Fluctuations meet function:

Molecular motors

Diego Frezzato, July 2018

Part of the course Fluctuations, kinetic processes and single molecule experiments

A molecular machine is a device made of a single (complex) molecule, or a

supramolecular complex, that transduces input energy into output energy; if the

output energy is mechanical work, the machine is usually called “motor”.

In cellular environment, these transductions are performed in accurate/precise way.

Operations of molecular machines can be

- cyclic (eg., motors and pumps)

- “one-shot” (some examples?)

Let us look here only at few traits of molecular machines. An excellent review:

D. Chowdhury, “Stochastic mechano-chemical kinetics of molecular motors: a

multidisciplinary enterprise from a physicist’s perspective”, Physics Reports 529,

1-197 (2013)

- Chemical “fuels”. Energy is released by localized chemical reactions, mainly

hydrolysis of nucleoside triphosphates (NTPs): ATP, Guanosine Triphosphate

(GTP). Also inorganic pyrophosphate (PPi) generated from hydrolysis of ATP to

AMP can be used.

(25°C) 30.5 kJ/molrG

- From the manipulated substrates themselves. For example, polymerases can extract

energy from the substrates in creating the polymers.

- From light absorption (photons)

- From spatial gradients of chemicals’ concentration (eg., H+ gradients), charge, etc.

Forms of input energy

Just a mention to some of the many machines/motors which operate in

the cellular environment

1) Enzymes for synthesis/manipulation/degradation- degradation of macromolecules

- template-dictated polymerisation

- helicases, topoisomerases, etc. (unwrappers, unzippers, untanglers of DNA)

- controllers (e.g. “quality controllers” of genome replication)

2) Translocation motor proteins- porters (intracellular cargo transport)

- sliders acting as “rowers” (make relative sliding of two filaments)

- depolymerases (kinesins which crash their track-filament from one end)

-pistons, hooks, springs via polymerizing/depolymerizing cytoskeletal filaments

(eg, dynamic filamentous proteins in prokaryotic cells)

- translocases across membranes

(a) conventional kinesin (transport of organelles),(b) myosin V (transport of vesicles),(c) cytoplasmic dynein (transport of mRNA)[figure taken from A. B. Kolomeisky et al, Annu.Rev. Phys. Chem. 58, 675 (2007)].

Details of the two kinesin “heads”[figure taken from S. M. Block,Cell 93, 5 (1998)].

3) Rotary motors

4) Ion pumps (active transport through membranes)

F0F1-ATPsynthase (for ATP production)

~ 95% of ATP is our produced by this molecular machine

In 75 years of life, 2000 tons of ATP are produced on average!

~ 10 nm

~ 20 proteins

~ 500 kDa mass

outer side

inner side

~ 100% efficiency!

3 molecules of ATP per cycle, with a transit of 815 H+

~ 3 cycles/sec (~10 ATP molecules per second), depending

on the difference of pH at the two sides of the membrane

For insights:

- Z. Ahmad, J. L. Cox, ATP synthase: the right size base model for nanomotors in nanomedicine, The

Scientific World Journal, ID 567398 (2014)

-Y. Q. Gao, W. Yang, M. Karplus, A structure-based model for the synthesis and hydrolysis of ATP by

F1-ATPase, Cell, Vol. 193, pag. 193 (2005)

- D. Okuno, R. Iino, H. Noji, Rotation and structure of F0F1-ATP synthase, J. Biochem., Vol. 140,

pag. 655 (2011)

https://www.evolutionnews.org/2013/05/atp_synthase_an_1/A nice animation:

D. Okuno, R. Iino, H. Noji, J. Biochem. 149, 655 (2011)

Such motor can operate in reverse (at low proton gradients): ATP

hydrolysis ! The F1 unit sufficies to catalize ATP hydrolysis.

Single-molecule observation of the F1 rotation upon ATP hydrolysis

D. Okuno, R. Iino, H. Noji, J. Biochem.

149, 655 (2011)

Cycle of ATP hydrolysis

catalized by the F1

Bacteriorhodopsin

In the archea of salty ambients, e.g. in Halobacterium salinarium

“Pump” of H+ ions towards the exterior of the cell membrane

retinal

hν

λmax~570 nm

It can produce pH differences in-out up to 4 units!

W. Kühlbrandt, Bacteriorhodopsin – The movie, Nature, Vol. 406, pag. 569 (2000)

Coupling between bacteriorhodopsin and ATP-synthase !

Radiant energy (hν) Chemical energy (ATP)

Bacterial flagella

In E. coli

~ 300 turns/sec

Exerts a torque of ~ 550 pN nm

Gives a speed of ~ 30 μm/sec

Efficiency ~ 60%

45 nm

~ 20 proteine

Operates by exploiting gradients of H+ or Na+

~ 100 μm length

Bacterium membranes

D. J. De Rosier, The turn of the screw: the bacterial flagellar motor, Cell, Vol. 93, pag. 17 (1998)

Kinesin I

Dynein

move on microtubules

Myosin V

moves on actin filaments (or it “displaces” actin filaments in the mechanism of muscle contraction)

Transpsport/move vescicles, cell’s nucleous, mRNA, cytoskeleton filaments,

signaling proteins, protein fragments, … Energy is supplied by ATP hydrolisis.

Cellular transportersR. D. Vale, The molecular motor toolbox for intracellular transport, Cell, Vol. 112, pag. 467 (2003)

α β

tubulin dimer

protofilamentsα β

- +

25

nm

(helical structure)

cytoskeleton

Input energy: hydrolisis of ATP

~ 100 steps before the detachment from the microtubule

~ 800 nm/sec (100 steps/sec) in vitro

Efficiency ~ 60%

To stop it (“stall”) it is required a force of ~6 pN in opposition

“hand-over-hand” motion of kines

“Hand over hand” motion for kinesin on microtubules, as proved by means of

FIONA technique (“Fluorescence Imaging One Nanometer Accuracy”).

[Figure taken from A. Yildiz et al., Science 303, 677 (2004)]

Lymn-Taylor cycle

for the muscle contraction

acti

n

myosi

n

The energy is supplied by the polimerization itself!

phosphorylated nucleotides(cytosine, uracil, guanine,adenine)

~ 12 proteins

From the DNA “template”, at need itproduces the various forms of RNA thatare involved in the cellular processes

Very high accuracy: ~ 1 error every 10000nucleotides!

DNA

RNA

RNA polymeraseJ. Gelles, R. Landick, RNA polymerase as a molecular motor, Cell, Vol. 93, pag. 13 (1998)

Richard P. Feynman

Caltech (December 1959)

http://www.its.caltech.edu/~feynman/plenty.html

The F1-ATPase motor coupled to inorganic rod made of Nickel

H. Hess, G. D. Bachand, V. Vogel, Chem. Eur. J. 10, 2110 (2004)

M. G. L. van den Heuvel, C. Dekker, Science 317, 333 (2007)

New trends: combination biologic-synthetic

An example… how to make the biologic molecular motors work for us!

Are molecular machines just a “nanoscale version” of man-made

macroscopic machines?

Similarities with macroscopic machines are only apparent: it is not just a

matter of length scale!

Typical scales of the molecular machines:

Length: nm

Time: ms

Forces: pN

Energy: kBT (= 4 10-21 J “involved” per molecule at 25°C)

First argument: the “Scallop theorem” from hydrodynamics

[E. Purcell, Am. J. Phys. 45, 3 (1977)]

On the contrary, molecular machines are quite agile!

Re c cu l

with uc the velocity of the object (and of the fluid sticked to it), lc a charateristic

length of the object, ρ the fluid density, η the (shear) viscosity of the fluid. For

water at 20°C, ρ = 103 kg m-3 and η = 0.001 Pa s.

Low Re means that viscous forces prevail.

In hydrodynamics, the dimensionless Reynolds Number (Re) compares the

magnitude of inertial and viscous forces for an object (or a fluid element itself)

moving in a fluid:

A “nano-scallop” would have Re 10-10 ! It cannot propel itself: perceived viscous

drag is so high that opening/closing make a balance. A nano-scallop just fluctuates.

A scallop can move by opening and closing to expell water.

Typical lengh is lc= 1 cm, and it moves by few times its length

per second. It results Re 102 : high value, mechanical force

prevails on viscous drag, hence propulsion occurs.

Second argument: look at the rate of energy exchanges

- typical input energy-rate : 10-16 – 10-17 J/s

- typical energy-exchange rate with the thermal bath via “collisions”: 10-8 J/s

There are 9 orders of magnitude of difference! Random perturbation from the

environment is much intense than the detailed energy input.

On the other hand, molecular machines find their way!

… so it is not only a matter of length scale: at the nanoscale one has a

peculiar scenario.

Let us look at the main features: what does a molecular machine feel?

“For molecules, moving in a straight line would seem to be as difficult as walking

in a hurricane is for us. Nonetheless, molecular motors are able to move, and

with almost deterministic precision”

[quotation taken from D. Astumian, P. Hänggi, Physics Today 55, 33 (2002)].

Essential and ubiquitous traits of molecular machines

Molecular machines operate under isothermal conditions (temperature

gradients are not sustained at molecular scales...)

Machines are able to “receive” the energy input in detailed way

- Mechano-chemical coupling: a localized event (eg., a chemical reaction or photon

absorption) generates a cascade of responses. How can we describe such a

coupling?

For cyclic machines, steady states far-from-equilibrium can be

reached/maintained under energy input

- A net drift is generated (for example, an average velocity of kinesins on

microtubules, an average angular velocity of the F1-ATPase rotor, etc).

Transduction from “scalar” energy into “vectorial” processes

- Although the trajectory of the machine (in abstract sense) is stochastic, on average

the motion is directed: there is a drift.

Sources of input energy

- Chemical “fuels”, enrgy from the manipulated substrates themselves, photons,

gradients of chemicals’ concentration or of electric charge.

Reaction coordinate

-The internal free-energy V(x) of the machine can be reduced (by means of proper

averages over fast-fluctuating variables) to a low-dimensional free energy

landscape on few essential degrees of freedom.

One of these degrees of freedom is the peculiar reaction coordinate along which the

specific action is performed. What distinguishes such a coordinate from the others?

Accurate and precise operation

- A single trajectory of a molecular machine (in its low-dimensional free-energy

landscape) is stochastic. However, trajectories deviate little from the average,

spread of cycles’ period is little, etc. How is it possible?

Release of waste products (waste chemicals and dissipated energy)

- For example, release of hydrolysis products.

- As for macroscopic finite-time (irreversible) transformations in isothermal

conditions, the free-energy transduction into work cannot be complete (the Second

Principle of Thermodynamics puts a limit!): part of the free-energy difference at

disposal is wasted as heat exchange with the thermal bath (ultimately: global

entropy production).

Average energy dissipation rate for objects

of different length-scale, operating under

steady-state conditions.

[figure taken from C. Bustamante et al,

Physics Today 58, 43 (2005)]

310 /Bk T s

In the essence:

what do we need to let a dead molecule alive & working?

All the three ingredients are necessary to let a machine working!

x

2) detailed energy input

1) fluctuationsof structuralvariables x

3) Breaking forward-backward symmetry

of the machine operation (“directionality”)

fluid environment

(also crawded)

mechano-chemical coupling

For example, fluctuations alone leave the molecule “dead”: it would fluctuate at

thermal equilibrium without any net average “drift” (no directed action)

The structural asymmetry [i.e., asymmetries in the energy landscape ] is not

sufficient to induce such a drift! For example, the “polarity” of an actin microtuble

is not sufficient to make kinesins moving, on average, in one direction…

( )V x

A “ratchet model” has been proposed as paradigm of browian motors: when the

nano-ratched is in contact with the thermal bath (random noise from collisions),

structural asymmetry of the teeth should “rectify” the fluctuations. That is not true!

What is missing? Only a “targeted” energy input can keep the machine out-of-

equilibrium to get a drift. The ratched idea must be revisited…

Pictorial representation of the “ratched model” with energy input

state at rest

“energized

state”

acti

vat

ion

The drift is originated by promotion to an “energized state” (eg., kinesin plus ATP

on the head domains), followed by relaxation back to the state at rest: the two

energy landscapes must be different (using the metaphor of the ratchet, the shape

of the teeth must be different at rest and in the energized state…)

A. B. Kolomeisky, M. E. Fisher, Molecular Motors: A Theorist’s Perspective,

Annu. Rev. Phys. Chem. 58, 675 (2007)

Generalization (end of the story …or the beginning for the modelling!)

Fluctuations of x on an energy landscape which is, by itself, stochastically

modulated by the energy input (chemical reactions, photon absorption, etc):

)Vc(x

configurational variables of the machine

set of parameters that specify the istantaneous shape of the energy landscape

The energy input modulates (stochastically) c

affects the fluctuations of x

average drift along the reaction coordinate (directed action of the machine)

Simulated trajectories could be compared with the real-time experimental

observations of the single machine during operation!

[ Recall the trajectories of the single F1 domain of the ATPase …]

1) Stochastic localized reactions (“chemical Langevin”) which modulate c

2) Brownian dynamics of x on the energy landscape ( )Vc

x

Stochastic trajectories of the operating machine could be generated by means of

a generalized Langevin equation which couples:

Difficulties: too many variables to handle, and too many unknown parameters!

Need to adopt a simpler approach: a discrete representation with the same kind of

phenomenology.

Full dynamics on a modulated energy landscape are replaced by a “kinetic

mechanism” made of a few elementary steps,

Make a list of Ns relevant “sites” (stable conformations)

…

1 23

45 6

Ns…

Sketch out a likely kinetic mechanism involving these sites as “species”.

Some of the elementary steps must involve the “energizing molecules” (eg., ATP).

5 + ATP 3k1

3k2

6

k-1

5k3

3

…

x set of continuum variables

1 2, ,...sNx x x

reference configurations of the relevant states (“sites”)

DIS

CR

ET

IZA

TIO

N

Finite number of relevat states (stable intermediates, or conformations

individuated by an educated guess)

Advantages:

- Direct pictorial representation of the chemical steps (see examples below);

- More “friendly” for chemists!

- Small number of parameters (kinetic constants) which could be measured

experimentally

- Simple calculations: under suitable conditions (eg., excess of “chemical fuel”, eg.

high ATP concentration) the steps of the mechanism may become of the first-order;

a “master-equation” is easily written and solved (see below)

Disadvantage

- Full (small-steps) stochastic trajectories of the single machine, x(t), are not

generated in such a coarse-grained perspective

The objective

Think to an ensemble of machines, describe the time-evolution of the populations of

each site.

Population of the site = probability of observing the machine in that given site

Some steps are bimolecular (eg., those involving ATP). In the excess of energizing

molecules, bimolecular steps can be reduced to pseudo uni-molecular steps (with

kinetic constants dependent on the fixed concentration of the chemical fuels…)

, ' , ' 'j j j j j i j j

i

K k k

populations of the

sites at time t (their

sum is equal to 1)

( )( )

d tt

dt

PK P

1

2

( )

( )( )

...

( )sN

P t

P tt

P t

P

Master Equation for first-order kinetics: transitions amongst Ns sites

s sN N kinetic matrix

Constraint to assure conservation

( at any time):

first-order kinetic constants

1( ) 1

sN

jjP t

Given the initial populations, P(0), the populations P(t) at any subsequent time are

obtained by applying standard numerical methods (…)

Fluctuations at thermal equilibrium (a “dead” molecule)

, ,j eq j i i eq i jP k P k ,lim ( )i eq it

P t P

detailed balance condition

thermal equilibrium populations

For activated fluctuations, detailed balance must be broken

, ,lim ( )i i ss eq it

P t P P

steady-state populations

If detailed balance is broken in proper way, a non-null “current” along the

reaction coordinate can be present even at the steady-state: each machine

experiences a drift along such a coordinate, i.e., the machine operates!

Abstract representation for a translocation motor

track

l-th segment on the track

1, 2, 3, ,, , ,...cl l l N lx x x x : specifies the conformations of sites 1, 2, 3, …, Nc

for the l-th segment

,diss jk : kinetic constant for the irreversible detachment from the track if the

motor is on site j

1/cN rIt can be demostrated theoretically that[R. D. Astumian, Science. 276, 917 (1997)]

0.39rFor kinesin at high ATP concentrations it is known that

3cN A likely scheme must consider at

least 3 intermediates per segment

of the track.

i

i

K M + ATP K M ATP

K M ATP K M ADP P

K M ADP P K M ADP

K M ADP K M

A minimal scheme for

Kinesin/Microtubule/ATP,ADP

Involving 4 intermediates

Randomness parameter:

D = diffusion coefficient along the track

d = length of the step

v = mean velocity at steady-state

2 /r D vd

experimentally achievable

Pathways of ATP hydrolysis with kinesin.

[from M. L. Moyer et al, Biochemistry 37, 800 (1998)]

A more elaborated scheme