deformation

any change in distance between two material points of a continuous medium, representing the change in shape or size of an object under the effect of a force.

how do deformations help study cells?

they directly reflect how cells mechanically interact with their environment

biological examples of cell deformation

- RBCs pass through a narrow capillary must undergo strong deformation - a cell moving through a dense matrix changes its shape to slip between surrounding fibers - the cell nucleus can compress or elongate in response to internal cytoskeletal forces

biological impacts of deformations

- excessive nuclear deformations can cause nuclear envelope rupture = genomic instability - cancer cells actively modulate their ability to deform = better tissue invasion skills

deformations of cell migration

When a cell migrates through a tissue, it elongates in the direction of motion, emits protrusions (lamellipodia, filopodia), and generates traction forces via its cytoskeleton. These forces cause internal deformations visible at the cytoplasmic and nuclear levels.

material point

occurs when a material deforms, each initial point of the material is moved to a new position -> then, the displacement vector is deformed

displacement vector depends on ___

considered point; it defines a displacement field throughout the material equation : ⃗u = ⃗x − X⃗

displacement field

a vector function defined over the initial domain of the material

how can the displacement field be tracked how experimentally?

through particle tracking, fluorescence microscopy imaging, or cell contour tracking

biological example of displacement field

traction force microscopy - a cell placed on a soft gel can exert tractions on it. mark the gel with fluorescent beads and track the displacements of the beads between the initial and deformed stations

mathematical modelling of traction force microscopy

A bead is initially at position X⃗ = (10 μm, 5 μm) in a gel. After traction by a cell, it is observed at ⃗x = (12 μm, 6.5 μm). The displacement vector is therefore 2 μm horizontally and 1.5 μm vertically.

rigid displacement vs deformation

relative change between the object's points

uniaxial elongation deformation

an object is stretched in one direction.

biological examples of uniaxial elongation

- a cell pulled between two micropipettes - an actin fiber under tension.

compression deformation

shortening in one direction.

biological examples of compression deformation

nucleus compressed in a microchannel

shear deformation

parallel sliding of layers

biological examples of shear compression

a cell subjected to fluid flow or on a soft substrate.

deformation in biology

— Epithelial cells elongate during morphogenetic processes. — The cytoskeleton undergoes internal shear during migration. — Living tissues can be locally compressed by surrounding forces.

small strain tensor

denoted ε. used to describe the local deformation of a material by expanding the displacement field around a given point

small, shape variation displacements can be approximated by ___

first derivatives of displacement

symmetric tensor

measures the relative variation of distance between two neighboring points of the material.

elongations or contractions are measured with

diagonal components (εxx, εyy, εzz) along the axes

shear deformation is measured with

off-diagonal components (εxy), signifies changes in angles between initial axes

when a cell migrates on a soft substrate...

it can locally stretch it - εxx > 0

a compressed cell nucleus experiences contraction in several directions

εii < 0

X

initial position coordinates

x

after deformation coordinates

small strain tensor is symmetric, true or false?

true

small strain tensors represent...

the shape variation of the material

deformation gradient tensor

signified by F describes how a small material volume transforms

if F = I

there is no deformation

if F does not = I

there is a significant local deformation of the material

biological examples of large deformation tensor

In tumors or embryonic tissues, cells exert significant traction on their environment (e.g., collagen, extracellular matrix). Large deformations of these matrices modify : — the distribution of forces around cells ; — the local tissue structure ; — the mechanical response of the cell via mechanosensitive sensors.

linear framework is no longer valid when

displacements are large

deformation gradient tensor describes ___

stretching, shear, and rotation of a small volume

deformation gradient tensor can be applied to

cell migration, growth, tissue remodeling

deformation invariants

certain quantities derived from the tensor that remain unchanged

when reference frame changes, ____ changes

tensor components

trace as a common invariant of small strain tensor

trace -> tr(ε) = εxx + εyy + εzz represents the relative volumetric deformation positive = expansion, negative = compression

determinant of F (large deformations) as a common invariant of small strain tensor

det(F) = local volume change ratio if det(F) > 1, the material has expanded if det(F) < 1, the material has contracted

biological example of strain tensor invariants

During an osmotic shock, a cell swells or shrinks depending on the concentration gradient. The volume change can be described by : — the trace of ε if deformations are small ; — or det(F) if deformations are large.

what allows quantification of cytoplasmic or nuclear swelling from images?

defining trace and the determinant of F in the small strain tensor

invariants are dependent or independent on the chosen reference frame?

independent

the trace of ε measures what?

the volume change (for small deformations)

the determinant of F gives what?

local volumetric enlargement factor (non linear case)

biological applications of invariants?

osmosis nuclear swelling growth