Week 4: Motility

Question 1

What is the definition of viscosity?

Answer 1

The constant of proportionality between the shear stress and the velocity gradient.

Question 2

What is shear stress?

Answer 2

F/A

F = force
A = area

Question 3

What is the velocity gradient?

Answer 3

v/d

v = velocity
d = plate separation

Question 4

What is the equation for viscosity relating shear stress and the velocity gradient?

Answer 4

F/A = viscosity * v/d

Question 5

Draw a diagram representing all the components used to find viscosity.

Answer 5

(21)

Question 6

What is the equation for viscosity?

Answer 6

n = K(pb - ps)t

n = dynamic viscosity
K = proportionality constant
pb = ball density
ps = sample density
t = ball rolling time

Question 7

How do particles at small length scales move? Why do they move like this?

Answer 7

At small length scales thermal energies are sufficient to move all particles in a random manner?

Question 8

How id the average kinetic energy related to temperature?

Answer 8

<0.5mv^2> = kT/2

Question 9

How does the RMS velocity relate to temperature?

Answer 9

<v^2>^0.5 = (kT/m)^0.5

Question 10

What is the equation for mean square displacement for microscopic translational diffusion?

Answer 10

<r^2> = 2nDt

n = number of dimensions
r = displacement
<r^2> = mean square displacement
D = diffusion coefficient
t = time

Question 11

What is the fluctuation dissipation theorem?

Answer 11

D =kT/f

kT = thermal energy
D = diffusion coefficient
f = frictional coefficient

Question 12

What is Stokes’ relation for a rigid sphere?

Answer 12

f = 6pina

n = solvent’s viscosity
a = radius of the spherical particle

Question 13

What is the Stokes-Einstein equation?

Answer 13

D = kT/(6pin*a)

Question 14

What is Fick’s first law?

Answer 14

The net flux of particles is proportional to the slope of the concentration function.

(22)

Question 15

What is Fick’s second law?

Answer 15

The time rate if change in concentration is proportional to the curvature of the concentration function.

(23)

Question 16

How does Fick’s second law behave?

Answer 16

The concentration has a Gaussian profile for an initial delta function concentration at the origin.

Question 17

What is the function for concentration from Fick’s second law?

Answer 17

(24)

Question 18

What is the expected result for the microscopic mean square displacement?

Answer 18

(25)

Question 19

When does laminar flow occur?

Answer 19

At high viscosities.

Low Reynolds numbers.

Question 20

When does turbulent flow ocur?

Answer 20

At low viscosities.

Hugh Reynolds numbers.

Question 21

What is Newtons law of motion in terms of external and frictional forces?

Answer 21

f(ext) + f(fric) = f(tot) = mass * acceleration

Question 22

What is the inertial term in Newton’s law of motion?

Answer 22

mass * acceleration

Question 23

What is the Reynolds number?

Answer 23

The ratio of the inertial force to the frictional force.

Question 24

What is the formula for the Reynolds number?

Answer 24

(26)

Question 25

What happens in a liquid when Reynold's number is small?

Answer 25

Friction dominates. Stirring produces the least possible response (laminar flow). The flow stops immediately after the external force stops.

Question 26

What happens in a liquid when Reynold's number is large?

Answer 26

Inertial effects dominate. The coffee keeps swirling after you stop stirring and the flow is turbulent.

Question 27

What is the scallop theorem?

Answer 27

A hypothetical microscopic swimmer trying to make progress by cycling between forward and backward strokes of its paddles. On the first stroke the paddles move backwards relative to the body at relative speed v, propelling the body backwards at speed u. On the second stroke the paddles move forward at relative speed v', propelling the body backwards at speed u'. The cycle repeats. The progress made on the first stroke is all lost on the second stroke. Reciprocal motion like this cannot give net progress in Low-Re number fluid mechanics.

Question 28

What is the Navier-Stokes equation for an incompressible fluid?

Answer 28

(27)

Question 29

If Re << 1, what can we do to the Navier-Stokes equation?

Answer 29

Time does not matter so we can neglect the inertial terms. The pattern of motion is the same, whether slow or fast, whether forward of backwards in time.

Question 30

Give a brief explanation as to how organisms move at low Reynold's numbers? Give two examples of organisms that do this.

Answer 30

they need non-reciprocal motion. Two examples are: 1) Cilia - used with microorganisms such as paramecium. 2) Flagellae - used on bacteria.

Question 31

With use of a diagram, explain how cilia and falgellae help organisms move through low Reynold's numbers.

Answer 31

(28) A rod dragged along its axis at velocity v feels a resisting force proportional to -v. Similarly a rod ragged perpendicular to its axis feels a resisting force also proportional to -v. However, the viscous friction coefficient for motion parallel to the axis is smaller than for perpendicular motion. The motion of the fluid created by a power stroke of a cilium is only partly undone by the backflow created by the recovery stroke.

Question 32

Consider a particle travelling at velocity v0. Derive an equation for the distance it will travel before coming to a rest.

Answer 32

(29)

Question 33

Give examples of microorganisms that crawl/swim.

Answer 33

Firbroblast Nematode sperm Neurophil Amoeba bacterium Paramecium

Question 34

Give examples of microorganisms that travel through extension and contraction.

Answer 34

Filopodium Acrosome Lamellipodium Cilium Spasmoneme Muscle

Question 35

Give examples of microorganisms that travel via internal transport.

Answer 35

Axon, slow Axon, fast Anaphase Axopodium Nitella Chromatophore Physarum Listeria

Question 36

What is the formula for delta G in transmembrane diffusion?

Answer 36

(30)

Question 37

What is the Flux or net rate of transport for transmembrane diffusion?

Answer 37

J = D(C2-C1)/I I = membrane thickness

Question 38

What is the equation for Normal Brownian lateral diffusion?

Answer 38

(31)

Question 39

What is the equation for anomalous lateral diffusion?

Answer 39

(32)

Question 40

How does anomalous lateral diffusion differ from normal Brownian lateral diffusion?

Answer 40

Slower diffusion, treated in viscous medium. Lipid microdomains, membrane proteins, two broad D distributions, interactions with support or skeletons. Obstacles or inhomogeneity constraining D measurements.

Question 41

What equation is used to measure D?

Answer 41

(33)

Question 42

What is the result of the beam waist being too small when measuring D?

Answer 42

D is underestimated.

Question 43

How is D measured? Use a diagram to help.

Answer 43

Laser beam is fed into the sample, with w being the radius of the beam waist as intensity is dropped to e^-2. Measure intensity versus time. Data fitted by correlation function formulated on the 2D fluorescence model. (34)

Question 44

How can viscosity be measured for a given fluid?

Answer 44

The classical approach is to use a U-tube or a capillary with a falling ball inside. In the latter case, the adjustable tilting angles enables us to control the time for the ball to fall, making the measurements cover a wide range of viscosities.

Question 45

Explain what Reynold's number implicates?

Answer 45

The ratio of the inertial force to the frictional force gives the Reynold's number.

Question 46

What are the main factors that cause anomalous membrane diffusion?

Answer 46

Lipid rafts or microdomains. Membrane proteins or channels. Environmental factors that affect membrane morphologies Phase behaviour such as T, pH, ions