# Intro to mechanobiology

Note about Intro to mechanobiology.

## Course information

- Time: Mon. 2, 3, 4 (9:10 ~ 12:10)
- Location: 工學院綜合大樓 213
- Lecturers: 趙本秀 ＆ 郭柏齡
- Textbooks:
- Introduction to cell mechanics and mechanobiology by Jacobs et al

- Ceiba for course outlines
- Grading:
- 30% HW
- 30% Quiz
- 30% Oral presentation
- 10% Final report

- G drive: https://drive.google.com/drive/u/1/folders/1Io2UKPcdtmzvJyWskBetsR5Zh_bLpffi

## 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 terbulent 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 stifness inside cartilage (from synovial surface to bone surface)

- tension, compression, fluid flow (hydrodynamic pressure, shear stress etc.), osmotic pressure, ion current

## Clinical relevence of machanobiology

- deafness: cochlear hair cells
- arteriosclerosis: endothelial and smooth muscle cells
- muscular dystropy and cardiomyopathy: myocytes and
*fibroblasts*- Congenital muscular dystropy 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 glucoma: 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 stuctures: 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 informations (e.g. messenger RNA)

## Information flow in machanobiology

**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 environmentby motor proteins and cyctoskeletons - determination of external physical cues (e.g. stifness) - cell growth and differentiation

### Lipid rafts in the plasma membrane

- Rich in cholesterol (less fluidity) and glycolipids
- Rich in cytoskeleton anchor complexes and mechanoreceptors

### 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 leakge 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): extrememly hydrophilic, providing compressive strength
- fibrous proteins (e.g. collagen): tensile strength

- Example of neonatal rat cardiomyocyte growth and developement
- Soft surface (100~300 Pa): round and undifferentiated
- Native environment in the heart (10 kPa): cylindral with the best aspect ratio (7:1) with sarcomere
- Stiff surfaces (glass): flat and polygonal

### Crosstalk of cell-cell junctions and cell-matrix junctions

- Cadherin and integrin pathways
- Cellular movement, differentialtion, and growth

## Crtilage tissue structure and homostasis

- chondrocyte and ECM interactions
- influenced by physical forces (pressure, shear stress)
- Spatial heterogeniety of ECM compostion (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 grug targets
- Enzyme-linked: e.g. EGFRs, JAK-STAT

### Relays and amplifiers:

- second messengers (Ca, IP3, DAG, cAMP, …), kinases 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 dissciate 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.

### Machanoreceptors and transduction

- Stretch-activated ion channels
- Integrins
- E-cadherins e.g. fluid shear stress -> TF (β-catenin) to nucleus
- Phisical 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
- Machano-sensing 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 chrmatin structure

# Solid Mechanics Primer

A crawing cell uses pseudood and forward attachment point to move forward.

## Rigid body approach

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

## Deformable cell

- Displacement field ($\Delta x$ indside the cell) is not uniform
- Displacement is related to mechanical peroperties (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 (dimless)
- 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 equlibrium (assuming littel 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 certian 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 indep. strains

## Consitutive 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 equlibrium: 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 (viscosty) 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
- Labellar flow: no inertia or time dependent terms involved

### Force balance inside a fluid element

- Pressure (nomal stress)
- Friction, viscous force

### General assumptions

- Steady flow: force balanced
- Newtonian fluid: viscosity deos not depnedent on shear rate
- Imcompressible: 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 adjecent 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 colliod 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 = $\eta\frac{v}{R^2}$

- Substitude 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

- Cortial bone
- Trabecular bone (spongy)
- Bone marrow
- Osteons: Concentric circles
- Osteocytes: with channels connecting each other and the blood vessels
- Osteoclast: A special microphage 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
- Peizoelectric 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 adjecent 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 colliod 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 = $\eta\frac{v}{R^2}$

- Substitude 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

- Cortial bone
- Trabecular bone (spongy)
- Bone marrow
- Osteons: Concentric circles
- Osteocytes: with channels connecting each other and the blood vessels
- Osteoclast: A special microphage 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
- Peizoelectric 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)