Lecture 11

• Energetics & Kinetics of cellular rxns • Regional stiffness & motion – AFM : Yeast; Myocytes • Mechano-electrical coupling – Electro-mechanical coupling Homework

Free energy landscapes

• Large activation barrier is reduced by the interaction ( with a small cost of deforming E). The barrier is reduced.

Mechanical model of enzyme

• E has a binding site with a shape, charge distribution, hydrophobicity, and H-binding sites, ~matching those on the substrate. To match perfectly, S (and possibly E) must deform. One bond (spring) may stretch close to breaking point. Bond can be broken by thermal energy, stabilizing the P, that no longer fits in the enzyme.

Getting rate eqns from rxn scheme:

• 1. Each node leads to a diffEq for #molecules in the corresponding state • 2. Find all arrows impinging on a node. The time derivative of the # in this state is positive for each arrow pointing toward the node, and negative for each pointing away

E



S

 [

S

]   

ES

 

P dP E P dt



E

 

k

1 [

S

]( 1 

fraction

..

of P ES

 ( 1 

P E

) 

P

..

ES

)

time

 ..

(

k

 1

fraction

..

of

 ..

k

2 )

time P ES unoccupied

..

occupied

@

quasi

..

ss

:

P ES



k

 1 

k

1 [

S k

2  ]

k

1 [

S

]

Define

:

K v M

max  (

k

 1 

k

2

k

1 

k

2 [

E

] )

v



K M

[

S

]  [

s

] 1 

v

1

v m

( 1 

K M

[

S

] ) 1/v 1/[S]

• 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

Koster, DA et al. Nature : , 2004

TOP1B removing supercoils

Model of TOP1B

Elasticity of cells

Nano versus macro elasticity Behaviour 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?

k sp



E surface k B T

Regional Elasticity

• Motion of beads inside cells measured by mean squared

x

2 displacement. 

E

1   2 • Material stiffness, E, and Poisson’s ratio determines overall stiffness of object, the surface stiffness. From Hertzian model of continuum mechanics.

nanoscale mapping of cells

• Regional (topographic) distribution of stiffness. • AFM Cantilever must be more (or at least as) compliant than the cell, I.e. impedance matching . k lever < k cell • If k lever > k cell then no motion fidelity because cell needs to overcome cantilever stiffness before it moves.

• If k lever < k cell then OK

Measuring spring constant with AFM

• Deflection image of trapped yeast • Bud scar shown

• Height map • Deflection Map • Force map

• Mica is infinitely stiff re:cantilever, so slope is 1. • F= k lever d • To account for drift, • F*= k lever (d-d 0 ) • Neglect tip surface adhesion.

Sample Height

• Cantilever k = 0.05 +- 0.01 nN/m • Yeast C.B. k = 0.06 nN/m • Mammalian C.B. k = 0.002 nN/m • Yeast have thick cell wall, chitin • Cantilever & C.W. are 2 springs in series • Noise (rms) of combination is 0.06 nm • Resonance of free cantilever is 3.7 KHz • Resonance of PZ tube scanner is 4.5 KHz

Do cells emit sound?

• Myocytes beat in culture • Insect muscles • eg., in vivo muscle, hair cells, flagella all oscillate, @ f’s 1 to 300 Hz; Ca waves.

• Single myofibrils • Coupled molecular motors theoretically up to 10 KHz.

yeast deflection mode images:

Pelling, AE, et al. Science, 305:1147, 2004 Color represents deflection Dried cells Live cells trapped in filter

Resonance of AFM

Lngmuir 19:4539, 2003

Source of sound

 w n 2 ~ Y (Resonance) • Arrhenius plot • Similar to activation energies for molecular motors, dynein, myosin, kinesin.

• Yeast has these w

E a

 w 0

e



E a

/

RT



activation

..

energy

 58 .

15

kj

/

mole

What is the origin of the sound?

• Motion : – Active metabolic process : Azide stops ATP production by mitochondria. Does not D Y, nor morphology.

– Mechanical resonance/ Brownian

Speeds

• Speed: 3 nm X 1 kHz = 3 m m/sec • myosin 0.2 to 8 m m/sec • MT proteins : 0.02 to 7 m m/sec • Other cell activities have 10X these speeds

and forces

• Force 3 nm X 0.06 N/m = 0.2 nN • When AFM force , no D in amplitude until F > 10 nN : • 10 nN too big for a single protein • Must be many proteins coordinated

Origin of Sonocytology

• Cooperativity is common, eg., muscle, hair cells, flagella all oscillate, but @ lower f’s 1 to 300 Hz; Ca waves. • Coupled molecular motors theoretically up to 10 KHz.

• Non-invasive w/o dyes or quantum dots • Communication; pumping?

• For softer cells, need refined cantilever. • Cancer cell sound differential?

How does muscle fatigue?

• Test of a ‘skinned’ muscle fiber from EDL of rat.

• Can activate by direct stimulation of any step in the cascade. AP in T system VS activation SR Ca++ release Force Pederson, TH: Science 305: 1144, 2004

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..

• Tactile sensation in us: Pacinian corpuscles • Gating by mechanical energy • What governs the transient behaviour?

C. Elegans mechanotransduction:

Goodman, MB, Science 306, 427, 2004 • Cellular anatomy is entirely described • First animal to be genetically coded • 12 proteins mediate the response and are coded by mec genes • Knocking out MEC 2,4 & 6 abolishes the current • Allele of MEC 10 reduces it ( substitutes a glutamate for a glycine). • Insert into Xenopus oocytes

EC mechanoregulation

Skeletal Muscle Organization The Muscle Fiber

• Hundreds of molecular motors • Homologous proteins • Gene Knockouts have shown many other functions for motor proteins

Homework

• What is the average

Comparative motors

ATP SYNTHASE — A MARVELLOUS ROTARY ENGINE OF THE CELL

< previous next >

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 p /3) equivalent to free energy from ATP hydrolysis • Can see rotation by attaching an actin filament

For rotary motion: I  d 2 d t 2   M  M    w L 2 4 I 1  m  L 2 3

Nature Reviews Molecular Cell Biology 2; 669-677 (2001)

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 JE

CURRENT 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 T NATURE 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.

Current is coulombs per second. How many charges in a coulomb?

For this you need Faraday's constant 96,500 Coulombs per mole of charged molecules, in this case potassium ions.

Q K Kflux 0.24

 10 12 18 moles sec

If work, W, is done on the particle during diffusion, then the time is increased as: W kT So say W = 10 KT, then tw = 20 ms So how fast can the motor go? Assuming a back-and-forth motion it would take at least 40 ms, so the max frequency = 250 Hz or 10 nM X 250 per second = 2.5 microns per second. (linear motion).

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

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