Seminar 1Tensor Model of Epithelial Morphogenesis

Author: Jan RozmanMentor: assoc. prof. dr. Primož Ziherl

June 7, 2017

University of LjubljanaFaculty of Mathematics and Physics

AbstractA formalism is presented whereby the tissue-level strain tensor describing epithelial morpho-

genesis can be decomposed into the individual contributions of cell-level processes: Cell division,size and shape change, cell network rearrangement and cell death. This allows us to quantify thecontribution of the individual processes to the development of the embryo. The formalism’s resultsare then compared to simulations as well as experimental measurements of the developing fruit flydorsal thorax.

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Contents1 Introduction 2

2 Derivation 22.1 Strain tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.2 Decomposition of morphogenetic deformation . . . . . . . . . . . . . . . . . . . . . . . 3

3 Application 73.1 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73.2 Dorsal thorax of fruit fly . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

4 Discussion 9

1 IntroductionThe epithelium is the outermost layer of tissue and consists of cells arranged into sheets which oftenform a single mono-layer. Its growth and deformation during embryonic development are known tobe the consequence of collective dynamics of cells [1]. However the influence of individual cellular pro-cesses on epithelial morphogenesis is not yet very well understood. While models analysing the effectof a single cellular process [2, 3] or concerning other types of tissue [5, 6] have existed beforehand, for-malisms allowing for the unambiguous decomposition of epithelial tissue growth into the contributionof individual processes are a relatively recent development [1, 7].

We first present one such formalism [1, 8] based on the network of links connecting the centers ofmass of neighbouring cells and relying on the distinct effects individual cellular processes have on theepithelial topology. The formalism gives a linear tensor balance equation between the strain tensordescribing tissue morphogenesis and the individual contributions of different processes. It also allowsfor a separate analysis of the isotropic and anisotropic effect of each process on epithelial growth.

The results of applying the formalism to experimentally studied systems and simulations are thendiscussed. Lastly the formalism is compared with another recent model attempting to separate ep-ithelial morphogenesis into its component parts.

2 Derivation

2.1 Strain tensor

Before adapting the formalism of strain tensors to the selected instances of epithelial tissue morpho-genesis it may be prudent to first very briefly summarize its use for normal, non-cellular, continuousmedia [9, 10]. We can define the deformation vector of a point at position R before and X afterdeformation as u = X −R. Let us now consider two points at infinitesimal distance dR = R′ −Rbefore and dX = X′ −X after deformation. We can then relate the initial and final distance betweenthe points with the deformation gradient [1]:

F0 = I +∇u, (1)

so that dX = F0dR; where I is the unit tensor.To quantify deformations in a way that is unaffected by rotations and rigid movements we now

define the Cauchy-Green strain tensor

G0 = 12(F0FT

0 − I), (2)

which gives the difference between the initial and final square of distance dX2 = dR2 + 2Gi,kdRidRk(Fig. 1). The information included in the strain tensor can be decomposed as follows: Its trace givesthe change in volume (the isotropic part of the deformation), while its deviator G− 1

3tr(G)I represents

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the anisotropic part of the deformation [1, 10]. It must be noted that classical strain tensor formalismdoes not include topological changes; for example, it can be used to describe the stretching of anelastic medium but not the formation of a hole in the medium. As most cellular processes have atopological effect on the tissue (Fig. 2b) this will present a key problem in adapting strain tensors sothat they can be used to study epithelial morphogenesis.

Figure 1: Strain tensor formalism which is used to quantify deformations of continuous mediacan be adapted to epithelial deformations during morphogenesis. However due to cellular processesdeformations in epithelial tissue also include topological changes, which are not included in classicalstrain tensor formalism. Adapted from Ref. [10].

2.2 Decomposition of morphogenetic deformation

We can understand epithelial morphogenesis as a deformation of the tissue driven by the complexinterplay of cell-level processes. As an example the development of the fruit fly dorsal thorax involvesapproximately 2.5 · 104 cell divisions, 2 · 105 cell rearrangements, and 3 · 103 cell deaths (Fig. 2a) [8].However, a formalism can still be developed that decomposes morphogenesis, given by a strain tensorof the epithelium Ge, into the contributions of individual processes: Cell division (D), rearrangement(R), size and shape change (S) and apoptosis or cell death (A) in the form of a simple balanceequation:

Ge = D + R + S + A. (3)

Here the left hand side corresponds to the total change of the epithelium on the level of cellular patchesand the right hand sides to the constituent processes.

The formalism is based on the network of vectors connecting centers of mass of neighboring cells,ljk (where j and k are indices denoting the neighboring cells). We first define the total texture of acell patch so that it corresponds with the variance of cell link lengths in all directions (Fig. 2c):

M =∑〈j,k〉

12wjkljk ⊗ ljk, (4)

where 〈j, k〉 runs over all pairs of neighbouring cells and the 1/2 factor both insures that all linksare counted once and gives a 1/2 weight to links running over the edge of the patch. The weightwj,k is equal to 1 unless the link goes over a four-fold vertex in which cases it equals 1/2, as it isexperimentally observed that four-fold vertices are unstable in the majority of epithelial tissue [11](the theoretical condition for four-fold vertex instability is that tension in all edges is equal [12]). Theweight 1/2 is chosen so that the number of links multiplied by their respective weights is the samebefore and after the four-fold vertex resolves into two three-fold vertices (Fig. 3).

As the links are not conserved during morphogenesis, with new links forming due to cell divisionor disappearing due to cell death, it is possible to separate the total change in texture between the

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Figure 2: Epithelial morphogenesis can be explained as a consequence of cell-level pro-cesses by adapting the classical strain tensor formalism. Panel (a) shows the initial and final shape ofa developing fruit fly thorax. These tissue scale evolution is a consequence of cell-scale events: division,rearrangements, cell deaths and changes in the size and shape of individual cells. These can in turnbe understood as changes in the network of links between neighboring cells (b). Information aboutthe network of links can be collected in a texture tensor M (c). Rather than the normal units of µm2,the texture tensor in the panel is conveniently expressed in units of initial cell edge length square. Byanalyzing how the texture tensor changes during morphogenesis, we can compute the epithelial straintensor Ge and decompose it into the contribution of each individual process. Adapted from Ref. [1].

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Figure 3: Special care must be paid to four-fold vertices, as their formation in epithelial tissueis in the vast majority of cases only temporary. In panel (a) the red and green link run over a four-foldvertex. After the vertex resolves into one of the two possible arrangements of three-fold vertices (b orc) only one of the two links remains. Links running over four-fold vertices are therefore given a weightof 1/2 as two such links eventually evolve into a single normal link over a cell edge.

initial (index i) and final (index f) state of the epithelium into a geometric (G) and a topologi-cal (T) part:

∆M = Mf −Mi

=∑〈jk〉

12w

cf,jklcf,jk ⊗ lcf,jk −

∑〈jk〉

12w

ci,jklci,jk ⊗ lci,jk︸ ︷︷ ︸

G

+∑〈jk〉

12w

af,jklaf,jk ⊗ laf,jk −

∑〈jk〉

12w

di,jkldi,jk ⊗ ldi,jk︸ ︷︷ ︸

T(5)

so that the first term includes only the change associated with conserved links (index c), whereas thesecond term represents the change of the network due to the appearance of new links (index a) andthe disappearance of some of the initial links (index d).

In the simple case where all links and weights are conserved and the deformation gradient Fh ishomogeneous, we can use the relation between the initial and final link lengths lf = F0li, to calculatethe total geometric change:

G = FhMciFT

h −Mc0. (6)

Conversely, the gradient of deformation can here be calculated directly from the initial and finaltexture, giving us F0 =

(∑〈jk〉

12wjklf,jk ⊗ lf,jk

) (∑〈jk〉

12wjkli,jk ⊗ lf,jk

)−1.

We now attempt to write a relation similar to Eq. (6) for the general case, first defining twoadditional quantities similar to the gradient of deformation in the homogenous case:

F1 =

∑〈jk〉

12wjkl

cf,jk ⊗ lcf,jk

∑〈ik〉

12wjkl

c0,jk ⊗ lcf,jk

−1

(7)

F2 =

∑〈jk〉

12wjkl

cf,ik ⊗ lci,jk

∑〈ik〉

12wikl

ci,jk ⊗ lci,jk

−1

, (8)

so that

G = F1MciFT

2 −Mci = F2Mc

iFT1 −Mc

i , (9)

where the second equality follows from Mci and Mc

f being symmetric. Based on experimental observa-tions and only when the amount of conserved links is significant, we can replace F1 and F2 in Eq. (9)by their geometric average F =

√F1F2, thereby restoring the symmetry of Eq. (6):

G = FMciFT −Mc

i . (10)

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Thus the total tensor of geometric change as well as the final texture are fully determined by the initialtexture and F. Before continuing, we must note that, for any tensor T in units of length square, wecan construct a symmetric dimensionless tensor by applying T̃ = 1

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[T (Mc

i )−1 + (Mci )−1 TT

]. As it

will be useful later, we also define Ψ = F[Mc

i ,FT], where the square bracket indicate the comutator.

By multiply (10) from the right with 12 (Mc

i )−1, we get:

12G (Mc

i )−1 = 12(FFT − I

)+ 1

2F{

(Mci )−1 [F,Mc

i ]}T

. (11)

As F corresponds closely to the gradient of deformation F0, the first term in Eq. (11), henceforthdenoted by G∗, is a good approximation of the Green strain tensor 1

2

(F0FT

0 − I)[9], allowing us to

use said term as our approximation for the strain tensor. By further making Eq. (5) and Eq. (11)symmetric as described above and inserting the former into the latter, we can write:

G∗ = G̃− Ψ̃ = ∆M̃− T̃− Ψ̃. (12)

As division is associated only with the appearance of new links, cell death only with the disappearanceof initial links and cell rearrangement with replacing one initial link with one new one (Fig. 2b), wecan write the tensor of topological change T as the sum of contributions of individual processes Tα:

T =∑α

Tα (13)

where α runs over the processes of division, death and rearrangement, giving the balance equation:

G∗ = ∆M̃−∑α

Tα − Ψ̃. (14)

By taking into account that the initial and final number of links ni and nf is not necessarily equal,we can rearrange the change in texture:

∆M̃ =(ninf

M̃f − M̃i

)+(nf − ninf

M̃f

)(15)

so that the final texture in the first term is renormalised to the initial number of links, meaning thatthe first term is associated only with changes in cell shape and size, while the second corresponds withthe change in the number of links due to topological processes. Denoting the change in link-numberdue to each process with ∆nα we can finally write the balance equation for the strain tensor as thesum of shape and size change S∗ and the topological contributions of individual processes P∗α:

G∗ = ninf

M̃f − M̃i − Ψ̃︸ ︷︷ ︸S∗

+∑α

∆nαnf

M̃f − T̃α︸ ︷︷ ︸P∗

α

(16)

As mentioned before, the trace and deviator of the strain tensor respectively give the isotropic andanisotropic deformation. By taking note of both the trace and the deviator being linear operators,the trace and deviator of each right-hand-side term in the above equation gives the isotropic andanisotropic contribution of the corresponding process to the total morphogenetic strain. Because thetime-scale of morphogenetic events is also important, we also define the average strain rate tensor,which will be used in experimental analysis. For any dimensionless tensor T we define its average rateover the observed period of embryonic development as 〈δT/δt〉, where δT is the change in the tensor’svalue between two successive images and δt is the corresponding time interval between them, givingus: ⟨

δG∗

δt

⟩=⟨δS∗

δt

⟩+∑α

⟨δP∗αδt

⟩. (17)

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The use of strain rate tensors rather than strain tensors also has an additional advantage: Theabove formalism relies on the conservation of sufficiently many (but not all) links, which is generallytrue between successive images but not necessarily on the time-scale of tissue morphogenesis (Fig. 4).

Figure 4: The tissue strain and strain rate tensor can be decomposed into individualcontributions of cellular processes. As the trace and the deviator of the strain tensor give theisotropic and anisotropic part of the cellular matter deformation, we can further quantify the isotropicand anisotropic effect of each process. Visually we can represent the isotropic and anisotropic part ofthe strain rate tensor by a circle and bar notation. The size of the circle corresponds to the isotropicpart of the dilation, with a white circle denoting a positive and a grey circle a negative value, while thesize and orientation of the bar represent the size and orientation of the anisotropic change. Adaptedfrom Ref. [1].

3 Application

3.1 Simulations

To test if the formalism correctly determines which process drives tissue deformation it was first usedon simulations with prescribed cell-level processes [1, 8]. The simulations were based on a cellularPotts model [13, 14]: The two-dimensional simulation space representing a planar mono-layer wasdivided into a grid and each field was assigned a number. All fields with the same number representa single cell, excluding one number reserved for the medium surrounding the cells. A simple energywas assumed for each cell, consisting of only surface tension with a preferential area. The externalmedium had no effect on system energy.

The system was evolved using a version of the Metropolis algorithm. At each time-step twoneighboring fields were chosen. If they belonged to different cells, meaning that they carried differentnumbers, the value of the first field was ascribed to the second. The move was automatically accepted

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if it decreased system energy, while the probability of acceptance decreased exponentially with thesize of an increase in energy.

The formalism was without further corrections applied to the special cases where only one of thecellular processes (division, rearrangement, apoptosis or size and shape change) was actively induced(Fig. 5a). In all four cases it correctly determined the dominant process and found its contributingtensor to be close in value to the total strain rate tensor. The small but non-zero contribution of theother processes found in some of the simulations were caused passively by energy relaxation.

3.2 Dorsal thorax of fruit fly

Having shown the derivation of the formalism and its application in simulations of known cellularprocesses, we can now discuss its use in analyzing the experimentally obtained images of an actualdeveloping tissue, the dorsal thorax of fruit fly, a naturally mono-layered epithelial tissue shown inthe yellow dashed box in Fig. 2a [1, 8].

Figure 5: The formalism was applied both to simulations and experimental images. Theaccuracy of Eq. (17) has been tested with simulation based on a cellular pots model for the specialcase where only one of the constituent processes is actively present, finding that the equation holdstrue in all four cases and the formalism correctly identifies the dominant process. The small, butnon-zero, parts of the remaining cell-process tensors in the simulations arise from the necessity ofallowing passive rearrangement and apoptosis to give realistic simulations (a). Panel (b) shows theanisotropic component of cell process tensors for rearrangement (R), division (D), shape change (B)and apoptosis (A), as well as total strain rate tensor (G). The tissue is highly heterogeneous in termsof cellular processes. Only rearrangement, division and shape change make a significant contributionto the anisotropic strain rate, while the contribution of apoptosis is mostly isotropic. Adapted fromRef. [1].

The dorsal thorax was observed for a period of 14 hours, with successive images made every 5

8

minutes [1]. It should be noted that the imaging process was sufficiently non-invasive so that theembryo could still develop into a functional adult organism [8]. The cell process contribution andtotal strain rate tensors were calculated over patches of tens of cells (shown colored in the bottomrow of Fig. 2a). The selected temporal and spatial step was sufficient to give tensor values that variedsmoothly both in space and time (Fig. 5b).

The analysis, based primarily on the strain rate tensor’s anisotropic component, shows the tissueto be highly heterogeneous in terms of cellular processes, while remaining symmetric along the midlineaxis. Cell division, rearrangement and shape change all have a significant contribution to the totalanisotropic component of strain rate in various regions of the epithelum. Interestingly, while cell deathsare numerous and spatially regulated, their contribution to total anisotropic strain rate is negligible.We also note that the cell division process tensor is well aligned with the average local orientation ofdivision.

4 DiscussionThe formalism presented above is a recent development in that it decomposes morphogenesis directlyinto the contribution of individual cell-level processes, rather than focusing on the effect of only oneprocess. It can also easily be generalized to some processes that have not been discussed here, such asthe melding of neighboring cells. The formalism has only been applied to planar deformations it shouldalso be valid in three dimensions without further correction [1]. It must however be noted that while itdecomposes morphogenesis into its component processes it does not give any direct information aboutthe physical forces behind them; effectively it analyzes epithelial strain but not epithelial stress.

The presented formalism is however not unique as another system of morphogenesis decompositionhas been proposed in Ref. [7]. Rather than directly relying on the network of links between neighboringcells it uses triangles centered on the threefold vertex junctions in the cellular network, so that trianglecorners correspond to cell centers (Fig. 6a), as the deformation of triangles can be unambiguouslydefined. Furthermore, the epithelium can be tiled using these triangles without gaps or overlap.Cellular processes are then described by the discontinuous deformations they cause in the networkof triangles (Fig. 6b) [7, 15]. In comparison to the above, this formalism is designed only for twodimensional epithelia. It also differs in the inclusion of an additional term in the balance equation (17)describing correlations between cell elongation and tissue rotation or cell area change, which couldlead to the two models giving different predictions when this term is significant.

Figure 6: An alternative formalism for decomposing morphogenesis can be developed,based on a network of triangles connecting the centers of neighboring cells. Triangles are used becausetheir deformation can be calculated unambiguously, and the network of triangles tiles the epitheliumwithout gaps and overlaps (a). Cellular processes are then analyzed as changes in the network oftriangles (b). Adapted from Ref. [7].

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