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The central component of the CPM is the definition of the Hamiltonian. The Hamiltonian is determined by the configuration of the cell lattice and perhaps other sub-lattices containing information such as the concentrations of chemicals. The original CPM Hamiltonian included adhesion energies, and volume and surface area constraints. We present a simple example for illustration:. Many extensions to the original CPM Hamiltonian control cell behaviors including chemotaxis , elongation and haptotaxis. Core GGH or CPM algorithm which defines the evolution of the cellular level structures can easily be integrated with intracellular signaling dynamics, reaction diffusion dynamics and rule based model to account for the processes which happen at lower or higher time scale.
Cellular Potts model The cellular Potts model is a lattice -based computational modeling method to simulate the collective behavior of cellular structures. The primary rule base has three components: rules for selecting putative lattice updates a Hamiltonian or effective energy function that is used for calculating the probability of accepting lattice updates.
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Multiscale and hybrid modeling using CPM Core GGH or CPM algorithm which defines the evolution of the cellular level structures can easily be integrated with intracellular signaling dynamics, reaction diffusion dynamics and rule based model to account for the processes which happen at lower or higher time scale. We show that the updated VirtualLeaf yields different results than the traditional vertex-based models for differential adhesion-driven cell sorting and for the neighborhood topology of soft cellular networks.
How cells form tissues, organs, and organisms remains one of the most intriguing and most central questions of biology. Recent theoretical approaches to study collective cell behavior are taking a prominent role in addressing these questions. Theoretical approaches provide deeper intuition about processes that typically are unfamiliar to the researchers by testing the physical plausibility of speculative hypotheses or by making predictions that can be tested experimentally. To support theoretical analysis of tissue formation, a large range of mathematical methods have been proposed.
These range from systems of partial differential equations models see, e. For example, Odell et al. As developmental biologists adopt biophysical methods and borrow principles of control theory to explain tissue formation, simulations will need to capture interactions between multiple cell types and the diverse forms of cell—cell communication those interactions encode Lander Thus, cell-based modeling approaches enable integration of the physics of collective cell behavior with diverse modes of subcellular biological regulation.
A large range of cell-based modeling techniques are available; they can be roughly classified into single-particle and multiparticle methods, and lattice-based and off-lattice techniques Merks Single-particle techniques are efficient computationally and have found wide application, but they also have limitations. Also, it can be important that subcellular compartments interact with their local environment relatively independently from one another. Although it is possible to simulate such problems using single-particle-based methods [see, e.
Multiparticle methods are also better suited for the simulation of local mechanisms responsible for collective behavior e. The CPM represents cells as usually connected domains of lattice sites on a regular lattice. Cells move on the lattice by randomly extending or retracting their domain to adjacent lattice sites, according to a Hamiltonian energy function that describes the contractile and viscoelastic structures that form each cell, the physical adhesive interactions between cells, and in some cases, extracellular materials. These represent the junctions between three or sometimes more cells in confluent tissues as point particles, connected using structural elements that represent the cell boundaries.
Where the CPM defines tissues as assemblages of cells with individual cells represented as collections of adjacent lattice sites, VM describes the tissue as a polygonal tessellation of junctionally connected cells with each cell represented by a series of nodes representing three-cell junctions. Like cells in CPMs, the dynamic movements of cells in VMs are driven by the physical properties of cell—cell interfaces, which are governed by a Hamiltonian function that usually includes interfacial tensions, cell adhesion, and cell area constraints. For instance, because the string-like elastic elements in their basic formulation describe cell—cell interfaces and require all cells to be interconnected, VMs are unsuitable for non-confluent tissues.
In this paper, we discuss further limitations of traditional VMs including a description of cell—cell interfaces as straight lines, b separation of membrane fluctuations and model dynamics, and c algorithms that represent cell neighbor changes with rule-based T1 and T2 transitions. In this paper, we introduce a variant of the VM in which these three limitations have been resolved. VirtualLeaf differs from traditional VMs in that a cell interfaces are represented by multiple nodes that allow membrane fluctuations; b tissue topology changes exclusively through cell division with no T1 or T2 transitions; and c tissue dynamics are advanced used a Metropolis algorithm that incorporates membrane fluctuations.
The present extension of VirtualLeaf introduces a new rule for cell—cell shear or sliding, from which T1 and T2 transitions emerge naturally, allowing the application of VirtualLeaf to problems of animal development. We will discuss two cases for which the updated VirtualLeaf yields different results than traditional VM.
First, we discuss simulations of differential adhesion-driven cell sorting and show that the new update rule for cell sliding facilitates complete cell sorting. Overview of the cell-based model. Nodes that connect three or more cells are shown in dark blue. The 2 connected nodes shown in light blue account for membrane flexibility.
Numbers represent cells. New vertices and edges are green, and red vertices and edges are to be removed.
Multiscale and hybrid modeling using CPM
Blue edges are moved by sliding. During a Monte Carlo step, VirtualLeaf attempts to move and slide all nodes once in a random order. The key novelty that makes the model applicable to animal tissues is that here we allow cells to move through the tissue. To this end, we introduce a sliding operator to further reduce the Hamiltonian Fig. For each node, we first attempt to move it.
If the node is of order 3 or higher, we also try to slide it see flowchart in Fig. After completion of one MCS, the descriptions of the cell membranes are refined if necessary, so as to keep an approximately even distribution of edge lengths. In other models i. In those cases, we apply an operator splitting approach in which the Monte Carlo steps are alternated with steps of the additional rules.
VirtualLeaf provides new insight into both problems. Classic experiments by Holtfreter reviewed in Steinberg 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 , 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. 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 , cell sorting has become a key benchmark problem for cell-based modeling methodology.
Differential adhesion-driven cell rearrangement in the VirtualLeaf. 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. The rule-based T1 transition is initiated if the length of an intercellular interface, i. In the extended VirtualLeaf, T1 transitions are represented by a combination of two sliding moves, where both moves are driven by the Hamiltonian Fig.
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.
Cellular Potts Models: Multiscale Extensions and Biological Applications - CRC Press Book
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. We will leave a full analysis of the sliding operator relative to the rule-based treatment of T1 transitions to future work.
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.
Krieg and coworkers demonstrated that the contradictory prediction can be attributed to differential cortical tension DCT , an alternative to DAH Harris , 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. Parameter study of interface-specific cortical tension.
All other parameters remain unchanged see Supporting Text S1. This figure shows the tissues after a simulation of , MCS Color figure online. 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.
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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. In the absence of cell rearrangements as, e. 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, cells, after which the topology of the tissue was examined. Comparison of straight walls and T1 transitions with flexible walls and sliding on cell morphology. Hexagonal networks can be found in the green region of the plot.
The sum of cortical tension and adhesion energy is smaller than 0 in the blue region, causing a soft network to occur. Simulations within the red region will be unstable. C, D, E Relative amounts of cells with n neighbors when the tissue is in equilibrium. The bars represent the averages and the error bars the standard deviations of 10 time points between generations 7 and 8. See Supporting Text S1 for detailed simulation descriptions. Bars in C-E are colored in the order red, blue, yellow, green Color figure online. 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.
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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. 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. 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.
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. We have validated the new model using simulations of differential adhesion-driven cell sorting and found that it can reproduce the key phenomena of differential adhesion-driven cell sorting, including cell mixing Fig.
In contrast, in traditional VMs, T1 transitions are initiated independently of the energy minimization process, as soon as the length of a cell—cell interface drops below a threshold. Because of this natural integration with the energy minimization algorithm, simulations with the sliding operator more quickly reached complete cell sorting Fig. A full quantitative comparison of the two approaches will be left for the future work and will give more insight into the causes underlying these differences.
Similarly, in simulations applying the new slide operator, increasing the probability of slides and movements by increasing the Boltzmann temperature, T , increases the speed of cell sorting Fig. Preliminary simulations with the new slide operator suggest that perhaps somewhat counterintuitively increasing the step size has little effect on the speed of cell sorting, because it affects only the node moves, but not the node slides that are responsible for cell rearrangement.
Interestingly, our preliminary results suggest that increasing the number of nodes used per cell increases the speed of cell rearrangement Video S A full quantitative analysis of the effect of these and the other parameters on the biological behavior of our model and its computational efficiency will be left to the future work.
As a first step, we plan to perform detailed comparisons with the CPM, which will require quantitative mapping of the model parameters in VirtualLeaf with those of the CPM. The deviations of cell—cell boundaries from simple lines imply that strain on these structures occurs both parallel and perpendicular to the junction. Deviations may reflect differential pressures between neighboring cells, the ability of the boundary to bend under compression e.
The presence of such irregular boundaries becomes more apparent when imaging tissues with higher magnification or when larger cells are sufficiently resolved by lower magnification objectives.
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Bubbly vertex dynamics Ishimoto and Morishita represents the cell—cell and cell—medium interfaces as curves instead of straight lines, where the curvature is due to pressure differences. This innovation changes the forces acting upon the vertices, and hence, it modifies the dynamics of the VM, but it does not allow buckling of cell boundaries. These adaptations allow simulations of tissues that may experience anisotropic tensions or whose mechanics may be shaped by both medioapical cortical dynamics and junctional contractility.
To better simulate such tissues, as a next step we will incorporate a bending stiffness [see, e. This will make it possible to explore the full parameter range between the maximally stiff, straight cell—cell interfaces in VMs, and the fully floppy cell—cell interfaces that we can currently represent in VirtualLeaf. In this generalization, the cells can be represented on any tessellation, which has the advantage that the method can be interfaced with a wider range of methods for continuum mechanics where using arbitrary meshes is useful, e.
The key innovation in their approach, which is shared with VirtualLeaf and VMs, is that cells are represented as polygons. This facilitates simulation of cortical cell tension. This possibly introduces similar lattice effects as those found for the CPM, but an advantage of the approach is that it facilitates collision detection.
Future extensions, e.
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In conclusion, with the present extension of a sliding operator, we introduce a new multiparticle method for cell-based modeling and simulation. The method can be categorized within a continuum of closely related multiparticle, Hamiltonian-based methods ranging from lattice-based to off-latice methods. Finally, the VM simplifies the representation of the tissue, by only representing tricellular junctions, connected by straight lines Weliky and Oster Skip to main content Skip to sections.
Advertisement Hide. Download PDF. Open Access. First Online: 29 March VirtualLeaf represents confluent tissues in two dimensions as a set of interconnected polygonal cells.