Gravity-Dependent Physical Processes

ASEN 5016 Lecture 21: Gravity-Dependent Physical Processes

OBJECTIVES

1. Identify primary gravity-dependent and -independent forces acting at the level of cells and/or molecules

2. Analyze and utilize applications of dimensionless numbers to assess relative forces

Is it really “Zero-g” up there?

"It is a mathematical fact that the casting of this pebble from my hand alters the centre of gravity of the universe."

Thomas Carlyle

Gravity can produce two effects on an object:

- deformation (weight, structural)

and/or

- displacement (motion, transport)

Structural

- stress and strain relationships dependent on physical properties

Transport

- motion dependent on density differences

Stokes Sedimentation – "creeping flow" (no inertia)

For Re < 1 à "creeping flow" à no slip stream

Reynolds Number

Re = inertial / viscous = r L V / μ

r = density

L = length

V = velocity

μ = viscosity

Swimming bacterium (Re = 10^-7) equivalent to a human swimming through asphalt! (Purcell)

Steady state (terminal) velocity à dV/dt = 0

Review of Newton’s Second Law of Motion

· m (d²z / dt²) = 0 (no accelerated motion)

· F (grav) = [4/3 p a³ r] g

· F (bouy) = [4/3 p a³ r₀] g

· F (drag) = f (dz/dt)

à m (d²z / dt²) = F (grav) – F (bouy) – F (drag)

S F = 0 = (4/3 p a³ r g) – (4/3 p a³ r ₀ g) – (6p μ a V)

V_sed = 2/9 (ρ_cell – ρ_fluid) (g/μ) a²

V_sed = sedimentation velocity (cm/sec)

ρ_cell = density of cell

ρ_fluid = density of fluid

g = gravity acceleration = 980 cm/sec²

μ = viscosity

a = effective Stokes’ radius = (3V/4π)¹^/3

Navier-Stokes “Creeping Flow” for a sedimenting bacterial cell: V_sed ~ 0.06 microns/sec

Diffusion

Brownian Motion (or Einstein’s “random walk”)

Volume of a sphere: v = 4/3 p r³

Effective Stokes radius for cylinder: a = (3v / 4p )¹^/3

RMS distance diffused:

<x>² = 2Dt

<x> = Root Mean Square diffusion distance

t = time

D = K_bT / (6π μ a)

D = Diffusion coefficient

K_b= 1.38x10^-23 J/Molecule° K

particles > ~2-5 mm in aqueous solution à "non Brownian"

Maxwell-Boltzman Equilibrium Distribution

Defines particle concentration gradient

N_h / N_o = e ^{- [ (VΔρ
g h)/(K}_b^{T) ]}

N_h / N_o = ratio of cell concentration at height h to 0

V = cell volume

Δρ = density difference (cell-fluid)

g = gravity (9.8 m/sec²)

h = height

K_b = Boltzman constant = 1.38 x 10^-23 J / (particle ° K)

T = temperature °K (°K = °C + 273)

· N_h / N_o = 1 – colloidal

· N_h / N_o < 1 – distributed

· N_h / N_o = 0 - fully sedimented

More info and particle distribution animation…

Gradient-driven diffusion becomes more complex.

Binary diffusion (one solute concentration gradient in a solvent) is governed by Fick’s law:

-j₁ = D dc₁/dx

where -j₁ is the flux of species 1 along the x axis as a function of concentration (c) and its diffusion coefficient (D).

Bi-directional diffusion (a concentration of one particle species migrating into an opposite gradient of another) is further characterized by the Stefan-Maxwell relationship for dilute gases, and does not require designating solute or solvent:

n-1

Ñy_i = S [y_i y_j/ D_ij ] (V_j-V_i)

j=1

where V is volume average velocity, D is the relevant diffusion coefficient, and y is a mole fraction. (A parallel form of this relationship can be used for liquids, but the diffusion coefficient values are no longer the same as for the binary case.)

Dimensionless Parameters

Damköhler (Da)	max reaction rate to max transport rate
Sherwood (Sh)	total mass transport to diffusion only
Peclet (Pe)	convection to diffusion
Schmidt (Sc)	momentum to diffusion
Reynolds (Re)	momentum to drag
Grashof (Gr)	buoyancy to viscous resistance

Peclet Number à convective motion / diffusive motion

Pe = (V_sed L) / D

V_sed = sedimentation velocity (cm/sec)

D = Diffusion Coefficient (cm² / sec)

L = Characteristic Length (cm)

· if Pe > 10 à convection dominant, neglect diffusion

· if Pe < 0.1 à diffusion dominant, neglect sedimentation

for a typical cell, 0.1 < Pe < 10

· prokaryotes, Pe < 1

· eukaryotes, Pe > 1

Other Factors

Intercellular and extracellular factors

Inside the cell…

· Nucleus / cytoplasm density difference

· Microfilaments – actin, myosin, etc.

· Microtubules – tubulin, dynein, kinesin, etc.

· Intermediate filaments – cytokeratins, skin, etc.

· Shear force

Outside the cell…

Transport phenomena

· Nutrient diffusion and uptake rate

· Waste excretion and dispersion

Boundary layer issues

Electrostatic surface forces

Bulk fluid mixing

Cell motility

Other forces?

Summary

1. Characterize the physical aspects of the biological system

2. Identify all forces acting on the system (g-dependent and –independent)

3. Draw a Free Body Diagram (intra- and extracellular components)

4. Correlate observed biological responses to specific physical factors

5. Isolate cause-and-effect gravity-dependent and independent relationships

6. Establish a cascade chain of events (beginning with the ‘gravity trigger’ and ending with the observed biological response)