Cellular Dynamics

Examples of Cell Migration

 Immune cells defending host

 Endothelial cells during angiogenesis

 Dermal fibroblasts during wound healing

 Tumor cell during metastasis

How do cells move

 Random motility

 Persistent random walk

 Taxis

Cell Migration Experiments

• usually take place on a protein coated surface

• a gel can be used to release chemoattractant (under-agarose assay or transient and steady-state profiles

• Time-lapse imaging (can look at single cell & population)

• single cell vs population data

• usually 2d movement

• mathematical modeling

• mean square displascement (not expected to know this for exams)

• important parameters— persistent random walk
  - root-mean square speed S
  - persistence time in a direction, P

Cell Migration Paths

Persistence in direction over short time periods

 Random direction over longer time periods

 As chemoattractant gradient increases, walk is much more directed

cell speed

The following is the sequence cells use to “walk”:

 Lamellipod extension

 Contractile force

 Relaxation

 Displacement of the cell body towards leading lamellipod

 Speed and persistence time can vary by orders of magnitude depending on cell function

Types of Taxis (don’t need to know)

 Taxis: Directed turning towardsgradient – what allows this?

 Orthokinesis: Change in movement speed as

concentration changes

 Klinokinesis: Change in turning frequency as concentration changes

 Orthotaxis: Speed is affected by movement direction

 Klinotaxis: Random turning affected by movement direction

cell flux equation exam questions will be simliar to hw questions

For orientation accuracy, what is more important?

 A – A higher concentration of ligand

 B – A higher concentration gradient of ligand

B

Orientation Accuracy

 What is more important, high concentration or gradient steepness?

 Orientation accuracy as a function of gradient steepness (same concentration)

 Orientation accuracy as a function of concentration (same relative gradient steepness)

too much binding results in biphasic response where the cell gets overwhlemed by binding too much

 What is adhesiveness?

 Ligand density * affinity

 How do you think cell speed depends on

substrate adhesiveness?

 A – Low adhesiveness is best

 B – Intermediate adhesiveness is best

 C – High adhesiveness is best

 D – Speed is independent of adhesiveness

B

Dimensional Analysis

 Can quickly determine the important aspects of problem

 Nondimensional = need all units to cancel out

 One strategy is to first arrange parameters of a phenomena to find a characteristic time constant ()

 Then compare the ratio of the time constants from different phenomena: Diffusion, Convection, Reaction

 Can compare various characteristics of system: Times, Forces

Correct Length Dimension

 Depends on the geometry of the problem

 Can have rectangular, cylindrical, or spherical coordinates

 Diffusion, convection, and reaction may or may not occur along the same dimension / direction

Reynolds Nymber

 Important in fluid flow

 Inertial Forces / Viscous Forces
Re =pVL/mu=pVD/mu

 Re < 2,300 = Laminar flow

 Most biological systems and devices

 Re > 4,000 = Turbulent flow

 Nondimensional Navier–Stokes:

Examples of Reynolds Number

Bullet, Boat, etc

 Length scale is of the

object moving through

the fluid

 Pipe, Channel, etc.

 Length scale is of the

diameter of the pipe

that the fluid flows

through

Convection

tau - L/V

Diffusion

tau=L²/D

Peclet Number (may have aquestion wehre theres an experiemnt and you have to determine if convection or difusion is dominatn)

convection/diffusion = VL/D

convection = t = L/V

diffusion = t = L²/D

Pe = (L²/D)/(L/V) =VL/D
 Pe small = Diffusion dominates (has a smaller time

constant)

 Pe large = Convection (bulk flow) dominates (has a

smaller time constant)

Damhohler Number, Da

 Reaction Rate / Convection Rate

 Reaction:tau = 1/kC..

• reaction rate ~ 1/tau

 Convection:Tau = L/v

• V = velocity

• L = length

• k = reaction rate

• C0 = initial reaction

• n = order of reaction

Da = convection/reaction?

 Da small = Reaction-limited, bulk flow is fast

 Da large = Reaction rate dominates and reaction

becomes convection-limited

Theile Modulus

 small = Reaction-limited, diffusion is fast

 large = Reaction rate dominates and reaction

becomes diffusion-limited.

What term dominates?
A ligand is diffusing in a BioMEMS device channel that is

flowing at 1 micron / s and the channel is 200 microns across

 A - Pe small, Diffusion dominates (has a smaller time constant)

 B - Pe large, Convection dominates (has a smaller time constant)

 C – Pe ~ 1, neither term dominates

C, choose an L and calculate—> convection and diffusion are equal

Time Scale of REaction

 Order of reaction, n, is the exponent of concentrationin the rate equation

 Zero order: k = k’C

 First order:kC

 Second order:kC²

 Units of the rate constant:1/((mol/L)^(n-1)s)

 Time constant of reaction: tau = 1kC..

 First order → tau ~ 1/k