Computational Study of the Impacts of Mechanical and Physical Cell Properties on Mitotic Cell Rounding in Developing Epithelia

Mitotic rounding (MR) during cell division is critical for the robust segregation of chromosomes into daughter cells and is frequently perturbed in cancerous cells. MR has been studied extensively in individual cultured cells, but the physical mechanisms regulating MR in intact tissues are still poorly understood. A cell undergoes mitotic rounding by simultaneously reducing adhesion with its neighbors, increasing actomyosin contraction around the cortex, and increasing the osmotic pressure of the cytoplasm. Whether these changes are purely additive, synergistic or impact separate aspects of MR is not clear. Specific modulation of these processes in dividing cells within a tissue is experimentally challenging, because of off-target effects and the difficulty of targeting only dividing cells. In this study, we analyze MR in epithelial cells by using a newly developed multi-scale, cell-based computational model that is calibrated using experimental observations from a model system of epithelial tissue growth, the Drosophila wing imaginal disc. The model simulations predict that increase in apical surface area of mitotic cells is solely driven by increasing cytoplasmic pressure. MR however is not achieved within biological constraints unless all three properties (cell-cell adhesion, cortical stiffness and pressure) are simultaneously regulated by the cell. The new multi-scale model is computationally implemented using a parallelization algorithm on a cluster of graphic processing units (GPUs) to make simulations of tissues with a large number of cells feasible. The model is extensible to investigate a wide range of cellular phenomena at the tissue scale.


Introduction
Epithelia are tissues composed of tightly adherent cells that provide barriers between internal cells of organs and the environment and are one of the four basic tissue types in the human body [1][2][3] (Fig. 1).Epithelial expansion driven by cell proliferation is a key feature throughout development, and it also occurs in hyperplasia, a precursor to cancer.Cell divisions during development must occur robustly, as mis-segregation of chromosomes leads to severe genetic abnormalities (aneuploidy) in daughter cells.
Over 90% of all human tumors are epithelially-derived [4], and the accumulation of genetic errors during cell division can lead to all of the hallmarks of cancer [5].In tissues, mitotic cells must become sufficiently round to avoid the mis-segregation of chromosomes all the while still remaining connected with their neighbors [6].A deeper understanding of the biophysical mechanisms governing the behavior of mitotic cells in epithelia will result in a better understanding of many diseases including cancer.
Epithelial cells entering mitosis rapidly undergo structural changes that result in the apical area of the cell becoming larger and rounder, in a process known as mitotic rounding (MR) [7,8].MR occurs in detached cells, cells adherent to a substrate as well as in epithelial cells within tissues [9][10][11].The beginning of MR in epithelia coincides with an increased polymerization of actomyosin at the cell cortex, which results in an increase in cortical tension and is necessary for MR [10,12].Simultaneously, intracellular pressure increases [10], and cells partially reduce adhesion to their neighbors and the substrate [12].Experiments that can specifically target only dividing peer-reviewed) is the author/funder.All rights reserved.No reuse allowed without permission.
The copyright holder for this preprint (which was not .http://dx.doi.org/10.1101/037820doi: bioRxiv preprint first posted online Jan. 25, 2016; cells and measure physical properties of individual cells within tissues are very challenging. Computational modeling coupled with experimentation has become a powerful tool for identifying the biophysical principles governing organogenesis [13].Biophysicallyderived computational models can complement current experimental methods by predicting the response of tissue to mechanical perturbations of individual cells.MR is investigated in this paper by using a novel multi-scale sub-cellular element model (SEM) called Epi-scale that simulates epithelial cells in growing tissues.Novel biologicallyrelevant features of the model include: i) separate representations of the sub-cellular elements, as well as cell-cell interactions; ii) a detailed description of cell properties during mitotic rounding; and iii) a systematic calibration of model parameters to provide accurate biological simulations of tissue growth.We performed parameter sweeps on the extent that a mitotic cell regulates cell-cell adhesion, membrane stiffness, or internal pressure and analyzed the impacts of such parameter variations on cross-sectional areas of mitotic cells at the apical surface as well as the roundness of mitotic cells.
Model simulations demonstrate that cell-cell adhesion and stiffness significantly impact roundness but do not increase cell area during MR.Solely increasing cell pressure during MR increases both cell area and roundness.However, the internal pressure increase needed to achieve experimentally observed levels of roundness leads to nonphysical levels of cell area expansion (Fig. S5).The model predicts that a cell must peer-reviewed) is the author/funder.All rights reserved.No reuse allowed without permission.
The copyright holder for this preprint (which was not .http://dx.doi.org/10.1101/037820doi: bioRxiv preprint first posted online Jan. 25, 2016; regulate all three cellular properties simultaneously to achieve physiological levels of area expansion and roundness without adversely affecting tissue integrity.
The paper is organized as follows.The Methods section describes the modeling background and model development.The Results section provides details of model calibration of single cell parameters using quantitative biophysical data.The calibrated model predicts emergent properties of epithelial topology.The model is then used to investigate new questions into the relative contributions of cell-cell adhesion, membrane stiffness and intracellular pressure impact the extent of mitotic rounding.The paper ends with a detailed discussion of the models' predictions in the more general biological context.It also describes future extensions of the computational model environment, for simulating epithelial tissue mechanics in greater biological detail.

Modeling background
Multiple computational approaches have been utilized to model various aspects of epithelial tissue dynamics, each with its particular focus and applications [14].For example, the cellular Potts modeling (CPM) approach has been used successfully to take into account cell adhesivity to study cell aggregation as well as cell morphogenesis [15,16].Finite element models (FEMs) have also been implemented to investigate epithelial cell behavior [17,18].Vertex based models (VBM) provided an efficient approach to study the regulation of cell topology, tissue-size regulation, tissue morphogenesis, and the role of cell contractility in determining tissue curvature [19][20][21][22][23][24].
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The copyright holder for this preprint (which was not .http://dx.doi.org/10.1101/037820doi: bioRxiv preprint first posted online Jan. 25, 2016; Cells are defined in VBMs by the several vertices representing the meeting points of shared cell-cell contacts (as reviewed in [19]).An implementation and comparison of the five popular cell-based modelling approaches for simulating the self-organization of multicellular tissues within a consistent computational framework, Chaste [25] (http://www.cs.ox.ac.uk/chaste), was recently described in [26].
The Subcellular Elements Model (SEM), developed initially by Newman's group [27] for simulating multi-cellular systems to encompass multiple length scales, is currently actively used as a general computational modeling approach.SEMs have been extended to predict how mechanical forces generated by cells are redistributed in a tissue and for studying tissue rheology, tissue fusion, thrombus formation, and cell-cell signaling.SEM was also used to study aspects of epithelial cell mechanics without making assumptions about cell shapes [28].Each cell in a SEM consists of a set of nodes representing a coarse-grained representation of subcellular components of biological cells.Node-node interactions are represented by energy potentials.Another SEM developed by Jamali et al. [29] represents the membrane and nucleus of the cell by nodes connected by overdamped springs.Gardiner et al. [30] described a SEM with locally-defined mechanical properties.Christely et al. [31] have developed an efficient computational implementation of the SEM simulating role of Notch signaling in cell growth and division, on GPU cluster to decrease computational time.A particular advantage of the SEM approach is that it can provide local representations of mechanical properties of individual cells which can be directly related to the experimental data [32].
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The copyright holder for this preprint (which was not .http://dx.doi.org/10.1101/037820doi: bioRxiv preprint first posted online Jan. 25, 2016; The Epi-scale computational model This paper represents detailed simulations obtained using novel multi-scale SEM called Epi-scale, of developing epithelia with a focus on the two-dimensional (2D) planar cell shapes near the apical surfaces of cells.This is a simplifying approximation that was used in many previous models of wing disc growth [22,23,[33][34][35].In particular, it is reasonable to use a 2D model for studying many epithelial processes in the Drosophila wing disc pouch because it consists of a single layer of cells and the essential structural components of those cells, including E-cadherins and actomyosin, are concentrated on the apical surface of the epithelia (Fig. 1c-d).E-cadherin is responsible for adhesion between two neighboring cells, and actomyosin, which is concentrated at the apical MR drives cell contractility.The nucleus and most of the cytoplasm are pushed up to the apical surface during cell division.Using a 2D approximation also allows us to model a large number of cells with high resolution and special attention to mechanical cell properties.However, future development of the Epi-scale simulation platform implemented on GPU clusters, will also enable 3D simulations with reasonable computational costs.
In what follows, we first describe different types of the subcellular elements that are used to simulate each cell, and the interactions between them.Then, the equations of motion of each subcellular element are provided.Finally, approaches for modeling cell's growth, transition to mitotic phase, and division are described.

Subcellular elements
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The copyright holder for this preprint (which was not .http://dx.doi.org/10.1101/037820doi: bioRxiv preprint first posted online Jan. 25, 2016; Epi-scale represents individual cells as collections of two types of interacting subcellular elements: internal nodes and membrane nodes (Fig. 2).The internal nodes account for the cytoplasm of the cell, and the membrane nodes represent both plasma membrane and associated contractile actomyosin cortex.The internal and membrane nodes are placed on a 2D plane, representing the apical surface of epithelia.
Interactions between internal and membrane nodes are modeled using potential energy functions as shown in Fig. 2a [31,36].The internal-internal nodes interactions represent the cytoplasmic pressure of a cell, and the interactions between membrane nodes of the same cell are used to model the cortical stiffness.Cell-cell adhesion is modeled by membrane-membrane nodes interactions between two neighboring cells.List of all potential functions used in the Epi-scale to model mechanical properties of cells and epithelial tissue and description of their biological relevance are provided in Table 1.
Epi-scale utilizes spring and Morse potential functions.Linear and torsional springs are used for modeling interactions  !" !!" and  !!"! , while Morse potential functions are used for modeling interactions  !" !" ,  !" !! , and  !!!!" (Fig. 2).Morse potential consists of two terms, generating short-range repulsive and long-range attractive forces [27].The following expression is a Morse potential function for the  !" !" : where  !" ,  !" ,  !" , and  !" are Morse parameters for the  !" !" which are carefully calibrated using specific experimental data.The same form of the potential with different sets of parameters is also used for  !" !" and  !" !!" (Table 2).These potential functions govern the motion of internal and membrane nodes inside the cells resulting in the deformation and rearrangement of cells within the tissue.

Equations of motion
Equations of motion differ for membrane nodes and internal nodes.For each internal or membrane node and at each time-point, displacement of a subcellular element is calculated based on the potential energy functions.The model assumes that nodes are in an overdamped regime [22,36,37] so that inertia forces acting on the nodes can be neglected.This leads to the following equations of motion describing movements of membrane and internal nodes, respectively: where  is the damping coefficient,  !! and    are positions of internal and membrane nodes, respectively.The dot represents a time derivative.
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The copyright holder for this preprint (which was not .http://dx.doi.org/10.1101/037820doi: bioRxiv preprint first posted online Jan. 25, 2016; Eqns.( 2) and ( 3) are discretized in time using forward Euler method and positions of nodes    and    are incremented at discrete times.The forward Euler discretization of the equation of motion of internal nodes (Eqn.( 2)) has the form: where ∆ is the time step size.The same discretization technique is used for the equation of motion of the membrane nodes.
The Epi-scale was computationally implemented on the cluster of Graphical Processing Units (GPUs).This enabled us to run large number of simulations with subcellular resolution at micro-scale with low computational cost and to study the impact of changes in individual cell physical properties on the tissue development at the macroscale.(See Supplementary Information (SI-S1) for details.)

Cell cycle
The Drosophila wing disc, which was used for calibrating the model, has a spatially uniform growth rate which decreases over time [38].The growth rate for cell ii is modeled by an exponentially decaying function fit to the specific experimental data [38], with a random term representing stochastic variation among cells: where  !!"# is the average growth rate of cells in the beginning of a simulation and  − !,  ! is a random number chosen from a uniform distribution in the range − !,  ! . ! is the decay constant of the growth rate.
Cells evolve through interphase and mitosis phases.Variable Cell Progress (  [0,1]) describes in the model cell's progress between the beginning of the interphase ( = 0) to the end of the cell division ( = 1). is updated based on cell's growth rate as follows: The number of internal nodes inside the cell increases as the cell grows (Fig. 2).It has been shown experimentally that epithelial cells undergoing mitosis increase their intracellular pressure by adjusting their osmolarity relative to their surroundings [39].
Additionally, the actomyosin cortex is enriched, and cellular adhesion to the substrate and to neighboring cells are downregulated [10,[40][41][42][43][44].Since these changes in mitotic cells occur concurrently, the relative impact on mitotic cells cannot easily be decomposed into separable effects in experiments.
To simulate MR, parameters regulating cell-cell adhesion, actomyosin cortex, and internal pressure of cells in the mitotic phase (M phase) are modified (Table 2), representing changes of cell physical properties during mitosis [9,10,43].Mitosis is modeled by a linear transition from the interphase parameter range to the mitotic parameter range determined based on the experimental observations.For example  !" , a Morse parameter used for representing cytoplasmic pressure on the membrane of the cell (see SI-S2.3), is varied from the interphase value ( !"#$% !" ) to the mitotic value ( !"# !" ), by using the following function of CP: where  !"# !! is the parameter value in the mitosis range.Similar linear variations of parameter values are used for representing enrichment of actomyosin cortex and reduction in cell-cell adhesion with neighboring cells in mitotic phase (see Table 1).
Cells in the mitotic (M) phase -which lasts approximately 30 minutes -divide into two daughter cells (Fig. 2).As CP becomes equal to 1, cytokinesis occurs that is modeled by separating internal and membrane elements of the mother cell into two sets representing daughter cells.The axis of division is implemented perpendicular to the cell's longest axis, following Hertwig's rule [45].New membrane elements are created along the cleavage plane for each daughter cell, and cell parameters for nodes of each daughter cell are set to values from the interphase range and CP is reset to zero for both daughter cells.
Membrane nodes in the beginning of a simulation are arranged in a circle for each cell, and internal nodes randomly placed within each cell (Fig. 3a).After initialization, internal nodes rapidly rearrange in a simulation, and cells self-organize into a polygonal network, similar to the experimentally observed cell packing geometry of epithelia (Fig. 3b).Cells constantly grow, divide and interact with each other resulting in a detailed simulation of the developing epithelial tissue (Fig. 3c-d).

Model Calibration
Before running predictive model simulations, the model parameters, described in the Epi-scale computational model section, were calibrated using experimental data for the third instar Drosophila wing disc, which is a powerful model for studying organ formation [46,47] (Fig. 4, SI Movie S4.1).Experimental values for epithelial cell lines were used to calibrate the model parameters when experimental data for Drosophila wing disc were not available.
Mechanical stiffness of the actomyosin cortex is primarily responsible for the modulus of elasticity  and the Poisson's ratio of the cells [48]. is experimentally obtained by applying known force to either side to stretch a cell, and measuring the cell's deformation based on the applied force [49,50].This experiment can be reproduced in silico by applying forces to membrane nodes on either side of a simulated cell, and measuring the deformation (Fig. 4a-a'', 4c).Parameters corresponding to cortical stiffness ( !"#$% !" and  !"#$% !" ) were calibrated to have  = 19 , which is within the biological range of 10 − 55  [49,50] measured for epithelial cells.
The cell-cell adhesive force  !"! , or the force needed to detach two adhered cells from each other, is dependent on the strength of cell-cell adhesions. !"! is experimentally obtained by measuring the force needed to detach two adhered cells from each other.This experiment can be reproduced in silico by applying forces to membrane nodes on either side of two adhered cells, and measuring the force needed to separate them (Fig. 4b-b").Parameters corresponding to cell-cell adhesion ( !"#$% !"! and  !"! ) were calibrated such that  !"! = 20 / (Fig. 4d), which is in the range of experimental results for epithelial MDCK cells [51] and E-Cadherin-transfected S180 cells [52].
Cells in the wing disc have spatially-uniform growth-rates that slow down as the tissue approaches its final size [38].The growth rate in the Epi-Scale model described by Eqn.
(5) was calibrated (Table 3) such that the number of cells in time as the tissue grows follows the experimental data [38] (Fig. 4e).Cells in mitosis deviate from cells in interphase in their area and roundness.The area and roundness of interphase and mitotic cells are calibrated based on the data from the wing disc (Fig. 5e-f).SI-S2.2 provides additional details about the methods used in this study to measure the size and roundness of cells in the imaginal wing disc pouch.

Tissue topology emerges from cell self-organization driven by cellular mechanics
After model calibration, simulations were run to determine whether this cellular-scale calibration was sufficient to recapitulate topological properties of the tissue (Fig. 6) [53,54].One metric for tissue topology is the distribution of cell neighbor numbers, or polygon class distribution.Based on the simulation results for studying the tissue growth by using the Epi-Scale model, the polygon class distribution approaches steady state after 35 hours (Fig. 5a, 6b).This distribution matches with the ones reported experimentally for the wing disc and other epithelial systems [32] (Fig. 6d) as well as obtained using other computational models for simulating growing tissues such as vertex based model [22].
Another way to quantify tissue topology is through evaluation of the three laws describing topological relationships: Euler's law, Lewis law, and Aboav-Weaire Law.
Euler's law states that, on average, cells forming a packed sheet should be hexagonal [54].The Lewis law states that cells with more neighbors should have a larger normalized area [55].The Aboav-Weaire law states that the average polygon class of each cell's neighbors decreases as the cell's polygon class increases [56].Simulation results obtained using the calibrated model show the average side of cells to be equal to 5.96 which is in a very good agreement with the Euler's law.The model simulations also satisfy two other laws as shown in Fig. 6c.

Impacts of adhesion, stiffness, and cytoplasmic pressure on MR
The Epi-scale model is suitable for generating and testing new hypothesis regarding mechanical mechanisms of the MR because it is capable of representing non-polygonal shapes of cells, and parameters representing mechanical cell properties in the model can be directly related to the properties of cells measured in experiments.Simulations were conducted to predict relative contributions of different cell properties to the relative peer-reviewed) is the author/funder.All rights reserved.No reuse allowed without permission.
size ( !"# / !"#$% ) and roundness () of mitotic cells.Cell-cell adhesion, cortical stiffness and internal pressure were individually varied in two sets of simulations (Table 4) to decouple the effects of these properties on the final size and shape of mitotic cells.
In the first set (Fig. 6, blue lines), only one of the three properties of mitotic cells is varied in each simulation, while the other two properties are kept constant and equal to interphase values.In the second set of simulations (Fig. 6, black lines), two of the three parameters are set to calibrated values for mitosis.The third parameter under investigation is varied with respect to its calibrated mitotic value.The extent of variation of each parameter ranges from 100% below to 100% higher the calibrated value.Each of these parameter sweeps can be interpreted conceptually as changing the relative degree to which a mitotic cell is regulating each property during the mitotic phase of the cell cycle.
Fig. 7a-c shows that variation of each of these three cell properties considerably affects the roundness of mitotic cells.As seen in Fig. 7a, decreasing only cell-cell adhesion during the M phase increases the roundness of mitotic cells.Increasing the cortical stiffness (Fig. 7b) and the internal pressure (Fig. 7c) will increase the roundness of mitotic cells.The levels of internal pressure needed to achieve "wild type" values of roundness result in unphysical levels of cell areas (SI Movie S4.2) with noticeable incidence of cell-cell rearrangements.Reducing adhesion leads to rounder cells, but only regulating adhesion levels during mitosis would require a complete loss of adhesion (100% below calibrated values) to reach wild-type levels of roundness.
peer-reviewed) is the author/funder.All rights reserved.No reuse allowed without permission.slightly decreases the size of mitotic cells, while increasing cytoplasmic pressure significantly increases it (Fig. 7f, blue line).Table 4 provides detailed information on the impact of variation of different cell properties on mitotic cell's roundness and ratio of maximum cross-sectional area during mitosis to area at interphase.The schematic diagram in Fig. 8 recaps the results shown in Fig. 7 and represents the impact of variation of each individual mechanical property on the final size and roundness of mitotic cells.For a cell to be both large and round requires not only modulating internal pressure but also reducing both adhesion and increasing stiffness co-currently.
In summary, the obtained results suggest that increase in cross-sectional area of mitotic cells is solely driven by increasing cytoplasmic pressure.Mitotic roundness however is not achieved within biological constraints unless all three properties (cell-cell adhesion, cortical stiffness and pressure) are simultaneously regulated by the cell.Without concurrent regulation reducing cell-cell adhesion and increasing cortical stiffness, unrealistic high levels of pressure increase would be required to enforce mitotic roundness, resulting in unphysical levels of cell areas.

Discussion
General models for investigating epithelial mechanotransduction, including MR, require coupling of biologically calibrated mechanical components capable of representing non-peer-reviewed) is the author/funder.All rights reserved.No reuse allowed without permission.
The copyright holder for this preprint (which was not .http://dx.doi.org/10.1101/037820doi: bioRxiv preprint first posted online Jan. 25, 2016; polygonal cell shapes, and simulating the membranes as well as cytoplasm of individual cells as separate entities.To accomplish this, a novel multi-scale sub-cellular model, called Epi-scale, was developed in this paper for simulating mechanical and adhesive properties of cells in the developing columnar epithelium of the wing disc which consists of a single layer of cells.The model approximates the tissue as a 2D surface since the majority of the contractile and adhesive forces are localized at the apical surface of the epithelium.Parameter ranges for the computational model were obtained by calibrating the model using single cell stretching experiments, double cell stretching experiments, as well as experimentally observed size distributions of cells during mitosis and interphase, and tissue growth rate of the Drosophila wing disc.Cell-cell adhesion and cell elasticity were calibrated using experimental data on epithelial tissues.The calibrated model was tested by successfully reproducing emergent properties of developing tissue such as the polygon class distributions. Epi-scale enables one to produce new hypotheses about the underlying biophysical mechanisms governing mitotic rounding of epithelial cells within the developing tissue micro-environment.In particular, results of the model simulations predict that robust mitotic rounding requires co-current changes in cell-cell adhesion, cortical stiffness and cytoplasmic pressure.Regulating only one of the cellular properties does not result in the experimentally observed levels of apical areal expansion and degree of mitotic roundness.The individual contributions of changes in these three mechanical properties to the mitotic cell roundness and area were characterized through detailed parameter sensitivity analysis (Fig. 7).
Changes in all three cell properties were shown to contribute to the degree of roundness in vivo, but internal pressure was shown to be the primary driver of mitotic cell area increase.Cell properties can be modulated experimentally in tissue as a whole.However, it is currently challenging to target only dividing cells in a tissue.Cellcell adhesion is dictated by the adhesive interactions of AJs, which can be modulated pharmacokinetically, or through genetic modification of E-cadherin molecules to alter their binding affinities.Cell stiffness can be adjusted by reducing the contractility of the cortex through pharmacological perturbations.Internal pressure of cells is primarily dictated by osmotic channels regulating the flow of water and ions through the cell membrane, and can be adjusted by modulating those channels, or by changing the osmolarity of the media.The computational model simulations provide insight into the individual contributions of cell properties to MR and can predict the consequences of dysregulation of mitotic cell rounding on the development and homeostasis of epithelial tissues.One experimental approach that could in future be used for testing the model predictions would be to regulate the expression of E-Cadherin, Myosin-II, and osmotic channel antagonists under a Cyclin B promotor, active during mitosis, resulting in modulation only in dividing cells [57,58].Alternatively, opto-genetic methods could be employed to selectively regulate individual cell properties [59].peer-reviewed) is the author/funder.All rights reserved.No reuse allowed without permission.

Fig. 2 .
Fig. 2.Diagram of the underlying physical basis of model simulations.(a) Intracellular

Fig. 3 .
Fig. 3. Initial conditions and sample simulation output.(a) Initial condition of a simulation 626peer-reviewed) is the author/funder.All rights reserved.No reuse allowed without permission.The copyright holder for this preprint (which was not .http://dx.doi.org/10.1101/037820doi: bioRxiv preprint first posted online Jan.25, 2016;

Fig. 8 .
Fig. 8. Schematic detailing the contribution of individual cellular mechanical properties peer-reviewed) is the author/funder.All rights reserved.No reuse allowed without permission.

Table 1 : 6 & 7
Potential energy functions in Epi-scale model ./ ** Other Morse parameters are equal to zero. Fig.

Table 3 :
Tissue scale parameters in Epi-scale

Table 4 : Impact of adhesion, stiffness, and cytoplasmic pressure on MR
These values are reported based on varying the desired parameter from 100% below the calibrated value to 100% above the calibrated value *