Molecular Machines

and Enzymes

Lecture 10:

Tsvi Tlusty, tsvi@unist.ac.kr

Sources:

Nelson – Biological Physics (CH.10)

Julicher, Ajdari, Prost – Modeling molecular motors (RMP 1997)

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.

I. Survey of molecular devices in cells

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.

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.

• “One-shot” machines

• Cyclic machines

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.

(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?

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

(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

All eukaryotic cells

contain cyclic motors

• Molecular forces were measured

by single molecule experiment.

Some important cyclic machines

II. Mechanical machines

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

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)

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

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?

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?

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?

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

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

• 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

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

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

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

Low load:

forward

High load:

backward

Modeling molecular machines:

qualitative picture

x

L

/f L /f L

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

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

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

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

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

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

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

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

III. Molecular implementation of

mechanical principles

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?

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.

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

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

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.

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

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

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.

• 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.

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.

VI. Kinetics of enzymes and

molecular machines

Michaelis-Menten rule describes

the kinetics of simple enzymes

𝑉 =[𝑆]

[𝑆] + 𝐾𝑀𝑉max

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.

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

Experimental test: Lineweaver-Burk graph

max

1 11

[ ]

MK

v v S

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

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]

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]

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.

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

Cyclic model for kinesin stepping

Summary

• 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