Why use mathematical modlling to study cells?

cells move in a complex environment

Mathematical modeling can be used to visualize cell

Migration Adhesion Traction Plasticity

Math modeling can be used to understand how cells react to their mechanical environment like:

With cellular rigidity, stress, and compression

Math modeling can also be used to understand how cells transmit forces on other cells like in:

Tissue (development, etc.) Embryogenesis Cancer

What three tools are necessary to describe cellular phenomena quantitatively?

vectors fields derivatives

Vectors are used to study

direction

Biological application of vectors

detect the direction in which the concentration increases

Forces are used to understand

mechanical forces and how they are generated

Gradients are used to understand

how something is polarized in the direction of a vector

Scalars quantify

simple values

Vectors quantify

oriented values

Fields quantify

values distributed in space

Scalars (s) do not depend on

direction in space

Examples of scalar

temperature mass molecular concentration pressure

Biological examples of scalar

Concentration of glucose around a cell Intracellular pressure Light intensity received by a photoreceptor

Vectors have

direction, sense, and a magnitude

Vectors are represented by

an arrow in space

Vectors are defined by its components

V1, V2, V3 in an orthonormal basis (E1, E2, E3)

The direction of a vector is that of

the line passing through both endpoints

The sense of a vector is from

the origin to the endpoint or vice versa

biological examples of a vector

force exerted by a cell on the matrix migration speed of a cell growth direction of an axon

A field is

a function that assigns a quantity (scalar or vector) to every point in space

Scalar field

c(⃗x) gives a scalar at each position ⃗x (e.g., temperature, concentration).

Vector field

⃗v(⃗x) gives a vector at each position ⃗x (e.g., fluid velocity, force).

Dot product of two vectors

⃗u · ⃗v = ∥⃗u∥ ∥⃗v∥ cos(θ)

θ in a dot product of two vectors

is the angle between them

The dot product

measures how "aligned" two vectors are

If two vectors point in the same direction, the dot product is

maximal

If two vectors are orthogonal, the dot product is

zero

Biological example of a dot product

The mechanical work done by a cell is proportional to the dot product between the force vector it exerts on the substrate and the resulting displacement vector.

dot product formula in 2D coordinates

⃗u · ⃗v = uxvx + uyvy

the cross product of two vectors is

a vector perpendicular to the plane containing ⃗u and⃗ v

the cross product is useful for interpreting

torques or moments

biological application of cross products

a cell exerts a force (microneedle) that causes an object to rotate

torque of a cell exerting a force can be modeled

by a cross product between the lever arm and the force

the tensor/outer product of two vectors is denoted by

⃗u ⊗ ⃗v

tensor products generate

matrices

tensor products describe

how one vector acts on another to transform directions into new ones

it is fundamental for defining quantities in mechanics such as

stress tensor deformation tensor

tensor products are NOT commutative

⃗u ⊗ ⃗v = ⃗v ⊗ ⃗u.

in cell mechanics, the stress tensor can be expressed as:

the sum of tensor products between force and direction vectors

biological application of tensor products

actin filaments under tension generate anisotropic stresses that can be represented by σ ∝ f ⊗⃗n, where ⃗ f is the force per filament and ⃗n the filament orientation.

dot product vs. cross product

dot product measures alignment between two vectors cross product quantifies rotational effects (torque)

how are matrices used to represent vectors

to represent linear transformations between vectors

matrix

an array of numbers arranged in rows and columns

matrices can represent

a linear transformation (rotation, stretching, shearing) a table of coefficients linking two quantities (stress, strain) a field of values (in the case of tensors)

matrices can be:

added multiplied by a scalar multiplied by a vector

condition to be able to compute the multiplication between matrices

matrix A has p columns matrix B must have p rows

transpose/symmetry

the diagonals can be transposed if A = At, then the matrix is symmetrical

biological example of matrix transposition / symmetry

the stress tensor in an elastic medium (such as the cytoskeleton) is a symmetric matrix -> forces are the same on both sides of a plane

biological example of identity and inverse matrices

when a cell undergoes a deformation, one can look for the inverse transformation that would return it to its original shape (passive elasticity)

matrices encode

stretching, shearing, or rotation effects

differential operators such as the gradient or divergence quantify

fields such as temperature, concentration, deformation

gradient of a scalar field

if f(x,y,z) is a scalar of molecular concentration... the gradient of f is a vector indicating in each direction how f varies locally

biological application of gradient of a scalar field

If f(x, y) represents a chemokine concentration, then ∇f indicates the direction in which the cell must move to “climb” the gradient

biological application of gradient of a scalar field pt. II

gradient of a growth factor concentration guides endothelial cells during angiogenesis

jacobian matrix

gradient of a vector field

biological interpretation of gradient of a vector field

this gradient encodes the local variation of the velocity field useful for understanding cytoplasmic deformation or stress within the actin network.

divergence of a vector field measures

outgoing flux of the field at a given point

is divergence of a vector a scalar, true or false?

true

biological interpretation of divergence of a vector

If ⃗v is a velocity field within the cytoplasm, then the divergence indicates whether there is dilation (> 0) or compression (< 0).

biological example of divergence of a matrix

The divergence of the stress tensor in the cytoskeleton gives the internal force field (traction force microscopy)

gradient with respect to divergence

gradient applies a derivative that increases the rank of objects (scalar → vector, vector → matrix).

divergence with respect to gradient

divergence applies an operation that decreases the rank (vector → scalar, matrix → vector).

what is a gradient?

measures how a quantity changes locally.

what is divergence?

measures how much of that quantity “flows out” from a point.

tensors

a mathematical object that represents physical quantities in different coordinate systems.

tensors are used to

generalize the notions of scalars and vectors model physical phenomena within the cell (mechanical stresses, deformations, fluxes)

what does a tensors' particularity allow for?

it transforms in a well-defined way when the reference frame changes.

tensor order 0

scalar, represented: T example: temperature, pressure

tensor order 1

vector example: cell velocity

tensor order 2

tensor example: stress tensor

biological example of tensors

necessary to understand how cells deform, exert forces, or mechanically respond to their environment (especially those of order 2)

what does a vector lack?

ability to describe complex directional effects such as: the distribution of forces on a cell surface local deformation state of the cytoplasm elasticity or viscosity of the nucleus

quantities the depend on the action and the application require what order of tensor?

second-order

tensors generalize:

scalars (rank 0) and vectors (rank 1)

second order tensors are represented by

matrices