Module 6: Cell Shape & Mechanics

A mammalian cell of diameter \(\sim 15\) μm contains roughly \(10^{10}\) actin monomers, \(10^{8}\) tubulin dimers, \(10^{7}\) intermediate-filament subunits, and a water content near 70-80% at a total crowded protein concentration of 300–400 mg/mL. Mechanically, this cytoplasm behaves as a viscoelastic solid with an apparent Young modulus \(E \sim 10^2\!-\!10^4\) Pa, a shear modulus roughly \(G \sim E/3\) (for near-incompressible material with \(\nu \approx 0.5\)), and a rich spectrum of relaxation times ranging from microseconds (single-filament thermal fluctuations) to tens of seconds (cortical remodelling).

The shape of an adherent cell is set by a mechanical equilibrium between the cortical tension \(T\) (a 2D tension in the actomyosin cortex, N/m) and an excess hydrostatic pressure \(\Delta P\) between cytoplasm and extracellular space. For a nearly spherical bleb of radius \(R\) the Young-Laplace law applies:

At the cortex, the actomyosin network is not a passive elastic sheet but an active contractile one: phosphorylated myosin II minifilaments generate a 2D active stress \(\sigma_{\text{a}}\) that is set by \([\text{ROCK}]\), \([\text{MLCK}]\), and the local concentration of myosin regulatory light chain. The steady-state shape obeys a force balance of the form

Donald Ingber introduced the tensegrity model of cell mechanics in a series of papers (Ingber 1993 J. Cell Sci., Ingber 2003 J. Cell Sci. I/II) inspired by Kenneth Snelson’s tensegrity sculptures and R. Buckminster Fuller’s terminology. The central claim is architectural: the cytoskeleton is pre-stressed; the three filament classes play complementary mechanical roles:

• Microtubules are stiff compression struts (persistence length \(\ell_p \sim 5\) mm, Young modulus \(\sim 1.2\) GPa). They resist axial compression from the contractile cortex and keep the cell from collapsing inward.
• Actin filaments and the actomyosin cortex are tension cables: they provide the centripetal pull against which the microtubule struts push out.
• Intermediate filaments(vimentin, keratin, lamins) are non-linear tension cables that strain-stiffen dramatically above \(\sim 50\%\) strain, providing catastrophic-failure resistance.

The key mechanical consequence is that cellular stiffness scales linearly with pre-stress:

Tensegrity also predicts the “action at a distance” observed in mechanotransduction: pulling on an integrin with a magnetic bead instantly distorts the nucleus tens of micrometres away, because the pre-stressed network transmits force along continuous load paths (Maniotis, Chen & Ingber 1997 PNAS).

Cells are neither purely elastic (like a spring) nor purely viscous (like honey). The standard rheological signatures are (i) stress relaxation under imposed strain, (ii) creep under imposed stress, and (iii) frequency-dependent complex moduli \(G'(\omega) + i G''(\omega)\). A minimal model capturing the cytoplasm’s short-time elasticity and long-time flow is the standard linear solid (SLS):

Fabry, Maksym, Butler, Glogauer, Navajas & Fredberg (2001 PRL) discovered that for many cell types, the storage and loss moduli over 5 decades of frequency (0.01–1000 Hz) follow a weak power law rather than an SLS:

The two frameworks are complementary. On short timescales relevant for mitosis, bleb retraction, or AFM indentation, the SLS captures the dominant single-relaxation-time response. On broad-frequency-band measurements (optical tweezers, magnetic cytometry), the power law is more faithful. Both reduce to pure elasticity at very high frequency and Newtonian flow at very low.

Atomic force microscopy (AFM) with a sharp tip (colloidal bead, pyramidal silicon, or CNT tip) presses into an adherent cell while a photodiode records cantilever deflection. The raw data is a force-indentation curve \(F(\delta)\); the apparent Young modulus is extracted by fitting Hertz contact theory(Radmacher 1996; Rotsch & Radmacher 2000). For a conical tip of half-angle \(\alpha\) on a half-space:

Fitting Hertz to an approach curve yields an instantaneous modulus \(E_0\); allowing the cantilever to dwell at fixed indentation and recording force relaxation yields the equilibrium modulus \(E_\infty\) and the relaxation time \(\tau\). A typical spread for adherent cells, compiled from the AFM literature, is:

• Neutrophils, leukocytes: \(E \sim 100\!-\!300\) Pa.
• Red-blood-cell membrane: \(E \sim 300\) Pa (Evans 1976).
• MCF-7 breast cancer cells: \(E \sim 500\!-\!800\) Pa.
• Fibroblasts, HeLa: \(E \sim 1\!-\!3\) kPa.
• Osteoblasts: \(E \sim 3\!-\!5\) kPa.
• Myoblasts: \(E \sim 4\!-\!10\) kPa.
• Smooth-muscle cells: \(E \sim 5\!-\!15\) kPa.

Malignant transformation softens cells significantly: Cross et al. (2007 Nat. Nanotechnol.) showed metastatic lung/pleural cancer cells are \(\sim 70\%\) softer than benign controls, a biomarker now under clinical evaluation.

Micropipette aspiration (Evans 1976, 1983) applies a calibrated suction pressure \(\Delta P\) to a glass micropipette of radius \(R_p \sim 1\!-\!5\) μm touching the cell surface and records the resulting length \(L\) aspirated into the pipette. For a cortex-like tension shell, the Young-Laplace relation gives

For a red blood cell, Evans demonstrated that aspiration in fact probes the 2D shear elasticity \(\mu\) of the spectrin-actin skeleton, not just a tension:

Micropipette aspiration remains a gold-standard method for cortex tension and for the mechanical phenotyping of RBCs in sickle-cell disease, malaria infection, and hereditary membrane disorders.

A cell adherent to a deformable elastic substrate exerts traction forces through focal adhesions. Dembo & Wang introduced traction-force microscopy (TFM) in 1999 (Biophys. J.): they cast a thin polyacrylamide or polydimethylsiloxane (PDMS) gel embedded with fluorescent marker beads, plated cells on top, imaged bead displacements \(\vec u(\vec r)\) before and after detaching the cell with trypsin, and solved the inverse elastic problem for the surface traction field \(\vec T(\vec r)\).

Typical single-cell traction magnitudes are \(\sim 1\!-\!10\) kPa localized at focal adhesions near the cell periphery, with total integrated force per cell \(\sim 10\!-\!100\) nN. Traction increases with substrate stiffness (cells pull harder on stiffer gels) and is abolished by ROCK or myosin-II inhibition.

Extensions include 2.5D TFM (Legant 2010) for z-force components, 3D TFM inside collagen gels (Koch 2012), and micropost arrays (Tan et al. 2003 PNAS) where each PDMS post acts as a calibrated cantilever spring. TFM underpins modern mechanobiology: it is how one measures the force that a single cell exerts on its microenvironment.

The focal adhesion (FA) is a layered, force-sensitive molecular assembly at the plasma membrane where integrin heterodimers (\(\alpha\beta\), e.g. \(\alpha_5\beta_1\) for fibronectin, \(\alpha_V\beta_3\) for vitronectin) engage the extracellular matrix and couple to the intracellular actin cytoskeleton through a plaque of >150 proteins. Chen, Mrksich, Huang, Whitesides & Ingber (1997, 2003) showed that FAs form only if the cell can spread above a critical size, establishing spreading area as a mechanical prerequisite.

Kanchanawong et al. (2010 Nature) used interferometric photo-activated localization microscopy (iPALM) to resolve the FA into three strata at nanometre z-resolution:

• Integrin signaling layer (0–20 nm above the membrane): paxillin, FAK.
• Force-transduction layer (20–40 nm): talin rod, vinculin.
• Actin-regulatory layer (40–90 nm): \(\alpha\)-actinin, VASP, zyxin; seamlessly merges into the actin stress fibre.

The key mechanosensor is the talin rod: del Rio et al. (2009 Science) stretched single talin molecules with magnetic tweezers and showed that eleven cryptic vinculin-binding sites open sequentially under forces of 5–25 pN. Vinculin binding stabilizes the adhesion and recruits actin-regulatory proteins, producing a positive feedback loop: tension recruits vinculin, vinculin recruits actin, actin generates more tension. This force-dependent reinforcement is the molecular basis of “catch-bond” adhesion strengthening.

In a landmark study, Engler, Sen, Sweeney & Discher (2006 Cell) plated naive human mesenchymal stem cells on polyacrylamide gels of controlled Young modulus and showed that substrate stiffness alone is sufficient to direct lineage:

• \(E \sim 0.1\!-\!1\) kPa (brain-like) → neurogenic (\(\beta\)3-tubulin\(^+\)).
• \(E \sim 8\!-\!17\) kPa (muscle-like) → myogenic (MyoD\(^+\)).
• \(E \sim 25\!-\!40\) kPa (pre-calcified bone) → osteogenic (RUNX2\(^+\)).

The differentiation was independent of growth factors and could be blocked by non-muscle myosin II inhibition (blebbistatin), implicating cytoskeletal contractility and force transmission through FAs.

The mechanistic link was completed by Dupont et al. (2011 Nature), who identified YAP/TAZ(Yes-associated protein / transcriptional co-activator with PDZ-binding motif) as mechanosensitive transcription cofactors. On stiff substrates, actomyosin tension drives YAP/TAZ into the nucleus where they partner with TEAD transcription factors to activate CTGF, CYR61, and other proliferation/lineage genes; on soft substrates, YAP/TAZ are sequestered in the cytoplasm and degraded.

The actomyosin cortex is a 100–500 nm thick meshwork of F-actin, myosin II minifilaments (hexamers of MYH9/10/14 heavy chains + regulatory and essential light chains), septins, cross-linkers (\(\alpha\)-actinin, filamin), and linkers to the plasma membrane (ezrin, radixin, moesin — the ERM proteins). Mechanical properties of the cortex are dominated by the active stress from myosin contractility, typically \(\sigma_a \sim 10^2\!-\!10^3\) Pa, and by viscous dissipation from cross-linker turnover.

A bleb is a transient spherical protrusion that forms when the plasma membrane detaches locally from the cortex and is inflated by cytoplasmic pressure (Charras & Paluch 2008 Nat. Rev. Mol. Cell Biol.). The bleb-cycle has three phases: expansion (pressure driven), static (new cortex reassembly by Arp2/3 and formins), and retraction (myosin II contraction). Blebbing underlies ameboid migration, apoptotic cell fragmentation, and mitotic rounding.

Cytokinesis is the topological separation of two daughter cells via a contractile actomyosin ring assembled at the cell equator under control of RhoA, Anillin, and the septin collar. Key ingredients are:

• Equatorial RhoA-GTP band established by Ect2 (RhoGEF) recruitment to the central spindle via MgcRacGAP/CYK4.
• Actin polymerization by mDia-family formins (not Arp2/3) produces long unbranched cables.
• Non-muscle myosin II bundles the actin and contracts the ring at \(\sim 0.5\) μm/s.
• Anillin (a scaffold binding actin, myosin, RhoA, and septins) couples the ring to the plasma membrane.
• Septins (SEPT2/7/9/11 in humans) form GTP-binding filaments that organise the cleavage furrow and mark the abscission site.

Furrow ingression is described by the equation of motion \(\eta\,\dot R = -\sigma_r + (T_r/R)\), with \(\sigma_r \sim \) ring tension per unit length (~100 nN/μm), \(\eta\) a cortical viscosity, and \(R\) the ring radius. A 15 μm-diameter mammalian cell closes cytokinesis in 5–10 min.