We validate the model extensions by looking at two classical problems: (a) differential adhesion cell sorting (Glazier and Graner 1993; Graner and Glazier 1992) and (b) cell packing in epithelial monolayers (Farhadifar et al. 2007). VirtualLeaf provides new insight into both problems.
Cell Sorting
Classic experiments by Holtfreter (reviewed in Steinberg 1996) have shown that cells of different embryonic tissues can phase separate. A number of closely related hypotheses have been proposed to explain this phenomenon. Steinberg (1963, 2007) has proposed the differential adhesion hypothesis. In this view, cell sorting is due to the interplay of differential adhesion and random cell motility, which progressively replaces weaker intercellular adhesions for stronger adhesions. In addition to differential adhesion, contraction of the cortical cytoskeleton contributes to the equilibrium configurations of mixed cell aggregates (Krieg et al. 2008), leading to the differential surface contraction (Harris 1976) aka differential interfacial tension (Brodland 2002) hypothesis.
Because of its importance for biological development and the possibility to predict the configuration corresponding with the energy minimum from the differential interfacial energies (Steinberg 1963), cell sorting has become a key benchmark problem for cell-based modeling methodology. Cell sorting has been reproduced in a practically all available cell-based models, including cellular automata (Antonelli et al. 1973), vertex-based models (Hutson et al. 2008), center-based models (Graner and Sawada 1993), and the Cellular Potts Model (Graner and Glazier 1992; Glazier and Graner 1993), but small differences are observed (Osborne et al. 2017): The kinetics of cell sorting differs between cell-based modeling methods as well as the extent to which the simulation gets trapped into local minima. Also, methodology relying on single particles to represent a cell may require unrealistically long interaction lengths or unrealistic cell motility models to achieve complete cell sorting (Osborne et al. 2017)
Following previous Cellular Potts and vertex-based approaches (Graner and Glazier 1992; Glazier and Graner 1993; Hutson et al. 2008), we assume that cell motility is governed by volume conservation and an adhesion energy defined at all cell–cell and cell–medium boundaries,
$$\begin{aligned} H=\lambda _A\sum _{c\in C}\left( A(c)-A_T(c)\right) ^2+\sum _{\mathbf {e}\in E}J\left( \mathbf {e}\rightarrow L,\mathbf {e}\rightarrow R\right) \Vert \mathbf {e}\Vert \end{aligned}$$
(2)
with A(c) and \(A_T (c)\) the actual area and resting areas of the cells. The adhesion energy is a sum over all edges \(\mathbf {e}\in E\) in the tissue, with parameter \(J(\mathbf {e}\rightarrow L,\mathbf {e}\rightarrow R)\) the adhesion energy per unit cell-cell interface separating the cell at the left (L) and the cell at the right (R) of the interface, where one cell can be the medium.
Sliding Operator Enables Complete Cell Sorting
Figure 2a–c and Videos S1–S3 show the simulation results for three typical settings of the adhesion parameter J. The simulations are initiated with a configuration of \(20\times 20\) cells of size \(10\times 10\), with mixed or segregated cell type assignments as shown in the first column of Fig. 2. The target area is set equal to the initial area, at \(A_T=100\). The step size for the Monte Carlo algorithm is \(\varDelta x=0.5\), \(l_\mathrm {min}=6\), and \(l_\mathrm {max}=8\). For these parameter settings, the nodes are moved randomly over a square of side 1 / 20 of that of the initial length of the cell–cell interfaces, and one cell–cell interface consists typically of one to two edges, such that sliding moves occur over half to a full cell–cell interface.
In Fig. 2a, the heterotypic adhesion, i.e., the adhesion between green and red cells, is stronger than the homotypic adhesion, i.e., \(J(\mathrm {green},\mathrm {green})=J(\mathrm {red},\mathrm {red})>J(\mathrm {red},\mathrm {green})\). The model evolves toward a checkerboard configuration, which maximizes the contact area between red and green cells. Figure 2b, c shows example simulations for which the homotypic adhesion is stronger than the heterotypic adhesion, that is, \(J(\mathrm {green},\mathrm {green})=J(\mathrm {red},\mathrm {red}) <J(\mathrm {red},\mathrm {green})\). In addition, in Fig. 2b the adhesion of the green cells with the surrounding medium is stronger than that of the red cells, i.e., \(J(\mathrm {green},\mathrm {ECM})< J(\mathrm {red},\mathrm {ECM})\). Cell sorting requires stochastic boundary movement; at \(T=0\) no energetically unfavorable moves are accepted, and the configuration gets stuck at the initial condition, whereas cell sorting is accelerated at higher temperatures (Fig. 2e). Altogether, in analogy with the Cellular Potts Model (Graner and Glazier 1992; Glazier and Graner 1993), the extended VirtualLeaf reproduces the key phenomena related to differential adhesion-driven cell rearrangement: cell sorting, checkerboard pattern formation, and engulfment.
In order to represent cell rearrangements, previous vertex-based simulations applied a rule-based T1 transitions. In these simulations, the T1 transition rearranges four adjacent cells as shown in Fig. 1b. The rule-based T1 transition is initiated if the length of an intercellular interface, i.e., an edge \(\mathbf {e}\) connecting a 3-connected node \(v_1\) with a second node \(v_2\), drops below a threshold, \(\Vert \mathbf {e}\Vert <\theta _\mathrm {T1}\). The T1 transition then deletes \(\mathbf {e}\) by fusing \(v_1\) and \(v_2\) and generates a new edge, \(\mathbf {e}_\perp \), perpendicular to \(\mathbf {e}\). In the absence of noise terms, vertex-based models based on such rule-based T1 transitions generally cannot achieve complete cell sorting, except in specific three-dimensional cases where almost complete cell sorting can be achieved (Hutson et al. 2008).
In the extended VirtualLeaf, T1 transitions are represented by a combination of two sliding moves, where both moves are driven by the Hamiltonian (Fig. 1c). As a first test of the extent to which the sliding operator changes the kinetics of cell sorting, in a second set of simulations we replaced it for rule-based T1 transitions. Figure 2d and Video S4 show a cell sorting experiment with only rule-based T1 transitions and a cellular temperature of \(T=10\), the same cellular temperature as that used in Fig. 2c. Without the sliding operator, cell sorting proceeds well over short times, with small clusters of green and red cells forming, but cell sorting remains incomplete. We have currently not investigated the causes of this in detail, but a potential factor is that the sliding operator is fully integrated in the energy minimization processes, in contrast to the rule-based treatment of T1 transitions. Also changing the threshold for rule-based T1 transitions, currently set at \(\theta _\mathrm {T1}=\varDelta x/2=0.25\), will likely speed up cell sorting. We will leave a full analysis of the sliding operator relative to the rule-based treatment of T1 transitions to future work.
Differential Cortical Tension
As an experimental test of the differential adhesion hypothesis, Krieg and coworkers (Krieg et al. 2008) have measured the adhesive forces between induced germline progenitor cells from early zebrafish embryos. The heterotypic adhesion forces between induced endodermal, mesodermal, and ectodermal cells were approximately equal, whereas the homotypic adhesion forces differed between germ layers. Mesodermal cells adhered most strongly to one another, followed by endodermal cells, and ectodermal cells had the weakest adhesive forces to one another. Based on these data, the authors estimated relative values of the adhesion parameters, J, in a Cellular Potts Model. Strikingly, in the Zebrafish germline progenitor aggregates the least coherent ectodermal cells sorted to the middle of the cellular aggregates. This finding contradicts the differential adhesion hypothesis (DAH), which predicts that the least cohesive cells move to the aggregate’s periphery, see, e.g., the CPM (Graner and Glazier 1992) and our own simulations (Fig. 3, top-left to bottom-right diagonal). Krieg and coworkers demonstrated that the contradictory prediction can be attributed to differential cortical tension (DCT), an alternative to DAH (Harris 1976), with the highest cortical tension occurring at cell–medium interfaces. To implicitly incorporate cortical tension effects into the Cellular Potts Model, Krieg and coworkers reinterpreted the CPM such that a high value of J corresponded with a high interfacial tension.
To test if VirtualLeaf could represent both DAH and DCT explicitly in the same model framework, we modified the Hamiltonian (Eq. 2) to add a cell-dependent cortical tension term that is only active at the tissue boundaries. The new Hamiltonian becomes
$$\begin{aligned} H= & {} \lambda _A\sum _{c\in C}\left( A(c)-A_T(c)\right) ^2+\sum _{\mathbf {e}\in E} J\left( \mathbf {e}\rightarrow L,\mathbf {e}\rightarrow R\right) \Vert \mathbf {e}\Vert \nonumber \\&+\,\lambda _\mathrm {cortical}\sum _{\left\{ c\in C|c\cap \partial C\right\} }\left( P(c)-P_T(c)\right) ^2 \end{aligned}$$
(3)
with \(\partial C\), the boundary of the tissue, \(\lambda _\mathrm {cortical}\), a parameter and \(P_T(c)\) a cell type- specific target perimeter. \(P(c)=\sum _{\mathbf {e}\in (c\rightarrow E)}\Vert \mathbf {e}\Vert \) is the perimeter of cell c, and \(P_T (c)\), a target perimeter. Note that the cortical tension term was only applied at the cell–medium interfaces, which would be equivalent to setting \(\lambda _\mathrm {cortical}=0\) at cell–cell interfaces. The adhesion parameters were set such that \(J(r,r)<J(g,g)<J(g,r)\), i.e., red cells are more coherent than green cells, and red–green interfaces are energetically unfavorable. We have also assumed increased line tension at the boundary of the cell aggregate due to myosin activity (Krieg et al. 2008), by setting \(J(l,M)=0\) and \(J(d,M)=0\), but this has little effect on the results.
Figure 3 shows a parameter study of this model. If the two cells have equal cortical tension at the boundary of the aggregate (top-left to bottom-right diagonal and Videos S5 and S6), the coherent red cells sort to the center, as expected in the absence of additional assumptions. The sorting order is reversed if \(P_T(r)>P_T(g)\), thus reducing the cortical tension of red cells relative to that of the green cells (Fig. 3, upright corner and Video S7).
Epithelial Cell Packing
The structure of multicellular tissues and the shape of the constituent cells are driven by the interplay of cell division, cell growth, intercellular frictional forces, and global tissue mechanics. Epithelial tissues of plants (Kim et al. 2014) and of animals (Farhadifar et al. 2007) can be represented by two-dimensional tessellations and are, therefore, a popular model system for studying morphogenesis and emergence of tissue form (Lewis 1926). In particular, the number of neighbors in many epithelial tissues shows a characteristic distribution: Hexagonal cells are the most frequent, followed by pentagonal and heptagonal cells. Although the experimentally observed distribution can arise due to random cell division alone (Gibson et al. 2006), the biophysics of cell packing, i.e., programs of cell rearrangement and patterning of interfacial tensions, allows tissues to assume alternative, often narrower, i.e., more hexagonal neighbor distributions (Farhadifar et al. 2007). In the absence of cell rearrangements (as, e.g., in plant tissues), mathematical simulations have shown that cells must divide over the center of mass and the division plane must follow shortest paths, thus forming equally sized, symmetrically shaped daughter cells (Sahlin and Jönsson 2010).
VirtualLeaf can Reproduce Key Features of Epithelial Cell Packing
To see if our model, in particular the flexible cell membranes and the sliding operator, could lead to different predictions for epithelial tissues, we here focus on the results described by Farhadifar et al. (2007) using the model implementation detailed in Staple et al. (2010). Their vertex-based model uses a Hamiltonian of the form,
$$\begin{aligned} H=\lambda _A\sum _{c\in C}\left( A(c)-A_T(c)\right) ^2+ \sum _{\mathbf {e}\in E}J \Vert \mathbf {e}\Vert + \lambda _\mathrm {cortical}\sum _{c\in C}\left( P(c)-P_T(c)\right) ^2, \end{aligned}$$
(4)
with \(P_T(c)=0\) and \(J=J\left( \mathbf {e}\rightarrow L,\mathbf {e}\rightarrow R\right) \) the same for all cell interfaces. In the absence of cell division, in this model two distinctive equilibrium cell shape patterns or ‘ground states’ can emerge depending on the parameters. For positive line tension, \(J>0\), or negative line tension, \(J<0\), with a sufficient high contractility, \(\lambda _\mathrm {cortical}\), the global energy minimum in the absence of cell divisions (ground state) of the vertex model is a regular, hexagonal tessellation with cellular areas smaller than the target area. The hexagonal tessellation resists compression, expansion, or shearing. The alternative global minimum is a ‘soft network,’ which occurs at negative line tensions combined with no, or relatively low contractility. The soft network is characterized by many, alternative irregular tessellations of equal pattern energy, with cellular areas equal to the target area. This soft-to-stiff transition is thought to reflect a soft matter phase transition that accompanies jamming of granular materials (Atia et al. 2018; Tlili et al. 2018; Bi et al. 2015, 2016).
Farhadifar et al. (2007) have shown that the cell packing deviates from these global equilibria if cell division is introduced. The authors picked one cell at random, doubled its target area, and relaxed the cellular configuration to the nearest equilibrium using a conjugate gradient method. They then divided the cell over a randomly oriented axis passing through the cell centroid, after which they relaxed the configuration again to its nearest equilibrium. This procedure was repeated until the tissue consisted of 10,000 cells, after which the topology of the tissue was examined.
To determine if our simulation methods could reproduce these results, we used a vertex-based special case of VirtualLeaf, in which there were no 2-connected nodes, i.e., the cell-cell interfaces could not buckle and topological changes occurred through rule-based T1 and T2 transitions. We replicated Farhadifar’s cell division algorithm with only minor modifications. We picked one cell at random, slowly increased its target area, and relaxed the tissue to steady state using the Metropolis algorithm. Once the actual area of this cell exceeded twice the target area of the other cells, we let the cell divide over a randomly oriented axis passing through the cell centroid and assigned the original target area to the daughter cells, and the procedure was repeated. Our simulations (Fig. 4a and Videos S8–S10 ) agree visually with the three cases reported previously (Farhadifar et al. 2007) and illustrate the key results of these simulations, displayed upon the ground state diagram by Farhadifar et al. (2007). Our vertex-based model replicates a typical ‘stiff’ network (Case I), located in the parameter region with a hexagonal ground state, producing cells of approximately uniform size. Furthermore, our model can replicate the outcome of cells with a higher cortical tension (Case II) producing cells with more variable areas and a tessellation that contains large polygons with nine sides or over. Lastly, our model can recapitulate the ‘soft network’ or ground state (Case III) where cells evolve irregular shapes equal to the target area.
After eight rounds of cell division, the distribution of polygon classes (\(P_n\), the fraction of polygons in the final tissue with n sides) in Case I agree, with only minor differences, with those reported in Farhadifar et al. (2007) (red bars in Fig. 4c). Both models reveal pentagon and hexagon-shaped cells dominate at \(P_5\approx 0.3\) and \(P_6\approx 0.3\) while heptagons are slightly less frequent at \(P_7\approx 0.2\), and tetragons and octagons are present at frequencies of \(P_4\approx P_8\approx 0.1\). Our model also has qualitative agreement in Case II and Case III with those reported for the vertex model although our model generated fewer 3-, 4-, 8-, and 9-sided cells. This difference can likely be attributed to the stochasticity in our simulations, which relaxes the configurations more quickly, similar to the effect of annealing reported in Farhadifar et al. (2007).
Flexible Membranes and Sliding Change Case III, but not Cases I and II
We next tested whether membrane flexibility and the membrane sliding operator could replace algorithm-based T1 transitions to generate a topology indicative of growing tissues. We investigated the performance of these model innovations for three specific cases (Fig. 4b and Videos S11–S13). For Case I and Case II, the simulations in the presence of sliding and membrane flexibility showed no obvious differences with simulations of the vertex model. For Case I (Fig. 4c) and for Case II (Fig. 4d), the distribution of neighbor numbers did not differ between straight membranes (red and blue bars) and flexible membranes (yellow and green bars). Interestingly, for Case III both the visual appearance (Fig. 4b) and the neighbor distribution (Fig. 4e) were strongly affected in the presence of sliding and membrane flexibility (green bars): The number of heptagons was higher than for the other simulation conditions, and the number of pentagons was reduced. In the absence of membrane flexibility, sliding did not have this effect (blue bars), whereas for membrane flexibility and with T1 transitions, we observed only a small effect (yellow bars).
In Case I and Case II, the line tension (Case I) or cortical tension (Case II) straightens cell boundaries, such that boundary flexibility has no effect. In Case III, the specific topology of ‘soft networks’ is due to the boundaries’ resistance to compression by adjacent cells. Adding additional nodes to the membranes makes them flexible and allows membranes to buckle (see the ‘bubbly’ boundaries in Fig. 4b and Video S13, Case III), which will likely reduce the number of T1 transitions. We did not understand in detail why the distribution of neighbor numbers was particularly strongly affected in the presence of sliding. A potential explanation is that T1 transitions may introduce spurious energy barriers or time delays between configurations of higher and lower energy, consistent with the incomplete cell sorting discussed in Sect. 3.1.1, whereas for sliding such effects are reduced.