A thermodynamically motivated model for stress-fiber reorganization

Quantifying the microstructural and biomechanical changes in the porcine ventricles during growth and remodelling

Cardiac tissue growth and remodelling (G & R) occur in response to the changing physiological demands of the heart after birth. The early shift to pulmonary circulation produces an immediate increase in ventricular workload, causing microstructural and biomechanical changes that serve to maintain overall physiological homoeostasis. Such cardiac G & R continues throughout life. Quantifying the tissue's mechanical and microstructural changes because of G & R is of increasing interest, dovetailing with the emerging fields of personalised and precision solutions. This study aimed to determine equibiaxial, and non-equibiaxial extension, stress-relaxation, and the underlying microstructure of the passive porcine ventricles tissue at four time points spanning from neonatal to adulthood. The three-dimensional microstructure was investigated via two-photon excited fluorescence and second-harmonic generation microscopy on optically cleared tissues, describing the 3D orientation, rotation and dispersion of the cardiomyocytes and collagen fibrils. The results revealed that during biomechanical testing, myocardial ventricular tissue possessed non-linear, anisotropic, and viscoelastic behaviour. An increase in stiffness and viscoelasticity was noted for the left and right ventricular free walls from neonatal to adulthood. Microstructural analyses revealed concomitant increases in cardiomyocyte rotation and dispersion. This study provides baseline data, describing the biomechanical and microstructural changes in the left and right ventricular myocardial tissue during G & R, which should prove valuable to researchers in developing age-specific, constitutive models for more accurate computational simulations. STATEMENT OF SIGNIFICANCE: There is a dearth of experimental data describing the growth and remodelling of left and right ventricular tissue. The published literature is fragmented, with data reported via different experimental techniques using tissues harvested from a variety of animals, with different gender and ages. This prevents developing a continuum of data spanning birth to death, so limiting the potential that can be leveraged to aid computational modelling and simulations. In this study, equibiaxial, non-equibiaxial, and stress-relaxation data are presented, describing directional-dependent material responses. The biomechanical data is consolidated with equivalent microstructural data, an important element for the development of future material models. Combined, these data describe microstructural and biomechanical changes in the ventricles, spanning G &R from neonatal to adulthood.

Cell orientation under stretch: A review of experimental findings and mathematical modelling

The key role of electro-chemical signals in cellular processes had been known for many years, but more recently the interplay with mechanics has been put in evidence and attracted substantial research interests. Indeed, the sensitivity of cells to mechanical stimuli coming from the microenvironment turns out to be relevant in many biological and physiological circumstances. In particular, experimental evidence demonstrated that cells on elastic planar substrates undergoing periodic stretches, mimicking native cyclic strains in the tissue where they reside, actively reorient their cytoskeletal stress fibres. At the end of the realignment process, the cell axis forms a certain angle with the main stretching direction. Due to the importance of a deeper understanding of mechanotransduction, such a phenomenon was studied both from the experimental and the mathematical modelling point of view. The aim of this review is to collect and discuss both the experimental results on cell reorientation and the fundamental features of the mathematical models that have been proposed in the literature.

Engineering Strategies to Move from Understanding to Steering Renal Tubulogenesis

Rebuilding the kidney in the context of tissue engineering offers a major challenge as the organ is structurally complex and has a high variety of specific functions. Recreation of kidney function is inherently connected to the formation of tubules since the functional subunit of the kidney, the nephron, is based on tubular structures. In vivo, tubulogenesis culminates in a perfectly shaped, patterned, and functional renal tubule via different morphogenic processes that depend on delicately orchestrated chemical, physical, and mechanical interactions between cells and between cells and their microenvironment. This review summarizes the current understanding of the role of the microenvironment in the morphogenic processes involved in in vivo renal tubulogenesis. We highlight the current state-of-the-art of renal tubular engineering and provide a view on the design elements that can be extracted from these studies. Next, we discuss how computational modeling can aid in specifying and identifying design parameters and provide directions on how these design parameters can be incorporated in biomaterials for the purpose of engineering renal tubulogenesis. Finally, we propose that a step-by-step reciprocal interaction between understanding and engineering is necessary to effectively guide renal tubulogenesis. Impact statement Tubular tissue engineering lies at the foundation of regenerating kidney tissue function, as the functional subunit of the kidney, the nephron, is based on tubular structures. Guiding renal tubulogenesis toward functional renal tubules requires in-depth knowledge of the developmental processes that lead to the formation of native tubules as well as engineering approaches to steer these processes. In this study, we review the role of the microenvironment in the developmental processes that lead to functional renal tubules and give directions how this knowledge can be harnessed for biomaterial-based tubular engineering using computational models.

Toward a Physiologically Relevant 3D Helicoidal-Oriented Cardiac Model: Simultaneous Application of Mechanical Stimulation and Surface Topography

Myocardium consists of cardiac cells that interact with their environment through physical, biochemical, and electrical stimulations. The physiology, function, and metabolism of cardiac tissue are affected by this dynamic structure. Within the myocardium, cardiomyocytes' orientations are parallel, creating a dominant orientation. Additionally, local alignments of fibers, along with a helical organization, become evident at the macroscopic level. For the successful development of a reliable in vitro cardiac model, evaluation of cardiac cells' behavior in a dynamic microenvironment, as well as their spatial architecture, is mandatory. In this study, we hypothesize that complex interactions between long-term contraction boundary conditions and cyclic mechanical stimulation may provide a physiological mechanism to generate off-axis alignments in the preferred mechanical stretch direction. This off-axis alignment can be engineered in vitro and, most importantly, mirrors the helical arrangements observed in vivo. For this purpose, uniaxial mechanical stretching of dECM-fibrin hydrogels was performed on pre-aligned 3D cultures of cardiac cells. In view of the potential development of helical structures similar to those in native hearts, the possibility of generating oblique alignments ranging between 0° and 90° was explored. Indeed, our investigations of cell alignment in 3D, employing both mechanical stimulation and groove constraint, provide a reliable mechanism for the generation of helicoidal structures in the myocardium. By combining cyclic stretch and geometric alignment in grooves, an intermediate angle toward favored direction can be achieved experimentally: while cyclic stretch produces a perpendicular orientation, geometric alignment is associated with a parallel one. In our 2D and 3D culture conditions, nonlinear cellular addition of the strains and strain avoidance concept reliably predicted the preferred cellular alignment. The 3D dECM-fibrin model system in this study shows that cyclical stretching supports cell survival and development. Using mechanical stimulation of pre-aligned heart cells, maturation markers are augmented in neonatal cardiomyocytes, while the beating culture period is prolonged, indicating an improved model function. We propose a simplified theoretical model based on numerical simulation and nonlinear strain avoidance by cells to explain oblique alignment angles. Thus, this work lays a possible rational basis for understanding and engineering oblique cellular alignments, such as the helicoidal layout of the heart, using approaches that simultaneously enhance maturation and function.

The influence of entropic crowding in cell monolayers

Cell-cell interaction dictates cell morphology and organization, which play a crucial role in the micro-architecture of tissues that guides their biological and mechanical functioning. Here, we investigate the effect of cell density on the responses of cells seeded on flat substrates using a novel statistical thermodynamics framework. The framework recognizes the existence of nonthermal fluctuations in cellular response and thereby naturally captures entropic interactions between cells in monolayers. In line with observations, the model predicts that cell area and elongation decrease with increasing cell seeding density-both are a direct outcome of the fluctuating nature of the cellular response that gives rise to enhanced cell-cell interactions with increasing cell crowding. The modeling framework also predicts the increase in cell alignment with increasing cell density: this cellular ordering is also due to enhanced entropic interactions and is akin to nematic ordering in liquid crystals. Our simulations provide physical insights that suggest that entropic cell-cell interactions play a crucial role in governing the responses of cell monolayers.

Cell reorientation on a cyclically strained substrate

Cyclic strain avoidance, the phenomenon of cell and cytoskeleton alignment perpendicular to the direction of cyclic strain of the underlying 2D substrate, is an important characteristic of the adherent cell organization. This alignment has typically been attributed to the stress-fiber reorganization although observations clearly show that stress-fiber reorganization under cyclic loading is closely coupled to cell morphology and reorientation of the cells. Here, we develop a statistical mechanics framework that couples the cytoskeletal stress-fiber organization with cell morphology under imposed cyclic straining and make quantitative comparisons with observations. The framework accurately predicts that cyclic strain avoidance stems primarily from cell reorientation away from the cyclic straining rather than cytoskeletal reorganization within the cell. The reorientation of the cell is a consequence of the cell lowering its free energy by largely avoiding the imposed cyclic straining. Furthermore, we investigate the kinetics of the cyclic strain avoidance mechanism and demonstrate that it emerges primarily due to the rigid body rotation of the cell rather than via a trajectory involving cell straining. Our results provide clear physical insights into the coupled dynamics of cell morphology and stress-fibers, which ultimately leads to cellular organization in cyclically strained tissues.

Modeling Receptor Motility along Advecting Lipid Membranes

This work aims to overview multiphysics mechanobiological computational models for receptor dynamics along advecting cell membranes. Continuum and statistical models of receptor motility are the two main modeling methodologies identified in reviewing the state of the art. Within the former modeling class, a further subdivision based on different biological purposes and processes of proteins’ motion is recognized; cell adhesion, cell contractility, endocytosis, and receptor relocations on advecting membranes are the most relevant biological processes identified in which receptor motility is pivotal. Numerical and/or experimental methods and approaches are highlighted in the exposure of the reviewed works provided by the literature, pertinent to the topic of the present manuscript. With a main focus on the continuum models of receptor motility, we discuss appropriate multiphyisics laws to model the mass flux of receptor proteins in the reproduction of receptor relocation and recruitment along cell membranes to describe receptor–ligand chemical interactions, and the cell’s structural response. The mass flux of receptor modeling is further supported by a discussion on the methodology utilized to evaluate the protein diffusion coefficient developed over the years.

Contact guidance as a consequence of coupled morphological evolution and motility of adherent cells

Adherent cells seeded on substrates spread and evolve their morphology while simultaneously displaying motility. Phenomena such as contact guidance, viz. the alignment of cells on patterned substrates, are strongly linked to the coupling of morphological evolution with motility. Here, we employ a recently developed statistical thermodynamics framework for modelling the non-thermal fluctuating response of cells to probe this coupling. This thermodynamic framework is first extended via a Langevin style model to predict temporal responses of cells to unpatterned and patterned substrates. The Langevin model is then shown to not only predict the different experimentally observed temporal scales for morphological observables such as cell area and elongation but also the interplay of morphology with motility that ultimately leads to contact guidance.

Scaffold Geometry-Imposed Anisotropic Mechanical Loading Guides the Evolution of the Mechanical State of Engineered Cardiovascular Tissues in vitro

Cardiovascular tissue engineering is a promising approach to develop grafts that, in contrast to current replacement grafts, have the capacity to grow and remodel like native tissues. This approach largely depends on cell-driven tissue growth and remodeling, which are highly complex processes that are difficult to control inside the scaffolds used for tissue engineering. For several tissue engineering approaches, adverse tissue growth and remodeling outcomes were reported, such as aneurysm formation in vascular grafts, and leaflet retraction in heart valve grafts. It is increasingly recognized that the outcome of tissue growth and remodeling, either physiological or pathological, depends at least partly on the establishment of a homeostatic mechanical state, where one or more mechanical quantities in a tissue are maintained in equilibrium. To design long-term functioning tissue engineering strategies, understanding how scaffold parameters such as geometry affect the mechanical state of a construct, and how this state guides tissue growth and remodeling, is therefore crucial. Here, we studied how anisotropic versus isotropic mechanical loading—as imposed by initial scaffold geometry—influences tissue growth, remodeling, and the evolution of the mechanical state and geometry of tissue-engineered cardiovascular constructs in vitro. Using a custom-built bioreactor platform and nondestructive mechanical testing, we monitored the mechanical and geometric changes of elliptical and circular, vascular cell-seeded, polycaprolactone-bisurea scaffolds during 14 days of dynamic loading. The elliptical and circular scaffold geometries were designed using finite element analysis, to induce anisotropic and isotropic dynamic loading, respectively, with similar maximum stretch when cultured in the bioreactor platform. We found that the initial scaffold geometry-induced (an)isotropic loading of the engineered constructs differentially dictated the evolution of their mechanical state and geometry over time, as well as their final structural organization. These findings demonstrate that controlling the initial mechanical state of tissue-engineered constructs via scaffold geometry can be used to influence tissue growth and remodeling and determine tissue outcomes.

The influence of boundary conditions and protein availability on the remodeling of cardiomyocytes

The heart muscle is capable of growing and remodeling in response to changes in its mechanical and hormonal environment. While this capability is essential to the healthy function of the heart, under extreme conditions it may also lead to heart failure. In this work, we derive a thermodynamically based and microscopically motivated model that highlights the influence of mechanical boundary conditions and hormonal changes on the remodeling process in cardiomyocytes. We begin with a description of the kinematics associated with the remodeling process. Specifically, we derive relations between the macroscopic deformation, the number of sarcomeres, the sarcomere stretch, and the number of myofibrils in the cell. We follow with the derivation of evolution equations that describe the production and the degradation of protein in the cytosol. Next, we postulate a dissipation-based formulation that characterizes the remodeling process. We show that this process stems from a competition between the internal energy, the entropy, the energy supplied to the system by ATP and other sources, and dissipation mechanisms. To illustrate the merit of this framework, we study four initial and boundary conditions: (1) a myocyte undergoing isometric contractions in the presence of either an infinite or a limited supply of proteins and (2) a myocyte that is free to dilate along the radial direction with an infinite and a limited supply of proteins. This work underscores the importance of boundary conditions on the overall remodeling response of cardiomyocytes, suggesting a plausible mechanism that might play a role in distinguishing eccentric vs. concentric hypertrophy.

Contact guidance via heterogeneity of substrate elasticity

Contact guidance, the widely-known phenomenon of cell alignment, is an essential step in the organization of adherent cells. This guidance is known to occur by, amongst other things, anisotropic features in the environment including elastic heterogeneity. To understand the origins of this guidance we employed a novel statistical thermodynamics framework, which recognises the non-thermal fluctuations in the cellular response, for modelling the response of the cells seeded on substrates with alternating soft and stiff stripes. Consistent with observations, the modelling framework predicts the existence of three regimes of cell guidance: (i) in regime I for stripe widths much larger than the cell size guidance is primarily entropic; (ii) for stripe widths on the order of the cell size in regime II guidance is biochemically mediated and accompanied by changes to the cell morphology while (iii) in regime III for stripe widths much less than the cell size there is no guidance as cells cannot sense the substrate heterogeneity. Guidance in regimes I and II is due to "molli-avoidance" with cells primarily residing on the stiff stripes. While the molli-avoidance tendency is not lost with decreasing density of collagen coating the substrate, the reduced focal adhesion formation with decreasing collagen density tends to inhibit contact guidance. Our results provide clear physical insights into the interplay between cell mechano-sensitivity and substrate elastic heterogeneity that ultimately leads to the contact guidance of cells in heterogeneous tissues. STATEMENT OF SIGNIFICANCE: Cellular morphology and organization play a crucial role in the micro-architecture of tissues and dictates their biological and mechanical functioning. Despite the importance of cellular organization in all facets of tissue biology, the fundamental question of how a cell organizes itself in an anisotropic environment is still poorly understood. We employ a novel statistical thermodynamics framework which recognises the non-thermal fluctuations in the cellular response to investigate cell guidance on substrates with alternating soft and stiff stripes. The propensity of cells to primarily reside on stiff stripes results in strong guidance when the period of the stripes is larger than the cell size. For smaller stripe periods, cells sense a homogeneous substrate and guidance is lost.

Efficient computational modelling of smooth muscle orientation and function in the aorta

Understanding the mechanical effects of smooth muscle cell (SMC) contraction on the initiation and the propagation of cardiovascular diseases such as aortic dissection is critical. Framed by elastic lamellar sheets in the lamellar unit, there are SMCs in the media with a distinct radial tilt, which indicates their contribution to the radial strength. However, the mechanical effects of this type of anisotropy have not been fully discussed. Therefore, in this study, we propose a constitutive framework that models the passive and active mechanics of the aorta, taking into account the dispersed nature of the aortic constituents by applying the discrete fibre dispersion method. We suggest an isoparametric approach by evaluating various numerical integration methods and introducing a non-uniform discretization of the unit hemisphere to increase its computational efficiency. Finally, the constitutive parameters are fitted to layer-specific experimental data and initial computational results are briefly presented. The radial tilt of SMCs is also analysed, which has a noticeable influence on the mechanical behaviour of the aorta. In the absence of sufficient experimental data, the results indicate that the active contribution of SMCs has a remarkable impact on the mechanics of the healthy aorta.

Mechanosensitive myosin II but not cofilin primarily contributes to cyclic cell stretch-induced selective disassembly of actin stress fibers

Cells adapt to applied cyclic stretch (CS) to circumvent chronic activation of proinflammatory signaling. Currently, the molecular mechanism of the selective disassembly of actin stress fibers (SFs) in the stretch direction, which occurs at the early stage of the cellular response to CS, remains controversial. Here, we suggest that the mechanosensitive behavior of myosin II, a major cross-linker of SFs, primarily contributes to the directional disassembly of the actomyosin complex SFs in bovine vascular smooth muscle cells and human U2OS osteosarcoma cells. First, we identified that CS with a shortening phase that exceeds in speed the inherent contractile rate of individual SFs leads to the disassembly. To understand the biological basis, we investigated the effect of expressing myosin regulatory light-chain mutants and found that SFs with less actomyosin activities disassemble more promptly upon CS. We consequently created a minimal mathematical model that recapitulates the salient features of the direction-selective and threshold-triggered disassembly of SFs to show that disassembly or, more specifically, unbundling of the actomyosin bundle SFs is enhanced with sufficiently fast cell shortening. We further demonstrated that similar disassembly of SFs is inducible in the presence of an active LIM-kinase-1 mutant that deactivates cofilin, suggesting that cofilin is dispensable as opposed to a previously proposed mechanism.

Development of an FEA framework for analysis of subject-specific aortic compliance based on 4D flow MRI

This paper presents a subject-specific in-silico framework in which we uncover the relationship between the spatially varying constituents of the aorta and the non-linear compliance of the vessel during the cardiac cycle uncovered through our MRI investigations. A microstructurally motivated constitutive model is developed, and simulations reveal that internal vessel contractility, due to pre-stretched elastin and actively generated smooth muscle cell stress, must be incorporated, along with collagen strain stiffening, in order to accurately predict the non-linear pressure-area relationship observed in-vivo. Modelling of elastin and smooth muscle cell contractility allows for the identification of the reference vessel configuration at zero-lumen pressure, in addition to accurately predicting high- and low-compliance regimes under a physiological range of pressures. This modelling approach is also shown to capture the key features of elastin digestion and SMC activation experiments. The volume fractions of the constituent components of the aortic material model were computed so that the in-silico pressure-area curves accurately predict the corresponding MRI data at each location. Simulations reveal that collagen and smooth muscle volume fractions increase distally, while elastin volume fraction decreases distally, consistent with reported histological data. Furthermore, the strain at which collagen transitions from low to high stiffness is lower in the abdominal aorta, again supporting the histological finding that collagen waviness is lower distally. The analyses presented in this paper provide new insights into the heterogeneous structure-function relationship that underlies aortic biomechanics. Furthermore, this subject-specific MRI/FEA methodology provides a foundation for personalised in-silico clinical analysis and tailored aortic device development. STATEMENT OF SIGNIFICANCE: This study provides a significant advance in in-silico medicine by capturing the structure/function relationship of the subject-specific human aorta presented in our previous MRI analyses. A physiologically based aortic constitutive model is developed, and simulations reveal that internal vessel contractility must be incorporated, along with collagen strain stiffening, to accurately predict the in-vivo non-linear pressure-area relationship. Furthermore, this is the first subject-specific model to predict spatial variation in the volume fractions of aortic wall constituents. Previous studies perform phenomenological hyperelastic curve fits to medical imaging data and ignore the prestress contribution of elastin, collagen, and SMCs and the associated zero-pressure reference state of the vessel. This novel MRI/FEA framework can be used as an in-silico diagnostic tool for the early stage detection of aortic pathologies.

Computational models of cardiac hypertrophy

Cardiac hypertrophy, defined as an increase in mass of the heart, is a complex process driven by simultaneous changes in hemodynamics, mechanical stimuli, and hormonal inputs. It occurs not only during pre- and post-natal development but also in adults in response to exercise, pregnancy, and a range of cardiovascular diseases. One of the most exciting recent developments in the field of cardiac biomechanics is the advent of computational models that are able to accurately predict patterns of heart growth in many of these settings, particularly in cases where changes in mechanical loading of the heart play an import role. These emerging models may soon be capable of making patient-specific growth predictions that can be used to guide clinical interventions. Here, we review the history and current state of cardiac growth models and highlight three main limitations of current approaches with regard to future clinical application: their inability to predict the regression of heart growth after removal of a mechanical overload, inability to account for evolving hemodynamics, and inability to incorporate known growth effects of drugs and hormones on heart growth. Next, we outline growth mechanics approaches used in other fields of biomechanics and highlight some potential lessons for cardiac growth modeling. Finally, we propose a multiscale modeling approach for future studies that blends tissue-level growth models with cell-level signaling models to incorporate the effects of hormones in the context of pregnancy-induced heart growth.

A thermodynamic transient cross-bridge model for prediction of contractility and remodelling of the ventricle

Cardiac hypertrophy is an adaption of the heart to a change in cardiovascular loading conditions. The current understanding is that progression may be stress or strain driven, but the multi-scale nature of the cellular remodelling processes have yet to be uncovered. In this study, we develop a model of the contractile left ventricle, with the active cell tension described by a thermodynamically motivated cross-bridge cycling model. Simulation of the transient recruitment of myosin results in correct patterns of ventricular pressure predicted over a cardiac cycle. We investigate how changes in tissue loading and associated deviations in transient force generation can drive restructuring of cellular myofibrils in the heart wall. Our thermodynamic framework predicts in-series sarcomere addition (eccentric remodelling) in response to volume overload, and sarcomere addition in parallel (concentric remodelling) in response to valve and signalling disfunction. This framework provides a significant advance in the current understanding of the fundamental sub-sarcomere level biomechanisms underlying cardiac remodelling. Simulations reveal that pathological tissue loading conditions can significantly alter actin-myosin cross-bridge cycling over the course of the cardiac cycle. The resultant variation in sarcomere stress pushes an imbalance between the internal free energy of the myofibril and that of unbound contractile proteins, initiating remodelling. The link between cross-bridge thermodynamics and myofibril remodelling proposed in this study may significantly advance current understanding of cardiac disease onset.

Computational modeling for cardiovascular tissue engineering: the importance of including cell behavior in growth and remodeling algorithms

Understanding cardiovascular growth and remodeling (G&R) is fundamental for designing robust cardiovascular tissue engineering strategies, which enable synthetic or biological scaffolds to transform into healthy living tissues after implantation. Computational modeling, particularly when integrated with experimental research, is key for advancing our understanding, predicting the in vivo evolution of engineered tissues, and efficiently optimizing scaffold designs. As cells are ultimately the drivers of G&R and known to change their behavior in response to mechanical cues, increasing efforts are currently undertaken to capture (mechano-mediated) cell behavior in computational models. In this selective review, we highlight some recent examples that are relevant in the context of cardiovascular tissue engineering and discuss the current and future biological and computational challenges for modeling cell-mediated G&R.

What factors determine the number of nonmuscle myosin II in the sarcomeric unit of stress fibers?

Actin stress fibers (SFs), a contractile apparatus in nonmuscle cells, possess a contractile unit that is apparently similar to the sarcomere of myofibrils in muscles. The function of SFs has thus often been addressed based on well-characterized properties of muscles. However, unlike the fixed number of myosin molecules in myofibrils, the number of nonmuscle myosin II (NMII) within the contractile sarcomeric unit in SFs is quite low and variable for some reason. Here we address what factors may determine the specific number of NMII in SFs. We suggest with a theoretical model that the number lies just in between the function of SFs for bearing cellular tension under static conditions and for promptly disintegrating upon forced cell shortening. We monitored shortening-induced disintegration of SFs in human osteosarcoma U2OS cells expressing mutants of myosin regulatory light chain that virtually regulates the interaction of NMII with actin filaments, and the behaviors observed were indeed consistent with the theoretical consequences. This situation-specific nature of SFs may allow nonmuscle cells to respond adaptively to mechanical stress to circumvent activation of pro-inflammatory signals as previously indicated, i.e., a behavior distinct from that of muscles that are basically specialized for exhibiting contractile activity.

Cellular Contact Guidance Emerges from Gap Avoidance

In the presence of anisotropic biochemical or topographical patterns, cells tend to align in the direction of these cues-a widely reported phenomenon known as "contact guidance." To investigate the origins of contact guidance, here, we created substrates micropatterned with parallel lines of fibronectin with dimensions spanning multiple orders of magnitude. Quantitative morphometric analysis of our experimental data reveals two regimes of contact guidance governed by the length scale of the cues that cannot be explained by enforced alignment of focal adhesions. Adopting computational simulations of cell remodeling on inhomogeneous substrates based on a statistical mechanics framework for living cells, we show that contact guidance emerges from anisotropic cell shape fluctuation and "gap avoidance," i.e., the energetic penalty of cell adhesions on non-adhesive gaps. Our findings therefore point to general biophysical mechanisms underlying cellular contact guidance, without the necessity of invoking specific molecular pathways.

Free-energy-based framework for early forecasting of stem cell differentiation

Commitment of stem cells to different lineages is inherently stochastic but regulated by a range of environmental bio/chemo/mechanical cues. Here, we develop an integrated stochastic modelling framework for predicting the differentiation of hMSCs in response to a range of environmental cues, including sizes of adhesive islands, stiffness of substrates and treatment with ROCK inhibitors in both growth and mixed media. The statistical framework analyses the fluctuations of cell morphologies over approximately a 24 h period after seeding the cells in the specific environment and uses the cytoskeletal free-energy distribution to forecast the lineage the hMSCs will commit to. The cytoskeletal free energy which succinctly parametrizes the biochemical state of the cell is shown to capture hMSC commitment over a range of environments while simple morphological factors such as cell shape, tractions on their own are unable to correlate with lineages hMSCs adopt.

A computational model to predict cell traction-mediated prestretch in the mitral valve

Prestretch is observed in many soft biological tissues, directly influencing the mechanical behavior of the tissue in question. The development of this prestretch occurs through complex growth and remodeling phenomena, which yet remain to be elucidated. In the present study it was investigated whether local cell-mediated traction forces can explain the development of global anisotropic tissue prestretch in the mitral valve. Towards this end, a model predicting actin stress fiber-generated traction forces was implemented in a finite element framework of the mitral valve. The overall predicted magnitude of prestretch induced valvular contraction after release of in vivo boundary constraints was in good agreement with data reported on valvular retraction after excision from the heart. Next, by using a systematic variation of model parameters and structural properties, a more anisotropic prestretch development in the valve could be obtained, which was also similar to physiological values. In conclusion, this study shows that cell-generated traction forces could explain prestretch magnitude and anisotropy in the mitral valve.

Entropic Forces Drive Cellular Contact Guidance

Contact guidance-the widely known phenomenon of cell alignment induced by anisotropic environmental features-is an essential step in the organization of adherent cells, but the mechanisms by which cells achieve this orientational ordering remain unclear. Here, we seeded myofibroblasts on substrates micropatterned with stripes of fibronectin and observed that contact guidance emerges at stripe widths much greater than the cell size. To understand the origins of this surprising observation, we combined morphometric analysis of cells and their subcellular components with a, to our knowledge, novel statistical framework for modeling nonthermal fluctuations of living cells. This modeling framework is shown to predict not only the trends but also the statistical variability of a wide range of biological observables, including cell (and nucleus) shapes, sizes, and orientations, as well as stress-fiber arrangements within the cells with remarkable fidelity with a single set of cell parameters. By comparing observations and theory, we identified two regimes of contact guidance: 1) guidance on stripe widths smaller than the cell size (w ≤ 160 μm), which is accompanied by biochemical changes within the cells, including increasing stress-fiber polarization and cell elongation; and 2) entropic guidance on larger stripe widths, which is governed by fluctuations in the cell morphology. Overall, our findings suggest an entropy-mediated mechanism for contact guidance associated with the tendency of cells to maximize their morphological entropy through shape fluctuations.

A microscopically motivated model for the remodeling of cardiomyocytes

We present a thermodynamically based model that captures the remodeling effects in cardiac muscle cells. This work begins with the formulation of the kinematics of a cardiomyocyte resulting from a prescribed macroscopic deformation and the reorganization of the internal structure. Specifically, relations between the macroscopic deformation and the number of sarcomeres, the sarcomere stretch, and the number of myofibrils in the cell are determined. The remodeling process is split into two separate phases-(1) elongation/shortening of the existing myofibrils by addition/detachment of sarcomeres and (2) formation of new myofibrils. The remodeling associated with each phase is modeled through a dissipation postulate. We show that remodeling is based on a competition between the internal energy, the entropy, the energy supplied to the system by ATP and other sources to drive the remodeling process, and dissipation mechanisms. While the variations in entropy associated with phase (1) are neglected, the substantial entropy loss associated with the formation of new myofibrils is determined. To illustrate the merit of the proposed framework, we compute the response of cardiomyocytes subjected to isometric axial stretch that are either free to deform or fixed in the transverse direction. We also examine the predictions of this model for cardiomyocytes subjected to various cyclic loadings. The proposed framework is capable of capturing a wide range of remodeling effects and agrees with experimental observations.

Transient active force generation and stress fibre remodelling in cells under cyclic loading

The active cytoskeleton is known to play an important mechanistic role in cellular structure, spreading, and contractility. Contractility is actively generated by stress fibres (SF), which continuously remodel in response to physiological dynamic loading conditions. The influence of actin-myosin cross-bridge cycling on SF remodelling under dynamic loading conditions has not previously been uncovered. In this study, a novel SF cross-bridge cycling model is developed to predict transient active force generation in cells subjected to dynamic loading. Rates of formation of cross-bridges within SFs are governed by the chemical potentials of attached and unattached myosin heads. This transient cross-bridge cycling model is coupled with a thermodynamically motivated framework for SF remodelling to analyse the influence of transient force generation on cytoskeletal evolution. A 1D implementation of the model is shown to correctly predict complex patterns of active cell force generation under a range of dynamic loading conditions, as reported in previous experimental studies.

Initial scaffold thickness affects the emergence of a geometrical and mechanical equilibrium in engineered cardiovascular tissues

In situ cardiovascular tissue-engineering can potentially address the shortcomings of the current replacement therapies, in particular, their inability to grow and remodel. In native tissues, it is widely accepted that physiological growth and remodelling occur to maintain a homeostatic mechanical state to conserve its function, regardless of changes in the mechanical environment. A similar homeostatic state should be reached for tissue-engineered (TE) prostheses to ensure proper functioning. For in situ tissue-engineering approaches obtaining such a state greatly relies on the initial scaffold design parameters. In this study, it is investigated if the simple scaffold design parameter initial thickness, influences the emergence of a mechanical and geometrical equilibrium state in in vitro TE constructs, which resemble thin cardiovascular tissues such as heart valves and arteries. Towards this end, two sample groups with different initial thicknesses of myofibroblast-seeded polycaprolactone-bisurea constructs were cultured for three weeks under dynamic loading conditions, while tracking geometrical and mechanical changes temporally using non-destructive ultrasound imaging. A mechanical equilibrium was reached in both groups, although at different magnitudes of the investigated mechanical quantities. Interestingly, a geometrically stable state was only established in the thicker constructs, while the thinner constructs' length continuously increased. This demonstrates that reaching geometrical and mechanical stability in TE constructs is highly dependent on functional scaffold design.

Thermodynamic Modeling of the Statistics of Cell Spreading on Ligand-Coated Elastic Substrates

Biological spread cells exist in a perpetually fluctuating state and therefore cannot be described in terms of a unique deterministic system. For modeling approaches to provide novel insight and uncover new mechanisms that drive cell behavior, a framework is required that progresses from traditional deterministic methods (whereby simulation of an experiment predicts a single outcome). In this study, we implement a new, to our knowledge, modeling approach for the analysis of cell spreading on ligand-coated substrates, extending the framework for nonequilibrium thermodynamics of cells developed by Shishvan et al. to include active focal adhesion assembly. We demonstrate that the model correctly predicts the coupled influence of surface collagen density and substrate stiffness on cell spreading, as reported experimentally by Engler et al. Low surface collagen densities are shown to result in a high probability that cells will be restricted to low spread areas. Furthermore, elastic free energy induced by substrate deformation lowers the probability of observing a highly spread cell, and, consequentially, lower cell tractions affect the assembly of focal adhesions. Experimentally measurable observables such as cell spread area and aspect ratio can be directly postprocessed from the computed homeostatic ensemble of (several million) spread states. This allows for the prediction of mean and SDs of such experimental observables. This class of cell mechanics modeling presents a significant advance on conventional deterministic approaches.

The homeostatic ensemble for cells

Cells are quintessential examples of out-of-equilibrium systems, but they maintain a homeostatic state over a timescale of hours to days. As a consequence, the statistics of all observables is remarkably consistent. Here, we develop a statistical mechanics framework for living cells by including the homeostatic constraint that exists over the interphase period of the cell cycle. The consequence is the introduction of the concept of a homeostatic ensemble and an associated homeostatic temperature, along with a formalism for the (dynamic) homeostatic equilibrium that intervenes to allow living cells to evade thermodynamic decay. As a first application, the framework is shown to accurately predict the observed effect of the mechanical environment on the in vitro response of smooth muscle cells. This includes predictions that both the mean values and diversity/variability in the measured values of observables such as cell area, shape and tractions decrease with decreasing stiffness of the environment. Thus, we argue that the observed variabilities are inherent to the entropic nature of the homeostatic equilibrium of cells and not a result of in vitro experimental errors.

Role of boundary conditions in determining cell alignment in response to stretch

The ability of cells to orient in response to mechanical stimuli is essential to embryonic development, cell migration, mechanotransduction, and other critical physiologic functions in a range of organs. Endothelial cells, fibroblasts, mesenchymal stem cells, and osteoblasts all orient perpendicular to an applied cyclic stretch when plated on stretchable elastic substrates, suggesting a common underlying mechanism. However, many of these same cells orient parallel to stretch in vivo and in 3D culture, and a compelling explanation for the different orientation responses in 2D and 3D has remained elusive. Here, we conducted a series of experiments designed specifically to test the hypothesis that differences in strains transverse to the primary loading direction give rise to the different alignment patterns observed in 2D and 3D cyclic stretch experiments ("strain avoidance"). We found that, in static or low-frequency stretch conditions, cell alignment in fibroblast-populated collagen gels correlated with the presence or absence of a restraining boundary condition rather than with compaction strains. Cyclic stretch could induce perpendicular alignment in 3D culture but only at frequencies an order of magnitude greater than reported to induce perpendicular alignment in 2D. We modified a published model of stress fiber dynamics and were able to reproduce our experimental findings across all conditions tested as well as published data from 2D cyclic stretch experiments. These experimental and model results suggest an explanation for the apparently contradictory alignment responses of cells subjected to cyclic stretch on 2D membranes and in 3D gels.

Growth and remodeling play opposing roles during postnatal human heart valve development

Tissue growth and remodeling are known to govern mechanical homeostasis in biological tissue, but their relative contributions to homeostasis remain unclear. Here, we use mechanical models, fueled by experimental findings, to demonstrate that growth and remodeling have different effects on heart valve stretch homeostasis during physiological postnatal development. Two developmental stages were considered: early-stage (from infant to adolescent) and late-stage (from adolescent to adult) development. Our models indicated that growth and remodeling play opposing roles in preserving tissue stretch and with time. During early-stage development, excessive tissue stretch was decreased by tissue growth and increased by remodeling. In contrast, during late-stage development tissue stretch was decreased by remodeling and increased by growth. Our findings contribute to an improved understanding of native heart valve adaptation throughout life, and are highly relevant for the development of tissue-engineered heart valves.

Computationally Informed Design of a Multi-Axial Actuated Microfluidic Chip Device

This paper describes the computationally informed design and experimental validation of a microfluidic chip device with multi-axial stretching capabilities. The device, based on PDMS soft-lithography, consisted of a thin porous membrane, mounted between two fluidic compartments, and tensioned via a set of vacuum-driven actuators. A finite element analysis solver implementing a set of different nonlinear elastic and hyperelastic material models was used to drive the design and optimization of chip geometry and to investigate the resulting deformation patterns under multi-axial loading. Computational results were cross-validated by experimental testing of prototypal devices featuring the in silico optimized geometry. The proposed methodology represents a suite of computationally handy simulation tools that might find application in the design and in silico mechanical characterization of a wide range of stretchable microfluidic devices.

Prediction of Cell Alignment on Cyclically Strained Grooved Substrates

Cells respond to both mechanical and topographical stimuli by reorienting and reorganizing their cytoskeleton. Under certain conditions, such as for cells on cyclically stretched grooved substrates, the effects of these stimuli can be antagonistic. The biophysical processes that lead to the cellular reorientation resulting from such a competition are not clear yet. In this study, we hypothesized that mechanical cues and the diffusion of the intracellular signal produced by focal adhesions are determinants of the final cellular alignment. This hypothesis was investigated by means of a computational model, with the aim to simulate the (re)orientation of cells cultured on cyclically stretched grooved substrates. The computational results qualitatively agree with previous experimental studies, thereby supporting our hypothesis. Furthermore, cellular behavior resulting from experimental conditions different from the ones reported in the literature was simulated, which can contribute to the development of new experimental designs.

Towards the understanding of cytoskeleton fluidisation-solidification regulation

The understanding of the self-regulation of the mechanical properties in non-sarcomeric cells, such as lung cells or cells during tissue development, remains an open research problem with many unresolved issues. Their behaviour is far from the image of the traditionally studied sarcomeric cells, since the crosstalk between the signalling pathways and the complexity of the mechanical properties creates an intriguing mechano-chemical coupling. In these situations, the inelastic effects dominate the cytoskeletal structure showing phenomena like fluidisation and subsequent solidification. Here, we proposes the inelastic contractile unit framework as an attempt to reconciles these effects. The model comprises a mechanical description of the nonlinear elasticity of the cytoskeleton incorporated into a continuum-mechanics framework using the eighth-chains model. In order to address the inelastic effect, we incorporate the dynamic of crosslinks, considering the [Formula: see text]-actinin and the active stress induced by the myosin molecular motors. Finally, we introduce a hypothesis that links the ability to fluidise and re-solidify as a consequence of the interaction between the active stress and the gelation state defined by the crosslinks. We validate the model with data obtained from experiments of drug-induced relaxation reported in the literature.

Emerging roles of mechanical forces in chromatin regulation

Cells are constantly subjected to a spectrum of mechanical cues, such as shear stress, compression, differential tissue rigidity and strain, to which they adapt by engaging mechanisms of mechanotransduction. While the central role of cell adhesion receptors in this process is established, it has only recently been appreciated that mechanical cues reach far beyond the plasma membrane and the cytoskeleton, and are directly transmitted to the nucleus. Furthermore, changes in the mechanical properties of the perinuclear cytoskeleton, nuclear lamina and chromatin are critical for cellular responses and adaptation to external mechanical cues. In that respect, dynamic changes in the nuclear lamina and the surrounding cytoskeleton modify mechanical properties of the nucleus, thereby protecting genetic material from damage. The importance of this mechanism is highlighted by debilitating genetic diseases, termed laminopathies, that result from impaired mechanoresistance of the nuclear lamina. What has been less evident, and represents one of the exciting emerging concepts, is that chromatin itself is an active rheological element of the nucleus, which undergoes dynamic changes upon application of force, thereby facilitating cellular adaption to differential force environments. This Review aims to highlight these emerging concepts by discussing the latest literature in this area and by proposing an integrative model of cytoskeletal and chromatin-mediated responses to mechanical stress.