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Course Notes of Introduction to mechanobiology.

Course information

Reference articles

Physical regulation of cells and tissues

  • cosmonaut: osteoporosis due to microgravity
  • Right arm bigger than the left (tennis player)
  • shape of liver cells (polygons) vs muscle cells (elongated cylinders)
  • Tissues / cells could sense physical cues
  • tension, compression, fluid flow (hydrodynamic pressure, shear stress etc.), osmotic pressure, ion current
    • Laminar flow (low Reynaud's number) vs turbulent flow => different endothelial response inside blood vessels
  • Sensed by mechanosensor complexes attached to both cytoskeletons inside cell and ECM outside, affecting gene expression, electrophysiology, etc.
  • Heterogenous structure and stiffness inside cartilage (from synovial surface to bone surface)

Clinical relevance of mechanobiology

  • deafness: cochlear hair cells
  • arteriosclerosis: endothelial and smooth muscle cells
  • muscular dystrophy and cardiomyopathy: myocytes and fibroblasts
  • Congenital muscular dystrophy due to mutations in dystrophin (part of anchor for cytoskeleton)
  • Sarcopenia in the elderly
  • Aspect ratio deviation in dilated / hypertrophic cardiomyopathy
  • Weight training : NADPH oxidase produces ROS in response to tension => muscle growth (young people) or death (old people)
  • Cartilage: chondrocytes
  • Spatial difference in cell alignment, ECM composition (stiffer towards the bone surface)
  • Running -> compression -> stimulates ECM and cell growth
  • Axial myopia and glaucoma: optical neurons, fibroblasts
  • Polycystic kidney disease (PCKD): epithelial cells of renal tubules
  • Cancer: cancer cells and cancer-associated fibroblast (CAF)
  • Adhesion to blood vessel walls: WBC (neutrophils, monocytes, macrophages)

Cellular biology 101

Cell structures, cell cycle and replication, central dogma and information flow, signal transduction and homeostasis.

  • DNA structures: H-bond => thermodynamically stable in antiparallel double helix, good for information storage
  • RNA structure: more reactive (2' hydroxyl group), not as stable as DNA, good for reaction catalysis (e.g. ribozymes) and carrying transient information (e.g. messenger RNA)

Information flow in mechanobiology

Outside-in vs inside-out

  • Outside-in: force transduction (directly or via ECM) - transducer (complex) - signal transduction cascade (amplifiers, filters, logical gates) - gene expression involving cell cycle, metabolism, and survival
  • Inside-out: Cells actively tug the external environment by motor proteins and cytoskeletons - determination of external physical cues (e.g. stiffness) - cell growth and differentiation

Lipid rafts in the plasma membrane

  • Rich in cholesterol (less fluidity) and glycolipids
  • Rich in cytoskeleton anchor complexes and mechanoreceptor

Cytoskeletons

  • Actin: stress fibers, with motor proteins (e.g. myosin), polarity(+)
  • Microtubule: compression fibers, consisting tubulin alpha and beta, polarity(+), with motor proteins (kinesin and dynein)
  • intermediate filaments (keratin filaments): study and less dynamic, buffer between the nucleus and the cell surface

Cell-cell junctions

  • Tight junctions: preventing leakage from apical side to basal side
  • Gap junctions (channeling two adjacent cells)
  • Desmosomes: cadherins (requires divalent cations), connected to keratin filaments (structure support)
    They know their spatial arrangement (basal vs apical)

Cell-matrix junctions

  • Reference book: mechanobiology of cell-cell and cell-matrix interactions
  • Basal lamina (ECM) - integrins (transmembrane part) - anchor complex - actin
  • ECM:
  • glycosaminoglycans (GAG): extremely hydrophilic, providing compressive strength
  • fibrous proteins (e.g. collagen): tensile strength
  • Example of neonatal rat cardiomyocyte growth and development
  • Soft surface (100~300 Pa): round and undifferentiated
  • Native environment in the heart (10 kPa): cylindrical with the best aspect ratio (7:1) with sarcomeres.
  • Stiff surfaces (glass): flat and polygonal

Crosstalk of cell-cell junctions and cell-matrix junctions

  • Cadherin and integrin pathways
  • Cellular movement, differentiation, and growth

Cartilage tissue structure and homeostasis

  • chondrocyte and ECM interactions
  • influenced by physical forces (pressure, shear stress)
  • Spatial heterogeneity of ECM composition (stiffness) and cell arrangement (clustering and orientation)

Mechanotransduction

Information flow

external stimulation -> outside-in -> processing -> inside-out -> cellular response (behavior)

Biological signal processing in a cell as a black box ?

  • Phenotype-dependence: same ligand + different context (cell type, receptor) = different response

Inside-out signaling

  • Altered protein function (activation/ deactivation): ms to secs
  • Altered gene expression (protein synthesis): hours to days
  • Central dogma: DNA -> mRNA -> protein (nowadays with a lot of regulations)
  • Gene expression level \(\approx\) mRNA content \(\approx\) protein activity (e.g. RNAseq)

Outside-in signaling

  • Extracellular signal -> transmembrane receptor -> intracellular relays, amplifiers, modulators...

Receptors

  • Ion-channel-coupled:
  • NMDA receptor: opens Ca channel when binds to glutamate
  • MET channel in cochlear hair cells: opens K channel when stretched
  • G-protein-coupled: a lot of targets
  • Enzyme-linked: e.g. EGFRs, JAK-STAT

Relays and amplifiers:

  • second messengers (Ca, IP3, DAG, cAMP, ...), kinase cascades
  • A complex network of signal transduction pathways => bioinformatics

Molecular switches

  • phosphorylation by kinase / dephosphorylation by phosphatase
  • GTP-binding: GDP->GTP by replacement. GTP -> GDP by phosphatase activity
  • GEF: GTP exchange factor
  • GAP: GTPase-activating protein

Signal transduction pathways (simplified)

GTP-linked receptors

  • Ligand binds to receptor, then the LR-complex binds to G protein

  • G protein replace GTP for GDP and dissociate from LR-complex

  • alpha subunit of G-protein dissociates from beta-gamma subunits

  • In Gs protein, the alpha subunit activates CA, converting ATP to cAMP (2nd messenger) for the cascade

  • In Gq protein, the beta-gamma subunits activates PCL-beta, cleaving a special phospholipid (PIP2) to IP3 and DAG, which in turn activate Ca release from ER and activate PKC for the cascade.

Mechanoreceptor and transduction

  • Stretch-activated ion channels
  • Integrins
  • E-cadherins e.g. fluid shear stress -> TF (β-catenin) to nucleus
  • Physical forces affects gene transcriptions (exp: movement of transcription factors after physical stress)
  • Stretching peptides -> exposure of folded AA residues -> signals (does not require a living cell)
  • Compression of chondrocytes: heterogenous, anisotropic strains and (probably) stress
  • Mechanosensing by adhesion site recruitment and stretching cytoskeletons -> substrate component and stiffness.
  • Chromatin deformation by force changes their relative positions and could alter gene expressions.
  • Force could change gene expression directly!
  • Osmotic loading of chondrocytes: changes in osmolarity => altered chromatin structure

Solid Mechanics Primer

A crawling cell uses pseudopod and forward attachment point to move forward.

Rigid body approach

  • Sum of moment = 0, Sum of external force = 0 (Newton 1st law)
  • Or applying Newton 2nd law
  • Free body diagram (reaction force: force exerted to the cell)
  • But cells are deformable: GG

Deformable cell

  • Displacement field (\(\Delta x\) inside the cell) is not uniform
  • Displacement is related to mechanical properties (stiffness)
  • Resolution down to molecular level is too much. Treat the cell as a continuum of infinitesimal elements (~100nm)

Stress

  • Scaled force, averaged by area (the same unit as pressure), affected by shape
  • A tensor described by two vectors
  • the force
  • the normal vector of the plane
  • \(\sigma_{xy}\) : On the yz plane (normal vector x), force with y direction
  • Normal stress: force parallel to the normal vector (tensile and compressive)
  • Shear Stress: force perpendicular to the normal vector
  • Mixed: decompose to the two above first

Strain

  • Displacement rescaled (normalized) by the original length
  • Averaged deformation (dimension-less)
  • Axial strain: \(\epsilon = \frac{\Delta L}{L}\), engineering strain, assuming \(\Delta L \ll L\)
  • Shear strain: \(\gamma = \frac{\delta}{L} = tan\theta \approx \theta\)
  • Transverse strain: Poisson's ratio \(\nu = -\epsilon_t / \epsilon_a > 0\)

Stress-Strain relationships

  • Linear (Young's modulus) -> nonlinear -> yield point (plastic change) -> ultimate -> break

Stress and strain fields

  • Average force (stress) adn average displacement (strain) in the cell
  • Force and torque equilibrium (assuming little acceleration and rotation)
  • For linearly elastic materials: 6 independent components
  • Experimental results: displacement field -> What we want: stress fields
  • Applying stress-strain relationships (Young's modulus, Poisson ratio, ...)

Stress on a (linearly elastic) material

  • Body force is insignificant to surface force
  • Large surface-to-volume ratio in small scales
  • Decompose surface forces on a small cube to tensors (x, y, z)

  • \(\sigma_{jj} = \lim_{A \rightarrow0}\frac{S_{jj}}{A}\), \(\tau_{ij} = \lim_{A \rightarrow0}\frac{S_{ij}}{A}\)

  • Equilibrium of stress

  • Take 1st Taylor expansion of surface forces related to certain directions
  • Sign convention: negative sides take negative values
  • Tensor representation: \(\sigma_{ij, j} = 0\) (Balance of volume forces)
  • \(\sigma_{ij} = \sigma_{ij}\) due to moment balance

Kinematics

  • Converting displacement to strain
  • Normal strain \(\epsilon_{ii} = \frac{du}{di}\)
  • Shear strain \(\theta \approx tan \theta = \frac{du}{dj}\)
  • \(\gamma = tan \theta \approx \theta\)
  • continuous shear strain \(\epsilon_{ij} = \gamma_{ij} / 2\)
  • Symmetry: \(\epsilon_{ij} = \epsilon_{ji}\), 6 independent strains

Constitutive equations

6 stress and 6 strains = 36 parameters

For a linearly elastic material under small strain (< 1%)

  • Young modulus E: \(\sigma_{jj} = E\epsilon_{jj}\)
  • Shear modulus G: \(\tau_{ij} = \sigma_{ij} = G\gamma_{ij} = 0.5G\epsilon_{ij}\)
  • Poisson ratio \(\nu\) : \(\nu = -\epsilon_{jj} / \epsilon_{ii}\). For biomaterials = 0.5
  • G = E / 2 (1 + ν) for small strain

Homogeneity

The scale we concern is much larger than the irregularities in the material.

e.g. A collagen gel is homogenous in the scale of mm, not nm.

Isotropy

In either direction, the response is the same.

Traction vector

  • Forces along the plane with xyz components
  • Force equilibrium: Traction forces = stresses * areas
  • Could be represented in matrix form
  • Use: displacement (beads) -> strain -> stress -> traction force
  • With Green function (complicated)

Large deformation (>1%)

Deformation gradient (F)

\(\vec{B} = F\vec{A}\) for \(\vec{A}\) deforms to \(\vec{B}\)

Principle directions of deformation

Find eigenvalues and eigenvectors of \(e = 0.5(FF^T-I)\)

  • eigenvectors: principle directions
  • eigenvalues: strain

Rheology

  • Fluid mechanics
  • Viscoelasticity: a subset

Fluid

  • Shear stress -> continual deformation (flow)
  • Defined by density (ρ) and viscosity (η)
  • Increased viscosity = harder to push sideways

Viscosity

  • 1 poise = 0.1 Ns/m^2
  • water = 0.001 Ns/m^2
  • Newtonian fluid : viscosity independent of shear stress
  • Linear flow profile
  • $\(\tau = \eta \frac{du}{dy}\)
  • The latter (\(\frac{du}{dy}\)) is called shear strain rate and velocity gradient

Stress balance inside a fluid

  • Internal friction (viscosity) and external force (stress)
  • Shear strain \(\gamma = \frac{\Delta x}{dy}\)
  • Shear strain rate (velocity gradient) \(\frac{du}{dy} = \frac{d}{dt}(\frac{\Delta x}{dy})\)

Microscopic model of viscosity

  • Particles move at different speeds at different layers
  • They also diffuse and bump the neighboring ones due to the speed difference.
  • Friction is proportional to velocity gradient (shear strain rate)

Non-Newtonian fluid

https://en.wikipedia.org/wiki/Non-Newtonian_fluid
* Viscosity is dependent on velocity gradient
* Blood: Binham fluid (flows only when the shear strain rate greater than the threshold)
* Ketchup: Shear thinning = pseudoplastic
* Corn starch with water: shear thickening = dilatant

Viscosity fluid's strain in response to oscillatory stress

https://en.wikipedia.org/wiki/Viscoelasticity

\(\sigma = \sigma_0cos(\omega t)\), \(\omega = 2 \pi f\)
* Similar to AC circuits
* Elastic: in-phase
* Viscosity: causing phase lag up to 90 degrees

Complex modulus (by Euler formula)

\(e^{ix} = cos(x) + isin(x)\)

  • Stress: \(\sigma^* = \sigma_0e^{i\omega t}\)
  • Strain: \(\epsilon^* = \epsilon_0e^{i(\omega t - \delta)}\)
  • Modulus: \(E^* = \frac{\sigma_0}{\epsilon_0}e^{i\delta} = E_1 + iE_2\)
  • Storage / elastic modulus: \(E_1 = \sigma_0cos\delta / \epsilon_0\)
  • Loss / damping modulus: \(E_2 = \sigma_0sin\delta / \epsilon_0\)
  • Complex shear modulus: \(G^* = \frac{\tau^*}{\gamma^*}\)

Hysteresis

  • Viscous component: transforms mechanical energy into heat
  • In biomaterials, the loop is repeatable and independent of loading rate

Creep

  • Strain increases when holding constant stress
  • Reorganization of molecules
  • Movement of water (in most biomaterials)

Stress relaxation

  • Stress decreases when holding constant strain
  • Reorganization of molecules

Viscoelasticity Models

https://en.wikipedia.org/wiki/Viscoelasticity#Constitutive_models_of_linear_viscoelasticity

  • Springs ( \(\sigma = E\epsilon\) ) and dashpots ( \(\sigma = \eta \frac{d\epsilon}{dt}\) )
  • Series: same stress, summing strain
  • Parallel: same strain, summing stress

Fluid mechanics

Reference
Nelson biological physics ch5

Difference between solid and fluid mechanics

  • Solid: inertia and acceleration, time-dependent, no convection (-), constant density
  • Fluid: Convection (+), may have variable density
  • Lamellar flow: no inertia or time dependent terms involved

Force balance inside a fluid element

  • Pressure (normal stress)
  • Friction, viscous force

General assumptions

  • Steady flow: force balanced
  • Newtonian fluid: viscosity does not dependent on shear rate
  • Incompressible: constant density

Acceleration of the fluid

\(\(dv(x, t) = v(x + dx, t + dt) - v(x, t)= \frac{\partial v}{\partial x}dx + \frac{\partial v}{\partial t}dt\)\)
\(\(dv(x, t) = \frac{\partial v}{\partial x}vdt + \frac{\partial v}{\partial t}dt\)\)
\(\(a(x, t) = \frac{dv(x, t)}{dt} = \frac{\partial v}{\partial t} + \frac{\partial v}{\partial x}v\)\)
Former term: solid mechanics, latter term: fluid convection

In 3D space:
\(\(a = \frac{\partial v}{\partial t} + v \cdot \nabla v\)\)

Net pressure force

Notice the negative number (force is in the opposite direction of pressure gradient)
\(\(\delta f_x^p \approx \frac{-\partial p}{\partial x} dx_1dx_2dx_3\)\)
\(\(\delta p \approx (-\nabla p) dx_1dx_2dx_3\)\)

Viscosity and Newtonian fluid

Shear stress: \(\tau = \eta \frac{du_1}{dx_2}\) : linear flow profile between parallel plates
Shear force:
\(\(f_x^v(x_1, x_2) = -\tau dx_1dx_3 = -\eta\frac{\partial v_1(x_2)}{\partial x_2} dx_1dx_3\)\)
\(\(f_x^v(x_1, x_2 + dx_2) = \eta\frac{\partial v_1(x_2 + dx_2)}{\partial x_2} dx_1dx_3\)\)
\(\(f_{x_1x_2}^v \approx \eta \frac{\partial^2v_1}{\partial x_2^2} dx_1dx_2dx_3\)\)
(Second Taylor expansion)
Net shear force:

\[\delta f_v = \eta \nabla^2 v dx_1dx_2dx_3\]

Incompressibility

Linear strain:
\(\(d\epsilon_x = \frac{\Delta x^\prime - \Delta x}{\Delta x} = \frac{dx (\frac{\partial v_x}{\partial x})dt}{dx} = \frac{\partial v_x}{\partial x}dt\)\)

Size change in 3D space:
\(\(dx_1dx_2dx_3(1 + (\frac{\partial v_1}{dx_1} + \frac{\partial v_2}{dx_2} + \frac{\partial v_3}{dx_3})dt)\)\)

Incompressible: \(\nabla \cdot v = 0\), i.e. divergence of velocity field = 0

Newton's second law

\(\(F = ma = \rho dx_1dx_2dx_3 Y_i + \delta f^p + \delta f^v\)\)
* 1st term: body force (e.g. gravity)

We get the Navier-Stoke equation:
\(\(\rho (\frac{\partial v}{\partial t} + (v \cdot \nabla) v) = \rho Y - \nabla p + \eta \nabla^2 v\)\)

  • \(\frac{\partial v}{\partial t}\): Solid acceleration
  • \((v \cdot \nabla) v\): fluid convective term
  • Y: body force
  • \(\nabla p\): pressure term
  • \(\eta \nabla^2 v\): viscous term

Microscopic model of fluid friction

Velocity gradient across adjacent layers plus particle diffusion => momentum exchange and frictional drag

Particle drift and friction law (in small Re)

\(F = \zeta v\), \(\zeta\): drag coefficient
* Stokes law (for spherical objects): \(\zeta = 6 \pi \eta R\)
* Electrophoresis: \(F = q\epsilon = \zeta v\)
* Sedimentation of colloid particles: \(f_s = -m_{net}g = \zeta v\)

Reynolds number (Re)

  • Dimensionless property
  • Fluid runs around a particle
  • acceleration = \(\frac{v^2}{R}\)
  • viscous force = \(\eta\frac{v}{R^2}\)
  • Substitute into Navier-Stoke equation: \(\frac{\rho v R}{\eta} = \frac{R^2}{\eta v}f_{ext} + 1\)

Large Re

  • Dominated by inertia
  • Fluid is mixed, turbulent flow with vortices
  • Examples: human in water, rockets
  • \(f_{ext} \approx \rho \frac{v^2}{R}\)

Small Re

  • \(\frac{\rho v R}{\eta} \ll 1\), \(f_{ext} \approx \frac{\eta v}{R^2}\), drifting velocity proportional to drag force.
  • Dominated by viscous drag, laminar flow (Re < 10)
  • Acceleration and inertia term extremely small, time reversible
  • Examples: bacteria in water, dyes in corn syrup
  • Reciprocal motion does not work (due to time reversibility)
  • Periodic movement (cilia) and rotational movement (flagella) break the symmetry of the drag coef. (and thus the drag force) and create propulsion.

Motion of fluid between parallel plates in small Re

  • Applying the Navier-Stoke equation, ignoring the acceleration and body force terms. Only the pressure and the drag terms interact.
  • No slip boundary condition (velocity = 0 and the walls)
  • Parabolic flow profile
  • Flow \(\propto pr^4\)

Response of osteocyte to fluid flow

  • Bone structure
  • Cortical bone
  • Trabecular bone (spongy)
  • Bone marrow
  • Osteons: Concentric circles
  • Osteocytes: with channels connecting each other and the blood vessels
  • Osteoclast: A special macrophage removing old bones

  • Osteoblast: will become osteocytes once the surrounding mineralized

  • Fluid mechanics in the bone

  • Tension, compression
  • Difference in hydrostatic pressure
  • Fluid flow and shear stress on the osteocytes
  • Piezoelectric collagen I ?

  • Stimulates osteocytes to produce more osteopontin

  • How to separate flow shear stress and convection of nutrients

  • Increase the flow shear stress by adding the viscosity (add dextran)

Divided by volume element (\(dx_1dx_2dx_3\)), dimension = force density:
\(\(\rho \frac{dv}{dt} = \rho Y - \nabla p + \eta \nabla^2 v\)\)

We get the Navier-Stoke equation:
\(\(\rho (\frac{\partial v}{\partial t} + (v \cdot \nabla) v) = \rho Y - \nabla p + \eta \nabla^2 v\)\)

Where
* \(\frac{\partial v}{\partial t}\): Solid acceleration
* \(v \cdot \nabla v\): fluid convective term
* Y: body force
* \(\nabla p\): pressure term
* \(\eta \nabla^2 v\): viscous term

Microscopic model of fluid friction

Velocity gradient across adjacent layers plus particle diffusion => momentum exchange and frictional drag

Particle drift and friction law (in small Re)

\(F = \zeta v\), \(\zeta\): drag coefficient
* Stokes law for spherical objects: \(\zeta = 6 \pi \eta R\)
* Electrophoresis: \(F = q\epsilon = \zeta v\)
* Sedimentation of colloid particles: \(f_s = -m_{net}g = \zeta v\)

Reynolds number (Re)

  • Dimensionless property
  • Fluid runs around a particle
  • acceleration = \(\frac{v^2}{R}\)
  • viscous force = \(\eta\frac{v}{R^2}\)
  • Substitute into Navier-Stoke equation: \(\frac{\rho v R}{\eta} = \frac{R^2}{\eta v}f_{ext} + 1\)

Large Re

  • Dominated by inertia
  • Fluid is mixed, turbulent flow with vortices
  • Examples: human in water, rockets
  • \(f_{ext} \approx \rho \frac{v^2}{R}\)

Small Re

  • \(\frac{\rho v R}{\eta} \ll 1\), \(f_{ext} \approx \frac{\eta v}{R^2}\), drifting velocity proportional to drag force.
  • Dominated by viscous drag, laminar flow (Re < 10)
  • Acceleration and inertia term extremely small, time reversible
  • Examples: bacteria in water, dyes in corn syrup
  • Reciprocal motion does not work (due to time reversibility)
  • Periodic movement (cilia) and rotational movement (flagella) break the symmetry of the drag coef. (and thus the drag force) and create propulsion.

Motion of fluid between parallel plates in small Re

  • Applying the Navier-Stoke equation, ignoring the acceleration and body force terms. Only the pressure and the drag terms interact.
  • No slip boundary condition (velocity = 0 and the walls)
  • Parabolic flow profile
  • Flow \(\propto pr^4\)

Response of osteocyte to fluid flow

  • Bone structure
  • Cordial bone
  • Trabecular bone (spongy)
  • Bone marrow
  • Osteons: Concentric circles
  • Osteocytes: with channels connecting each other and the blood vessels
  • Osteoclast: A special macrophage removing old bones

  • Osteoblast: will become osteocytes once the surrounding mineralized

  • Fluid mechanics in the bone

  • Tension, compression
  • Difference in hydrostatic pressure
  • Fluid flow and shear stress on the osteocytes
  • Piezoelectric collagen I ?

  • Stimulates osteocytes to produce more osteopontin

  • How to separate flow shear stress and convection of nutrients

  • Increase the flow shear stress by adding the viscosity (add dextran)

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