Analysis of initial cell spreading using mechanistic contact formulations for a deformable cell model

SimuCell3D: three-dimensional simulation of tissue mechanics with cell polarization

The three-dimensional (3D) organization of cells determines tissue function and integrity, and changes markedly in development and disease. Cell-based simulations have long been used to define the underlying mechanical principles. However, high computational costs have so far limited simulations to either simplified cell geometries or small tissue patches. Here, we present SimuCell3D, an efficient open-source program to simulate large tissues in three dimensions with subcellular resolution, growth, proliferation, extracellular matrix, fluid cavities, nuclei and non-uniform mechanical properties, as found in polarized epithelia. Spheroids, vesicles, sheets, tubes and other tissue geometries can readily be imported from microscopy images and simulated to infer biomechanical parameters. Doing so, we show that 3D cell shapes in layered and pseudostratified epithelia are largely governed by a competition between surface tension and intercellular adhesion. SimuCell3D enables the large-scale in silico study of 3D tissue organization in development and disease at a great level of detail.

Chemical etching of Ti-6Al-4V biomaterials fabricated by selective laser melting enhances mesenchymal stromal cell mineralization

Porous titanium scaffolds fabricated by powder bed fusion additive manufacturing techniques have been widely adopted for orthopedic and bone tissue engineering applications. Despite the many advantages of this approach, topological defects inherited from the fabrication process are well understood to negatively affect mechanical properties and pose a high risk if dislodged after implantation. Consequently, there is a need for further post-process surface cleaning. Traditional techniques such as grinding or polishing are not suited to lattice structures, due to lack of a line of sight to internal features. Chemical etching is a promising alternative; however, it remains unclear if changes to surface properties associated with such protocols will influence how cells respond to the material surface. In this study, we explored the response of bone marrow derived mesenchymal stem/stromal cells (MSCs) to Ti-6Al-4V whose surface was exposed to different durations of chemical etching. Cell morphology was influenced by local topological features inherited from the SLM fabrication process. On the as-built surface, topological nonhomogeneities such as partially adhered powder drove a stretched anisotropic cellular morphology, with large areas of the cell suspended across the nonhomogeneous powder interface. As the etching process was continued, surface defects were gradually removed, and cell morphology appeared more isotropic and was suggestive of MSC differentiation along an osteoblastic-lineage. This was accompanied by more extensive mineralization, indicative of progression along an osteogenic pathway. These findings point to the benefit of post-process chemical etching of additively manufactured Ti-6Al-4V biomaterials targeting orthopedic applications.

Stability of asymmetric cell division: A deformable cell model of cytokinesis applied to C. elegans

Cell division during early embryogenesis is linked to key morphogenic events such as embryo symmetry breaking and tissue patterning. It is thought that the physical surrounding of cells together with cell intrinsic cues act as a mechanical "mold," guiding cell division to ensure these events are robust. To quantify how cell division is affected by the mechanical and geometrical environment, we present a novel computational mechanical model of cytokinesis, the final phase of cell division. Simulations with the model reproduced experimentally observed furrow dynamics and describe the volume ratio of daughter cells in asymmetric cell divisions, based on the position and orientation of the mitotic spindle. For dividing cells in geometrically confined environments, we show how the orientation of confinement relative to the division axis modulates the volume ratio in asymmetric cell division. Further, we quantified how cortex viscosity and surface tension determine the shape of a dividing cell and govern bubble-instabilities in asymmetric cell division. Finally, we simulated the formation of the three body axes via sequential (a)symmetric divisions up until the six-cell stage of early C. elegans development, which proceeds within the confines of an eggshell. We demonstrate how model input parameters spindle position and orientation provide sufficient information to reliably predict the volume ratio of daughter cells during the cleavage phase of development. However, for egg geometries perturbed by compression, the model predicts that a change in confinement alone is insufficient to explain experimentally observed differences in cell volume. This points to an effect of the compression on the spindle positioning mechanism. Additionally, the model predicts that confinement stabilizes asymmetric cell divisions against bubble-instabilities.

Long-term adherent cell dynamics emerging from energetic and frictional interactions at the interface

Cell adhesion plays an important role in a wide range of biological situations, including embryonic development, cancer invasion, and wound healing. Although several computational models describing adhesion dynamics have been proposed, models applicable to long-term, large-length-scale cell dynamics are lacking. In this study we investigated possible states of long-term adherent cell dynamics in three-dimensional space by constructing a continuum model of interfacial interactions between adhesive surfaces. In this model a pseudointerface is supposed between each pair of triangular elements that discretize cell surfaces. By introducing a distance between each pair of elements, the physical properties of the interface are given by interfacial energy and friction. The proposed model was implemented into the model of a nonconservative fluid cell membrane where the cell membrane dynamically flows with turnover. Using the implemented model, numerical simulations of adherent cell dynamics on a substrate under flow were performed. The simulations not only reproduced the previously reported dynamics of adherent cells, such as detachment, rolling, and fixation on the substrate, but also discovered other dynamic states, including cell slipping and membrane flow patterns, corresponding to behaviors that occur on much longer timescales than the dissociation of adhesion molecules. These results illustrate the variety of long-term adherent cell dynamics, which are more diverse than the short-term ones. The proposed model can be extended to arbitrarily shaped membranes, thus being useful for the mechanical analysis of a wide range of long-term cell dynamics where adhesion is essential.

Data-driven spatio-temporal modelling of glioblastoma

Mathematical oncology provides unique and invaluable insights into tumour growth on both the microscopic and macroscopic levels. This review presents state-of-the-art modelling techniques and focuses on their role in understanding glioblastoma, a malignant form of brain cancer. For each approach, we summarize the scope, drawbacks and assets. We highlight the potential clinical applications of each modelling technique and discuss the connections between the mathematical models and the molecular and imaging data used to inform them. By doing so, we aim to prime cancer researchers with current and emerging computational tools for understanding tumour progression. By providing an in-depth picture of the different modelling techniques, we also aim to assist researchers who seek to build and develop their own models and the associated inference frameworks. Our article thus strikes a unique balance. On the one hand, we provide a comprehensive overview of the available modelling techniques and their applications, including key mathematical expressions. On the other hand, the content is accessible to mathematicians and biomedical scientists alike to accommodate the interdisciplinary nature of cancer research.

Mechanisms of frustrated phagocytic spreading of human neutrophils on antibody-coated surfaces

Complex motions of immune cells are an integral part of diapedesis, chemotaxis, phagocytosis, and other vital processes. To better understand how immune cells execute such motions, we present a detailed analysis of phagocytic spreading of human neutrophils on flat surfaces functionalized with different densities of immunoglobulin G (IgG) antibodies. We visualize the cell-substrate contact region at high resolution and without labels using reflection interference contrast microscopy and quantify how the area, shape, and position of the contact region evolves over time. We find that the likelihood of the cell commitment to spreading strongly depends on the surface density of IgG, but the rate at which the substrate-contact area of spreading cells increases does not. Validated by a theoretical companion study, our results resolve controversial notions about the mechanisms controlling cell spreading, establishing that active forces generated by the cytoskeleton rather than cell-substrate adhesion primarily drive cellular protrusion. Adhesion, on the other hand, aids phagocytic spreading by regulating the cell commitment to spreading, the maximum cell-substrate contact area, and the directional movement of the contact region.

Delineating mechanisms and design principles of Caenorhabditis elegans embryogenesis using in toto high-resolution imaging data and computational modeling

The nematode (roundworm) Caenorhabditis elegans is one of the most popular animal models for the study of developmental biology, as its invariant development and transparent body enable in toto cellular-resolution fluorescence microscopy imaging of developmental processes at 1-min intervals. This has led to the development of various computational tools for the systematic and automated analysis of imaging data to delineate the molecular and cellular processes throughout the embryogenesis of C. elegans, such as those associated with cell lineage, cell migration, cell morphology, and gene activity. In this review, we first introduce C. elegans embryogenesis and the development of techniques for tracking cell lineage and reconstructing cell morphology during this process. We then contrast the developmental modes of C. elegans and the customized technologies used for studying them with the ones of other animal models, highlighting its advantage for studying embryogenesis with exceptional spatial and temporal resolution. This is followed by an examination of the physical models that have been devised—based on accurate determinations of developmental processes afforded by analyses of imaging data—to interpret the early embryonic development of C. elegans from subcellular to intercellular levels of multiple cells, which focus on two key processes: cell polarization and morphogenesis. We subsequently discuss how quantitative data-based theoretical modeling has improved our understanding of the mechanisms of C. elegans embryogenesis. We conclude by summarizing the challenges associated with the acquisition of C. elegans embryogenesis data, the construction of algorithms to analyze them, and the theoretical interpretation.

Fusion Dynamics of Hybrid Cell-Microparticle Aggregates: A Jelly Pearl Model

We study the fusion of homogeneous cell aggregates and of hybrid aggregates combining cells and microparticles. In all cases, we find that the contact area does not vary linearly over time, as observed for liquid drops, but rather it follows a power law in t^2/3. This result is interpreted by generalizing the fusion model of soft viscoelastic solid balls to viscoelastic liquid balls, akin to jelly pearls. We also explore the asymmetric fusion between a homogeneous aggregate and a hybrid aggregate. This latter experiment allows the determination of the self-diffusion coefficient of the cells in a tissue by following the spatial distribution of internalized particles in the cells.

Phase field model for cell spreading dynamics

We suggest a 3D phase field model to describe 3D cell spreading on a flat substrate. The model is a simplified version of a minimal model that was developed in Winkler (Commun Phys 2:82, 2019). Our model couples the order parameter u with 3D polarization (orientation) vector field [Formula: see text] of the actin network. We derive a closed integro-differential equation governing the 3D cell spreading dynamics on a flat substrate, which includes the normal velocity of the membrane, curvature, volume relaxation rate, a function determined by the molecular effects of the subcell level, and the adhesion effect. This equation is easily solved numerically. The results are in agreement with the early fast phase observed experimentally in Dobereiner (Phys Rev Lett 93:108105, 2004). Also we find agreement with the universal power law (Cuvelier in Curr Biol 17:694-699, 2007) which suggest that cell adhesion or contact area versus time behave as [Formula: see text] in the early stage of cell spreading dynamics, and slow down at the next stages.

Quantitative modeling identifies critical cell mechanics driving bile duct lumen formation

Biliary ducts collect bile from liver lobules, the smallest functional and anatomical units of liver, and carry it to the gallbladder. Disruptions in this process caused by defective embryonic development, or through ductal reaction in liver disease have a major impact on life quality and survival of patients. A deep understanding of the processes underlying bile duct lumen formation is crucial to identify intervention points to avoid or treat the appearance of defective bile ducts. Several hypotheses have been proposed to characterize the biophysical mechanisms driving initial bile duct lumen formation during embryogenesis. Here, guided by the quantification of morphological features and expression of genes in bile ducts from embryonic mouse liver, we sharpened these hypotheses and collected data to develop a high resolution individual cell-based computational model that enables to test alternative hypotheses in silico. This model permits realistic simulations of tissue and cell mechanics at sub-cellular scale. Our simulations suggest that successful bile duct lumen formation requires a simultaneous contribution of directed cell division of cholangiocytes, local osmotic effects generated by salt excretion in the lumen, and temporally-controlled differentiation of hepatoblasts to cholangiocytes, with apical constriction of cholangiocytes only moderately affecting luminal size.

The initial step in bile duct development is the formation of a biliary lumen, a process which involves several cellular mechanisms, such as cell division and polarization, and secretion of fluid. However, how these mechanisms are orchestrated in time and space is difficult to understand. Here, we built a computational model of biliary lumen formation which represents every cell and its function in detail. With the model we can simulate the effect of biophysical aspects that affect duct formation. We have tested the individual and combined effects of directed cell division, apical constriction, and osmotic effects on lumen expansion by varying the parameters that control their relative strength. Our simulations suggest that successful bile duct lumen formation requires the simultaneous contribution of directed cell division of cholangiocytes, local osmotic effects generated by salt excretion in the lumen, and temporally-controlled differentiation of hepatoblasts to cholangiocytes, with apical constriction of cholangiocytes only moderately affecting luminal size.

Modeling Epiblast Shape in Implanting Mammalian Embryos

An indispensable prerequisite of mammalian development is successful morphogenesis in the epiblast, the embryonic tissue that gives rise to all differentiated cells of the adult mammal. The right control of both epiblast morphogenesis and the events that regulate its shape in particular during implantation is henceforth of tremendous importance. However, monitoring the process of development in implanting human embryos is ethically and technically challenging, making it difficult to troubleshoot when things go wrong, as it is unfortunately the case with over 30% of pregnancy failures. Although modern in vitro techniques have proven very insightful lately, more tools are needed in the quest to elucidate mammalian and human development. Mathematical and computational modeling position themselves as helpful complementary tools in the biologist's toolbox, enabling the exploration of the living in silico, beyond the boundaries set by ethical concerns and the potential limitations of wet lab techniques. Here, we show how mathematical modeling and computer simulations can be used to emulate and investigate mechanisms driving epiblast shape changes in mouse and human embryos during implantation.

Biomechanical model of cells probing the myosin-II-independent mechanosensing mechanism

Mechanosensing of cells to extracellular matrix (ECM) is highly active and plays a crucial role in various physiological processes. Growing numbers of studies provide evidence that cell sensitivity to ECM stiffness is a complex stress-strain feedback process. However, the mechanisms that rule this process are still not fully known. Here, an alternative mechanosensing scheme of cells, which is different from the previous myosin-II-based mechanisms, is proposed by employing the tension in cortical cytoskeletons (CSKs) as a force module to probe the substrate. The molecular mechanotransduction from cortical CSKs, through actin filaments and focal adhesions, and finally to the substrate, is mechanically modeled to scale the dynamic traction forces of cells. The developed model captures the characteristic spread of cells with respect to ECM stiffness whereby the spread is fully developed on a stiff substrate but not on a soft one. Furthermore, durotactic migration of cells on an elastic-gradient substrate is successfully modeled by the current method. The cells are concluded to migrate, actuated by the polarized traction forces from the stiffness gradient of the substrate and the stiffness matching between cells and substrate. Finally, the cells are proposed to actively target the preferred substrate by following a rule of mechanical matching between cells and substrate. This study provides a theoretical tool to advance our knowledge regarding the passive mechanical properties and the active sensing of cells, and further promotes the discovery of mechanosensing mechanisms as well as the material design for embryonic development and tissue homeostasis.

spheresDT/Mpacts-PiCS: Cell Tracking and Shape Retrieval in Membrane-labeled Embryos

Motivation

Uncovering the cellular and mechanical processes that drive embryo formation requires an accurate read out of cell geometries over time. However, automated extraction of 3D cell shapes from time-lapse microscopy remains challenging, especially when only membranes are labeled.

Results

We present an image analysis framework for automated tracking and three-dimensional cell segmentation in confocal time lapses. A sphere clustering approach allows for local thresholding and application of logical rules to facilitate tracking and unseeded segmentation of variable cell shapes. Next, the segmentation is refined by a discrete element method simulation where cell shapes are constrained by a biomechanical cell shape model. We apply the framework on Caenorhabditis elegans embryos in various stages of early development and analyze the geometry of the 7- and 8-cell stage embryo, looking at volume, contact area and shape over time.

Availability and implementation

The Python code for the algorithm and for measuring performance, along with all data needed to recreate the results is freely available at 10.5281/zenodo.5108416 and 10.5281/zenodo.4540092. The most recent version of the software is maintained at https://bitbucket.org/pgmsembryogenesis/sdt-pics.

Supplementary information

Supplementary data are available at Bioinformatics online.

Computational modelling unveils how epiblast remodelling and positioning rely on trophectoderm morphogenesis during mouse implantation

Understanding the processes by which the mammalian embryo implants in the maternal uterus is a long-standing challenge in embryology. New insights into this morphogenetic event could be of great importance in helping, for example, to reduce human infertility. During implantation the blastocyst, composed of epiblast, trophectoderm and primitive endoderm, undergoes significant remodelling from an oval ball to an egg cylinder. A main feature of this transformation is symmetry breaking and reshaping of the epiblast into a “cup”. Based on previous studies, we hypothesise that this event is the result of mechanical constraints originating from the trophectoderm, which is also significantly transformed during this process. In order to investigate this hypothesis we propose MG# (MechanoGenetic Sharp), an original computational model of biomechanics able to reproduce key cell shape changes and tissue level behaviours in silico. With this model, we simulate epiblast and trophectoderm morphogenesis during implantation. First, our results uphold experimental findings that repulsion at the apical surface of the epiblast is essential to drive lumenogenesis. Then, we provide new theoretical evidence that trophectoderm morphogenesis indeed can dictate the cup shape of the epiblast and fosters its movement towards the uterine tissue. Our results offer novel mechanical insights into mouse peri-implantation and highlight the usefulness of agent-based modelling methods in the study of embryogenesis.

Distribution and propagation of mechanical stress in simulated structurally heterogeneous tissue spheroids

The mechanical microenvironment of cells has been associated with phenotypic changes that cells undergo in three-dimensional spheroid culture formats. Radial asymmetry in mechanical stress - with compression in the core and tension at the periphery - has been analyzed by representing tissue spheroids as homogeneous visco-elastic droplets under surface tension. However, the influence of the granular microstructure of tissue spheroids in the distribution of mechanical stress in tissue spheroids has not been accounted for in a generic manner. Here, we quantify the distribution and propagation of mechanical forces in structurally heterogeneous multicellular assemblies. For this, we perform numerical simulations of a deformable cell model, which represents cells as elastic, contractile shells surrounding a liquid incompressible cytoplasm, interacting by means of non-specific adhesion. Using this model, we show how cell-scale properties such as cortical stiffness, active tension and cell-cell adhesive tension influence the distribution of mechanical stress in simulated tissue spheroids. Next, we characterize the transition at the tissue-scale from a homogeneous liquid droplet to a heterogeneous packed granular assembly.

Intercellular Adhesion Stiffness Moderates Cell Decoupling as a Function of Substrate Stiffness

The interplay between cell-cell and cell-substrate interactions is complex yet necessary for the formation and healthy functioning of tissues. The same mechanosensing mechanisms used by the cell to sense its extracellular matrix also play a role in intercellular interactions. We used the discrete element method to develop a computational model of a deformable cell that includes subcellular components responsible for mechanosensing. We modeled a three-dimensional cell pair on a patterned (two-dimensional) substrate, a simple laboratory setup to study intercellular interactions. We explicitly modeled focal adhesions and adherens junctions. These mechanosensing adhesions matured, becoming stabilized by force. We also modeled contractile stress fibers that bind the discrete adhesions. The mechanosensing fibers strengthened upon stalling. Traction exerted on the substrate was used to generate traction maps (along the cell-substrate interface). These simulated maps are compared to experimental maps obtained via traction force microscopy. The model recreates the dependence on substrate stiffness of the tractions' spatial distribution, contractile moment of the cell pair, intercellular force, and number of focal adhesions. It also recreates the phenomenon of cell decoupling, in which cells exert forces separately when substrate stiffness increases. More importantly, the model provides viable molecular explanations for decoupling: mechanosensing mechanisms are responsible for competition between different fiber-adhesion configurations present in the cell pair. The point at which an increasing substrate stiffness becomes as high as that of the cell-cell interface is the tipping point at which configurations that favor cell-substrate adhesion dominate over those favoring cell-cell adhesion. This competition is responsible for decoupling.

Modeling of Mechanosensing Mechanisms Reveals Distinct Cell Migration Modes to Emerge From Combinations of Substrate Stiffness and Adhesion Receptor-Ligand Affinity

Mesenchymal cell migration is an integral process in development and healing. The process is regulated by both mechanical and biochemical properties. Mechanical properties of the environment are sensed through mechanosensing, which consists of molecular responses mediated by mechanical signals. We developed a computational model of a deformable 3D cell on a flat substrate using discrete element modeling. The cell is polarized in a single direction and thus moves along the long axis of the substrate. By modeling discrete focal adhesions and stress fibers, we implement two mechanosensing mechanisms: focal adhesion stabilization by force and stress fiber strengthening upon contraction stalling. Two substrate-associated properties, substrate (ligand) stiffness and adhesion receptor–ligand affinity (in the form of focal adhesion disassembly rate), were varied for different model setups in which the mechanosensing mechanisms are set as active or inactive. Cell displacement, focal adhesion number, and cellular traction were quantified and tracked in time. We found that varying substrate stiffness (a mechanical property) and adhesion receptor–ligand affinity (a biochemical property) simultaneously dictate the mode in which cells migrate; cells either move in a smooth manner reminiscent of keratocytes or in a cyclical manner reminiscent of epithelial cells. Mechanosensing mechanisms are responsible for the range of conditions in which a cell adopts a particular migration mode. Stress fiber strengthening, specifically, is responsible for cyclical migration due to build-up of enough force to elicit rupture of focal adhesions and retraction of the cellular rear. Together, both mechanisms explain bimodal dependence of cell migration on substrate stiffness observed in the literature.

The role of actin protrusion dynamics in cell migration through a degradable viscoelastic extracellular matrix: Insights from a computational model

Actin protrusion dynamics plays an important role in the regulation of three-dimensional (3D) cell migration. Cells form protrusions that adhere to the surrounding extracellular matrix (ECM), mechanically probe the ECM and contract in order to displace the cell body. This results in cell migration that can be directed by the mechanical anisotropy of the ECM. However, the subcellular processes that regulate protrusion dynamics in 3D cell migration are difficult to investigate experimentally and therefore not well understood. Here, we present a computational model of cell migration through a degradable viscoelastic ECM. This model is a 2D representation of 3D cell migration. The cell is modeled as an active deformable object that captures the viscoelastic behavior of the actin cortex and the subcellular processes underlying 3D cell migration. The ECM is regarded as a viscoelastic material, with or without anisotropy due to fibrillar strain stiffening, and modeled by means of the meshless Lagrangian smoothed particle hydrodynamics (SPH) method. ECM degradation is captured by local fluidization of the material and permits cell migration through the ECM. We demonstrate that changes in ECM stiffness and cell strength affect cell migration and are accompanied by changes in number, lifetime and length of protrusions. Interestingly, directly changing the total protrusion number or the average lifetime or length of protrusions does not affect cell migration. A stochastic variability in protrusion lifetime proves to be enough to explain differences in cell migration velocity. Force-dependent adhesion disassembly does not result in faster migration, but can make migration more efficient. We also demonstrate that when a number of simultaneous protrusions is enforced, the optimal number of simultaneous protrusions is one or two, depending on ECM anisotropy. Together, the model provides non-trivial new insights in the role of protrusions in 3D cell migration and can be a valuable contribution to increase the understanding of 3D cell migration mechanics.

A quantitative high-resolution computational mechanics cell model for growing and regenerating tissues

Mathematical models are increasingly designed to guide experiments in biology, biotechnology, as well as to assist in medical decision making. They are in particular important to understand emergent collective cell behavior. For this purpose, the models, despite still abstractions of reality, need to be quantitative in all aspects relevant for the question of interest. This paper considers as showcase example the regeneration of liver after drug-induced depletion of hepatocytes, in which the surviving and dividing hepatocytes must squeeze in between the blood vessels of a network to refill the emerged lesions. Here, the cells' response to mechanical stress might significantly impact the regeneration process. We present a 3D high-resolution cell-based model integrating information from measurements in order to obtain a refined and quantitative understanding of the impact of cell-biomechanical effects on the closure of drug-induced lesions in liver. Our model represents each cell individually and is constructed by a discrete, physically scalable network of viscoelastic elements, capable of mimicking realistic cell deformation and supplying information at subcellular scales. The cells have the capability to migrate, grow, and divide, and the nature and parameters of their mechanical elements can be inferred from comparisons with optical stretcher experiments. Due to triangulation of the cell surface, interactions of cells with arbitrarily shaped (triangulated) structures such as blood vessels can be captured naturally. Comparing our simulations with those of so-called center-based models, in which cells have a largely rigid shape and forces are exerted between cell centers, we find that the migration forces a cell needs to exert on its environment to close a tissue lesion, is much smaller than predicted by center-based models. To stress generality of the approach, the liver simulations were complemented by monolayer and multicellular spheroid growth simulations. In summary, our model can give quantitative insight in many tissue organization processes, permits hypothesis testing in silico, and guide experiments in situations in which cell mechanics is considered important.

Integrated lung tissue mechanics one piece at a time: Computational modeling across the scales of biology

The lung is a delicately balanced and highly integrated mechanical system. Lung tissue is continuously exposed to the environment via the air we breathe, making it susceptible to damage. As a consequence, respiratory diseases present a huge burden on society and their prevalence continues to rise. Emergent function is produced not only by the sum of the function of its individual components but also by the complex feedback and interactions occurring across the biological scales - from genes to proteins, cells, tissue and whole organ - and back again. Computational modeling provides the necessary framework for pulling apart and putting back together the pieces of the body and organ systems so that we can fully understand how they function in both health and disease. In this review, we discuss models of lung tissue mechanics spanning from the protein level (the extracellular matrix) through to the level of cells, tissue and whole organ, many of which have been developed in isolation. This is a vital step in the process but to understand the emergent behavior of the lung, we must work towards integrating these component parts and accounting for feedback across the scales, such as mechanotransduction. These interactions will be key to unlocking the mechanisms occurring in disease and in seeking new pharmacological targets and improving personalized healthcare.

Discrete element modelling of tomato tissue deformation and failure at the cellular scale

Bruise damage in fruit results from cell wall failure and inter-cellular separation. Despite the importance of the micro-mechanics of plant tissue with respect to its integrity, it remains largely unquantified and poorly understood, due to many difficulties during experimental characterization. In this article, a 3D micro-mechanical plant tissue model that is able to model cell rupture and inter-cellular debonding and thus provide more insight into the micro-mechanics was developed. The model is based on the discrete element method (DEM) and represents the tissue as a mass-spring system. Each plant cell is represented as a deformable visco-elastoplastic triangulated mesh under turgor pressure. To model cell wall rupture, it is assumed that a spring connection in the wall breaks at a certain critical stretch ratio and that a ruptured cell is turgorless. The inter-cellular contact model assumes brittle fracture between a cell's node and an adjacent cell's triangle when their bond distance exceeds a critical value. A high-speed tomato fruit cell compression test was simulated and the modelled force-strain curve compares well with the experimental data, including for strains above the elastic limit. By varying the shape of the cell in the compression simulation it was shown that the force-strain curve is highly dependent on the cell shape and thus parameter fitting procedures based on a spherical cell model will be inaccurate. Furthermore, the wall stiffness and thickness showed a positive linear relationship with the force at cell bursting. Besides simulating compression tests of single cells, we also simulated tensile and compression tests on small tissue specimens. Realistic tissue structures of tomato mesocarp tissue were generated by a novel method using DEM simulations of deformable cells in a shrinking cylinder. The cell area, volume and anisotropy distributions of the virtual tissue compared well with micro-CT images of real tomato mesocarp tissue (normalized root mean square error values smaller than 3%). The tissue compression and tensile test simulations demonstrated an important influence of the inter-cellular bonding energy and tissue porosity on the tissue failure characteristics and elastic modulus.

Quantitative cell-based model predicts mechanical stress response of growing tumor spheroids over various growth conditions and cell lines

Model simulations indicate that the response of growing cell populations on mechanical stress follows the same functional relationship and is predictable over different cell lines and growth conditions despite experimental response curves look largely different. We develop a hybrid model strategy in which cells are represented by coarse-grained individual units calibrated with a high resolution cell model and parameterized by measurable biophysical and cell-biological parameters. Cell cycle progression in our model is controlled by volumetric strain, the latter being derived from a bio-mechanical relation between applied pressure and cell compressibility. After parameter calibration from experiments with mouse colon carcinoma cells growing against the resistance of an elastic alginate capsule, the model adequately predicts the growth curve in i) soft and rigid capsules, ii) in different experimental conditions where the mechanical stress is generated by osmosis via a high molecular weight dextran solution, and iii) for other cell types with different growth kinetics from the growth kinetics in absence of external stress. Our model simulation results suggest a generic, even quantitatively same, growth response of cell populations upon externally applied mechanical stress, as it can be quantitatively predicted using the same growth progression function.

The Effect of Cortical Elasticity and Active Tension on Cell Adhesion Mechanics

We consider a cell as an elastic, contractile shell surrounding a liquid incompressible cytoplasm and with nonspecific adhesion. We perform numerical simulations of this model to study the mechanics of cell-cell separation. By variation of parameters, we are able to recover well-known limits of the Johnson-Kendall-Roberts theory, the Derjaguin-Muller-Toporov model, adhesive vesicles with surface tension (Brochard-Wyart and de Gennes derivation), and thin elastic shells. We further locate biological cells on this parameter space by comparison to existing experiments on S180 cells. Using this model, we show that mechanical parameters can be obtained that are consistent with both dual pipette aspiration and micropipette aspiration, a problem not successfully tackled so far. We estimate a cortex elastic modulus of E_c ≈ 15 kPa, an effective cortex thickness of t_c ≈ 0.3 μm, and an active tension of γ ≈ 0.4 nN/μm. With these parameters, a Johnson-Kendall-Roberts-like scaling of the separation force is recovered. Finally, the change of contact radius with applied force in a pull-off experiment was investigated. For small forces, a scaling similar to both the Brochard-Wyart and de Gennes derivation and the Derjaguin-Muller-Toporov model is found.

Multivalent Flexible Nanogels Exhibit Broad-Spectrum Antiviral Activity by Blocking Virus Entry

The entry process of viruses into host cells is complex and involves stable but transient multivalent interactions with different cell surface receptors. The initial contact of several viruses begins with attachment to heparan sulfate (HS) proteoglycans on the cell surface, which results in a cascade of events that end up with virus entry. The development of antiviral agents based on multivalent interactions to shield virus particles and block initial interactions with cellular receptors has attracted attention in antiviral research. Here, we designed nanogels with different degrees of flexibility based on dendritic polyglycerol sulfate to mimic cellular HS. The designed nanogels are nontoxic and broad-spectrum, can multivalently interact with viral glycoproteins, shield virus surfaces, and efficiently block infection. We also visualized virus-nanogel interactions as well as the uptake of nanogels by the cells through clathrin-mediated endocytosis using confocal microscopy. As many human viruses attach to the cells through HS moieties, we introduce our flexible nanogels as robust inhibitors for these viruses.

A phenomenological model for cell and nucleus deformation during cancer metastasis

Cell migration plays an essential role in cancer metastasis. In cancer invasion through confined spaces, cells must undergo extensive deformation, which is a capability related to their metastatic potentials. Here, we simulate the deformation of the cell and nucleus during invasion through a dense, physiological microenvironment by developing a phenomenological computational model. In our work, cells are attracted by a generic emitting source (e.g., a chemokine or stiffness signal), which is treated by using Green’s Fundamental solutions. We use an IMEX integration method where the linear parts and the nonlinear parts are treated by using an Euler backward scheme and an Euler forward method, respectively. We develop the numerical model for an obstacle-induced deformation in 2D or/and 3D. Considering the uncertainty in cell mobility, stochastic processes are incorporated and uncertainties in the input variables are evaluated using Monte Carlo simulations. This quantitative study aims at estimating the likelihood for invasion and the length of the time interval in which the cell invades the tissue through an obstacle. Subsequently, the two-dimensional cell deformation model is applied to simplified cancer metastasis processes to serve as a model for in vivo or in vitro biomedical experiments.

Linking stem cell function and growth pattern of intestinal organoids

Intestinal stem cells (ISCs) require well-defined signals from their environment in order to carry out their specific functions. Most of these signals are provided by neighboring cells that form a stem cell niche, whose shape and cellular composition self-organize. Major features of this self-organization can be studied in ISC-derived organoid culture. In this system, manipulation of essential pathways of stem cell maintenance and differentiation results in well-described growth phenotypes. We here provide an individual cell-based model of intestinal organoids that enables a mechanistic explanation of the observed growth phenotypes. In simulation studies of the 3D structure of expanding organoids, we investigate interdependences between Wnt- and Notch-signaling which control the shape of the stem cell niche and, thus, the growth pattern of the organoids. Similar to in vitro experiments, changes of pathway activities alter the cellular composition of the organoids and, thereby, affect their shape. Exogenous Wnt enforces transitions from branched into a cyst-like growth pattern; known to occur spontaneously during long term organoid expansion. Based on our simulation results, we predict that the cyst-like pattern is associated with biomechanical changes of the cells which assign them a growth advantage. The results suggest ongoing stem cell adaptation to in vitro conditions during long term expansion by stabilizing Wnt-activity. Our study exemplifies the potential of individual cell-based modeling in unraveling links between molecular stem cell regulation and 3D growth of tissues. This kind of modeling combines experimental results in the fields of stem cell biology and cell biomechanics constituting a prerequisite for a better understanding of tissue regeneration as well as developmental processes.

Non-uniform force allocation for area preservation in spring network models

In modeling of elastic objects in a flow such as red blood cells, white blood cells, or tumor cells, several elastic moduli are involved. One of them is the area conservation modulus. In this paper, we focus on spring network models, and we introduce a new way of modeling the area preservation modulus. We take into account the current shape of the individual triangles and find the proportional allocation of area conservation forces, which would for individual triangles preserve their shapes. The analysis shows that this approach tends to regularize the triangulation. We demonstrate this effect on individual triangles as well as on the complete triangulations. Copyright © 2015 John Wiley & Sons, Ltd.

Immersed Boundary Models for Quantifying Flow-Induced Mechanical Stimuli on Stem Cells Seeded on 3D Scaffolds in Perfusion Bioreactors

Perfusion bioreactors regulate flow conditions in order to provide cells with oxygen, nutrients and flow-associated mechanical stimuli. Locally, these flow conditions can vary depending on the scaffold geometry, cellular confluency and amount of extra cellular matrix deposition. In this study, a novel application of the immersed boundary method was introduced in order to represent a detailed deformable cell attached to a 3D scaffold inside a perfusion bioreactor and exposed to microscopic flow. The immersed boundary model permits the prediction of mechanical effects of the local flow conditions on the cell. Incorporating stiffness values measured with atomic force microscopy and micro-flow boundary conditions obtained from computational fluid dynamics simulations on the entire scaffold, we compared cell deformation, cortical tension, normal and shear pressure between different cell shapes and locations. We observed a large effect of the precise cell location on the local shear stress and we predicted flow-induced cortical tensions in the order of 5 pN/μm, at the lower end of the range reported in literature. The proposed method provides an interesting tool to study perfusion bioreactors processes down to the level of the individual cell's micro-environment, which can further aid in the achievement of robust bioprocess control for regenerative medicine applications.

The universal growth rate behavior and regime transition in adherent cell colonies

In this work, we used five cell lineages, cultivated in vitro, to show they follow a common functional form to the growth rate: a sigmoidal curve, suggesting that competition and cooperation (usual mechanisms for systems with this behavior) might be present. Both theoretical and experimental investigations, on the causes of this behavior, are challenging for the research field; since the sigmoidal form to the growth rate seems to absorb important properties of such systems, e.g., cell deformation and statistical interactions. We shed some light on this subject by showing how cell spreading affects the radius behavior of the growing colonies. Doing numerical time derivatives of the experimental data, we obtained the growth rates. Using reduced variables for the time and rates, we obtained the collapse of all colonies growth rates onto one curve with sigmoidal shape. This suggests a universal-type behavior, with regime transition related to a morphological transition of adherent cell colonies.

Controlling seepage in discrete particle simulations of biological systems

It is now commonplace to represent materials in a simulation using assemblies of discrete particles. Sometimes, one wishes to maintain the integrity of boundaries between particle types, for example, when modelling multiple tissue layers. However, as the particle assembly evolves during a simulation, particles may pass across interfaces. This behaviour is referred to as 'seepage'. The aims of this study were (i) to examine the conditions for seepage through a confining particle membrane and (ii) to define some simple rules that can be employed to control seepage. Based on the force-deformation response of spheres with various sizes and stiffness, we develop analytic expressions for the force required to move a 'probe particle' between confining 'membrane particles'. We analyse the influence that particle's size and stiffness have on the maximum force that can act on the probe particle before the onset of seepage. The theoretical results are applied in the simulation of a biological cell under unconfined compression.

Cell Invasion Dynamics into a Three Dimensional Extracellular Matrix Fibre Network

The dynamics of filopodia interacting with the surrounding extracellular matrix (ECM) play a key role in various cell-ECM interactions, but their mechanisms of interaction with the ECM in 3D environment remain poorly understood. Based on first principles, here we construct an individual-based, force-based computational model integrating four modules of 1) filopodia penetration dynamics; 2) intracellular mechanics of cellular and nuclear membranes, contractile actin stress fibers, and focal adhesion dynamics; 3) structural mechanics of ECM fiber networks; and 4) reaction-diffusion mass transfers of seven biochemical concentrations in related with chemotaxis, proteolysis, haptotaxis, and degradation in ECM to predict dynamic behaviors of filopodia that penetrate into a 3D ECM fiber network. The tip of each filopodium crawls along ECM fibers, tugs the surrounding fibers, and contracts or retracts depending on the strength of the binding and the ECM stiffness and pore size. This filopodium-ECM interaction is modeled as a stochastic process based on binding kinetics between integrins along the filopodial shaft and the ligands on the surrounding ECM fibers. This filopodia stochastic model is integrated into migratory dynamics of a whole cell in order to predict the cell invasion into 3D ECM in response to chemotaxis, haptotaxis, and durotaxis cues. Predicted average filopodia speed and that of the cell membrane advance agreed with experiments of 3D HUVEC migration at r(2) > 0.95 for diverse ECMs with different pore sizes and stiffness.

The effect of the physical properties of the substrate on the kinetics of cell adhesion and crawling studied by an axisymmetric diffusion-energy balance coupled model

In this paper an analytical approach to study the effect of the substrate physical properties on the kinetics of adhesion and motility behavior of cells is presented. Cell adhesion is mediated by the binding of cell wall receptors and substrate's complementary ligands, and tight adhesion is accomplished by the recruitment of the cell wall binders to the adhesion zone. The binders' movement is modeled as their axisymmetric diffusion in the fluid-like cell membrane. In order to preserve the thermodynamic consistency, the energy balance for the cell-substrate interaction is imposed on the diffusion equation. Solving the axisymmetric diffusion-energy balance coupled equations, it turns out that the physical properties of the substrate (substrate's ligand spacing and stiffness) have considerable effects on the cell adhesion and motility kinetics. For a rigid substrate with uniform distribution of immobile ligands, the maximum ligand spacing which does not interrupt adhesion growth is found to be about 57 nm. It is also found that as a consequence of the reduction in the energy dissipation in the isolated adhesion system, cell adhesion is facilitated by increasing substrate's stiffness. Moreover, the directional movement of cells on a substrate with gradients in mechanical compliance is explored with an extension of the adhesion formulation. It is shown that cells tend to move from soft to stiff regions of the substrate, but their movement is decelerated as the stiffness of the substrate increases. These findings based on the proposed theoretical model are in excellent agreement with the previous experimental observations.

Unraveling liver complexity from molecular to organ level: challenges and perspectives

Biological responses are determined by information processing at multiple and highly interconnected scales. Within a tissue the individual cells respond to extracellular stimuli by regulating intracellular signaling pathways that in turn determine cell fate decisions and influence the behavior of neighboring cells. As a consequence the cellular responses critically impact tissue composition and architecture. Understanding the regulation of these mechanisms at different scales is key to unravel the emergent properties of biological systems. In this perspective, a multidisciplinary approach combining experimental data with mathematical modeling is introduced. We report the approach applied within the Virtual Liver Network to analyze processes that regulate liver functions from single cell responses to the organ level using a number of examples. By facilitating interdisciplinary collaborations, the Virtual Liver Network studies liver regeneration and inflammatory processes as well as liver metabolic functions at multiple scales, and thus provides a suitable example to identify challenges and point out potential future application of multi-scale systems biology.