The effect of noisy flow on endothelial cell mechanotransduction: a computational study

Primary cilium: a paradigm for integrating mathematical modeling with experiments and numerical simulations in mechanobiology

Primary cilia are non-motile, solitary (one per cell) microtubule-based organelles that emerge from the mother centriole after cells have exited the mitotic cycle. Identified as a mechanosensing organelle that responds to both mechanical and chemical stimuli, the primary cilium provides a fertile ground for integrative investigations of mathematical modeling, numerical simulations, and experiments. Recent experimental findings revealed considerable complexity to the underlying mechanosensory mechanisms that transmit extracellular stimuli to intracellular signaling many of which include primary cilia. In this invited review, we provide a brief survey of experimental findings on primary cilia and how these results lead to various mathematical models of the mechanics of the primary cilium bent under an external forcing such as a fluid flow or a trap. Mathematical modeling of the primary cilium as a fluid-structure interaction problem highlights the importance of basal anchorage and the anisotropic moduli of the microtubules. As theoretical modeling and numerical simulations progress, along with improved state-of-the-art experiments on primary cilia, we hope that details of ciliary regulated mechano-chemical signaling dynamics in cellular physiology will be understood in the near future.

Cellular mechanosensitivity to substrate stiffness decreases with increasing dissimilarity to cell stiffness

Computational modelling has received increasing attention to investigate multi-scale coupled problems in micro-heterogeneous biological structures such as cells. In the current study, we investigated for a single cell the effects of (1) different cell-substrate attachment (2) and different substrate modulus [Formula: see text] on intracellular deformations. A fibroblast was geometrically reconstructed from confocal micrographs. Finite element models of the cell on a planar substrate were developed. Intracellular deformations due to substrate stretch of [Formula: see text], were assessed for: (1) cell-substrate attachment implemented as full basal contact (FC) and 124 focal adhesions (FA), respectively, and [Formula: see text]140 KPa and (2) [Formula: see text], 140, 1000, and 10,000 KPa, respectively, and FA attachment. The largest strains in cytosol, nucleus and cell membrane were higher for FC (1.35[Formula: see text], 0.235[Formula: see text] and 0.6[Formula: see text]) than for FA attachment (0.0952[Formula: see text], 0.0472[Formula: see text] and 0.05[Formula: see text]). For increasing [Formula: see text], the largest maximum principal strain was 4.4[Formula: see text], 5[Formula: see text], 5.3[Formula: see text] and 5.3[Formula: see text] in the membrane, 9.5[Formula: see text], 1.1[Formula: see text], 1.2[Formula: see text] and 1.2[Formula: see text] in the cytosol, and 4.5[Formula: see text], 5.3[Formula: see text], 5.7[Formula: see text] and 5.7[Formula: see text] in the nucleus. The results show (1) the importance of representing FA in cell models and (2) higher cellular mechanical sensitivity for substrate stiffness changes in the range of cell stiffness. The latter indicates that matching substrate stiffness to cell stiffness, and moderate variation of the former is very effective for controlled variation of cell deformation. The developed methodology is useful for parametric studies on cellular mechanics to obtain quantitative data of subcellular strains and stresses that cannot easily be measured experimentally.

Shear-induced force transmission in a multicomponent, multicell model of the endothelium

Haemodynamic forces applied at the apical surface of vascular endothelial cells (ECs) provide the mechanical signals at intracellular organelles and through the inter-connected cellular network. The objective of this study is to quantify the intracellular and intercellular stresses in a confluent vascular EC monolayer. A novel three-dimensional, multiscale and multicomponent model of focally adhered ECs is developed to account for the role of potential mechanosensors (glycocalyx layer, actin cortical layer, nucleus, cytoskeleton, focal adhesions (FAs) and adherens junctions (ADJs)) in mechanotransmission and EC deformation. The overriding issue addressed is the stress amplification in these regions, which may play a role in subcellular localization of mechanotransmission. The model predicts that the stresses are amplified 250-600-fold over apical values at ADJs and 175-200-fold at FAs for ECs exposed to a mean shear stress of 10 dyne cm(-2). Estimates of forces per molecule in the cell attachment points to the external cellular matrix and cell-cell adhesion points are of the order of 8 pN at FAs and as high as 3 pN at ADJs, suggesting that direct force-induced mechanotransmission by single molecules is possible in both. The maximum deformation of an EC in the monolayer is calculated as 400 nm in response to a mean shear stress of 1 Pa applied over the EC surface which is in accord with measurements. The model also predicts that the magnitude of the cell-cell junction inclination angle is independent of the cytoskeleton and glycocalyx. The inclination angle of the cell-cell junction is calculated to be 6.6° in an EC monolayer, which is somewhat below the measured value (9.9°) reported previously for ECs subjected to 1.6 Pa shear stress for 30 min. The present model is able, for the first time, to cross the boundaries between different length scales in order to provide a global view of potential locations of mechanotransmission.

Computational models of the primary cilium and endothelial mechanotransmission

In endothelial cells (ECs), the mechanotransduction of fluid shear stress is partially dependent on the transmission of force from the fluid into the cell (mechanotransmission). The role of the primary cilium in EC mechanotransmission is not yet known. To motivate a framework towards quantifying cilia contribution to EC mechanotransmission, we have reviewed mechanical models of both (1) the primary cilium (three-dimensional and lower-dimensional) and (2) whole ECs (finite element, non-finite element, and tensegrity). Both the primary cilia and whole EC models typically incorporate fluid-induced wall shear stress and spatial geometry based on experimentally acquired images of cells. This paper presents future modelling directions as well as the major goals towards integrating primary cilium models into a multi-component EC mechanical model. Finally, we outline how an integrated cilium-EC model can be used to better understand mechanotransduction in the endothelium.

Aortic valve: mechanical environment and mechanobiology

The aortic valve (AV) experiences a complex mechanical environment, which includes tension, flexure, pressure, and shear stress forces due to blood flow during each cardiac cycle. This mechanical environment regulates AV tissue structure by constantly renewing and remodeling the phenotype. In vitro, ex vivo and in vivo studies have shown that pathological states such as hypertension and congenital defect like bicuspid AV (BAV) can potentially alter the AV's mechanical environment, triggering a cascade of remodeling, inflammation, and calcification activities in AV tissue. Alteration in mechanical environment is first sensed by the endothelium, which in turn induces changes in the extracellular matrix, and triggers cell differentiation and activation. However, the molecular mechanism of this process is not understood very well. Understanding these mechanisms is critical for advancing the development of effective medical based therapies. Recently, there have been some interesting studies on characterizing the hemodynamics associated with AV, especially in pathologies like BAV, using different experimental and numerical methods. Here, we review the current knowledge of the local AV mechanical environment and its effect on valve biology, focusing on in vitro and ex vivo approaches.

Finite-element modeling of viscoelastic cells during high-frequency cyclic strain

Mechanotransduction refers to the mechanisms by which cells sense and respond to local loads and forces. The process of mechanotransduction plays an important role both in maintaining tissue viability and in remodeling to repair damage; moreover, it may be involved in the initiation and progression of diseases such as osteoarthritis and osteoporosis. An understanding of the mechanisms by which cells respond to surrounding tissue matrices or artificial biomaterials is crucial in regenerative medicine and in influencing cellular differentiation. Recent studies have shown that some cells may be most sensitive to low-amplitude, high-frequency (i.e., 1-100 Hz) mechanical stimulation. Advances in finite-element modeling have made it possible to simulate high-frequency mechanical loading of cells. We have developed a viscoelastic finite-element model of an osteoblastic cell (including cytoskeletal actin stress fibers), attached to an elastomeric membrane undergoing cyclic isotropic radial strain with a peak value of 1,000 µstrain. The results indicate that cells experience significant stress and strain amplification when undergoing high-frequency strain, with peak values of cytoplasmic strain five times higher at 45 Hz than at 1 Hz, and peak Von Mises stress in the nucleus increased by a factor of two. Focal stress and strain amplification in cells undergoing high-frequency mechanical stimulation may play an important role in mechanotransduction.