Review

. 2024 Jan 26;2(1):100040.

doi: 10.1016/j.mbm.2024.100040. eCollection 2024 Mar.

Advances in modeling cellular mechanical perceptions and responses via the membrane-cytoskeleton-nucleus machinery

Hongyuan Zhu^{1

2}, Run Miao^{1

2}, Jin Wang^{1

2}, Min Lin^{1

2}

Affiliations

¹ The Key Laboratory of Biomedical Information Engineering of Ministry of Education, School of Life Science and Technology, Xi'an Jiaotong University, Xi'an 710049, PR China.
² Bioinspired Engineering and Biomechanics Center (BEBC), Xi'an Jiaotong University, Xi'an 710049, PR China.

PMID: 40395451
PMCID: PMC12082147
DOI: 10.1016/j.mbm.2024.100040

Review

Advances in modeling cellular mechanical perceptions and responses via the membrane-cytoskeleton-nucleus machinery

Hongyuan Zhu et al. Mechanobiol Med. 2024.

. 2024 Jan 26;2(1):100040.

doi: 10.1016/j.mbm.2024.100040. eCollection 2024 Mar.

Authors

Hongyuan Zhu^{1

2}, Run Miao^{1

2}, Jin Wang^{1

2}, Min Lin^{1

2}

Affiliations

¹ The Key Laboratory of Biomedical Information Engineering of Ministry of Education, School of Life Science and Technology, Xi'an Jiaotong University, Xi'an 710049, PR China.
² Bioinspired Engineering and Biomechanics Center (BEBC), Xi'an Jiaotong University, Xi'an 710049, PR China.

PMID: 40395451
PMCID: PMC12082147
DOI: 10.1016/j.mbm.2024.100040

Abstract

Mechanical models offer a quantitative framework for understanding scientific problems, predicting novel phenomena, and guiding experimental designs. Over the past few decades, the emerging field of cellular mechanobiology has greatly benefited from the substantial contributions of new theoretical tools grounded in mechanical models. Within the expansive realm of mechanobiology, the investigation of how cells sense and respond to their microenvironment has become a prominent research focus. There is a growing acknowledgment that cells mechanically interact with their external surroundings through an integrated machinery encompassing the cell membrane, cytoskeleton, and nucleus. This review provides a comprehensive overview of mechanical models addressing three pivotal components crucial for force transmission within cells, extending from mechanosensitive receptors on the cell membrane to the actomyosin cytoskeleton and ultimately to the nucleus. We present the existing numerical relationships that form the basis for understanding the structures, mechanical properties, and functions of these components. Additionally, we underscore the significance of developing mechanical models in advancing cellular mechanobiology and propose potential directions for the evolution of these models.

Keywords: Cellular mechanobiology; Cytoskeleton; Mechanical models; Mechanosensitive receptors; Nucleus.

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Conflict of interest statement

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Figures

**Fig. 1**
Modeling outside-in force transmission of a motile cell through the membrane-cytoskeleton-nucleus machinery. The yellow arrows showed the direction of forces provided by the cell. The dashed boxes marked with Roman numerals denote the typical structures around membrane, cytoskeleton and nucleus respectively. The snapshots marked with the same Roman numerals on both sides are the mathematical views of the corresponding structures. (**I, II**) Schematics of typical mechanical models of adhesions mediated by mechanosensitive receptors on cell membrane, cadherin (I) and integrin (II) respectively. (**III, IV**) Schematics of typical mechanical models of actomyosin cytoskeleton structures, filopodia (III) and lamellipodia (IV) respectively. (**V, VI**) Schematics of typical mechanical models of nuclear mechanical property (V) and nuclear mechanotransduction (VI) respectively. (I-VI) are adopted from references [50,83,105,111,117,169] respectively.

**Fig. 2**
Models for cadherin-mediated cell–cell interactions. A. A representative MD model for cadherin. **(I)** The scheme of the MD model depicting extracellular interactions in a 24 E-cadherin (CDH1) adherens junction. The insert shows the two types of cadherin interactions. **(II)** The representative force–displacement curves of two opposing cadherin monomers in an individual *tans*-dimer when stretching the two intracellular ends of the 24 E-cadherin junction. The light and dark green curves are the simulation results, while purple and black lines are linear fitting. There are two obvious elongation phases in a single force–displacement curve. **(III)** Statistical results of spring constants of cadherin *trans*-dimer, which are calculated from the slopes of the fitting lines as shown in II. $k_{s 1}$ and $k_{s 2}$ denote the spring constants of unbending and unbinding phases respectively. Purples dots are from isolated monomers, while green dots marked S1h-j are from independent replicate simulations of the 24 E-cadherin junction. The simulation results indicate *cis* interactions increase the stiffness of adherens junction. B. A representative MC model for cadherin. **(I)** The scheme of the MC model depicting cadherin clustering. Left: the lateral view of two cadherin-decorated membranes. Green and red solid ellipses are free EC1 domains of cadherins on the bottom and top membranes respectively. Blue solid ellipses are bound EC1 domains of cadherins. Right: the 2D lattice depicting top view of cell membranes. Green and red dipoles denote cadherins on top and bottom layers respectively. Blue dipoles denote the *trans*-bound cadherin dimers. The *cis* interactions only occurs between dipoles with the same directions. **(II)**. The snapshots show the cadherin patterns formed in different *trans* and *cis* binding affinities. Left: strong *trans* interaction but weak *cis* binding. Medium: both *trans* and *cis* interactions are strong enough. Right: strong *cis* interaction but weak *trans* binding. A is adopted from reference [50]. B is adopted from reference [61].

**Fig. 3**
Models for integrin-mediated cell-ECM interactions. A. A representative MD model for cadherin. **(I)** Scheme of conformational changes during integrin activation. i-iii denote integrins in bent closed, extended closed, and extended open (activated) states respectively. **(II)** The simulated integrin conformations by the MD models under different scenarios ITK (integrin-talin-kindlin), IT (integrin-talin), IK(integrin-kindlin) and I (integrin alone). The simulation resuts showed the distinct roles of talin and kindling in integrin activation. B. A representative MD model for cadherin. **(I)** Scheme of the MC model depicting integrin mediated cell-ECM interactions. In the model, the cell membrane and substrate are set as 3D networks consisted of springs with fixed elastic constants. The cell membrane and substrate are spaced by a glycocalyx layer with a fixed spring constant. Integrins are set as springs diffusing on the cell membrane, which reversibly binding to ligands on the substrate. **(II)** The integrin pattern obtained by the MC model at different relative thickness of glycocalyx. The model showed that increasing glycocalyx thickness assists the integrin clustering by a “kinetic trap” effect. C. An integrated model for integrin-initiated mechanotranduction incorporating: the spatial MC model considering kinetics of activation, binding and clustering **(I)**, the spring model considering force-dependent disassembly of integrin **(II)**, and the particle model for spatial arrangement of integrin and FAK molecules **(III)**. The simulation results demonstrated that: the lifetime of integrin clusters is longer on stiffer substrate **(IV)**, and the residence time for FAK in focal adhesion increases with increasing number of clustered integrins **(V)**. These conclusions together explained how substrate stiffness converted to FAK phosphorylation **(VI)**. A is adopted from reference [75]. B is adopted from reference [83]. C is adopted from reference [86].

**Fig. 4**
Models for filopodia contraction. A. A motor-clutch model depicting cellular contraction generated by filopodia. **(I)** The schematic illustration of the model. The actin filaments bind to the substrate by the molecular clutches. The myosin motors drive the actin filament slipping from the substrate. The substrate and clutches are elongated during loading. **(II)** The simulation results by the model predicted that traction force (dashed gray line) decreases while actin retrograde flow rate (solid black) increase as substrate stiffness increasing. B. A cell migration simulator based on motor-clutch model. **(I)** The schematic illustration of the model. In the model, the actomyosin filaments are modeled as motor-clutch modules, and the central cell body is attached to the substrate by a set of clutch molecules. The join force provided by actomyosins drive the cell migration. **(II)** Simulating cell migration on a stiffness-gradient substrate alternating soft and stiff regions. **(III)** The simulated average traction forces of modules as a function of substrate stiffness. The traction force is larger on the soft region of the stiffness-gradient substrate. **(IV)** The simulated tracks of the cells migrating on the stiffness-gradient substrate. A is adopted from reference [105], B is adopted from reference [114].

**Fig. 5**
Models for lamellipodia protrusion. A. A population–kinetics model for propelling leading edge of motile cell by dendritic nucleation. **(I)** The schematic illustration of the model. Near the leading edge, the new filament are nucleated on the existing filament and push the membrane advance. **(II)** Two reproductive patterns of actin filaments predicted by the model: when $ψ / 2 < θ \leq ψ$ , filaments reproduce in two orientations (top), when $θ > ψ$ , generating filaments in three orientations become possible (bottom). (**III**) The comparison between theoretical prediction (line) and electron micrograph statistics (histogram) of filament orientations at the leading edge. B. A stochastic model simulating the lamellipodia protrusion against applied loads. **(I)** The schematic illustration showing filament growth at a steady state against an intermediate load (middle, gray arrow), after increasing the load (top, blue arrow) and decreasing the load (bottom, red arrow). The barbed end and pointed end of filament are marked as red and blue respectively. Changing the load affects the filament density and angle distribution. **(II)** Simulated filament distributions under the three load conditions. **(III)** Quantification of filament numbers, barbed, and pointed end density under the three load condition. The simulation results suggested that increasing load leads a nucleation peak and an increase in filament density, while decreasing load increases capping, and decreases nucleation and thus filament density. A is adopted from reference [117]. B is adopted from reference [91].

**Fig. 6**
Models for stress fiber dynamics. A. A bio-chemo-mechanical model depicting assembly and contraction of stress fibers. **(I)** The model scheme illustrating a 2D cell consisting of a network of stress fibers. **(II)** The simulated distribution of stress fibers activation level ( $η$ ) in cells on the micro-posts with varied stiffness ( $k_{E}$ ). **(III)** The time evolution of normalized traction force provided by the cells on micro-posts with varied stiffness. The simulation results demonstrate that the activation level of stress fiber and thus cellular traction force increase with the support stiffness. B. An extended model depicting stress fiber behaviors in cells on ligand-patterned substrates. **(I)** The model scheme illustrating in a 2D cell on ligand-coated substrate. Both stress fibers and focal adhesions are considered in the representative volume element (RVE) of the cell. **(II)** The model scheme illustrating remodeling of a stress fiber subject to tensile strain ( $ε_{n}$ ). **(III)** The model scheme illustrating stretches of the cell on a Y-shaped ligand pattern (gray area). The partial of the cell perimeter ( $ω r_{c}$ ) in the free standing state is stretched to $L_{S}$ in the adhered state. **(IV)** The simulated distribution of focal adhesions in the spread cell on Y-shaped ligand pattern in the lowest free energy configuration. **(V)** The simulated distribution of stress fiber alignment in the spread cell on Y-shaped ligand pattern in the lowest free energy configuration. These results demonstrate that the cell is spreading toward the state of the minimal free energy through adjusting both the contractility of its active stress fiber and the deformation of its passive components. C. An extended model depicting stress fiber behaviors in cells under cyclic stretching. **(I)** The model scheme illustrating a cell adhering to a substrate subjected to cyclic stretch. The cell exchanges substances with the nutrient bath. **(II)** The model scheme illustrating the 2D cell with stress fiber components in unbound and polymerized state. **(III)** The simulated orientation distribution of stress fibers in cells subjected to cyclic stretching with varied loading frequency. The simulation results indicate that increasing cyclic loading results reorientations of cell and its stress fibers. D. An extended model depicting cell pairs on micro-post arrays. **(I)** The model scheme illustrating two cells adhering to micro-post connect to each other by cadherin mediated cell–cell junction. **(II)** The predicted stress fiber distribution in the different kinds of cell pairs. ETC: endothelial cell. FB: fibroblast. MSC: mesenchymal stem cell. SMC: smooth muscle cell. **(III-IV)** The predicted average traction ( $F_{avg}$ ) and junction tugging force ( $| F_{J} |$ ) of different kinds of cell pairs as functions of micro-post stiffness ( $k_{p}$ ). A is adopted from reference [127]. B is adopted from reference [132]. C is adopted from reference [135]. D is adopted from reference [136].

**Fig. 7**
Models for cortex mechanics. A. The model scheme illustrating an active gel network consisted of myosin motors, actin filaments and crosslinkers. The actin filaments are continuously polymerizing and depolymerizing at rate $k_{p}$ and $k_{d}$ respectively. B. A model based on active gel theory for depicting cortex dynamics during cytokinesis. Left is the simulated evolutions of cell shape and cortex thickness over time. Right is the microscopy images of a dividing cell. The color bar ( $ζ / ζ_{\max}$ ) indicates the myosin activity signal. C. A 3-tier composite model depicting the membrane tension propagation. **(I)**. The model scheme illustrating mechanical representations of membrane-cortex structure. The membrane is modeled as a series of linear springs (gray area). The adhesive linker between membrane and cortex is modeled as frictional components (blue area). The cortex is modeled as a thin layer of active gel (green area). **(II)** The simulated tension transmission as a function of cortex-membrane fiction for different targets of force application. The results summarizing three modes of membrane tension propagation observed in the experiments. A is adopted from reference [143]. B is adopted from reference [147]. C is adopted from reference [153].

**Fig. 8**
Models for mechanical properties of nucleus. A. A model for nuclear compliance measurement. **(I)** The model scheme illustrating how nuclear compliance is measured by aspiration and imaging. **(II)** Nuclear compliances of different cells as a function of time. B. A model for vibrational response of nucleus. **(I)** The model scheme illustrating the structure of a cell residing in ECM (Up). The cytoskeleton is modeled as a composite of contractile myosin in parallel with elastic microtubule and in series with elastic actin (Down). **(II)** The phase diagram of cytoskeletal stiffness ( $\bar{k}$ ) and initial strain generated by cytoskeletal filaments ( $ε_{0}^{*}$ ). The diagram is divided into torsional vibration-dominated regime ( $f_{tors} > f_{tors}$ , dark area) and translational vibration-dominated regime ( $f_{tors} < f_{tors}$ , light area). The natural frequencies of torsional and translational of the nucleus are also marked for a range of cell types. A is adopted from reference [161]. B is adopted from reference [166].

**Fig. 9**
Models for nuclear mechanotransduction. A. A model depicting direct force-induced stretching of chromatin. The scheme shows that the external force propagates to nucleus through integrin mediated focal adhesion and cytoskeleton, which stretches the chromatin and upregulate the transcript of DHFR gene. B. A model depicting mechanics-driven nuclear localization of transcription factor. **(I)** The model scheme illustrating cadherin disturbs integrin binding and regulate the cellular traction force ( $F_{trac}$ ) transmitting to the nucleus. The nucleus is regarded as an incompressible viscoelastic sphere. **(II)** The model scheme illustrating the flattening nucleus by actomyosin cytoskeleton opens the nuclear pore and faciliates the active transport of transcription factor YAP into nucleus. **(III)** The relationships between YAP nucleus/cytoplasm ratio ( $R_{NC}$ ) and nuclear flattening ( $λ_{N}$ ), as calculated by the model (line) and obtained by experiments on substrates with different ligand and stiffness (dots). A is adopted from reference [11]. B is adopted from reference [111].

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References

1. Romani P., et al. Crosstalk between mechanotransduction and metabolism. Nat. Rev. Mol. Cell Biol. 2021;22(1):22–38. - PubMed
1. Wagh K., et al. Mechanical regulation of transcription: recent advances. Trends Cell Biol. 2021;31(6):457–472. - PMC - PubMed
1. Belardi B., et al. Cell–cell interfaces as specialized compartments directing cell function. Nat. Rev. Mol. Cell Biol. 2020;21(12):750–764. - PubMed
1. Kanoldt V., Fischer L., Grashoff C. Unforgettable force – crosstalk and memory of mechanosensitive structures. Biol. Chem. 2019;400(6):687. - PubMed
1. Pegoraro A.F., Janmey P., Weitz D.A. Mechanical properties of the cytoskeleton and cells. Cold Spring Harbor Perspect. Biol. 2017;9(11) - PMC - PubMed

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Advances in modeling cellular mechanical perceptions and responses via the membrane-cytoskeleton-nucleus machinery

Affiliations

Advances in modeling cellular mechanical perceptions and responses via the membrane-cytoskeleton-nucleus machinery

Authors

Affiliations

Abstract

Conflict of interest statement

Figures

References

Publication types

LinkOut - more resources

Full Text Sources

Miscellaneous