Why use mathematical modlling to study cells?
cells move in a complex environment
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| Term | Definition |
|---|---|
| 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 |