters

Previous studi es o f mode l epitheli a have shown that the stresses in an epithe lium are hi ghl y sensiti ve to its current fabric f6 1, and that thi s fabric can change dramati ca ll y if a pal ch is deformed 17]. E xpe rimenta l [16 - 201 a nd computer 11 3,2 1] studi es o f ce ll masses show that if the mass is de formed quickl y, the ce ll s de form in a manner s imilar to the mass. If the mass is the n he ld in it s de formed slate, the cell s annea l within the mass- that is, they reshape and rearra nge within lhe mass so that indi vidual ce ll s become essenti a ll y isotropi c as they were in the o ri g ina l, unde fonned mass. The force required to ho ld the mass in it s deformed shape decays to a reduced va lue as the cell s anneal. Deta il s o f thi s process ha ve recentl y been e lucidated 12 1] using computer simulati o ns similar to those used here.

To investi gat e changes in ce ll shape. it is necessa ry 10

de fin e fabric parameters that arc we ll -behaved, es pec ially w ith respect 10 angular changes o f the coordinate system, and that lend themse lves to phys ica l inlerpretati on. Here, a set o f parameters thal sati s fi es these conditio ns

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CELL RESHAPING IN EMBRYONI EPITHELIA 91

and that represents cell shape, orientation and cell edge den ity i derived. The parameters are imilar to the kinds often used to describe other cellular materials [22].

Con ider a rectangular patch of n cells as shown chematically in Fig. 3. Following Chen and Brodland

[7], an interfacial tension -y is assumed to act along each cell boundary and to embody the mechanical effects of the cell membrane and its associated networks, contrac­tion of microfilament bundles, cell-cell adhesions and any other force-generating structures within the cell. The cytoplasm and it associated networks together with any organelles present are assumed to resist this motion as if they had an effective viscosity of 11- [7]. In the case of an epithelium, this viscosity is as umed to al o embody the properties of the basement membrane and extra-cellular matrix.

To characterize the geometry of the cells in the patch, a cell edge density p is defined such that

B p=A, (I)

where B is the sum of the lengths of all internal cell boundaries plus half of the perimeter of the patch, and A is the area of the patch. Half of the perimeter of the model is u ed because these edges are assumed to be shared between the patch shown and a mirror image patch that adjoins them. Physically, p corresponds to the area density of the cell edges. It is related to the ratio of the number of cells n per unit area A by the formula [141

p=ff-;., (2)

where f is a form factor that imulations how has a value of approximately 1.92 for isotropic patche with nearly uniform cell sizes. The value off increa es slightly with the shape factor K defined below, rising to 1.97 when K = 2. Although cell density may be more easily

y Composite Cell

FIGURE 3 A rectangular patch of epithelial cells. The fabric of the patch is defined by the shape K* and orientation a* of the elliptical composite cell shown, and by the cell densi ty p.

measured in experiments, cell edge density is more useful in the development of constitutive equations. Here, the latter is chosen as the primary descriptor.

For each cell, i, a shape factor

K; = (3)

is defined, where /~ax and /:nin are, respectively, the largest and smallest principal moments of inertia ( econd moments of area) of the cell. This is a generalisation of the cell shape parameter defined by Chen and Brodland [7] . In practice, the maximum and minimum moments of inertia of the ith cell are calculated from the centroidal moments / 1 and 1;

. u " and product of inertia /~ of that cell using standard transformation techniques such as Mohr's circle [23]. The process by which the maximum and minimum moments of inertia are found is represented by the two functions

/~ax = lmax (I:X, I~Y ' l~y) and

/:~in = lmin (1:X, I~Y' I~). (4)

Phy ically, the shape factor, K; corresponds to the ratio between the major and minor axes of an ellip e having the same moments of inertia as the cell. Thus it is a type of aspect ratio. The average cell shape factor, R is defined as

J 11

R= - LK;. 11

1= 1

(5)

In calculating this average and all other parameters that are based on a sum over the cells in a patch, any cells that contact the border of the patch are excluded since they often do not have representative shapes.

A composite cell (Fig. 3) is defined as an imaginary cell whose moments and product of inertia are the average of the centroidal moment and product of the individual cells; i.e. a cell whose mom nts and product of inertia are given by

For such a cell, th principal moments of inertia are

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92 G.W. BRODLAND AND J.H. VELDHUIS

and its shape factor is given by

K* = (8)

The parameters i< and K* are frame indifferent and invariant with respect to changes in scale.

The angle to the major axis of the composite cell is called a*, where a counterclockwise angle from the horizontal is considered positive, and -1rj2 < a ~ 1rj2. In general,

( 21" ) a* = arctan xy . r -r XX yy

However, should /~ = /~, then

{

1! 4'

a*= 0, -1!

4'

r >o xy

r =o xy

r <o xy

(9)

(10)

Together, the three parameters, p, K* and a* provide a set of parameters, representing cell edge density and cell shape and orientation, respectively, that are well-behaved with respect to coordinate transformations. The concepts used here to describe planar aggregates could be extended to three-dimensional aggregates without difficulty.

To non-dimensionalize the reporting of force data, a ratio of the forces generated by interfacial tensions to those produced by viscous forces is defined by the parameter

'YP Q = 2eJL8' 00

where 'Y is the interfacial tension along the junction bet­ween two cells, e is the true strain rate of patch elongation and 5 is the thickness of the cell patch. For an epithelium, 5 is the thickness of the epithelium and 'Y is the total force generated at each boundary. For a three-dimensional mass, 5 is assigned a unit value so that it is consistent with 'Y which, in that case, is calculated per unit of mass thickness. A dimensionless time parameter is defined by

where t is time.

. ~~

ypt 'T = 2JL5'

The Finite Element Formulation

(12)

To investigate the mechanical interactions that occur between cells in a tissue, the finite element method [24,25] as implemented in a cell-based formulation by Chen and Brodland [7] is used. Each n-sided cell (Fig. 2(b)) is broken

into n triangular elements. These area elements model the cytoplasm together with any networks of intermediate filaments or other cytoplasmic components of mechanical importance. These networks are sufficiently labile that they produce an equivalent viscosity JL at the low strain rates (of the order of 10-4/s) typical of morphogenetic movements. The viscosity JL is also assumed to embody the mechanical effects of the basement membrane and its interactions with the cells.

Elsewhere [7,26], the authors have discussed the forces produced by circumferential microfilament bundles, apical mats of microtubules, the cell membrane and its associated networks, and cell-cell adhesions, and have argued that they can be resolved into equivalent nodal forces and approximated by an equivalent interfacial tension 'Y (Fig. 2). This interfacial tension is modelled using a constant-force rod element along each cell boundary. Together, the assumptions of a constant interfacial tension 'Y and a cytoplasmic viscosity JL give rise to a "first order" model of epithelia that is appropriate at the strain rates characteristic of morphogenetic movements.

An updated Lagrangian formulation is used, and since none of the materials in the model are viscoelastic, stresses depend only on the current geometry and strain rates; parameters that embody the strain history of the materials are not required.

RESULTS

Cell Annealing

One of the distinctive characteristics of biological cells in aggregates is that they tend to anneal under the action of their interfacial tensions [ 16,18,19). To model this, an anisotropic starting configuration consisting of a patch of 200 cells was generated by constructing a Voronoi tessella­tion on a W by W region, annealing it fully, and then multiplying the x-coordinates of each point by a factor that yielded a starting K* of Ko = 2.0 (Fig. 4(a)). The result was a patch that was 1.97W long in the x-direction, that had cells which were strongly aligned and approximately twice as wide as they were tall (a* = -0.335° and i< = 2.035). The interfacial tension 'Y along each cell edge was then allowed to act until a state of equilibrium was reached (Fig. 4(b)). As expected [7], a honeycomb-like pattern results .

Figure 5(a) shows how K* changes as the patch anneals. It would not be surprising if the more a cell is out-of­round, the greater the forces that drive it to become more rounded. Thus, it would be reasonable to expect the K*- T

curve to exhibit an exponential decay according to the first part of the following equation

{

Koo + (Ko - Koo)e -A.J."''"""'•T, K* > Kcutoff & ?j- ~ 0 (a)

K* ( 'T) = K*, K* ~ Kcutoff & !!JC..d ~ 0t (b) * T Koo +(KO - Kex>)e -A,"'enomgT' d;T > 0 (C)

tThat is if the driving forces would otherwise cause K* to decrease

(13)

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CELL RESHAPING TN EMBRYON IC EPITHELIA 93

y L = 1.97 W

w

(a)

(b) FIGURE 4 Annealing of a rectangular patch . (a) The initial configuration, with Ko = 2.0. (b) The annealed patch. When the figure is viewed from a low angle so that the vertical dimension is strongly foreshortened, the columnar pattern of the cells. characteristic of locking. is evident.

where ,\descending is a decay parameter and Koo i th asymptote of the decay. A series of similar annealing te ts were carried out for patches with Ko = 1.5 and 2.5. The annealing curves were shifted horizontally by various Torrse• values to enhance the fitting of a single exponential curve. The best visual fit to the three K* - Tcurves shown is obtained by setting A = 0.07 and Koo = I. Discrepancies between the simulation and Eq. ( 13(a)) ari e in part because the initial configurations were not equilibrium

(a)

'tQffset

2.5 Ko= 2.5 -• ~ KO= 2.0 -Cl)

~ 2 Ul 0 K 0= 1.5 Q. E 0

1.5 0 C'll Q. Q. Kcutoff = 1.25 C'll ~

0 25 50 75 100

Time (-r)

configurations, and rapid local adju tments occur during the first few time steps causing the initial parts of each curve to descend at an augmented rate.

To understand the horizontal asymptotes predicted by the simulations, it is necessary to consider the process of annealing- a process in which potential energy is released as contraction of cell boundaries deforms viscous cytoplasm. Graphically, this can be understood as progre sion down a free energy hypersurface (Fig. 6). If the system has the geometry and net energy E corre ponding to point A in Fig. 6, the surface in the neighbourhood of A can be con tructed by setting its directional derivatives with respect to incremental displacements u; equal to the net driving force/;, in that direction; i.e. by etting

aE -=J;. au;

( 14)

As suggested by the figure, the sy tern may be drawn into a local minimum, such as Point . Unless an external perturbation acts, the system will not be able to escape this and reach the global minimum, Point D.

A typical Voronoi patch i initially isotropic and will anneal to apprmdmately K* = 1.02 and iC = 1.25 . Thus, although the annealed patch is very nearly isotropic, individual cell s are not free to become completely round, bur instead be om lock din loca l minima in which typical individual ells are approximately 1.25 time as long as they arc wide. When an anisotropi pat h anneal , its cells also be ome locked in local minima when they reach a certain length to width ratio . However, sin e they remain aligned during annealing, K* does not descend below a cutoff value shown a Kculoff = 1.25 (Eq. 13(b)) in Fig. 5(a). The slight reduction in K* that occurs in simulations that start below K* = 1.25 are apparently due to local, initial adjustment of the type noted above. In tests where a patch is stretching sufficiently quickly that K* increases with 'T

(b)

--+11-~ 't Offset

1

Ko= 2.0 & Eqn (16) - 0.8 0 -Ul Ul Cl)

0.6 "" .. tJ)

0.4 0 25 50 75 100

Time (t )

FIGURE 5 (a) Kappa composite versus time. (b) Stress versus time. All quantities shown are dimensionless.

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94 G.W. BRODLAND AND J.H. VELDHUlS

FIGURE 6 A schematic of the free energy surface.

(not shown, but similar to Ref. [7]), the e ffect of annea ling is less pronounced, and a va lue of Aasccnding = 0.04 (Eq. 13(c)) provides a good fit to the simul ati ons.

T he si mul ati ons revea l ano ther phenomena that produces loca l minima during annea ling of pre-stretched cell patches . Observing Fig. 4(b) from a low viewing angle so that the verti ca l ax is of the graph is strongly fores hortened revea ls that the cell s tend to line up in columns normal to the directi on of e longati on. This phenomena also occurs when ce ll sheets are stretched (Fig. 3(b) of Ref. [7]). To understand why thi s happens, cons ider a triple j uncti on along the boundary between two typica l co lu mns of ce ll s. The cell angles arc typica ll y at approximately 120° and there is no imbalance to move that node up or down so that one of its sides becomes shorter and a rearra ngement ultimate ly occurs. Thi s phenomenon can not be significantl y ame liorated by increasing the range of cell sizes nor by temporary modificati ons to the computer code to encourage ce ll s to move inwa rds from the top and bottom edges of the patch.

A number of these characteri stics are observed during annea ling of two-di mensional fcrrolluid foams 127]. Thi s is not surprisi ng since these foams are nearl y ana logous to the cell mode l used here, differing onl y because the gas in the foam is inv iscid. Like the model ce ll s, these foams lose relatively litt le tota l edge length during annea ling and they tend to lock in config urati ons that correspond to loca l, but not g loba l, min ima. When a loca l bubble rearrangement occurs in a foam, further re laxa ti on occurs in the edges o f adjacen t bubbles and thi s may trigger one or more additiona l rearrange ments. Foams with a larger range of bubble sizes tend to be more subject to cascades of bubble rearrangemcnts. Figure 7 shows that cell rcarrangemcnts are also typica ll y fo llowed by a slight exponenti al- like decay in K* (labelled c). as the rea rranged ce ll s and the ir neighbours undergo further loca l annea ling. This e ffect is particu larly ev ident towards the ri ght of the fi gure where the ce ll rearrangemen ts are more wide ly separated in time. Both the model ce ll s and the fcnolluid bubbles are rather highly constrained, in that they consist of di screte unit s,

- 15 a: 0 -tn - 10 c Q)

E Q) Cl

5 c ea ... ... ea Q)

a: 0 a; 0 0 20 40

Time {'t )

1.5

1.4

1.3

-·~ -~ tn 0 c. E 0 0 ea c. c. ea ~

FIGU RE 7 Kappa composite and cell rearrangement versus time. The simulation that generated thi s data used Fig. 4(a) as the starting config uration. but used a step size 8 ti rnes larger than those used in the other sirnulations. The larger step size allowed some of the local minima to be overshot, accounting for the lower K* va lue that is achieved, and it produced changes in cell shape that are more ev ident in the graph. The use of such a large step size cannot be justified in general. The exponential decays in K* fo llowing cell rearrangement are labe lled with an e. Note that the initial part of the K* curve ha> been truncated so that the K* ax is can be shown magnified.

they ha ve indi vidual volume constraints, and they must remain close packed. Rea l cell s are not so ri gorously constra ined. Instead, they need not take the shape of exact ri ght polyhedral prisms and they tend to undergo constant movement relati ve to each other. In addition, mitos is facilitates rean angement by providing frequent mecha­nical and topologica l disruptions and by increasing the range of ce ll s izes. As a result, the loca l minima shown in Fig. 6 may not be significant in real ti ssues.

Recent studies of the viscoelasti c properti es of cell masses show that some of their mechanical characteri stics can be c lose ly approx imate ly by the sum of two decay ing exponenti als and a constant r 19]. rather than by one ex ponenti al and a constant as used in Eq. ( 13). The curve fo r Ko = 2.0 in Fig. 5(a) also corre lates well (R 2 = 0 .9999) with the two-exponent equation

K( T * ) = 1.428+ 0.33e o.so-r* + 0 .33e Oll -r* (1 5)

where

T* = T - TQff;~t · ( 16)

It is unc lea r whethe r there is a causa l basis for the strong corre lation between the numerica l result s and the form of Eq. ( 15). although it may be the result of cell reshaping towards K = I interacting with a locking mechani sm. Alternati vely, the difference may ari se because the cell masses studied experimenta ll y are three-dimensiona l while the present model is planar. In any case. it is not difficult to generali se Eq . ( 13) by adding additional exponenti al terms, as needed .

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CELL RESHAPING IN EMBRYONJ EPITHELIA 95

Figure 5(b) shows a dimension1e s stress - time curve for the Ko = 2.0 annealing simulation. Dimensionle stress S correlates (R 2 = 0.9999) with,

Sy = 2Fx = 0.48 + 0.26e - 0.50T* + 0.26e - O.ItT* ' (17) ypL

where Fx is the compressive force required along the top edge of the cell patch to prevent it from moving upwards, and L is the length of the patch in the x-direction. Statistical mechanics-based equations show that there is a causal relationship between cell hape and stres in mod~! epithelia [6]. That the same two time constants appear m Eqs. (15) and (17) and that the coefficients in front of the corresponding exponential terms are proportional to eac.h other, indicates that the transient portion of Sy ts proportional to the transient portion of K*, and confirms this relationship.

Bulk Deformation

To complement the studies of cell patch stretching published elsewhere [7], many combinations of patch extension, narrowing and shear were investigated. Figure 8 shows one of these test , consisting of the shear of an isotropic, L by L patch (Fig. 8, upper left inset) . Because the deformations are large, the direction of instantaneous principal stretch rotates. The nodes along the edges of the patch are constrained to remain in a straight line and the upper edge of the rhombus is moved in the x-direction at a constant velocity (3L. The displacement d of the upper edge at any moment is given by

...... ~ -s ·c;; &. E 0 0

3

2.5

2

ea 1.5 Cl. Q.

~ 1

0

d = (3Lt

Eqn (20} (P = 0)

0.5 1

P=O

P= 10

p =50

1.5

Displacement (d/L)

( 18)

2

FIGURE 8 Kappa composite versus displacement for pure shear.

Figure 8 shows K* for three different ratios of interfacial tension forces to viscous forces

( 19)

When no intetfacial tensions act (P = 0), the cells elon­gate and rotate passively approximately like the principal diagonal of the rhombus. It is then possible to estimate the shapes of the cells by caJculating the length of the major axis of the rhombus and of the original square, noting that cells must narrow by the same ratio as they elongate since their volume is constant. The result is given by

As hown in Fig. 8, the finite element si mutations give a value of K* that is quite close to that predicted by Eq. (20). When interfacial tensions are present (P = l 0 and P = 50), cell elongation is allayed by elf-rounding as soon as the cells begin to stretch. The actual shapes obtained depend on a balance between the principal elongation produced by shear and the rounding produced by annealing. Although this is similar to the case of pure elongation [7], here, the rate and direction of principal elongation are constantly changing. The ellip es in the insets along the top of Fig. 8 show the shape and orientations of the composite cell for the case where P = 10.

Figure 9 haws a square patch that has an anisotropic fabric (Ko = 2.03 at an orientation of ao = 59 .7°) in its initial configuration and that is then stretched in t11e x-direction at a fixed true strain rate e. Figure 9 shows how K* changes as the patch is stretched until it reache 5 times it original length. When there are no interfacial tensions (Q = 0, a defined by Eq. (11 )), the cells deform as if they were drawn on a rubber sheet that was then stretched . Initially, the cells are oriented more in line with they-axis than the x-axi and so patch tretching causes them to be ome wider and somewhat shorter during the initial stages of deformation, causing K* t decrease. Soon, however, the long axes of the cells have rotated sufficiently that they are oriented more towards the x-axis than they-axis, and K* increases again with A.

When interfacial tensions are present (Q = 20 and 100, as defined by q. ( 11 )), annealing contributes to the initial rounding of the cells and causes the cells to reach a lower minimum K* value (Fig. 9). By the time the patch has stretched to A = 5, the curve for Q = I 00 has reached a steady stat value of approximately 1.8, which is compatible with the values reported in hen and Brodland r7] for tretching of an i otropic patch. Thus, in thi s case, by A = 5, the fabric is independent of the details of the configuration from which it tarted. For the case where Q = 20, the interfacial tensions are smaller, and the fabric does not reach a steady state value by A = 5.

The insets along the top of Fig. 9 show the composite cell shape and orientation for the Q = 100 case. For that

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96 G.W. BRODLAND AND J.H. VELDHUIS

-... ~ -~ en 0 Q. E 0 0 ns Q. Q. ns ~

' .. y

~X 0

3

2.5

2

1.5

1

1 2 3 Stretch (A.)

4 5

FIGURE 9 Kappa composi te versus stretch for a patch wi th Ko = 2.03 and a 0 = 59 .7°.

case, the initial cell rounding e ffect is suffic iently strong that , in the absence of ex ternal forces, it would cause the patch to lengthen in the x-direction more quickl y than the specified strain rate. This g ives ri se to a stress in the x-direction (not shown) which is initi all y compress ive ((]'x = - 42.7). As the patch is stretched, the cell s align with the s tre tch direc tion and (]'> asymptotica ll y approaches a value of u, = 90, which is in agreement with previous simulations and analytica l ca lculations for Q = 100 [61 .

DIS USSION AND CONCLUSIONS

The computer simulations reported here are based on a first-order model for epitheli a. This model assumes that the mechanica l e ffect of the surface-re lated components of the cell can be approx imated by an interfac ial tension 'Y and that the internal components (and any basement membrane) can be approximated by a viscos ity p,. Although cell membrane systems 128 - 301 and the ce ll cytoplasm 131 ,32] are known to be viscoelasti c and to demonstrate other hystereti c e ffects, at the rates of interest here, the present first-order model seems appropri ate. The model is corroborated by recent studies of ce ll sorting and ti ssue engulfment in which differences in cell surface tensions [33 1 dri ve cell rearrangements and do so at rates consistent with the morphogenetic movements of interest here. The compute r model used here successfull y models the complex sequence of cell - cell interactions and rearrangements tha t charac te ri se these phe nom ena

f 14,34] . Simu lations based on thi s model have also shown that modest variations in interfacial tension from one cell edge to another, as might occur due to random cell surface phenomena or stra in-rate e ffects have neglig ible impact on the outcomes.

The parameters K*, a* and p, corresponding to cell shape and orien tation and cell edge density, respective ly, are found to provide descriptions of the cellular fabri c of the epithelium that are intuitive and sensitive to small tex tural changes. They also exhibit good invariance characteristics with respect to patch size and coordinate sca le and orientation. When at least 200 cell s are used in the simu lati ons, the parameter va lues are typica ll y repeatable within approximately 2%, because cell - ce ll variations are averaged out and edge e ffects arc limited.

The simul ations show that ce ll s in a ti ssue require time to reshape and that thi s causes the ti ssue to exhibit viscoelastic characteristi cs that arc ev idenced in at least three ways. If the shape of a cell patch is he ld fixed, the cell s in it annea l and the stresses required to maintain the patch geometry decay to a plateau value . If an initially isotropic patch deforms at a constan t rate in terms of true strain , the cell shapes and orientations will asymptoticall y approach those associated wi th that steady state strain rate and direction . If transient loads or deform ations are applied, the cell fabri c will lag the de form ation hi story. It is by these mea ns that ce ll s with no inherent viscoelas tic properties exhibit viscoelastic properties as an assem­blage. These findin gs are consistent with a wide range of experiments [ 16 - 20] and support earl y observations by Steinberg and others that ti ssues exhibit viscoelastic liquid

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CELL RESHAPING IN EMBRYONIC EPITHELIA 97

~roperties. However, data from the present simulations are Insufficient to resolve whether one time constant as in Eq. 03) is sufficient or whether two as in Eq. (15) are necessary. However, it seems significant that the decay portion of K* and Sy, like that of real tissues [19,33] can be represented so accurately using a model with two.

Like other model studies of cell systems, the present one reveals how cells with specific properties would behave under various load or deformation conditions. Inasmuch as the model embodies the real system it endeavours to represent, it can reveal fundamental characteristics of that system. The present model includes first order material approximations for both the interior and the perimeter of the cell. The interior properties accommodate the cytoplasm and its networks of filaments While the perimeter properties include the cell membrane and its associated networks and cell-cell adhesion ~ystems. Model properties can be changed and used to mvestigate how the behaviour of the real system would change if its properties were altered. Alternatively, the properties of the real system can be inferred from the properties of the model system that its behaviour most closely matches [26,35]. If differences in behaviour between the model and real systems cannot be resolved by adjusting the numerical values of the model parameters, this would indicate that the real system has characteristics not embodied in the model. Models may be unique in their ability to make the incisive determination that present understanding as embodied in a model is insufficient to explain observed characteristics of the real system.

The findings of the present model are consistent with qualitative observations of cells. However, further measurements must be taken of real cell systems before quantitative comparisons can be made. In particular, measurements of local deformation [I] must be com­plemented by detailed studies of cell shape and by measurements of sub-cellular mechanical properties in the same animal system and at the same location and stage of development.

To measure the interfacial tension 'Y between cells is a difficult challenge that apparently remains to be achieved. Although recent studies have proved successful in measuring the surface tension between cells and a surrounding medium [36], the interfacial tension between cells is more difficult to measure. The simulations suggest that 'Y could be determined by matching [37] experimen­tally measured curves of cell shape K* or stress Sy with those from simulations (Fig. 5). In general, the time scale of the experiments would have to be multiplied by a factor F to match the simulation results, which are shown in terms ofT. Equation (12) shows that this factor is

(21)

If any four of the terms in this equation are known, then the fifth can be found. Although the experimental data published to date do not provide sufficient information that

this approach can be used to calculate 'Y for any particular type of epithelium or cell aggregate, Eq. (21) identifies the data that must be measured for its calculation.

The present study contrasts with classical continuum­based constitutive models [25,38-41] in that it is part of a strategy to derive the properties of a tissue from the properties of its sub-cellular components. The cells of interest here are much more free to rearrange than those modelled in traditional continuum formulations, and the strain rates are much lower. Even so, one of the ultimate objectives of the present work is to lead to the development of cell-based constitutive equations that can be used to investigate morphogenetic movements and the causes of malformation-type birth defects. Reliable constitutive equations are crucial to that work.

The present study shows that cell aggregates exhibit characteristics that would not be predicted by continuum models. In particular, cells in a two-dimensional patch can become locked into configurations that are not optimal for the individual cells, and this locking can augment the resistance of the patch to further deformation. Locking of this type can be identified by the presence of columnar arrangements of cells normal to the direction of patch elongation. Locking can be understood in terms of local minima in a global quasi­energy surface, and can be reduced in numerical simulations by judicious choice of the minimum side length. Recent simulations of tissues whose cells undergo mitosis [14] show that mitosis is effective in preventing locking because it continuously perturbs the cellular geometry of the patch.

In conclusion, finite element-based computer simu­lations have made it possible to bring the methods of mechanics to bear on the study of cell reshaping and rearrangement in epithelia. The simulations show how cell reshaping lags the deformation history, how it is allayed by cell rearrangement and why it causes the epithelium as a whole to exhibit viscoelastic mechanical properties. This is an important step toward understanding how the properties of embryonic tissues arise from their cellular nature and sub-cellular components. It is also a crucial step towards the development of cell-based constitutive equations for epithelia.

Acknowledgements

This research was funded by a Research Grant to G.W. Brodland from the Natural Sciences and Engineering Research Council of Canada (NSERC).

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