Lecture 10• Background for cell propulsion• Fluid dynamics• Enzyme kinetics • How do animals swim?: • 1. pushing fluid backward by limb action; • 2. pushing fluid forward by resistance of

body.• I.e fish starting from release will accelerate

until the backward & forward momentum (of the fluid) balance. Viscosity is only significant at the boundary layer.

Cell Propulsion• Small scale phenomenon: slow velocities

driven by surface forces: pressure and viscous stress. Fluid resistance is significant, and balances propulsive force.

• Motion of a body depends on the ratio of viscous and inertial effects: Reynold’s number: Small for cells, large for almost all animals. Cellular world is ruled by friction.

• Reynold’s number quantifies the relative magnitudes of frictional and inertial forces

/

/

/

2L

UL

vR

Cellular Motors

• Molecular motors must move (swim) in fluids, where most of the work is dissipated

• What forces must they overcome?

• Where do the motors get their fuel?

• How do they exhaust spent fuel?

• What is the efficiency?

Creature R

Bacteria 10-4

Spermatozoa 10-2

Flying Insects

Birds

104

105

Oscillatory muscles

Synchronous Asynchronous

Stretch activation

Stretch- activated currents

Sliding filamentds

Myosin

• 5.3 pN for each myosin molecule

• 100 molecules per filament.

• Each filament has c.s.a. of 1.8 X 10 –15 m2 in the relaxed muscle.

Strain in solids and fluids

Gd

z

A

f)(

fA

d

d

v

A

f

Sample fluid properties

Fluid m (kg m-3) Pa-S fcrit (N)

Air 1 2 X 10-5 4 X 10-10

Water 1000 0.0009 8 X 10-10

Olive Oil 900 0.08 7 X 10-6

Glycerine 1300 1 0.0008Corn Syrup 1000 5 0.03

When f > fcrit- inertial forces dominate

Swimming: is it worth it?

• Cilium with velocity, v, length, d, time scale:

• Diffusion time scale :

• Swimming time, ts should be < tD

Ddt

vdt

D

s

/

/2

d

Dv

Viscous flow

• Newtonian fluids are isotropic

• What is a viscous fluid?

• When f< fcrit

/

/

2

crit

o

f

dAvf vo fA

d

Shear

Planar geometry

• I.e., 1 m cilium, D = 10-5 cm2/sec,

• so v> 103 m /sec:

• stirring and swimming is not energetically favorable for nutrition.

Comparative motors

ATP SYNTHASE — A MARVELLOUS ROTARY ENGINE OF THE CELL < previous  next >

Rotary Cellular Motors• The rotary mechanism of ATP synthase , Stock D, Gibbons C, Arechaga I,

Leslie AGW, Walker JECURRENT OPINION IN STRUCTURAL BIOLOGY ,10 (6): 672-679 DEC 2000

•  • 2. ATP synthase - A marvellous rotary engine of the cell, Yoshida M,

Muneyuki E, Hisabori TNATURE REVIEWS MOLECULAR CELL BIOLOGY 2 (9): 669-677 SEP 2001

•  • 3. The gamma subunit in chloroplast F-1-ATPase can rotate in a

unidirectional and counter-clockwise manner Hisabori T, Kondoh A, Yoshida M FEBS LETTERS 463 (1-2): 35-38 DEC 10 1999

•  • 4. Constructing nanomechanical devices powered by biomolecular motors.C.

Montemagno, G Bachand, Nanotechnology 10: 225-2312, 1999.

ATP SYNTHASE — A MARVELLOUS ROTARY ENGINE OF THE CELL < previous  next >

F1 ATPase: A rotary motor

• Can either make or break ATP, hence is reversible

• Torque of 40 pN-nM; work in 1/3 rev. is 80 pn-nM (40 * 2/3) equivalent to free energy from ATP hydrolysis

• Can see rotation by attaching an actin filament

For rotary motion:

I2t

d

d

2 M M

w L2

4

I1

3m L2

Nature Reviews Molecular Cell Biology 2; 669-677 (2001)ATP SYNTHASE — A MARVELLOUS ROTARY ENGINE OF THE CELL

< previous  next >

Current is coulombs per second. How many charges in a coulomb?For this you need Faraday's constant 96,500 Coulombs per mole ofcharged molecules, in this case potassium ions.

Q K Kflux0.24

96 50010 12 2.5 10 18 moles

sec

If work, W, is done on the particle during diffusion, then the time is increased as:

So say W = 10 KT, then tw = 20 ms t w t d e

W

kT

So how fast can the motor go? Assuming a back-and-forth motionit would take at least 40 ms, so the max frequency = 250 Hz or10 nM X 250 per second = 2.5 microns per second. (linear motion).

Elasticity

Nano versus macro elasticityBehaviour relative to kT: Stretch a rubber band and a

string of paper clips. Significant for The nanometer-scale monomers of a

macromolecule, but not for a string of paper clips. The retracting force exerted by a stretched rubber band is entropic. It increases disorder.

Do most polymers have persistence lengths longer than their total (contour) length?

• When L>> the chain has many bends and is always crumpled in solution – the FJC model applies, with each link approximated as 2 and perfectly flexible joints.

• To count all possible curved states in a smooth-bending rod in solution- it’s a WLC- supercoiling is possible.

• Promoters have different abilities to uncoil

• Twisting DNA torsional buckling instability

• Unwinding and causes local denaturation

• Many motors are needed: RNA plymerase, DNA polymerase: 100 nucleotides/sec.

• Forces (pN) can stop transcription

Mechano - regulation

• Growth, proliferation, protein synthesis, gene expression, homeostasis.

• Transduction process- how?• Single cells do not provide enough material. • MTC can perturb ~ 30,000 cells and is

limited.• MTS is more versatile- more cells, longer

periods, varied waveforms..

Markov Chains

• A dynamic model describing random movement over time of some activity

• Future state can be predicted based on current probability and the transition matrix

Transition Probabilities

Win Lose

Win 3/4 1/2

Lose 1/4 1/2

Sum 1 1

Today’s Game Outcome

Tom

orro

w’s

Gam

e O

utco

me

Need a P forToday’s game

Grades Transition Matrix

11Sum

1/21/4Bad

1/23/4Good

BadGood

This Semester

Nex

t S

emes

ter

Grade Tendencies To predict future:

Start with now:What are the gradeprobabilities for thissemester?

WinLose

1/4

1/2

1/23/4

16/54/32/14/34/1

16/114/12/14/34/3

4/1

4/3

2/14/1

2/14/3

1,

1,

2221

1211

1

ilose

iwin

i

ii

P

P

P

aa

aaA

APP

Markov Chain

Intial ProbabilitySet independently

Computing Markov Chains

% A is the transition probability

A= [.75 .5

.25 .5]

% P is starting Probability

P=[.1

.9]

for i = 1:20

P(:,i+1)=A*P(:,i)

end