Introduction into the Theory
of Biological Nanomachines
Alexander S. Mikhailov/Holger Flechsig

Lectures 5 & 6

External Work R

Chemostat XµXjXSystemjY

Chemostat YµY

Q
Thermal BathExternal work R

T0

Molecular machines are single-molecule chemical engines.
All of them represent enzymes.
Now we want to understand how mechanical work can beproduced by enzymes.

Enzymes are single-molecule protein catalysts

Enzyme protein

Enzymes are single-molecule protein catalysts

Active center

Enzyme protein

Enzymes are single-molecule protein catalysts

SubstrateActive center

Enzyme protein

Enzymes are single-molecule protein catalysts

Active center

Enzyme protein

Enzymes are single-molecule protein catalysts

Active center

Enzyme protein

Enzymes are single-molecule protein catalysts

ProductActive center

Enzyme protein

Cyclic conformational changes in enzymes
Substrate

Cyclic conformational changes in enzymes

Cyclic conformational changes in enzymes

Cyclic conformational changes in enzymes

Cyclic conformational changes in enzymes

Cyclic conformational changes in enzymes

Cyclic conformational changes in enzymes

ProductCyclic conformational changes in enzymes

Cyclic conformational changes in enzymesCyclic conformational changes in enzymes
ESE
Two conformational motions
make a cycle.They correspond totransitions from free enzyme E
to substrate-enzyme complex ESand back.

Cyclic conformational changes in enzymes
ESE
Two conformational motions
make a cycle.They correspond totransitions from free enzyme E
to substrate-enzyme complex ES
and back.

Here, we have assumed that product P isimmediately released and have not included
a separate product-enzyme state EP.

Conformational changes within an enzymic cycle

ESESEEEESEPEP
EP1
If there are several substrates and (intermediate) products, even more conformational changes may take place.

Thus, an enzyme can repeatedly change its shape in each next catalytic
turnover cycle.
These shape changes are induced by binding and dissociation of ligands(substrates and products) and, generally, also by chemical changes in theligand state in the enzyme.
They represent internal conformational motions within a protein. These internalmotions can be used to generate mechanical work.

Ligand-induced mechanochemical motions are relaxational motions

ESEEP

When a ligand L binds to a protein P, additional interactions
between the ligand and the protein are established. Thus, a newphysical system LP is formed. The equilibrium conformation ofligand-protein complex LP is different from that of a freeprotein P. Therefore, just after ligand binding, this system is
not in its correct equilibrium state. The process ofconformational relaxation begins. It continues until theequilibrium state of the complex LP is reached.
Similarly, if the ligand is initially present and the complex LP isin the equilibrium state, but then the ligand dissociates, the freeprotein P is first not in its equilibrium state. It undergoes
conformational relaxation (=mechanochemical motion) towards
its own equilibrium state.
When the state of the ligand in the complex is changed, LP ->1LP , this also generally changes the equilibrium conformation
2of the complex, inducing next conformational relaxation
process within it.

Ligand-induced mechanochemical motions in proteins

Adenylate Kinase

Calcium pump ATPase

Cdc42

HCV helicase

LysineSucrose Phosphotase

Ligand-induced mechanochemical motions in proteins

Adenylate Kinase

Calcium pump ATPase

Cdc42

HCV helicase

LysineSucrose Phosphotase

Functional mechanochemical motions in enzymes

Transport of ligands tooptimal positions
within an enzyme.

Complex enzymes

Transport of ions andother particles acrossa membrane

Ion pumps, active
channels, transporters

Operations with otherbiomolecules (DNA,RNA)Transport of particles
o v e r fi l a m e n t s a n d
microtubules, generationof mechanical forces

Biomolecularmanipulators

MotorsWhenever cyclic functional mechanochemical motions are involved, the enzyme
acts as a biological nanomachine!
For motors, such cyclic motions need be further transformed into study translationalor rotational motions.
How to convert a machine into a motor?

Whenever cyclic functional mechanochemical motions are involved,
the enzyme acts as a biological nanomachine!
For motors, such cyclic motions need be further transformed into
steady translational or rotational motions.

Machine proteins typically have a domain structure

flexible hinge

Open stateMachine proteins typically have a domain structure

ligandflexible hinge

Open stateMachine proteins typically have a domain structure

ligandflexible hinge

Open state

Closed stateMachine proteins typically have a domain structure

Closed state
If the ligand is removed, the protein undergoes reverse relaxation to its
initial open state.

Machine proteins typically have a domain structure

flexible hinge

Open state

Closed stateMachine proteins typically have a domain structure

flexible hinge

Open state

Closed state

If the ligand is removed, the protein undergoes reverse relaxation to its
initial open state.

substrate
product

Open state

Closed stateA toy dimer model of a protein machine
Dimer
contracts

Substrate introduces
additional short spring
Product is formed

Dimer : two beads
connected by elastic spring

Product is released,dimer returns to its original shapeMacroscopic mechanical model: energy
We assume that product is instantaneously released once it has been formed. Then, thedimer has only two states: s = 0 without the ligand (substrate) and s = 1 with the ligand(substrate). The mechanical elastic energy of the dimer in these two states is (x is thedistance between the beads)
Es =1()

2
1
1
E(x ,s) = k0 x − l0 + skc (x − lc )22s =0Stiﬀness
Stiﬀnessand natural length and natural lengthof the dimer spring of the ligand spring()

2
1
E s=0 = k0 x − l02
21E s=1 = C + k1 x − l12()xl1

k1 = k0 + kc
l1 =k0l0 + kc lc
k0 + kc(11 k0l0 + kc lc
2
2
C = k0l0 + kc lc −22 k +k 2()

(0))

2l0

Macroscopic mechanical model: dynamics
Our toy machine is in a viscous fluid. On length scales below tens of nanometers and timescale above picoseconds, the dynamics is dominated by viscous friction forces andinertial eﬀects are negligible.

dx∂E(x ,s)= −γdt∂xEs =1

The dimer performs relaxation motion in agiven parabolic potential that depends on
the ligand state s.

dx= −γ k0(x − l0 ) if s = 0dtdx= −γ k1(x − l1 ) if s = 1
dt

Transition s = 0→ s = 1 (substrate binding)takes place when x = l0 − ε ,Transition s = 1→ s = 0 (combined product formation and release)takes place when x = l1 + ε .

s =0xl1

l1 + ε l0 − εl0()

21
E s=0 = k0 x − l0221
E s=1 = C + k1 x − l12()

Molecular oscillations
The toy machine performs periodic oscillations. Because equations of motion are linear,
they can be easily solved. Thus, the profile of oscillations and their period can bedetermined. We have

⎛ 1 1 ⎞ ⎛ l0 − l1 − ε ⎞T = γ ⎜ + ⎟ ln ⎜⎟ε⎠⎝ k0 k1 ⎠ ⎝Es =1

The oscillations are autonomous and stable withrespect to perturbations. We have previously
discussed autonomous self-oscillations inmacroscopic open reactors. Now, we have similar
self-oscillations in a single model molecule.

s =0xs =1l1l1 + ε l0 − εl0dx= −γ k0(x − l0 ) if s = 0dt

s =0dx= −γ k1(x − l1 ) if s = 1dt

Energetics of the toy machine (macroscopic limit)
Energies of all states of this machine can be easily determined (we assume here that ε isvery small).
CB
A

A*D
freefreeE A = E0(x = l0 )+ E substrate= E substrate(before substrate binding)()

21boundE B = E1(x = l0 )+ E
= C + k1 l0 − l1 + E substrate(just after substrate binding)2
boundbound
EC = E1(x = l1 )+ E substrate= C + E substrate(after relaxation to equilibrium state with bound substrate)boundsubstrate

()

2
1
free
free
E D = E0(x = l1 )+ E product= k0 l1 − l0 + E product(just after the reaction and product release)
2
free
free
E A* = E0(x = l0 )+ E product= E product(after relaxation to equilibrium state of free dimer)free
E substrate+ E0

EB

3.5Assuming that no dissipation is involved in substrate binding

3.0and product release (E A = E B and EC = E D ), we find

2.5()

()

()()

222111
ΔE = k1 l0 − l1 + k0 l1 − l0 = k0 + k1 l0 − l1222This is the (minimal) diﬀerence in the chemical
potentials of substrate and product, needed tooperate such a machine.

2.0
1.5
1.0
0.50.0EC
EDfreeE product+ E0

When thermal fluctuations are included, the model is stochastic
Dimer
contracts

Substrate introduces
additional short spring
Product is formed

Dimer : two beads
connected by elastic spring

Product is released,
dimer returns to its original shape

All transitions are reversible and their rate constants should satisfy
the conditions of detailed balance.

Properties of stochastic toy machine
Thermal fluctuating forces need to be included into equations of motiondx= −γ k0(x − l0 )+ ξ (t ), if s = 0dtdx= −γ k1(x − l1 )+ ξ (t ), if s = 1dt(ξ (t )ξ (t ′ ) = 2γ kBTδ t − t ′Es =1)

ΔE1Transitions (binding of substrate, release of product)
will be possible within some windows (l0 ± ε ) and (l1 ± ε ).
Conversion from substrate to product will be reversible,
taking place inside a window within the cycle.

s =0
ΔE0x
l1l0

If energy differences ΔE0 and ΔE1 are not large
as compared to kBT , this machine can also operate
in the opposite direction. It will be then converting"products" into "substrates" if product concentration
is high enough.

How to make a motor using a cyclic machine?
Mechanical Ratchets

Ratchets are mechanical devices used to convert cyclicmotion into steady rotation or translation.
Molecular motors are also using ratchet mechanisms.

How to make a motor using a cyclic machine?
Mechanical Ratchets

Ratchets are mechanical devices used to convert cyclic
motion into steady rotation or translation.
Molecular motors are also using ratchet mechanisms.

How to convert a molecular machine into a motor?

This machine cyclically changes
its length. Using it, we want toconstruct a motor that would
transport a filament.

ΔlΔlThis ratchet motor transports the filament in the left direction bydistance Δl = l0 − l1 in each its cycle.

How to convert a molecular machine into a motor?

This machine cyclically changes
its length. Using it, we want toconstruct a motor that would
transport a filament.

ΔlΔlThis ratchet motor transports the filament in the left direction by
distance Δl = l0 − l1 in each its cycle.

Inchworm motion mechanism

This is the ratchet motion: the worm just cyclicallychanges its shape.

Inchworm motion mechanism

This is the ratchet motion: the worm just cyclicallychanges its shape.

We want to make an inchworm from our toy molecular machineΔlΔlAfter one cycle, the machine moves forward by step Δl = l0 − l1 .
There are molecular motors, such as HCV helicase, that indeed use this translocation mechanism.

We want to make an inchworm from our toy molecular machineΔlΔlAfter one cycle, the machine moves forward by step Δl = l0 − l1 .
There are molecular motors, such as HCV helicase, that indeed use this translocation mechanism.

Brownian ratchets

Richard Feynman (1918-1988)

Suppose that the ratchet is very small and the wheel performs
Brownian rotational motion. Will they ratchet rectify thermal
fluctuations and generate mechanical work?
If this were possible, perpetual mobile could have been obtained.
R. Feynman showed that this cannot be the case. Not only thecogwheel, but also the handle will undergo thermal fluctuations.
As a result, no directed rotational motion will take place.

Brownian ratchets

Richard Feynman (1918-1988)

Suppose that the ratchet is very small and the wheel performs
Brownian rotational motion. Will they ratchet rectify thermal
fluctuations and generate mechanical work?
If this were possible, perpetual mobile could have been obtained.
R. Feynman showed that this cannot be the case. Not only thecogwheel, but also the handle will undergo thermal fluctuations.
As a result, no directed rotational motion will take place.

Brownian ratchets

Then, R. Feynman considered the ratchet connected
to two thermal bathes at diﬀerent temperatures,Richard Feynman (1918-1988)

Brownian ratchets

Richard Feynman (1918-1988)

Then, R. Feynman considered the ratchet connected
to two thermal bathes at diﬀerent temperatures, andshowed that it can operate as a motor, producing
mechanical work.

Thermal BathT2

Brownianratchet

Thermal BathT1

Motion in ratchet potentials

VFext

Fext

XdX
∂V
γ=−+ Fext
dt∂X

A small force is enough to move a particle to the right, but, under
the same force in the opposite direction, the motion is blocked.

Motion in ratchet potentials

V

Fext

Fext

XdX
∂V
γ=−+ Fext
dt∂X

A small force is enough to move a particle to the right, but, under
the same force in the opposite direction, the motion is blocked.

Thermal Brownian motion in a ratchet potentialVξ (t )XdX∂Vγ=−+ ξ (t )dt∂X
The answer is NO. Even if the potential has a ratchet form, the equilibrium
Boltzmann distribution for the particle coordinate holds. There is no flow in thisequilibrium distribution.

⎛ V(X ) ⎞p(X ) = Aexp ⎜ −⎟⎝ kBT ⎠Otherwise, work could have been extracted from thermal fluctuations andPerpetuum Mobile could have been made.

Periodic external forceVFext (t )XSuppose for a while that thermal forces are absent, but a periodic external force
is applied:

Fext (t )F0−F0dX∂V
γ=−+ Fext (t )dt∂XT0timePeriodic external force

VdX∂V
γ=−+ Fext (t )dt
∂XX
Fext (t )F0T0time−F0Within an interval of force amplitudes
F + > F0 > F −and for sufficiently long periods T , this force will leadto particle translocation into a new trough im each oscillation cycle.

Hence, steady translational motion
can be induced by such periodicforce!

Fluctuating external force

VXdX∂Vγ=−+ Fext (t )dt∂XAnalysing this system, it can be noted that the rectangular shape of oscillations is notimportant, the same eﬀect can be found for harmonic or other oscillation shapes.
Moreover, it is also not important that the oscillations are strictly periodic and theiramplitude is constant.
Actually, statistically persistent translocation of the particle can be produced even by afluctuating external force, provided that some conditions are satisfied:Such fluctuating force should not be too strong or too weak. Moreover, it should not
too rapidly change.
Note, however, that such fluctuating external force will still have diﬀerent statisticalproperties than the thermal noise.

Brownian ratchet
VXdX∂V
γ=−+ Fext (t )+ ξ (t )dt
∂XNon-thermal
fluctuating force

Thermal noise

Fext (t ) = 0

When some statistical conditions are satisfied, non-biased fluctuating external force
can induce steady translational motion in an asymmetric (ratchet) potential!

Can molecular machines behave as Brownian ratchets?
dX∂Vγ=−+ Fext (t )+ ξ (t )dt
∂X-+Polar Filament

A machine is weakly bound to the filament and can move along it, experiencing a periodic ratchet
potential.
It is subject to equilibrium thermal fluctuations, which cannot themselves induce persistenttranslational motion.
Additional interactions with the filament are caused by active irregular conformational changes in themachine, powered by the free energy brought under conversion of substrate into product.
As a result, non-thermal forces acting on the machine arise.
If some conditions are satisfied, such actively fluctuating machine is able to persistently translocate
itself along the filament.

Two motor operation modes
Brownian ratchets:
generation of non-thermal
active noise

Deterministic ratchets:ordered conformationaloscillations

EnzymeCharacteristic energy of ordered conformational oscillations in deterministic ratchets
must be significantly larger than the thermal energy kBT. Therefore, the difference Δµ = µ s − µ pin chemical potentials of substrate and product must also be large as compared to kBT.
In contrast to this, Brownian ratchets may operate even when Δµ ∼ kBT.

The entire cell can be viewed as a factory that contains an elaborate network of interlocking
assembly lines, each of which is composed of a set of large protein machines. .... Like themachines invented by humans to deal efficiently with the macroscopic world, these protein
assemblies contain highly coordinated moving parts.