Computational model for cell migration in three-dimensional matrices

Modeling extracellular matrix reorganization in 3D environments

Extracellular matrix (ECM) remodeling is a key physiological process that occurs in a number of contexts, including cell migration, and is especially important for cellular form and function in three-dimensional (3D) matrices. However, there have been few attempts to computationally model how cells modify their environment in a manner that accounts for both cellular properties and the architecture of the surrounding ECM. To this end, we have developed and validated a novel model to simulate matrix remodeling that explicitly defines cells in a 3D collagenous matrix. In our simulation, cells can degrade, deposit, or pull on local fibers, depending on the fiber density around each cell. The cells can also move within the 3D matrix. Different cell phenotypes can be modeled by varying key cellular parameters. Using the model we have studied how two model cancer cell lines, of differing invasiveness, modify matrices with varying fiber density in their vicinity by tracking the metric of fraction of matrix occupied by fibers. Our results quantitatively demonstrate that in low density environments, cells deposit more collagen to uniformly increase fibril fraction. On the other hand, in higher density environments, the less invasive model cell line reduced the fibril fraction as compared to the highly invasive phenotype. These results show good qualitative and quantitative agreement with existing experimental literature. Our simulation is therefore able to function as a novel platform to provide new insights into the clinically relevant and physiologically critical process of matrix remodeling by helping identify critical parameters that dictate cellular behavior in complex native-like environments.

Cellular response to substrate rigidity is governed by either stress or strain

Cells sense the rigidity of their substrate; however, little is known about the physical variables that determine their response to this rigidity. Here, we report traction stress measurements carried out using fibroblasts on polyacrylamide gels with Young's moduli ranging from 6 to 110 kPa. We prepared the substrates by employing a modified method that involves N-acryloyl-6-aminocaproic acid (ACA). ACA allows for covalent binding between proteins and elastomers and thus introduces a more stable immobilization of collagen onto the substrate when compared to the conventional method of using sulfo-succinimidyl-6-(4-azido-2-nitrophenyl-amino) hexanoate (sulfo-SANPAH). Cells remove extracellular matrix proteins off the surface of gels coated using sulfo-SANPAH, which corresponds to lower values of traction stress and substrate deformation compared to gels coated using ACA. On soft ACA gels (Young's modulus <20 kPa), cell-exerted substrate deformation remains constant, independent of the substrate Young's modulus. In contrast, on stiff substrates (Young's modulus >20 kPa), traction stress plateaus at a limiting value and the substrate deformation decreases with increasing substrate rigidity. Sustained substrate strain on soft substrates and sustained traction stress on stiff substrates suggest these may be factors governing cellular responses to substrate rigidity.

Modeling of adhesion, protrusion, and contraction coordination for cell migration simulations

Cell migration is a highly complex, dynamical biological phenomenon that involves precise spatio-temporal coordination of distinctive sub-processes including adhesion, protrusion, and contraction of the cell. Observations of individual tumor cell migration reveal that cells generally exhibit either mesenchymal-type or amoeboid-type migration modes in native like environments. However, it has also been observed that some migrating cells are capable of morphologically adapting to their environment by modifying their type of migration. Recent studies suggest in fact that changes in biophysical and biomechanical properties of tumor cells can reversibly control their transition from one type of migration to the other. These changes may be caused by internal cell biomechanical mechanisms as well as mechanical and topological properties of the extracellular matrix. In order to understand the complex transition between the two modes and the role played by internal cellular mechanics during migration, we have developed a novel axisymmetric hyperviscoelastic cell model to simulate the dynamical behavior of a migrating cell. Numerical results from our study quantitatively demonstrate that the biomechanical properties of the cell may play an important role in the amoeboid-mesenchymal transition during migration. Our study will therefore not only help in creating a new platform for simulating cellular processes but will also provide insights into the role of sub-cellular mechanics in regulating various modes of migration during tumor invasion and metastasis.

Analytic study of three-dimensional single cell migration with and without proteolytic enzymes

Cell motility is a fundamental physiological process that regulates cellular fate in healthy and diseased systems. Cells cultured in 3D environments often exhibit biphasic dependence of migration speed with cell adhesion. Much is not understood about this very common behavior. A phenomenological model for 3D single-cell migration that exhibits biphasic behavior and highlights the important role of steric hindrance is developed and studied analytically. Changes in the biphasic behavior in the presence of proteolytic enzymes are investigated. Our methods produce a framework to determine analytic formulae for the mean cell speed, allowing general statements in terms of parameters to be explored, which will be useful when interpreting future experimental results. Our formula for mean cell speed as a function of ligand concentration generalizes and extends previous computational models that have shown good agreement with in vitro experiments.

Tumor cell migration in complex microenvironments

Tumor cell migration is essential for invasion and dissemination from primary solid tumors and for the establishment of lethal secondary metastases at distant organs. In vivo and in vitro models enabled identification of different factors in the tumor microenvironment that regulate tumor progression and metastasis. However, the mechanisms by which tumor cells integrate these chemical and mechanical signals from multiple sources to navigate the complex microenvironment remain poorly understood. In this review, we discuss the factors that influence tumor cell migration with a focus on the migration of transformed carcinoma cells. We provide an overview of the experimental and computational methods that allow the investigation of tumor cell migration, and we highlight the benefits and shortcomings of the various assays. We emphasize that the chemical and mechanical stimulus paradigms are not independent and that crosstalk between them motivates the development of new assays capable of applying multiple, simultaneous stimuli and imaging the cellular migratory response in real-time. These next-generation assays will more closely mimic the in vivo microenvironment to provide new insights into tumor progression, inform techniques to control tumor cell migration, and render cancer more treatable.

Computational modelling and analysis of mechanical conditions on cell locomotion and cell-cell interaction

Between other parameters, cell migration is partially guided by the mechanical properties of its substrate. Although many experimental works have been developed to understand the effect of substrate mechanical properties on cell migration, accurate 3D cell locomotion models have not been presented yet. In this paper, we present a novel 3D model for cells migration. In the presented model, we assume that a cell follows two main processes: in the first process, it senses its interface with the substrate to determine the migration direction and in the second process, it exerts subsequent forces to move. In the presented model, cell traction forces are considered to depend on cell internal deformation during the sensing step. A random protrusion force is also considered which may change cell migration direction and/or speed. The presented model was applied for many cases of migration of the cells. The obtained results show high agreement with the available experimental and numerical data.

Inherent interfacial mechanical gradients in 3D hydrogels influence tumor cell behaviors

Cells sense and respond to the rigidity of their microenvironment by altering their morphology and migration behavior. To examine this response, hydrogels with a range of moduli or mechanical gradients have been developed. Here, we show that edge effects inherent in hydrogels supported on rigid substrates also influence cell behavior. A Matrigel hydrogel was supported on a rigid glass substrate, an interface which computational techniques revealed to yield relative stiffening close to the rigid substrate support. To explore the influence of these gradients in 3D, hydrogels of varying Matrigel content were synthesized and the morphology, spreading, actin organization, and migration of glioblastoma multiforme (GBM) tumor cells were examined at the lowest (<50 µm) and highest (>500 µm) gel positions. GBMs adopted bipolar morphologies, displayed actin stress fiber formation, and evidenced fast, mesenchymal migration close to the substrate, whereas away from the interface, they adopted more rounded or ellipsoid morphologies, displayed poor actin architecture, and evidenced slow migration with some amoeboid characteristics. Mechanical gradients produced via edge effects could be observed with other hydrogels and substrates and permit observation of responses to multiple mechanical environments in a single hydrogel. Thus, hydrogel-support edge effects could be used to explore mechanosensitivity in a single 3D hydrogel system and should be considered in 3D hydrogel cell culture systems.

'Run-and-tumble' or 'look-and-run'? A mechanical model to explore the behavior of a migrating amoeboid cell

Single cell migration constitutes a fundamental phenomenon involved in many biological events. Amoeboid cells are single cell organisms that migrate in a cyclic manner like worms. In this paper, we propose a 3D finite element model of an amoeboid cell migrating over a 2D surface. In particular, we focus on the mechanical aspect of the problem. The cell is able to generate cyclic active deformations, such as protrusion and contraction, in any direction. The progression of the cell is governed by a tight synchronization between the adhesion forces, which are alternatively applied at the front and at the rear edges of the cell, and the protrusion-contraction phases of the cell body. Finally, two important aspects have been taken into account: (1) the external stimuli in response to which the cell migrates (e.g. need to feed, morphogenetic events, normal or abnormal environment cues), (2) the heterogeneity of the 2D substrate (e.g. obstacles, rugosity, slippy regions) for which two distinct approaches have been evaluated: the 'run-and-tumble' strategy and the 'look-and-run' strategy. Overall, the results show a good agreement with respect to the experimental observations and the data from the literature (e.g. velocity and strains). Therefore, the present model helps, on one hand, to better understand the intimate relationship between the deformation modes of a cell and the adhesion strength that is required by the cell to crawl over a substrate, and, on the other hand, to put in evidence the crucial role played by mechanics during the migration process.

Kinetic Monte Carlo and cellular particle dynamics simulations of multicellular systems

Computer modeling of multicellular systems has been a valuable tool for interpreting and guiding in vitro experiments relevant to embryonic morphogenesis, tumor growth, angiogenesis and, lately, structure formation following the printing of cell aggregates as bioink particles. Here we formulate two computer simulation methods: (1) a kinetic Monte Carlo (KMC) and (2) a cellular particle dynamics (CPD) method, which are capable of describing and predicting the shape evolution in time of three-dimensional multicellular systems during their biomechanical relaxation. Our work is motivated by the need of developing quantitative methods for optimizing postprinting structure formation in bioprinting-assisted tissue engineering. The KMC and CPD model parameters are determined and calibrated by using an original computational-theoretical-experimental framework applied to the fusion of two spherical cell aggregates. The two methods are used to predict the (1) formation of a toroidal structure through fusion of spherical aggregates and (2) cell sorting within an aggregate formed by two types of cells with different adhesivities.

Biochemical and mechanical extracellular matrix properties dictate mammary epithelial cell motility and assembly

Biochemical and mechanical cues of the extracellular matrix have been shown to play important roles in cell-matrix and cell-cell interactions. We have experimentally tested the combined influence of these cues to better understand cell motility, force generation, cell-cell interaction, and assembly in an in vitro breast cancer model. MCF-10A non-tumorigenic mammary epithelial cells were observed on surfaces with varying fibronectin ligand concentration and polyacrylamide gel rigidity. Our data show that cell velocity is biphasic in both matrix rigidity and adhesiveness. The maximum cell migration velocity occurs only at specific combination of substrate stiffness and ligand density. We found cell-cell interactions reduce migration velocity. However, the traction forces cells exert onto the substrate increase linearly with both cues, with cells in pairs exerting higher maximum tractions observed over single cells. A relationship between force and motility shows a maximum in single cell velocity not observed in cell pairs. Cell-cell adhesion becomes strongly favored on softer gels with elasticity ≤ 1250 Pascals (Pa), implying the existence of a compliance threshold that promotes cell-cell over cell-matrix adhesion. Finally on gels with stiffness similar to pre-malignant breast tissue, 400 Pa, cells undergo multicellular assembly and division into 3D spherical aggregates on a 2D surface.

Cell-biomaterial mechanical interaction in the framework of tissue engineering: insights, computational modeling and perspectives

Tissue engineering is an emerging field of research which combines the use of cell-seeded biomaterials both in vitro and/or in vivo with the aim of promoting new tissue formation or regeneration. In this context, how cells colonize and interact with the biomaterial is critical in order to get a functional tissue engineering product. Cell-biomaterial interaction is referred to here as the phenomenon involved in adherent cells attachment to the biomaterial surface, and their related cell functions such as growth, differentiation, migration or apoptosis. This process is inherently complex in nature involving many physico-chemical events which take place at different scales ranging from molecular to cell body (organelle) levels. Moreover, it has been demonstrated that the mechanical environment at the cell-biomaterial location may play an important role in the subsequent cell function, which remains to be elucidated. In this paper, the state-of-the-art research in the physics and mechanics of cell-biomaterial interaction is reviewed with an emphasis on focal adhesions. The paper is focused on the different models developed at different scales available to simulate certain features of cell-biomaterial interaction. A proper understanding of cell-biomaterial interaction, as well as the development of predictive models in this sense, may add some light in tissue engineering and regenerative medicine fields.

Bio-inspired materials for parsing matrix physicochemical control of cell migration: a review

Cell motility is ubiquitous in both normal and pathophysiological processes. It is a complex biophysical response elicited via the integration of diverse extracellular physicochemical cues. The extracellular matrix directs cell motility via gradients in morphogens (a.k.a. chemotaxis), adhesive proteins (haptotaxis), and stiffness (durotaxis). Three-dimensional geometrical and proteolytic cues also constitute key regulators of motility. Therefore, cells process a variety of physicochemical signals simultaneously, while making informed decisions about migration via intracellular processing. Over the last few decades, bioengineers have created and refined natural and synthetic in vitro platforms in an attempt to isolate these extracellular cues and tease out how cells are able to translate this complex array of dynamic biochemical and biophysical features into functional motility. Here, we review how biomaterials have played a key role in the development of these types of model systems, and how recent advances in engineered materials have significantly contributed to our current understanding of the mechanisms of cell migration.

Cell-oriented modeling of angiogenesis

Due to its significant involvement in various physiological and pathological conditions, angiogenesis (the development of new blood vessels from an existing vasculature) represents an important area of the actual biological research and a field in which mathematical modeling proved particularly useful in supporting the experimental work. In this paper, we focus on a specific modeling strategy, known as "cell-centered" approach. This type of mathematical models work at a "mesoscopic scale," assuming the cell as the natural level of abstraction for computational modeling of development. They treat cells phenomenologically, considering their essential behaviors to study how tissue structure and organization emerge from the collective dynamics of multiple cells. The main contributions of the cell-oriented approach to the study of the angiogenic process will be described. From one side, they have generated "basic science understanding" about the process of capillary assembly during development, growth, and pathology. On the other side, models were also developed supporting "applied biomedical research" for the purpose of identifying new therapeutic targets and clinically relevant approaches for either inhibiting or stimulating angiogenesis.

Integrins in cell migration

Integrin-based adhesion has served as a model for studying the central role of adhesion in migration. In this article, we outline modes of migration, both integrin-dependent and -independent in vitro and in vivo. We next discuss the roles of adhesion contacts as signaling centers and linkages between the ECM and actin that allows adhesions to serve as traction sites. This includes signaling complexes that regulate migration and the interplay among adhesion, signaling, and pliability of the substratum. Finally, we address mechanisms of adhesion assembly and disassembly and the role of adhesion in cellular polarity.

Viscoelastic gel-strip model for the simulation of migrating cells

Migrating tumor cells can exhibit both mesenchymal- and amoeboid-type behaviors. Recent studies have shown that both cellular and extracellular structural and mechanical variables control the transition of tumor cells from one mode to the other and provide them with morphological plasticity. The mesenchymal-mode migration is characterized by strong adhesion and proteolytic machinery to navigate through complex extracellular matrices. The amoeboid-mode migration is characterized by little or no adhesion and strong actomyosin contraction to squeeze through the matrices. While adhesion dependent migration has been computationally and experimentally studied in both 2D and 3D environments, quantitative models of amoeboid motion in native environments are lacking. In order to address this major gap in our understanding and to probe the mesenchymal to amoeboid transitions quantitatively and comprehensively, we have developed an axisymmetric viscoelastic gel-strip model of a single cell to investigate a cell migrating in native-like environments. In this model, cell migration and morphology are governed by internal stresses as well as external forces. The internal stresses are controlled by F-actin density distribution, protrusion strength, and contraction strength. The external forces are controlled by adhesion strength and steric resistance from the extracellular matrix. Our model predicts that the transition of the cell migration mode from mesenchymal- to amoeboid-type, and vice versa, is closely related to the loss of adhesion as well as increased contraction strength of the cells. Our results indicate that amoeboid migration is more suited for low-resistance environment while mesenchymal migration is preferred in high-resistance environment, which would explain the versatile behaviors of tumor cells in complex environments.

A numerical model for durotaxis

Cell migration is a phenomenon that is involved in several physiological processes. In the absence of external guiding factors it shares analogies with Brownian motion. The presence of biochemical or biophysical cues, on the other hand, can influence cell migration transforming it in a biased random movement. Recent studies have shown that different cell types are able to recognise the mechanical properties of the substratum over which they move and that these properties direct the motion through a process called durotaxis. In this work a 2D mathematical model for the description of this phenomenon is presented. The model is based on the Langevin equation that has been modified to take into account the local mechanical properties of the substratum perceived by the cells. Numerical simulations of the model provide individual cell tracks, whose characteristics can be compared with experimental observations directly. The present model is solved for two important cases: an isotropic substratum, to check that random motility is recovered as a subcase, and a biphasic substratum, to investigate durotaxis. The degree of agreement is satisfactory in both cases. The model can be a useful tool for quantifying relevant parameters of cell migration as a function of the substratum mechanical properties.

Engineering strategies to recapitulate epithelial morphogenesis within synthetic three-dimensional extracellular matrix with tunable mechanical properties

The mechanical properties (e.g. stiffness) of the extracellular matrix (ECM) influence cell fate and tissue morphogenesis and contribute to disease progression. Nevertheless, our understanding of the mechanisms by which ECM rigidity modulates cell behavior and fate remains rudimentary. To address this issue, a number of two and three-dimensional (3D) hydrogel systems have been used to explore the effects of the mechanical properties of the ECM on cell behavior. Unfortunately, many of these systems have limited application because fiber architecture, adhesiveness and/or pore size often change in parallel when gel elasticity is varied. Here we describe the use of ECM-adsorbed, synthetic, self-assembling peptide (SAP) gels that are able to recapitulate normal epithelial acini morphogenesis and gene expression in a 3D context. By exploiting the range of viscoelasticity attainable with these SAP gels, and their ability to recreate native-like ECM fibril topology with minimal variability in ligand density and pore size, we were able to reconstitute normal and tumor-like phenotypes and gene expression patterns in nonmalignant mammary epithelial cells. Accordingly, this SAP hydrogel system presents the first tunable system capable of independently assessing the interplay between ECM stiffness and multi-cellular epithelial phenotype in a 3D context.

From molecular signal activation to locomotion: an integrated, multiscale analysis of cell motility on defined matrices

The adhesion, mechanics, and motility of eukaryotic cells are highly sensitive to the ligand density and stiffness of the extracellular matrix (ECM). This relationship bears profound implications for stem cell engineering, tumor invasion and metastasis. Yet, our quantitative understanding of how ECM biophysical properties, mechanotransductive signals, and assembly of contractile and adhesive structures collude to control these cell behaviors remains extremely limited. Here we present a novel multiscale model of cell migration on ECMs of defined biophysical properties that integrates local activation of biochemical signals with adhesion and force generation at the cell-ECM interface. We capture the mechanosensitivity of individual cellular components by dynamically coupling ECM properties to the activation of Rho and Rac GTPases in specific portions of the cell with actomyosin contractility, cell-ECM adhesion bond formation and rupture, and process extension and retraction. We show that our framework is capable of recreating key experimentally-observed features of the relationship between cell migration and ECM biophysical properties. In particular, our model predicts for the first time recently reported transitions from filopodial to “stick-slip” to gliding motility on ECMs of increasing stiffness, previously observed dependences of migration speed on ECM stiffness and ligand density, and high-resolution measurements of mechanosensitive protrusion dynamics during cell motility we newly obtained for this study. It also relates the biphasic dependence of cell migration speed on ECM stiffness to the tendency of the cell to polarize. By enabling the investigation of experimentally-inaccessible microscale relationships between mechanotransductive signaling, adhesion, and motility, our model offers new insight into how these factors interact with one another to produce complex migration patterns across a variety of ECM conditions.

Three-dimensional traction force microscopy: a new tool for quantifying cell-matrix interactions

The interactions between biochemical processes and mechanical signaling play important roles during various cellular processes such as wound healing, embryogenesis, metastasis, and cell migration. While traditional traction force measurements have provided quantitative information about cell matrix interactions in two dimensions, recent studies have shown significant differences in the behavior and morphology of cells when placed in three-dimensional environments. Hence new quantitative experimental techniques are needed to accurately determine cell traction forces in three dimensions. Recently, two approaches both based on laser scanning confocal microscopy have emerged to address this need. This study highlights the details, implementation and advantages of such a three-dimensional imaging methodology with the capability to compute cellular traction forces dynamically during cell migration and locomotion. An application of this newly developed three-dimensional traction force microscopy (3D TFM) technique to single cell migration studies of 3T3 fibroblasts is presented to show that this methodology offers a new quantitative vantage point to investigate the three-dimensional nature of cell-ECM interactions.

Biased three-dimensional cell migration and collagen matrix modification

Various tumours can be resected even for cure with complete removal. Surgical excision with a wide margin to ensure complete removal has therefore been suggested as the primary treatment for such lesions. The histological examination of the three-dimensional (3D) excision margins (3D histology) constitutes an important part of the quality control mechanisms in tumour surgery. Understanding histologically relevant properties of the constituents of the microenvironment in tumours and the circumferential portion of non-lesional tissue is therefore critical. Accompanied by the increasing availability of high performance computers in recent decades, there has been a strong movement aiming at the development of mathematical models whose implementations allow in silico simulations of biological reaction networks. Due to its relevance in numerous biological and pathological processes, there have been various attempts to model biased migration of single cells. The model introduced in this paper plays a prominent role insofar as it covers the under-represented 3D case. Moreover, it is uniformly formulated for both two and three dimensions. The velocity of each cell is characterised by a generalised Langevin equation, a stochastic differential equation, where chemotaxis as well as contact guidance are considered to simulate selected aspects of interactions between carcinoma cell groups and fibroblast-like cells. Algorithmic and numeric aspects of the developed equations are detailed in this paper and the results of the simulations are illustrated in different manners to emphasise specific features. A simple test scenario as well as a geometry based on segmentation data of a real histological slide has served for verification of the software. The resulting morphologies closely resemble that of desmoplastic stromal reaction readily identifiable in histological slides of infiltrating carcinoma, and the images can be interpreted in the context of 3D histology.

A low resistance microfluidic system for the creation of stable concentration gradients in a defined 3D microenvironment

The advent of microfluidic technology allows control and interrogation of cell behavior by defining the local microenvironment with an assortment of biochemical and biophysical stimuli. Many approaches have been developed to create gradients of soluble factors, but the complexity of such systems or their inability to create defined and controllable chemical gradients has limited their widespread implementation. Here we describe a new microfluidic device which employs a parallel arrangement of wells and channels to create stable, linear concentration gradients in a gel region between a source and a sink well. Pressure gradients between the source and sink wells are dissipated through low resistance channels in parallel with the gel channel, thus minimizing the convection of solute in this region. We demonstrate the ability of the new device to quantitate chemotactic responses in a variety of cell types, yielding a complete profile of the migratory response and representing the total number of migrating cells and the distance each cell has migrated. Additionally we show the effect of concentration gradients of the morphogen Sonic hedgehog on the specification of differentiating neural progenitors in a 3-dimensional matrix.

Cancer cell stiffness: integrated roles of three-dimensional matrix stiffness and transforming potential

While significant advances have been made toward revealing the molecular mechanisms that influence breast cancer progression, much less is known about the associated cellular mechanical properties. To this end, we use particle-tracking microrheology to investigate the interplay among intracellular mechanics, three-dimensional matrix stiffness, and transforming potential in a mammary epithelial cell (MEC) cancer progression series. We use a well-characterized model system where human-derived MCF10A MECs overexpress either ErbB2, 14-3-3ζ, or both ErbB2 and 14-3-3ζ, with empty vector as a control. Our results show that MECs possessing ErbB2 transforming potential stiffen in response to elevated matrix stiffness, whereas non-transformed MECs or those overexpressing only 14-3-3ζ do no exhibit this response. We further observe that overexpression of ErbB2 alone is associated with the highest degree of intracellular sensitivity to matrix stiffness, and that the effect of transforming potential on intracellular stiffness is matrix-stiffness-dependent. Moreover, our intracellular stiffness measurements parallel cell migration behavior that has been previously reported for these MEC sublines. Given the current knowledge base of breast cancer mechanobiology, these findings suggest that there may be a positive relationship among intracellular stiffness sensitivity, cell motility, and perturbed mechanotransduction in breast cancer.

Marrow-derived stem cell motility in 3D synthetic scaffold is governed by geometry along with adhesivity and stiffness

Design of 3D scaffolds that can facilitate proper survival, proliferation, and differentiation of progenitor cells is a challenge for clinical applications involving large connective tissue defects. Cell migration within such scaffolds is a critical process governing tissue integration. Here, we examine effects of scaffold pore diameter, in concert with matrix stiffness and adhesivity, as independently tunable parameters that govern marrow-derived stem cell motility. We adopted an "inverse opal" processing technique to create synthetic scaffolds by crosslinking poly(ethylene glycol) at different densities (controlling matrix elastic moduli or stiffness) and small doses of a heterobifunctional monomer (controlling matrix adhesivity) around templating beads of different radii. As pore diameter was varied from 7 to 17 µm (i.e., from significantly smaller than the spherical cell diameter to approximately cell diameter), it displayed a profound effect on migration of these stem cells-including the degree to which motility was sensitive to changes in matrix stiffness and adhesivity. Surprisingly, the highest probability for substantive cell movement through pores was observed for an intermediate pore diameter, rather than the largest pore diameter, which exceeded cell diameter. The relationships between migration speed, displacement, and total path length were found to depend strongly on pore diameter. We attribute this dependence to convolution of pore diameter and void chamber diameter, yielding different geometric environments experienced by the cells within.

Computational modeling of epithelial tissues

There is an extensive literature on the computational modeling of epithelial tissues at all levels from subcellular to whole tissue. This review concentrates on behavior at the individual cell to whole tissue level, and particularly on organizational aspects, and provides an indication of where information from other areas, such as the modeling of angiogenesis, is relevant. The skin, and the lining of all of the body cavities (lung, gut, cervix, bladder etc) are epithelial tissues, which in a topological sense are the boundary between inside and outside the body. They are thin sheets of cells (usually of the order of 0.5 mm thick) without extracellular matrix, have a relatively simple structure, and contain few types of cells. They have important barrier, secretory and transport functions, which are essential for the maintenance of life, so homeostasis and wound healing are important aspects of the behavior of epithelial tissues. Carcinomas originate in epithelial tissues.There are essentially two approaches to modeling tissues--to start at the level of the tissue (i.e., a length scale of the order of 1 mm) and develop generalized equations for behavior (a continuum approach); or to start at the level of the cell (i.e., a length scale of the order of 10 µm) and develop tissue behavior as an emergent property of cellular behavior (an individual-based approach). As will be seen, these are not mutually exclusive approaches, and they come in a variety of flavors.

Epithelial and mesenchymal phenotypic switchings modulate cell motility in metastasis

The most ominous stage of cancer progression is metastasis, or the dissemination of carcinoma cells from the primary site into distant organs. Metastases are often resistant to current extirpative therapies and even the newest biological agents cure only a small subset of patients. Therefore a greater understanding of tumor biology that integrates properties intrinsic to carcinomas with tissue environmental modulators of behavior is needed. In no aspect of tumor progression is this more evident than the acquisition of cell motility that is critical for both escape from the primary tumor and colonization. In this overview, we discuss how this behavior is modified by carcinoma cell phenotypic plasticity that is evidenced by reversible switching between epithelial and mesenchymal phenotypes. The presence or absence of intercellular adhesions mediate these switches and dictate the receptivity towards signals from the extracellular milieu. These signals, which include soluble growth factors, cytokines, and extracellular matrix embedded with matrikines and matricryptines will be discussed in depth. Finally, we will describe a new mode of discerning the balance between epithelioid and mesenchymal movement.

Advances in the biology of bone metastasis: how the skeleton affects tumor behavior

It is increasingly evident that the microenvironment of bone can influence the cancer phenotype in many ways that favor growth in bone. The ability of cancer cells to adhere to bone matrix and to promote osteoclast formation are key requirements for the establishment and growth of bone metastases. Several cytokine products of breast cancers (e.g. PTHrP, IL-11, IL-8) have been shown to act upon host cells of the bone microenvironment to promote osteoclast formation, allowing for excessive bone resorption. The increased release of matrix-derived growth factors, especially TGF-β, acts back upon the tumor to facilitate further tumor expansion and enhance cytokine production, and also upon osteoblasts to suppress bone formation. This provides a self-perpetuating cycle of bone loss and tumor growth within the skeleton. Other contributing factors favoring tumor metastasis and colonization in bone include the unique structure and stiffness of skeletal tissue, along with the diverse cellular composition of the marrow environment (e.g. bone cells, stromal fibroblasts, immune cells), any of which can contribute to the phenotypic changes that can take place in metastatic deposits that favor their survival. Additionally, it is also apparent that breast cancer cells begin to express different bone specific proteins as well as proteins important for normal breast development and lactation that allow them to grow in bone and stimulate bone destruction. Taken together, these continually emerging areas of study suggest new potential pathways important in the pathogenesis of bone metastasis and potential areas for targeting therapeutics.

Matrix rigidity induces osteolytic gene expression of metastatic breast cancer cells

Nearly 70% of breast cancer patients with advanced disease will develop bone metastases. Once established in bone, tumor cells produce factors that cause changes in normal bone remodeling, such as parathyroid hormone-related protein (PTHrP). While enhanced expression of PTHrP is known to stimulate osteoclasts to resorb bone, the environmental factors driving tumor cells to express PTHrP in the early stages of development of metastatic bone disease are unknown. In this study, we have shown that tumor cells known to metastasize to bone respond to 2D substrates with rigidities comparable to that of the bone microenvironment by increasing expression and production of PTHrP. The cellular response is regulated by Rho-dependent actomyosin contractility mediated by TGF-ß signaling. Inhibition of Rho-associated kinase (ROCK) using both pharmacological and genetic approaches decreased PTHrP expression. Furthermore, cells expressing a dominant negative form of the TGF-ß receptor did not respond to substrate rigidity, and inhibition of ROCK decreased PTHrP expression induced by exogenous TGF-ß. These observations suggest a role for the differential rigidity of the mineralized bone microenvironment in early stages of tumor-induced osteolysis, which is especially important in metastatic cancer since many cancers (such as those of the breast and lung) preferentially metastasize to bone.

An optical method to quantify the density of ligands for cell adhesion receptors in three-dimensional matrices

The three-dimensional matrix that surrounds cells is an important insoluble regulator of cell phenotypes. Examples of such insoluble surfaces are the extracellular matrix (ECM), ECM analogues and synthetic polymeric biomaterials. Cell-matrix interactions are mediated by cell adhesion receptors that bind to chemical entities (adhesion ligands) on the surface of the matrix. There are currently no established methods to obtain quantitative estimates of the density of adhesion ligands recognized by specific cell adhesion receptors. This article presents a new optical-based methodology for measuring ligands of adhesion receptors on three-dimensional matrices. The study also provides preliminary quantitative results for the density of adhesion ligands of integrins alpha(1)beta(1) and alpha(2)beta(1) on the surface of collagen-based scaffolds, similar to biomaterials that are used clinically to induce regeneration in injured skin and peripheral nerves. Preliminary estimates of the surface density of the ligands of these two major collagen-binding receptors are 5775 +/- 2064 ligands microm(-2) for ligands of alpha(1)beta(1) and 17 084 +/- 5353 ligands microm(-2) for ligands of alpha(2)beta(1). The proposed methodology can be used to quantify the surface chemistry of insoluble surfaces that possess biological activity, such as native tissue ECM and biomaterials, and therefore can be used in cell biology, biomaterials science and regenerative medical studies for quantitative description of a matrix and its effects on cells.

Protein-engineered biomaterials: nanoscale mimics of the extracellular matrix

BACKGROUND

Traditional materials used as in vitro cell culture substrates are rigid and flat surfaces that lack the exquisite nano- and micro-scale features of the in vivo extracellular environment. While these surfaces can be coated with harvested extracellular matrix (ECM) proteins to partially recapitulate the bio-instructive nature of the ECM, these harvested proteins often exhibit large batch-to-batch variability and can be difficult to customize for specific biological studies. In contrast, recombinant protein technology can be utilized to synthesize families of 3 dimensional protein-engineered biomaterials that are cyto-compatible, reproducible, and fully customizable.

SCOPE OF REVIEW

Here we describe a modular design strategy to synthesize protein-engineered biomaterials that fuse together multiple repeats of nanoscale peptide design motifs into full-length engineered ECM mimics.

MAJOR CONCLUSIONS

Due to the molecular-level precision of recombinant protein synthesis, these biomaterials can be tailored to include a variety of bio-instructional ligands at specified densities, to exhibit mechanical properties that match those of native tissue, and to include proteolytic target sites that enable cell-triggered scaffold remodeling. Furthermore, these biomaterials can be processed into forms that are injectable for minimally-invasive delivery or spatially patterned to enable the release of multiple drugs with distinct release kinetics.

GENERAL SIGNIFICANCE

Given the reproducibility and flexibility of these protein-engineered biomaterials, they are ideal substrates for reductionist biological studies of cell-matrix interactions, for in vitro models of physiological processes, and for bio-instructive scaffolds in regenerative medicine therapies. This article is part of a Special Issue entitled Nanotechnologies - Emerging Applications in Biomedicine.

A model of fibroblast motility on substrates with different rigidities

To function efficiently in the body, the biological cells must have the ability to sense the external environment. Mechanosensitivity toward the extracellular matrix was identified as one of the sensing mechanisms affecting cell behavior. It was shown experimentally that a fibroblast cell prefers locomoting over the stiffer substrate when given a choice between a softer and a stiffer substrate. In this article, we develop a discrete model of fibroblast motility with substrate-rigidity sensing. Our model allows us to understand the interplay between the cell-substrate sensing and the cell biomechanics. The model cell exhibits experimentally observed substrate rigidity sensing, which allows us to gain additional insights into the cell mechanosensitivity.

Computational study of proteolysis-driven single cell migration in a three-dimensional matrix

Cell migration is a fundamental process that is crucial to a variety of physiological events. While traditional approaches have focused on two-dimensional (2D) systems, recent efforts have shifted to studying migration in three-dimensional (3D) matrices. A major distinction that has emerged is the increased importance of cell-matrix interactions in 3D environments. In particular, cell motility in 3D matrices is more dependent on matrix metalloproteinases (MMPs) to degrade steric obstacles than in 2D systems. In this study, we implement the effects of MMP-mediated proteolysis in a force-based computational model of 3D migration, testing two matrix ligand-MMP relationships that have been observed experimentally: linear and log-linear. The model for both scenarios predicts maximal motility at intermediate matrix ligand and MMP levels, with the linear case providing more physiologically compelling results. Recent experimental results suggesting MMP influence on integrin expression are also integrated into the model. While the biphasic behavior is retained, with MMP-integrin feedback peak cell speed is observed in a low ligand, high MMP regime instead of at intermediate ligand and MMP levels for both ligand-MMP relationships. The simulation provides insight into the expanding role of cell-matrix interactions in cell migration in 3D environments and has implications for cancer research.

Lentivirus immobilization to nanoparticles for enhanced and localized delivery from hydrogels

Hydrogels can provide a controllable cell microenvironment for numerous applications in regenerative medicine, and delivery of gene therapy vectors can be employed to enhance their bioactivity. We investigated the delivery of lentiviral vectors from hydrogels, and employed the immobilization of lentivirus to hydroxylapatite (HA) nanoparticles as a means to retain and stabilize vectors within hydrogels, and thereby increase delivery efficiency. Entrapment of the vector alone within the hydrogel maintained the activity of the virus more effectively compared to the absence of a hydrogel, and release was slowed with an increasing solid content of the hydrogel. Association of the lentivirus with HA increased the activity of the complexes, with HA increasing the virus activity for 72 hours. Cells seeded onto lentivirus-HA-loaded hydrogels had a decreased number of infected cells outside of the hydrogel relative to the absence of HA. In vivo studies with collagen hydrogels loaded with lentivirus and HA-lentivirus demonstrated sustained and localized transgene expression for at least 4 weeks, with increased expression using the lentivirus-HA complex. This strategy of nanoparticle immobilization to stabilize and retain vectors is broadly applicable to hydrogels, and may provide a versatile tool to combine gene therapy and biomaterials for applications in regenerative medicine.

An agent-based model identifies MRI regions of probable tumor invasion in a patient with glioblastoma

We present an application of a previously developed agent-based glioma model (Chen et al 2009 Biosystems 95 234-42) for predicting spatio-temporal tumor progression using a patient-specific MRI lattice derived from apparent diffusion coefficient (ADC) data. Agents representing collections of migrating glioma cells are initialized based upon voxels at the outer border of the tumor identified on T1-weighted (Gd+) MRI at an initial time point. These simulated migratory cells exhibit a specific biologically inspired spatial search paradigm, representing a weighting of the differential contribution from haptotactic permission and biomechanical resistance on the migration decision process. ADC data from 9 months after the initial tumor resection were used to select the best search paradigm for the simulation, which was initiated using data from 6 months after the initial operation. Using this search paradigm, 100 simulations were performed to derive a probabilistic map of tumor invasion locations. The simulation was able to successfully predict a recurrence in the dorsal/posterior aspect long before it was depicted on T1-weighted MRI, 18 months after the initial operation.

Cell Guidance by 3D-Gradients in Hydrogel Matrices: Importance for Biomedical Applications

Concentration gradients of soluble and matrix-bound guidance cues in the extracellular matrix direct cell growth in native tissues and are of great interest for design of biomedical scaffolds and on implant surfaces. The focus of this review is to demonstrate the importance of gradient guidance for cells as it would be desirable to direct cell growth onto/into biomedical devices. Many studies have been described that illustrate the production and characterization of surface gradients, but three dimensional (3D)-gradients that direct cellular behavior are not well investigated. Hydrogels are considered as synthetic replacements for native extracellular matrices as they share key functions such as 2D- or 3D-solid support, fibrous structure, gas- and nutrition permeability and allow storage and release of biologically active molecules. Therefore this review focuses on current studies that try to implement soluble or covalently-attached gradients of growth factors, cytokines or adhesion sequences into 3D-hydrogel matrices in order to control cell growth, orientation and migration towards a target. Such gradient architectures are especially desirable for wound healing purposes, where defined cell populations need to be recruited from the blood stream and out of the adjacent tissue, in critical bone defects, for vascular implants or neuronal guidance structures where defined cell populations should be guided by appropriate signals to reach their proper positions or target tissues in order to accomplish functional repair.

Evolution of cell motility in an individual-based model of tumour growth

Tumour invasion is driven by proliferation and importantly migration into the surrounding tissue. Cancer cell motility is also critical in the formation of metastases and is therefore a fundamental issue in cancer research. In this paper we investigate the emergence of cancer cell motility in an evolving tumour population using an individual-based modelling approach. In this model of tumour growth each cell is equipped with a micro-environment response network that determines the behaviour or phenotype of the cell based on the local environment. The response network is modelled using a feed-forward neural network, which is subject to mutations when the cells divide. With this model we have investigated the impact of the micro-environment on the emergence of a motile invasive phenotype. The results show that when a motile phenotype emerges the dynamics of the model are radically changed and we observe faster growing tumours exhibiting diffuse morphologies. Further we observe that the emergence of a motile subclone can occur in a wide range of micro-environmental growth conditions. Iterated simulations showed that in identical growth conditions the evolutionary dynamics either converge to a proliferating or migratory phenotype, which suggests that the introduction of cell motility into the model changes the shape of fitness landscape on which the cancer cell population evolves and that it now contains several local maxima. This could have important implications for cancer treatments which focus on the gene level, as our results show that several distinct genotypes and critically distinct phenotypes can emerge and become dominant in the same micro-environment.

A mathematical approach to bone tissue engineering

Tissue engineering is becoming consolidated in the biomedical field as one of the most promising strategies in tissue repair and regenerative medicine. Within this discipline, bone tissue engineering involves the use of cell-loaded porous biomaterials, i.e. bioscaffolds, to promote bone tissue regeneration in bone defects or diseases such as osteoporosis, although it has not yet been incorporated into daily clinical practice. The overall success of a particular bone tissue engineering application depends strongly on scaffold design parameters, which do away with long and expensive clinical protocols. Computer simulation is a useful tool that may reduce animal experiments and help to identify optimal patient-specific designs after concise model validation. In this paper, we present a novel mathematical approach to bone regeneration within scaffolds, based on a multiscale framework. Results are presented over an actual scaffold microstructure, showing the potential of computer simulation, and how it can aid in the task of making bone tissue engineering a reality in clinical practice.

Prestress and adhesion site dynamics control cell sensitivity to extracellular stiffness

This study aims at improving the understanding of mechanisms responsible for cell sensitivity to extracellular environment. We explain how substrate mechanical properties can modulate the force regulation of cell sensitive elements primarily adhesion sites. We present a theoretical and experimental comparison between two radically different approaches of the force regulation of adhesion sites that depends on their either stationary or dynamic behavior. The most classical stationary model fails to predict cell sensitivity to substrate stiffness whereas the dynamic model predicts extracellular stiffness dependence. This is due to a time dependent reaction force in response to actomyosin traction force exerted on cell sensitive elements. We purposely used two cellular models, i.e., alveolar epithelial cells and alveolar macrophages exhibiting respectively stationary and dynamic adhesion sites, and compared their sensitivity to theoretical predictions. Mechanical and structural results show that alveolar epithelial cells exhibit significant prestress supported by evident stress fibers and lacks sensitivity to substrate stiffness. On the other hand, alveolar macrophages exhibit low prestress and exhibit sensitivity to substrate stiffness. Altogether, theory and experiments consistently show that adhesion site dynamics and cytoskeleton prestress control cell sensitivity to extracellular environment with an optimal sensitivity expected in the intermediate range.

Positively and negatively modulating cell adhesion to type I collagen via peptide grafting

The biophysical interactions between cells and type I collagen are controlled by the level of cell adhesion, which is dictated primarily by the density of ligands on collagen and the density of integrin receptors on cells. The native adhesivity of collagen was modulated by covalently grafting glycine-arginine-glycine-aspartic acid-serine (GRGDS), which includes the bioactive RGD sequence, or glycine-arginine-aspartic acid-glycine-serine (GRDGS), which includes the scrambled RDG sequence, to collagen with the hetero-bifunctional coupling agent 1-ethyl-3-(3-dimethylaminopropyl) carbodiimide. The peptide-grafted collagen self-assembled into a fibrillar gel with negligible changes in gel structure and rheology. Rat dermal fibroblasts (RDFs) and human smooth muscle cells demonstrated increased levels of adhesion on gels prepared from RGD-grafted collagen, and decreased levels of adhesion on RDG-grafted collagen. Both cell types demonstrated an increased ability to compact free-floating RGD-grafted collagen gels, and an impaired ability to compact RDG-grafted gels. RDF migration on and within collagen was increased with RDG-grafted collagen and decreased with RGD-grafted collagen, and dose-response experiments indicated a biphasic response of RDF migration to adhesion. Smooth muscle cells demonstrated similar, though not statistically significant, trends. The ability to both positively and negatively modulate cell adhesion to collagen increases the versatility of this natural biomaterial for regenerative therapies.

Computational and mathematical modeling of angiogenesis

Over the past two decades, a number of mathematical and computational models have been developed to study different aspects of angiogenesis that span the spatial and temporal scales encompassed by this complex process. For example, models have been built to investigate how growth factors and receptors signal endothelial cell proliferation, how groups of endothelial cells assemble into individual vessels, and how tumors recruit the ingrowth of whole microvascular networks. A prudent question to pose is: "what have we learned from these models?" This review aims to answer this question as it pertains to angiogenesis in the context of normal physiological growth, tumorigenesis, wound healing, tissue engineering, and the design of therapeutic strategies. We also provide a framework for parsing angiogenesis models into categories, according to the type of modeling approach used, the spatial and temporal scales simulated, and the overarching question being posed to the model. Finally, this review introduces some of the simplification strategies and assumptions used in model building, discusses model validation, and makes recommendations for application of modeling approaches to unresolved questions in the field.

Modelling cellular signalling systems

Cell signalling pathways and networks are complex and often non-linear. Signalling pathways can be represented as systems of biochemical reactions that can be modelled using differential equations. Computational modelling of cell signalling pathways is emerging as a tool that facilitates mechanistic understanding of complex biological systems. Mathematical models are also used to generate predictions that may be tested experimentally. In the present chapter, the various steps involved in building models of cell signalling pathways are discussed. Depending on the nature of the process being modelled and the scale of the model, different mathematical formulations, ranging from stochastic representations to ordinary and partial differential equations are discussed. This is followed by a brief summary of some recent modelling successes and the state of future models.

Cancer cell motility: optimizing spatial search strategies

Aberrantly regulated cell motility is a hallmark of cancer cells. A hybrid agent-based model has been developed to investigate the synergistic and antagonistic cell motility-impacting effects of three microenvironment variables simultaneously: chemoattraction, haptotactic permission, and biomechanical constraint or resistance. Reflecting distinct cell-specific intracellular machinery, the cancer cells are modeled as processing a variety of spatial search strategies that respond to these three influencing factors with differential weights attached to each. While responding exclusively to chemoattraction optimizes cell displacement effectiveness, incorporating permission and resistance components becomes increasingly important with greater distance to the chemoattractant source and/or after reducing the ligand's effective diffusion coefficient. Extending this to a heterogeneous population of cells shows that displacement effectiveness increases with clonal diversity as characterized by the Shannon index. However, the resulting data can be fit best to an exponential function, suggesting that there is a level of population heterogeneity beyond which its added value to the cancer system becomes minimal as directionality ceases to increase. Possible experimental extensions and potential clinical implications are discussed.