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
][
][
:..@
......)1(
......
)()1]([
121
1
211
][ 211
Skkk
SkP
ssquasi
occupiedtimeoffractionPP
unoccupiedtimeoffractionP
PkkPSkdt
dP
PESSE
ES
EES
E
ESESE
kkkS
)][
1(11
][
][
][
)(
:
2max
1
21
S
K
vv
sK
Sv
Ekv
k
kkK
Define
M
m
M
M
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 cellsNano 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?
Regional Elasticity
2
2
1
EE
x
Tkk
surface
Bsp
• Motion of beads inside cells measured by mean squared displacement.
• 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 . klever < kcell
• If klever > kcell then no motion fidelity because cell needs to overcome cantilever stiffness before it moves.
• If klever < kcell 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= klever d
• To account for drift,
• F*= klever (d-d0)
• Neglect tip surface adhesion. Sample Height
Def
lect
ion
• 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
Dried cells Live cells trapped in filter
Color represents deflection
Resonance of AFM
Lngmuir 19:4539, 2003
Source of sound
n ~ Y (Resonance)
• Arrhenius plot
• Similar to activation energies for molecular motors, dynein, myosin, kinesin.
• Yeast has these
molekjenergyactivationE
e
a
RTEa
/15.58..
/0
What is the origin of the sound?
• Motion :– Active metabolic process : Azide stops ATP
production by mitochondria. Does not Y, nor morphology.
– Mechanical resonance/ Brownian
Speeds
• Speed: 3 nm X 1 kHz = 3 m/sec
• myosin 0.2 to 8 m/sec
• MT proteins : 0.02 to 7 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 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.
Pederson, TH: Science 305: 1144, 2004
AP in T system
VS activation
SR Ca++ release
Force
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 TheMuscle 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 * 2/3) 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 >
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.
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).
• 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.
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