Agent-based modeling traction force mediated compaction of cell-populated collagen gels using physically realistic fibril mechanics

Compressive instabilities enable cell-induced extreme densification patterns in the fibrous extracellular matrix: Discrete model predictions

We present a new model and extensive computations that explain the dramatic remodelling undergone by a fibrous collagen extracellular matrix (ECM), when subjected to contractile mechanical forces from embedded cells or cell clusters. This remodelling creates complex patterns, comprising multiple narrow localised bands of severe densification and fiber alignment, extending far into the ECM, often joining distant cells or cell clusters (such as tumours). Most previous models cannot capture this behaviour, as they assume stable mechanical fiber response with stress an increasing function of fiber stretch, and a restriction to small displacements. Our fully nonlinear network model distinguishes between two types of single-fiber nonlinearity: fibers that undergo stable (supercritical) buckling (as in previous work) versus fibers that suffer unstable (subcritical) buckling collapse. The model allows unrestricted, arbitrarily large displacements (geometric nonlinearity). Our assumptions on single-fiber instability are supported by recent simulations and experiments on buckling of individual beams with a hierarchical microstructure, such as collagen fibers. We use simple scenarios to illustrate, for the first time, two distinct compressive-instability mechanisms at work in our model: unstable buckling collapse of single fibers, and snap-through of multiple-fiber groups. The latter is possible even when single fibers are stable. Through simulations of large fiber networks, we show how these instabilities lead to spatially extended patterns of densification, fiber alignment and ECM remodelling induced by cell contraction. Our model is simple, but describes a very complex, multi-stable energy landscape, using sophisticated numerical optimisation methods that overcome the difficulties caused by instabilities in large systems. Our work opens up new ways of understanding the unique biomechanics of fibrous-network ECM, by fully accounting for nonlinearity and associated loss of stability in fiber networks. Our results provide new insights on tumour invasion and metastasis.

Modeling the extracellular matrix in cell migration and morphogenesis: a guide for the curious biologist

The extracellular matrix (ECM) is a highly complex structure through which biochemical and mechanical signals are transmitted. In processes of cell migration, the ECM also acts as a scaffold, providing structural support to cells as well as points of potential attachment. Although the ECM is a well-studied structure, its role in many biological processes remains difficult to investigate comprehensively due to its complexity and structural variation within an organism. In tandem with experiments, mathematical models are helpful in refining and testing hypotheses, generating predictions, and exploring conditions outside the scope of experiments. Such models can be combined and calibrated with in vivo and in vitro data to identify critical cell-ECM interactions that drive developmental and homeostatic processes, or the progression of diseases. In this review, we focus on mathematical and computational models of the ECM in processes such as cell migration including cancer metastasis, and in tissue structure and morphogenesis. By highlighting the predictive power of these models, we aim to help bridge the gap between experimental and computational approaches to studying the ECM and to provide guidance on selecting an appropriate model framework to complement corresponding experimental studies.

Hybrid cellular Potts and bead-spring modeling of cells in fibrous extracellular matrix

The mechanical interaction between cells and the extracellular matrix (ECM) is fundamental to coordinate collective cell behavior in tissues. Relating individual cell-level mechanics to tissue-scale collective behavior is a challenge that cell-based models such as the cellular Potts model (CPM) are well-positioned to address. These models generally represent the ECM with mean-field approaches, which assume substrate homogeneity. This assumption breaks down with fibrous ECM, which has nontrivial structure and mechanics. Here, we extend the CPM with a bead-spring model of ECM fiber networks modeled using molecular dynamics. We model a contractile cell pulling with discrete focal adhesion-like sites on the fiber network and demonstrate agreement with experimental spatiotemporal fiber densification and displacement. We show that at high network cross-linking, contractile cell forces propagate over at least eight cell diameters, decaying with distance with power law exponent n= 0.35 - 0.65 typical of viscoelastic ECMs. Further, we use in silico atomic force microscopy to measure local cell-induced network stiffening consistent with experiments. Our model lays the foundation for investigating how local and long-ranged cell-ECM mechanobiology contributes to multicellular morphogenesis.

Click chemistry functionalization of self-assembling peptide hydrogels

Self-assembling peptide (SAP) hydrogels provide a fibrous microenvironment to cells while also giving users control of biochemical and mechanical cues. Previously, biochemical cues were introduced by physically mixing them with SAPs prior to hydrogel assembly, or by incorporating them into the SAP sequence during peptide synthesis, which limited flexibility and increased costs. To circumvent these limitations, we developed "Click SAPs," a novel formulation that can be easily functionalized via click chemistry thiol-ene reaction. Due to its high cytocompatibility, the thiol-ene click reaction is currently used to crosslink and functionalize other types of polymeric hydrogels. In this study, we developed a click chemistry compatible SAP platform by addition of a modified lysine (lysine-alloc) to the SAP sequence, enabling effective coupling of thiol-containing molecules to the SAP hydrogel network. We demonstrate the flexibility of this approach by incorporating a fluorescent dye, a cellular adhesion peptide, and a matrix metalloproteinase-sensitive biosensor using the thiol-ene reaction in 3D Click SAPs. Using atomic force microscopy, we demonstrate that Click SAPs retain the ability to self-assemble into fibers, similar to previous systems. Additionally, a range of physiologically relevant stiffnesses can be achieved by adjusting SAP concentration. Encapsulated cells maintain high viability in Click SAPs and can interact with adhesion peptides and a matrix metalloproteinase biosensor, demonstrating that incorporated molecules retain their biological activity. The Click SAP platform supports easier functionalization with a wider array of bioactive molecules and enables new investigations with temporal and spatial control of the cellular microenvironment.

Nonlinear Mechanical Properties of Prestressed Branched Fibrous Networks

Random fiber networks constitute the solid skeleton of many biological materials such as the cytoskeleton of cells and extracellular matrix of soft tissues. These random networks show unique mechanical properties such as nonlinear shear strain-stiffening and strain softening when subjected to preextension and precompression, respectively. In this study, we perform numerical simulations to characterize the influence of axial prestress on the nonlinear mechanical response of random network structures as a function of their micromechanical and geometrical properties. We build our numerical network models using the microstructure of disordered hexagonal lattices and quantify their nonlinear shear response as a function of uniaxial prestress strain. We consider three different material models for individual fibers and fully characterize their influence on the mechanical response of prestressed networks. Moreover, we investigate both the influence of geometric disorder keeping the network connectivity constant and the influence of the randomness in the stiffness of individual fibers keeping their mean stiffness constant. The effects of network connectivity and bending rigidity of fibers are also determined. Several important conclusions are made, including that the tensile and compressive prestress strains, respectively, increase and decrease the initial network shear stiffness but have no effect on the maximal shear modulus. We discuss the findings in terms of microstructural properties such as the local strain energy distribution.

Adhesion modulates cell morphology and migration within dense fibrous networks

One of the most fundamental abilities required for the sustainability of complex life forms is active cell migration, since it is essential in diverse processes from morphogenesis to leukocyte chemotaxis in immune response. The movement of a cell is the result of intricate mechanisms, that involve the coordination between mechanical forces, biochemical regulatory pathways and environmental cues. In particular, epithelial cancer cells have to employ mechanical strategies in order to migrate through the tissue's basement membrane and infiltrate the bloodstream during the invasion stage of metastasis. In this work we explore how mechanical interactions such as spatial restriction and adhesion affect migration of a self-propelled droplet in dense fibrous media. We have performed a systematic analysis using a phase-field model and we propose a novel approach to simulate cell migration with dissipative particle dynamics modelling. With this purpose we have measured in our simulation the cell's velocity and quantified its morphology as a function of the fibre density and of its adhesiveness to the matrix fibres. Furthermore, we have compared our results to a previousin vitromigration assay of fibrosarcoma cells in fibrous matrices. The results show good agreement between the two methodologies and experiments in the literature, which indicates that these minimalist descriptions are able to capture the main features of the system. Our results indicate that adhesiveness is critical for cell migration, by modulating cell morphology in crowded environments and by enhancing cell velocity. In addition, our analysis suggests that matrix metalloproteinases (MMPs) play an important role as adhesiveness modulators. We propose that new assays should be carried out to address the role of adhesion and the effect of different MMPs in cell migration under confined conditions.

Force chains in cell-cell mechanical communication

Force chains (FCs) are a key determinant of the micromechanical properties and behaviour of heterogeneous materials, such as granular systems. However, less is known about FCs in fibrous materials, such as the networks composing the extracellular matrix (ECM) of biological systems. Using a finite-element computational model, we simulated the contraction of a single cell and two nearby cells embedded in two-dimensional fibrous elastic networks and analysed the tensile FCs that developed in the ECM. The role of ECM nonlinear elasticity on FC formation was evaluated by considering linear and nonlinear, i.e. exhibiting 'buckling' and/or 'strain-stiffening', stress-strain curves. The effect of the degree of cell contraction and network coordination value was assessed. We found that nonlinear elasticity of the ECM fibres influenced the structure of the FCs, facilitating the transition towards more distinct chains that were less branched and more radially oriented than the chains formed in linear elastic networks. When two neighbouring cells contract, a larger number of FCs bridged between the cells in nonlinear networks, and these chains had a larger effective rigidity than the chains that did not reach a neighbouring cell. These results suggest that FCs function as a route for mechanical communication between distant cells and highlight the contribution of ECM fibre nonlinear elasticity to the formation of FCs.

Stretch-activated current in human atrial myocytes and Na+ current and mechano-gated channels' current in myofibroblasts alter myocyte mechanical behavior: a computational study

Background

The activation of stretch-activated channels (SACs) in cardiac myocytes, which changes the phases of action potential repolarization, is proven to be highly efficient for the conversion of atrial fibrillation. The expression of Na⁺ current in myofibroblasts (Mfbs) regenerates myocytes’ action potentials, suggesting that Mfbs play an active role in triggering cardiac rhythm disturbances. Moreover, the excitation of mechano-gated channels (MGCs) in Mfbs depolarizes their membrane potential and contributes to the increased risk of post-infarct arrhythmia. Although these electrophysiological mechanisms have been largely known, the roles of these currents in cardiac mechanics are still debated. In this study, we aimed to investigate the mechanical influence of these currents via mathematical modeling. A novel mathematical model was developed by integrating models of human atrial myocyte (including the stretch-activated current, Ca²⁺–force relation, and mechanical behavior of a single segment) and Mfb (including our formulation of Na⁺ current and mechano-gated channels’ current). The effects of the changes in basic cycle length, number of coupled Mfbs and intercellular coupling conductance on myocyte mechanical properties were compared.

Results

Our results indicated that these three currents significantly regulated myocyte mechanical parameters. In isosarcometric contraction, these currents increased segment force by 13.8–36.6% and dropped element length by 12.1–31.5%. In isotonic contraction, there are 2.7–5.9% growth and 0.9–24% reduction. Effects of these currents on the extremum of myocyte mechanical parameters become more significant with the increase of basic cycle length, number of coupled Mfbs and intercellular coupling conductance.

Conclusions

The results demonstrated that stretch-activated current in myocytes and Na⁺ current and mechano-gated channels’ current in Mfbs significantly influenced myocyte mechanical behavior and should be considered in future cardiac mechanical mathematical modeling.

An Agent-Based Discrete Collagen Fiber Network Model of Dynamic Traction Force-Induced Remodeling

Microstructural properties of extracellular matrix (ECM) promote cell and tissue homeostasis as well as contribute to the formation and progression of disease. In order to understand how microstructural properties influence the mechanical properties and traction force-induced remodeling of ECM, we developed an agent-based model that incorporates repetitively applied traction force within a discrete fiber network. An important difference between our model and similar finite element models is that by implementing more biologically realistic dynamic traction, we can explore a greater range of matrix remodeling. Here, we validated our model by reproducing qualitative trends observed in three sets of experimental data reported by others: tensile and shear testing of cell-free collagen gels, collagen remodeling around a single isolated cell, and collagen remodeling between pairs of cells. In response to tensile and shear strain, simulated acellular networks with straight fibrils exhibited biphasic stress-strain curves indicative of strain-stiffening. When fibril curvature was introduced, stress-strain curves shifted to the right, delaying the onset of strain-stiffening. Our data support the notion that strain-stiffening might occur as individual fibrils successively align along the axis of strain and become engaged in tension. In simulations with a single, contractile cell, peak collagen displacement occurred closest to the cell and decreased with increasing distance. In simulations with two cells, compaction of collagen between cells appeared inversely related to the initial distance between cells. These results for cell-populated collagen networks match in vitro findings. A demonstrable benefit of modeling is that it allows for further analysis not feasible with experimentation. Within two-cell simulations, strain energy within the collagen network measured from the final state was relatively uniform around the outer surface of cells separated by 250 μm, but became increasingly nonuniform as the distance between cells decreased. For cells separated by 75 and 100 μm, strain energy peaked in the direction toward the other cell in the region in which fibrils become highly aligned and reached a minimum adjacent to this region, not on the opposite side of the cell as might be expected. This pattern of strain energy was partly attributable to the pattern of collagen compaction, but was still present when mapping strain energy divided by collagen density. Findings like these are of interest because fibril alignment, density, and strain energy may each contribute to contact guidance during tissue morphogenesis.

Multi-cell ECM compaction is predictable via superposition of nonlinear cell dynamics linearized in augmented state space

Cells interacting through an extracellular matrix (ECM) exhibit emergent behaviors resulting from collective intercellular interaction. In wound healing and tissue development, characteristic compaction of ECM gel is induced by multiple cells that generate tensions in the ECM fibers and coordinate their actions with other cells. Computational prediction of collective cell-ECM interaction based on first principles is highly complex especially as the number of cells increase. Here, we introduce a computationally-efficient method for predicting nonlinear behaviors of multiple cells interacting mechanically through a 3-D ECM fiber network. The key enabling technique is superposition of single cell computational models to predict multicellular behaviors. While cell-ECM interactions are highly nonlinear, they can be linearized accurately with a unique method, termed Dual-Faceted Linearization. This method recasts the original nonlinear dynamics in an augmented space where the system behaves more linearly. The independent state variables are augmented by combining auxiliary variables that inform nonlinear elements involved in the system. This computational method involves a) expressing the original nonlinear state equations with two sets of linear dynamic equations b) reducing the order of the augmented linear system via principal component analysis and c) superposing individual single cell-ECM dynamics to predict collective behaviors of multiple cells. The method is computationally efficient compared to original nonlinear dynamic simulation and accurate compared to traditional Taylor expansion linearization. Furthermore, we reproduce reported experimental results of multi-cell induced ECM compaction.

Collective behaviors of multiple cells interacting through an ECM are prohibitively complex to predict with a mechanistic computational model due to its highly nonlinear dynamics and high dimensional space. We introduce a methodology where nonlinear dynamics of single cells are superposed to predict collective multi-cellular behaviors through a developed linearization method. We represent nonlinear single cell dynamics with linear state equations by augmenting the independent state variables with a set of auxiliary variables. We then transform the linear augmented state equations to a low-dimensional latent model and superpose the linear latent models of individual cells to predict collective behaviors that emerge from multi-cellular interactions. The method successfully reproduced experimental results of cell-induced ECM compaction.

Mechanical response of collagen networks to nonuniform microscale loads

As force is applied to fibrous proteins such as collagen or fibrin, the fibers respond by bending, stretching, or buckling, which together bring about a nonlinear relationship between force and displacement. The nonlinearity is typically understood in terms of strain stiffening in uniform extension or shear, but there remains a critical lack of data on how fibrous materials respond to other more complicated loadings. Here we study the mechanics of collagen networks in response to nonuniform loads applied on the local scale of the fibers. For this, we use particles made of an active hydrogel that undergoes a temperature-induced phase transition causing a large decrease in volume. We embed these particles in networks of fibrous collagen and use them as microactuators to apply controlled microscale loading. The resulting fiber displacements propagate over a long range with radial displacements u scaling as r^-n with n ≈ 1. By contrast, we find linear homogeneous materials have n ≈ 2, in agreement with classical linear elastic theory. Our experimental data supports the notion that the long range displacements result from buckling of fibers in compression and local straightening of fibers in tension, in agreement with previous studies. Surprisingly, global network anisotropy appears to have only a modest effect on the displacement propagation. These insights into the microscale mechanics demonstrate that the decay power n provides a useful metric to quantify the mechanics of fibrous materials. We therefore suggest it is a means to compare new theories with experimental data.

Cell Contractility Facilitates Alignment of Cells and Tissues to Static Uniaxial Stretch

During animal development and homeostasis, the structure of tissues, including muscles, blood vessels, and connective tissues, adapts to mechanical strains in the extracellular matrix (ECM). These strains originate from the differential growth of tissues or forces due to muscle contraction or gravity. Here we show using a computational model that by amplifying local strain cues, active cell contractility can facilitate and accelerate the reorientation of single cells to static strains. At the collective cell level, the model simulations show that active cell contractility can facilitate the formation of strings along the orientation of stretch. The computational model is based on a hybrid cellular Potts and finite-element simulation framework describing a mechanical cell-substrate feedback, where: 1) cells apply forces on the ECM, such that 2) local strains are generated in the ECM and 3) cells preferentially extend protrusions along the strain orientation. In accordance with experimental observations, simulated cells align and form stringlike structures parallel to static uniaxial stretch. Our model simulations predict that the magnitude of the uniaxial stretch and the strength of the contractile forces regulate a gradual transition between stringlike patterns and vascular networklike patterns. Our simulations also suggest that at high population densities, less cell cohesion promotes string formation.

Emergence of Collagen Orientation Heterogeneity in Healing Infarcts and an Agent-Based Model

Spatial heterogeneity of matrix structure can be an important determinant of tissue function. Although bulk properties of collagen structure in healing myocardial infarcts have been characterized previously, regional heterogeneity in infarct structure has received minimal attention. Herein, we quantified regional variations of collagen and nuclear orientations over the initial weeks of healing after infarction in rats, and employed a computational model of infarct remodeling to test potential explanations for the heterogeneity we observed in vivo. Fiber and cell orientation maps were generated from infarct samples acquired previously at 1, 2, 3, and 6 weeks postinfarction in a rat ligation model. We analyzed heterogeneity by calculating the dot product of each fiber or cell orientation vector with every other fiber or cell orientation vector, and plotting that dot product versus distance between the fibers or cells. This analysis revealed prominent regional heterogeneity, with alignment of both fibers and cell nuclei in local pockets far exceeding the global average. Using an agent-based model of fibroblast-mediated collagen remodeling, we found that similar levels of heterogeneity can spontaneously emerge from initially isotropic matrix via locally reinforcing cell-matrix interactions. Specifically, cells that sensed fiber orientation at a distance or remodeled fibers at a distance by traction-mediated reorientation or aligned deposition gave rise to regionally heterogeneous structures. However, only the simulations in which cells deposited collagen fibers aligned with their own orientation reproduced experimentally measured patterns of heterogeneity across all time points. These predictions warrant experimental follow-up to test the role of such mechanisms in vivo and identify opportunities to control heterogeneity for therapeutic benefit.

Computational modeling of cardiac fibroblasts and fibrosis

Altered fibroblast behavior can lead to pathologic changes in the heart such as arrhythmia, diastolic dysfunction, and systolic dysfunction. Computational models are increasingly used as a tool to identify potential mechanisms driving a phenotype or potential therapeutic targets against an unwanted phenotype. Here we review how computational models incorporating cardiac fibroblasts have clarified the role for these cells in electrical conduction and tissue remodeling in the heart. Models of fibroblast signaling networks have primarily focused on fibroblast cell lines or fibroblasts from other tissues rather than cardiac fibroblasts, specifically, but they are useful for understanding how fundamental signaling pathways control fibroblast phenotype. In the future, modeling cardiac fibroblast signaling, incorporating -omics and drug-interaction data into signaling network models, and utilizing multi-scale models will improve the ability of in silico studies to predict potential therapeutic targets against adverse cardiac fibroblast activity.

Discrete Element Framework for Modelling Extracellular Matrix, Deformable Cells and Subcellular Components

This paper presents a framework for modelling biological tissues based on discrete particles. Cell components (e.g. cell membranes, cell cytoskeleton, cell nucleus) and extracellular matrix (e.g. collagen) are represented using collections of particles. Simple particle to particle interaction laws are used to simulate and control complex physical interaction types (e.g. cell-cell adhesion via cadherins, integrin basement membrane attachment, cytoskeletal mechanical properties). Particles may be given the capacity to change their properties and behaviours in response to changes in the cellular microenvironment (e.g., in response to cell-cell signalling or mechanical loadings). Each particle is in effect an ‘agent’, meaning that the agent can sense local environmental information and respond according to pre-determined or stochastic events. The behaviour of the proposed framework is exemplified through several biological problems of ongoing interest. These examples illustrate how the modelling framework allows enormous flexibility for representing the mechanical behaviour of different tissues, and we argue this is a more intuitive approach than perhaps offered by traditional continuum methods. Because of this flexibility, we believe the discrete modelling framework provides an avenue for biologists and bioengineers to explore the behaviour of tissue systems in a computational laboratory.

Modelling is an important tool in understanding the behaviour of biological tissues. In this paper we advocate a new modelling framework in which cells and tissues are represented by a collection of particles with associated properties. The particles interact with each other and can change their behaviour in response to changes in their environment. We demonstrate how the propose framework can be used to represent the mechanical behaviour of different tissues with much greater flexibility as compared to traditional continuum based methods.