molecular machines and enzymestlusty/courses/livingmatter/...molecular machines and enzymes lecture...
TRANSCRIPT
![Page 1: Molecular Machines and Enzymestlusty/courses/LivingMatter/...Molecular Machines and Enzymes Lecture 10: Tsvi Tlusty, tsvi@unist.ac.kr Sources: Nelson –Biological Physics (CH.10)Julicher,](https://reader030.vdocument.in/reader030/viewer/2022040818/5e63c1b4e43f3c4b524c0993/html5/thumbnails/1.jpg)
Molecular Machines
and Enzymes
Lecture 10:
Tsvi Tlusty, [email protected]
Sources:
Nelson – Biological Physics (CH.10)
Julicher, Ajdari, Prost – Modeling molecular motors (RMP 1997)
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Outline
I. Survey of molecular devices in cells.
II. Mechanical machines.
III. Molecular implementation of mechanical principles.
IV. Kinetics of enzymes and molecular machines.
Biological Q: How do molecular machines convert scalar chemical energy
into directed motion, a vector?
Physical idea: Mechano-chemical coupling by random walk of the motor
on a free energy landscape defined by the geometry of the motor.
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I. Survey of molecular devices in cells
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Artificial and natural molecular machines
• A molecular machine, or a nanomachine: a molecule or a an
assembly of a few molecules, that performs mechanical
movements as a consequence of specific stimuli.
The Nobel Prize in Chemistry 2016:
Sauvage, Stoddart and Feringa
"for the design and synthesis of molecular machines".
The most complex molecular machines
are proteins found within cells.
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Classification of molecular devices in cells
(A) Enzymes: biological catalysts that enhance reaction rate.
(B) Machines: reverse the natural flow of a chemical or
mechanical process by coupling it to another process.
• “One-shot” machines: use free energy for a single motion.
• Cyclic machines: repeat motion cycles. Process some source of free
energy such as food molecules, absorbed sunlight, concentration
difference across a membrane, electrostatic potential difference.
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• “One-shot” machines
• Cyclic machines
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Cyclic machines are essential to Life
• Motors: transduce free energy into directional motion
(case study: kinesin)
• Pumps: transduce free energy to create concentration gradients
across membranes (ion channels).
• Synthases: transduce free energy to drive a chemical reaction and
then synthesize some products (ATP synthase).
Let’s look at a few representative classes of the molecular devices.
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(A) Enzymes enhance chemical reactions
• Even if ∆G<0, a reaction might still be very slow due energetic barriers.
• Good for energy storage, but how to release the energy when needed?
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Enzymes display saturation kinetics
• Most enzymes are made of proteins.
• Ribozymes consist of RNA.
• Ribosome: complex of proteins with RNA.
𝑉 =[𝑆]
[𝑆] + 𝐾𝑀𝑉max
speedup by 1012
Vmax=107 molec/sec
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(B) One-shot motors assist in cell
locomotion and spatial organization
● Translocation
Several mechanisms can rectify (make
unidirectional) the diffusive motion of
protein through the pore.
●Polymerization
Actin polymerization from one end of an
actin bundle provides force to propel a
Listeria bacterium through the cell surface
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All eukaryotic cells
contain cyclic motors
• Molecular forces were measured
by single molecule experiment.
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Some important cyclic machines
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II. Mechanical machines
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Macroscopic machines can be
described by an energy landscape
load
“motor”U = − w Rmoto
r
2
perfect
axis
The movement of macroscopic machines is
totally determined by the energy landscape.
1 2totU w R w R
2motorU w R
1loadU w R
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Overdamping macroscopic machines
cannot step past energy barriers
load
“motor”
U = − w Rmoto
r
2
perfect
axistot motor loadU U U
2motorU w R
1loadU w R
imperfect axis
(e.g. friction)
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Some machines have more degrees of freedom
• The angles are constrained to “valleys.
• Slipping: imperfections (e.g. irregular
tooth) may lead to hopping.
Gears: the angular variables
both decrease as w2 lifts w1.
• The machine stops if:
– w2 = w1 .
– Slipping rate too large.
constraint: 2n
N
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Microscopic machines can step past energy barriers
• A nanometric rod makes a one-way trip to the
right driven by random thermal fluctuations.
• Moving to the left is impossible by sliding bolts,
which can move down to allow rightward motion;
• It can move against external "load" f.
• Where does the work against f comes from?
• What happens if one makes the shaft into a circle?
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Can one extract work from thermal fluctuations?
• Rod cannot move at all unless 𝜀~𝑘𝐵 𝑇.
• But then bolts will spontaneously retract
from time to time…
• Leftward thermal kick at just such a
moment and rod steps leftwards.
• Where does the work against f comes from?
• What happens if one makes the shaft into a circle?
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Can we save the second law of thermodynamics?
• Bolts are tied down on the left side,
then released as they emerge on the right.
• Where does the work against f comes from?
• What happens if one makes the shaft into a circle?
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Can we save the second law of thermodynamics?
• Where does the work against f comes from?
-- Potential energy stored in the compressed
springs on the left.
• What happens if one makes the shaft into a circle?
-- All springs are released and the motion stops.
This is a toy model for model
for molecular translocation
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Q: Can the small load be lifted if T1 =T2 =T ?
A: May look quite possible, but 2nd law of thermodynamics:
(Heat cannot be converted to work spontaneously) prohibits this.
Pawl
φ T1
T2
Ratchet
Load
Spring
Feynman’s ratchet and pawl
φ
total ratchetU U MR
MR
ratchetU
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• load = 0:
• But also
no net rotations.
• Q: what if load > 0?
• A: work heat
1
pawl pulled over tooth=exp
by vanesB
PTk
2
pawl jumps=exp
spontaneouslyB
Pk T
pawl is up and wheel enough energy Rate Rate
can turn backwards freely to turn wheel forward
Pawl
φ T1
T2
Ratchet
Load
Spring
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Ratchet can be made reversible if T1 >T2
• Forward rotation:
– Heat from bath 1:
• Backward rotation:
– Heat from bath 2:
• Reversibility requires detailed balance
• The efficiency is
forward 0
1
expB
MRR
k T
1 work pawl Q MR MR
backward 0
2
expB
Rk T
2 work realeased , to the vanes Q MR MR
1 2forward backward
1 2
Q QR R
T T
1 2
1 1
2
1
1 maximal (Carnot)
Q QMR
Q Q
T
T
φ
total ratchetU U MR
MR
2
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Ratchet is irreversible if T1=T2=T
• Forward rotation:
• Backward rotation:
Ratchet moves backward
forward 0
1
expB
MRR
k T
backward 0
2
expB
Rk T
φ
total ratchetU U MR
MR
2
forward backward 0 exp exp 1B B
d MRR R
dt k T k T
ω
M
ωω
T
1 BMR k T
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Summary: necessary conditions for directional motion
• Asymmetric potential:
• Break detailed balance:
U
Pawl
φ
Ratchet
Load
Spring 1T
2T
1 2T T
forward backwardR R
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Low load:
forward
High load:
backward
Modeling molecular machines:
qualitative picture
x
L
/f L /f L
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Modeling molecular machines:
qualitative picture
• To move right need a thermal fluctuation of
• If then the leftwards motion is negligible.
• If rod diffuses freely between steps with
• What happens when
x
L
E f L
Bk T
0f
2
step
step
and velocity L L L
t vD t D
0?f
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x
L
Mathematical framework: Smoluchowski equation
• Think of large set of identical rods:
• What is the P(x,t) = prob. ratchet is in (x,x+dx)?
• Ratchet diffuse according to Fick’s law:
• Rods drift in the force field:
• The total current is:
D
Pj D
x
v
B
D Uj P
k T x
B
D
k T
Einstein’s relation
D v
B
U D Uj j j D P
x k T x
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x
L
Fokker-Planck and Smoluchowski equations
• Number conservation (continuity equation):
• With current :
Fokker-Planck equation
• At steady-state this is Smoluchowski’s equation
D v
B
U D Uj j j D P
x k T x
0j P
x t
0B
P P P UD
t x x k T x
0 const.B
P P P Uj D
t x k T x
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x
L
Steady-state is solution to Smoluchowski equation
• No net current:
• Gives equilibrium Boltzmann distribution:
0B
P P Uj D
x k T x
1 ( )( , ) exp .
B B
P U U xP x t
P k T Z k T
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x
L
The general steady state solution
• Steady-state
const.B
P P Uj D
x k T x
0 0
( )Define ( ) ( ) ( ) where ( ) exp
B
U xP x P x p x P x
k T
0
( )exp
B B
P P U p j p j U xP
x k T x x D x D k T
0
0
( )( ) exp
( ) ( )( ) exp exp
x
B
x
B B
j U yp x C dy
D k T
U x j U yP x C dy
k T D k T
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x
L
Solving the molecular ratchet
• Steady-state
• Sawtooth potential
• “Perfect” ratchet – because S-S bond strong
• Therefore: (1) 0 protein moves rightwards.
(2) ( ) 0 because it cannot jump back.
j
P L
0
0
( )( ) exp
( ) ( )( ) exp exp
x
B
x
B B
j U yp x C dy
D k T
U x j U yP x C dy
k T D k T
( ) for 0 xU x f x L
Bk T
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x
L
Solving the molecular ratchet
• Let’s scale the variables:
• Boundary condition
; ; ;B B B
f x f Lu F
k T k T k T
0 0
( )( ) exp e 1
( ) e ( ) where B
x u
u u
B
u u
j U y j L j Lp x C dy C du C e
D k T D F D F
j LP x p x C B e B
D F
( ) 0
( ) 1
F F
F u
P L C B e B C B Be
P x B e
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x
L
• Normalization in [0,L]
• Time to move on bolt
• Average speed
; ; ;B B B
f x f Lu F
k T k T k T
2
0 0
2
( ) 1 1 1
1 1
L F
F u
F
j LP x dx e du
D F
j Le F
D F
2
2 1F
D Fj
L e F
2
2
1 1FL e F
j D F
2
1F
L D Fv j L
L e F
v
D/L
B
f LF
k T
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III. Molecular implementation of
mechanical principles
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What’s missing for molecular machines?
• How can we apply these macroscopic ideas to single
molecules?
• What is the cyclic machine that eats chemical energy?
• Where are the experiments?
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Reaction coordinate simplifies
description of a chemical reactions
• Chemical reactions can be regarded as transitions between
different molecular configurations. To move between
configurations atoms rearrange their relative positions.
• The “states” are locally stable points in a multi-dimensional
configuration space (local minima of free energy).
• Thermal motion can push molecules to transit between states.
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Chemical reactions are random walks on free energy
landscape in space of molecular configurations
• There exists a path in configuration space that joins two minimums
while climbing the free energy landscape as little as possible.
• Chemical reaction is approximately 1D walk along this path, which is
called the reaction coordinate.
2 2H H H H
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Even big macromolecules can be described
by one or two reaction coordinates
• Rate of transition through a barrier‡ ‡
‡
‡
Rate exp
Rate exp
B
B
G E TS
G
k T
E
k T
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Enzyme catalysis: Conceptual model of
enzyme activity (Haldane 1930)
(1) Binding site of enzyme E matches substrate S.
(2) E and S deform to match perfectly and form ES state.
One bond in S (“spring” in S) is easily broken by deformation.
(3) Thermal fluctuation break bond and get EP state.
A new bond forms (“upper spring”), stabilizing product P.
(4): P does not perfectly fit to binding site, so it readily unbinds,
returning E to its original state.
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Enzymes reduce activation energy by binding
tightly to the substrate's transition state.
• Interaction free energy with large
binding free energy (dip) with
entropic cost aligning S and E.
• Binding free energy partly offset
by deformation of E, but net
effect is to reduce the barrier
• Net free energy landscape (sum of
three curves): Enzyme reduced
∆G‡ but not total ∆G
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Enzymes are simple cyclic machines
• Enzyme work by random-walking
down 1D free energy landscape.
‡
Barrier:
Net descent each work cycle:
G f L
G f L
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Mechanochemical motors move by random
walking on a 2D landscape
• E catalyzes the reaction S at high chemical
potential μS into P with low μP .
• E has another binding site which can attach to a
periodic “track”. e.g., kinesin which converts
ATP to ADP+Pi and can bind to a microtubule.
• 2D free energy landscape with 2 variables:
– Reaction coordinate: # of remaining S.
– location of machine on track.
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• tilted along chemical axis by free energy of
reaction.
• load force tilts of the surface along position.
• Furrow couples chemical energy to mechanical
motion. System random walks in furrow.
• Correspondence between potential energy
surface and kinetic mechanism of motor.
Potential energy is periodic in reaction coordinate
and position, reflecting cyclic nature of enzymatic
turnovers and motor cycles.
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Breaking the symmetry induces motion
• Mechanochemical cycle: landscape with 2 directions,
reaction coordinate and displacement.
• Landscape asymmetric in displacement +
concentrations of S and P out of equilibrium
= directed net motion.
• Deterministic free energy landscape: coupling of
variables is tight and we can follow a single reaction
coordinate one valley.
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VI. Kinetics of enzymes and
molecular machines
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Michaelis-Menten rule describes
the kinetics of simple enzymes
𝑉 =[𝑆]
[𝑆] + 𝐾𝑀𝑉max
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Fre
eenerg
y
Reaction coordinate
E+SES
EP
E+P at high P
E+P at low P
∆G'
12
1
Simplified cyclic process: .k S k
kE S ES E P
• low P: EP→E+P is quick process.
neglect shortly lived state EP.
• ∆G' >> other barriers and kBT
ES→EP is irreversible.
MM kinetics originate from the energy
landscape of the enzyme
• Different Initial and final states starting but enzyme returns to initial state.
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12
1
.k S k
kE S ES E P
MM kinetics originate from the energy
landscape of the enzyme
1 1 2
1 2
1
Steady state:
[ ][ ][ ] ( )[ ] 0
[ ][ ][ ] ,
[ ]
MM constant
M
M
d Ek S E k k ES
dt
S EES
K S
k kK
k
Fre
eenerg
y
Reaction coordinate
E+SES
EP
∆G'
22
max max 2
[ ][ ][ ][ ]
[ ]
[ ] with [ ]
[ ]
M
M
k S Ed Pv k ES
dt K S
Sv v v k E
K S
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Experimental test: Lineweaver-Burk graph
max
1 11
[ ]
MK
v v S
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Two-headed kinesin is a tightly coupled, perfect ratchet
• tightly coupled molecular motor ~ enzyme:
1D random walk along a valley, a step per reaction.
• P kept low motion is irreversible.
• A tightly coupled molecular motor with an irreversible step
move at a speed determined by MM kinetics.
• Kinesin is bound to microtubule ~ 100% duty ratio.
• Kinesin is highly processive: makes many steps before falling
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Kinesin is highly processive, even with load
• kinesin motility assay: optical tweezers
apparatus pulls a bead backwards.
• Feedback circuit continuously moves the
trap following the kinesin at constant
force 6.5 pN, with 2 mM ATP.
[Viscscher et al. 1999]
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Kinesin obeys MM kinetics
with load-dependent parameters
• Kinesin rarely moves backward “perfect ratchet” limit.
• Kinesin is tightly coupled:
exactly one step per one ATP molecule (even under load).
[Data from Visscher et al. 1999]
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Structural clues for kinesin as perfect ratchet
• Microtubule:
– β-subunit has a binding site for kinesin.
– Regular space at 8 nm intervals.
– Microtubule is polar (+/- ends).
• Kinesin binding is stereospecific:
bound kinesin points in one direction.
+ -
Kinesin heads:
neck
Nucleotide binding site
neck
linker
motor domain
• ADP at nucleotide-binding site: neck linker
flops between two different conformations.
• ATP at site: neck linker binds tightly to kinesin
head and points to the + end.
![Page 55: Molecular Machines and Enzymestlusty/courses/LivingMatter/...Molecular Machines and Enzymes Lecture 10: Tsvi Tlusty, tsvi@unist.ac.kr Sources: Nelson –Biological Physics (CH.10)Julicher,](https://reader030.vdocument.in/reader030/viewer/2022040818/5e63c1b4e43f3c4b524c0993/html5/thumbnails/55.jpg)
Structural clues for kinesin as perfect ratchet
• Kinesin binds ADP strongly.
• Kinesin without bound nucleotide binds microtubules strongly.
• Complex Microtubule·Kinesin·ADP is only weakly bound.
• Complex Microtubule·Kinesin·ATP is strongly bound; Only one head could
bind microtubule when only ADP exists;
• Binding ATP to one head stimulates another head to release its ADP
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Cyclic model for kinesin stepping
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Summary
![Page 58: Molecular Machines and Enzymestlusty/courses/LivingMatter/...Molecular Machines and Enzymes Lecture 10: Tsvi Tlusty, tsvi@unist.ac.kr Sources: Nelson –Biological Physics (CH.10)Julicher,](https://reader030.vdocument.in/reader030/viewer/2022040818/5e63c1b4e43f3c4b524c0993/html5/thumbnails/58.jpg)
• Necessary conditions for directional motion:
• Break spatial inversion symmetry
• Break thermal equilibrium.
• Smoluchowski equation for micro-machines:
• Enzymes are simple cyclic machines;
work by random walking on 1D free energy landscape.
• Saturation kinetics: Michaelis-Menten rule
• Mechanochemical motors move by random-walking on 2D
landscape reaction coordinate vs. spatial displacement.
• Two-headed kinesin is a tightly coupled motor.
U
forward backwardR R
0 const.B
P P P Uj D
t x k T x
max
[ ]
[ ]M
Sv v
K S