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

Lectures 1 & 2

Amazing Mechanical Machines

YouTube: Amazing molecular machines.mp4

Bacterium E. coli

0.5m
0.5 µmA factory of single-moleculenanoscale machines: motors, ion
pumps, enzymes,... Many thousandsof such machines are packed inside avolume of about 11 µm
m33.. The factory
is self-regulated and is able toreproduce itself.

100 nm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.

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.

Real-time imaging of walking myosin motors

Myosin motors walking over actin filaments. High-speed atomic force
microscopy (AFM). Scan size 130 nm x 65 nm, scan rate 146.7 ms per frame

Kodera, Yamamoto, Ishikawa & Ando „Video imaging of walking myosin V
by high-speed atomic force microscopy“ Nature 468, 72 (2010)

Artistic animation

Macroscopic Machines
and Motors

Molecular Machines andMotors•Characteristic sizes: centimeters tometers•Characteristic sizes: nanometers to fractionsof a micrometer•

Characteristic times: seconds to minutes•

Characteristic times: milliseconds to seconds•

Material basis: metals•

Material basis: biomolecules•

Energy is continuously supplied•

Energy is discretely supplied in chemical form
with ATP molecules•

Ordered, coordinated motions ofmechanical parts•

Ordered, coordinated intramolecular motionsof protein domains•

Operate under well-definedenvironmental conditions•

Operate under strong fluctuations inenvironment•

Robustness against damage: low•

High robustness against perturbations

Machine operation is autonomous and self-organizedMachine operation is autonomous and self-organizedIndustrial revolution: Factory in England, 18th century

Gerhard Ertl,
Nobel Prize in Chemistry 2008

Plan of this lecture course

1.
2.
3.
4.

Self-organisation in non-equilibrium systems
Stochastic thermodynamics of nanomachines
Mechanochemical motions in proteins, molecular ratchets
Methods of computational modelling for proteins (all-atom
MD, Go models, elastic networks)
5. Examples of structurally resolved modelling for protein
machines
6. Allosteric self-regulation of machine cycles

Fundamentals of Statistical Mechanics
S = kB lnW
Entropy is a statistical measure of disorder:
W is the number of microscopic states that all correspond toa given macroscopic state

dS
≥0
dt
The Second Law of Thermodynamics:
Entropy increases with time until the state of thermal
equilibrium is reached.

All systems tend to reach the state with the maximal
possible disorder?

Ludwig Boltzmann (1844-1906)

Do biological systems contradict the laws of physics?

Biological organisms seem to retain and even increase
their order over time. Does this contradict the Second
Law of Thermodynamics?

The Second Law is only applicable to closed systems. It
does not hold for a biological organism that permanently
exchanges substances and energy with its environment.
Even if such an organism is in a steady state, it is not thestate of thermal equilibrium. The flows persist in thissteady state.
„To maintain a closed system at equilibrium, no work isneeded, but work cannot be also performed by a systemin such a state. A dammed mountain lake contains muchpotential energy, but, in absence of outgoing flow, itcannot power a motor or a turbine. To generate work,
the system needs to be under a transition to an
equilibrium site. To keep the system over a long time
under such transition, one has to engineer it like a water
power station, supplying new material whose energy isused to produce work.“

Ludwig von Bertalanﬀy (1901-1972)

In 1926 (at the age of 25!), vonBertalannfy had finished his studyof philosophy and art history with adoctoral degree at the University of
Vienna and, 2 years later, publishedhis first book on theoreticalbiology, Critical Theory of Form
Development, soon followed by
two volumes of Theoretical
Biology.

What is Life?

dS
=σ −h
dt
Entropy
production

Entropyexportσ ≥ 0, but h = jsout − jsin may have any sign.

Thus, biological organisms feed on „negative entropy“, i.e.
they import order.

Erwin Schrödinger (1877-1961)

In emigration in Ireland, he had given in1943 several lectures in the Dublin Institutefor Advanced Studies which were published
one year later as a book with the title What
is Life? The Physical Aspect of a Living
Cell. The lectures were attended by anaudience of about four hundred, both
physicists and biologists.

Thermodynamic free energy is defined as

F = E −TS
EnergyTemperature

Suppose that (internal) energy E is not consumed or released inside an open system and temperature is fixed.
Than, the balance equation for free energy is

dF
dS
F
= −T
= −Tσ + jinF − joutdt
dt
In the steady state we have

Hence

dF
=0
dt

F
jinF = Tσ + jout
≥ Tσ

The influx of free energy should counterbalance the effect of persistent entropy production
in a non-equilibrium steady state.
A biological organism imports „order“ with the flux of free energy coming with its nutrients,
even just to maintain its steady state.

External Work R

SystemQ

Thermal BathT0This closed system undergoes relaxation to theequilibrium state with temperature T0. It can perform
work only transiently, during the relaxation process.

Ilya Prigogine (1917-2003)
Nobel Prize in Chemistry 1977
He was born in Russia and emigrated asa child with his parents first to Germany
and then to Belgium. In 1941 he hasreceived a doctoral degree from Free
University of Brussels and, in 1947, (atthe age of 30!) published his first book A
Thermodynamic Study of Irreversible
Phenomena.

External Work R

Thermal Bath 1 Q
1
T1

System

Q2

Thermal Bath 2
T2

This open system cannot reach any thermal equilibrium
state. In its steady state, a flow of energy is passing
through the system. Work can be persistently
performed.

Ilya Prigogine (1917-2003)
Nobel Prize in Chemistry 1977
He was born in Russia and emigrated asa child with his parents first to Germany
and then to Belgium. In 1941 he hasreceived a doctoral degree from Free
University of Brussels and, in 1947, (atthe age of 30!) published his first book,
A Thermodynamic Study of Irreversible
Phenomena.

External Work R

Thermal Bath 1 Q
1System

Q2

Thermal Bath 2

T1

T1 > T2

T2

Suppose that the system is in a steady state (but not at thermal equilibrium!). Even though the state is stationary, heat
Q would continue to arrive from the bath with the higher temperature and become released into the colder bath.
Because dQ = TdS, we have jins =and jins =1 dQ1(this is the rate of entropy removal from the hot bath)
T1 dt

1 dQ2
(this is the rate of entropy uptake by the cold bath).
T2 dt

Thus, the Second Law of thermodynamics yields
dS
1 dQ1 1 dQ2s= σ + jins − jout
=σ +
−
dt
T1 dt T2 dt
The First Law of thermodynamics implies that a balance equation for the internal energy should holddE dQ1 dQ2 dR
=
−
+
dt
dt
dt dt
where R is the work produced by the system.
If the system changes its volume V ,
dR
dV
= −p
dt
dt
where p is pressure.

External Work R

Thermal Bath 1 Q
1
T1
Assuming a steady state,
dE dS
=
=0
dt dt

System

Q2

Thermal Bath 2

T1 > T2

T2

From the Second Law we get
dQ2
T dQ= T2σ + 2 1
dt
T1 dt

Substituting this into the First Law, we find the rate of work generation (or power)⎛ T2 ⎞ dQ1
dR= −T2σ + ⎜ −1⎟
dt⎝ T1 ⎠ dt

Thus, the system can operate as a heat engine and generate mechanicalwork!
Most work is performed if the engine operates near thermal equilibrium, i.e.
where the entropy production rate is very small.

The efficiency of the engine is defined as the ratio of work generation (power)
to the heat flow from the hot bathη=dR /dtdQ1 /dt

When σ → 0, we find the maximal possible efficiency of the heat engine

η=

T2
T1

−1

External Work R

Chemostat XµX

jX
SystemjY

Chemostat YµYµ X > µY

Q
Thermal BathT0

Now, we consider an open chemical reactor where molecules X (substrate) are converted
into molecules Y (product).
This reactor is connected to two external reservoirs(chemostats) where concentrations of molecules X or Y are maintained constant.The
reactor is also connected to an external heat bath. The chemical potential in the chemostat
for X molecules is higher than in the chemostat for Y molecules.

External Work R

Chemostat XµX

jX
SystemjY

We assume that, in the considered chemical reaction,
one X molecule is converted into one Y molecule

Chemostat Y
µY

X →Y

Q

µ X > µY

Thermal BathThe First Law of thermodynamics yields

T0

dE
dR dQ
= µ X j X − µY jY +
−
dt
dt dt
where j X is the number of molecules X arriving per unit time into the reactor
and jY is the number of molecules Y leaving it per unit time.

The Second Law of thermodynamics yieldsdS
1 dQ= σ + s X j X − sY jY −dt
T0 dt
Here s X and sY are amounts of entropy per molecule X or Y in teh respective chemostats.
According to thermodynamics, we havesX = −∂µX∂T, sX = −∂µX∂TExternal Work R

Chemostat X

jX
SystemµX

From the Second Law
jY

Chemostat Y(Qµ X > µY)

dQ
= T0σ +T0 s X − sY j
dt

µY

Thermal BathThen, the Second Law yields

T0

(

) (

)

dR
= −T0σ + ⎡⎣T0 s X − sY − µ X − µY ⎤⎦ jdt

In the steady state,
dE dS== 0 and j X = jY = j
dt dt

Thus, the system can operate as a chemical engine and generatemechanical work!
Most work is performed if the engine operates near thermal equilibrium,
i.e. where the entropy production rate is very small.

The efficiency of the chemical engine is defined as the ratio of work generation (power)
to the (free) energy flow from the substrate chemostatη=(dR /dtµ X − µY j)

When σ → 0, we find the maximal possible efficiency of the chemical engine

T0

∂
η=
µ X − µY −1µ X − µY ∂T()

Persistent oscillations are possible in open chemical reactors

Example: the Brusselator model

A → X , 2X +Y → 3X , B + X → D +Y , X → E
Limit cycle

Substrate

Amplitude and shape of such self-oscillations do not depend on
initial conditions. The oscillations are not induced by external forces,
but autonomously generated by a system itself.

Product

Self-organised dissipative patterns in reaction-diﬀusion systems

Numerical simulations.
Philip Maini et al (2012)

Alan Turing (1912 - 1954)

Experimental observations .
Patrick De Kepper et al (2009)

time
These autonomously formed spatial structures are possible in distributed openchemical reactors. They exist only as long as the energy or material flow
passing through a system is maintained. Like in self-oscillations, theiramplitude and shape are determined by the system itself, not externallyimposed.

A mathematical genius, the founderof modern computer science (theTuring machine). His article, The
Chemical Basis of Morphogenesis(1952), has laid foundations for thetheory of non-equilibrium patternformationinchemicalandbiological systems.

200 µm

Photo-emission electron microscopy video of wave patterns under CO oxidation on Pt(110) catalytic surface. Real
time. S. Nettesheim, … G. Ertl, Phys. Rev. Lett. 65, 3013 (1990)

8 nm

Fast-STM videos of oxygen atoms adsorbed on Ru(001) surface, T = 300 K. Real time, 6 frames per second.
J. Wintterlin, …., and G. Ertl, Surf. Sci. 394, 159 (1997)

A toy model of molecular turbine (motor/pump)

So far, macroscopic engines were considered.
Now, we want to see what becomes diﬀerent when
we go to nanoscale…

Watermill

As an illustration, I use a toy model of molecularturbine proposed by my friend, prof Jose Maria
Sancho, from the Barcelona university.
R. Perez-Carrasco, J. M. Sancho, Europhys. Lett. 100, 40001(2012); Phys. Rev. E 88, 042705 (2013)This machine can be used as a motor or as a pump (ifit made to rotate in the opposite direction by an
external force

Watermill/ wheel pumpA toy model of molecular turbine

PistonWhen external force is relatively small,the turbine operates as a motor(mechanical engine). The pressure
diﬀerence between two compartments
pushes the piston in the right direction.
When the end of the channel isreached, the piston is replaced by a
new one at the opposite side and thecycle is repeated. In each cycle,
mechanical work is produced.

FE

Ideal gasA toy model of molecular turbine

FE

When external force is relatively small,the turbine operates as a motor(mechanical engine). The pressure
diﬀerence between two compartments
pushes the piston in the right direction.
When the end of the channel isreached, the piston is replaced by a
new one at the opposite side and thecycle is repeated. In each cycle,
mechanical work is produced.

A toy model of molecular turbine

FE

When external force is strong, theturbine operates as a pump. The piston
is dragged to the left until it reaches
the end of the channel. Then is it
replaced by another piston on the
opposite side and the same motion isrepeated again. In each cycle, a certain
number of particles is transported
against their gradient.

A toy model of molecular turbine

FE

When external force is strong, theturbine operates as a pump. The piston
is dragged to the left until it reaches
the end of the channel. Then is it
replaced by another piston on the
opposite side and the same motion isrepeated again. In each cycle, a certain
number of particles is transported
against their gradient.

The macroscopic theory

ρ1

ρ2

The total force acting on the piston is
F = (p1 − p2 )A − FEwhere p1 and p2 are pressuresin the two compartments.
For the ideal gas p = kBT ρ ,

FE

A
D

where ρ is the density. Therefore,()

F = kBTA ρ1 − ρ2 − FE
If γ is the effective friction coefficient of the piston,
the piston velocity therefore is()

F kBTA ρ1 − ρ2 − FE
v= =
γ
γ

The velocity vanishes and changes
its sign at the external stall force

Fstall = kBTAΔρ

When FE < Fstall , the turbine operates as a motor,
when FE > Fstall , the turbine operates as a pump.

ρ1

ρ2

Now we calculate the flux of particles J. This is the number of particles
transported through the channel per unit time.

If the turbine operates as a motor (FE < Fstall ), the number of particles
transported in each step is N + = V0 ρ1 where V0 = AD is the channel volume.

FE

A
D

If T is the step time, we haveVρ
J + = 0 1 = vAρ1 if FE < Fstall .
T
For the pump,Vρ
J − = 0 2 = vAρ2 if FE > Fstall .
T

The (Gibbs) free energy transported with each particle
in the case of ideal gas is

ρ1
Δg = µ1 − µ2 = kBT ln ρ1 − kBT ln ρ2 = kBT lnρ2where µ1 and µ2 are chemical potentials of particles in two compartments.
The energy fluxes associated with particle flows in two regimes therefore are
P + = ΔgJ + , P − = ΔgJ −MotorFE < Fstall

The mechanical power (work per unit time) is
PM = FE v =(1
F F −Fγ E stall)

The Gibbs free energy used to generate
this work per unit time is
P + = J + ( µ1 − µ2 ) = Aρ1v( µ1 − µ2 )The motor efficiency is

η=

PM
P

+

=

FE

Aρ1( µ1 − µ2 )

Physics of molecular machi(

1
PM = FE Fstall − F
γ

η=

PM
P

=
+

FE

)

Aρ1( µ1 − µ2 )(1
P = Aρ2( µ1 − µ2 ) FE − Fstallγ−)

P − Aρ2( µ1 − µ2 )
η=
=
PM
FE

Fig. 5: Turbine(top)
and eﬃciency(bottom)vs. external
Dashedpower
curvescorrespondto themacroscopic
theory Fig. 6: Turbine power (top) an
force.The motor regime is on the left side and the pump regime concentration diﬀerence. Th
Perez-Carrasco,
M. Sancho,Lett.
100,pump
40001 (2012)
on the right one. The shaded zone R.
corresponds
to J.theforce Europhys.
side andtheregime ogap where power and eﬃciency are null. The deterministic corresponds to the gradient ga

MotorFE < Fstall

The mechanical power (work per unit time) is
PM = FE v =(1
F F −Fγ E stall)

The Gibbs free energy used to generate
this work per unit time is
P + = J + ( µ1 − µ2 ) = Aρ1v( µ1 − µ2 )The motor efficiency is

η=

PM
P

=
+

FE > Fstall

Pump

The mechanical power (work per unit time)used to operate the pump is
PM = FE v =(1
F F −Fγ E stall E)

The generated Gibbs free energy used to transport particles
against the gradient per unit time isP − = J − ( µ1 − µ2 ) = Aρ2v( µ1 − µ2 )The pump efficiency isFEAρ1( µ1 − µ2 )P − Aρ2v( µ1 − µ2 ) Aρ2( µ1 − µ2 )
η=
=
=
PM
vFE
FE

Both eﬃciencies reach their maxima at the stall force!

Physics of molecular machi(

1
PM = FE Fstall − F
γ

η=

PM
P

=
+

FE

)

Aρ1( µ1 − µ2 )(1
P = Aρ2( µ1 − µ2 ) FE − Fstallγ−)

P − Aρ2( µ1 − µ2 )
η=
=
PM
FE

Fig. 5: Turbine(top)
and eﬃciency(bottom)vs. external
Dashedpower
curvescorrespondto themacroscopic
theory Fig. 6: Turbine power (top) an
force.The motor regime is on the left side and the pump regime concentration diﬀerence. Th
Perez-Carrasco,
M. Sancho,Lett.
100,pump
40001 (2012)
on the right one. The shaded zone R.
corresponds
to J.theforce Europhys.
side andtheregime ogap where power and eﬃciency are null. The deterministic corresponds to the gradient ga

What becomes diﬀerent at nanoscales?
ρ1ρ2
A. The piston will be subject to thermal fluctuations that aﬀect its motion.
B. The number of particles transported per a step may be small and stochastic
motions of individual particles have to be resolved.
FE

A

The piston motion is now described by a stochastic Langevin equationdXγ= F + ξ (t )dt
It includes a random thermal force ξ (t ).
According to the Langevin theory of Brownian motion;
this noise has Gaussian statistics, its mean value is zeroand its correlation function isξ (t )ξ (t ′ ) = 2γ kBTδ (t − t ′ )The frequency spectrum of thermal noise is flatand therefore it is also called "white".

FD ∼ kBT

When thermal fluctuations are taken into account,the mean flux of particles through the channel is⎡⎡⎛ FD ⎞ ⎤⎛ FD ⎞ ⎤ρ2 ⎢exp ⎜
−1
−
ρ
1−
exp
−
⎥ 1⎢
⎟
⎜
⎟⎥
k
T
k
T
⎝
⎠
⎝
FA ⎢⎣
⎥⎦
⎢⎣
B
B ⎠⎥
⎦
J=
⎛ FD ⎞⎛ FD ⎞γexp ⎜⎟ + exp ⎜ − k T ⎟ − 2
k
T
⎝ B ⎠
⎝ B ⎠

where F = Fstall − FE

This leads to two principal effects:
1. Within an interval of forces (gray region in the figure)the nanoturbine does not persistently operate as a motor or a pump,Solidcurvescorrespond
Fig. 5: Turbine power(top)
and eﬃciency(bottom) vs. external it
Fig.
6: Turbinerandomly
power (top)and eﬃciency(bottom)vs. particle
alternates
betweenthe two
regimesorce.The motor
regimeis onthethermal
left side fluctuations
and the pump regime
to thetheorywithn the right one. The shaded zone corresponds to the force
ap where power and eﬃciency are null. The deterministic
ower and eﬃciency (dashed line) are always greater thanhe stochastic ones (solid line) especially near the force gap.
1 = 40 mM, the rest of the fixed parameters in the figures are
he same as in fig. 2.

concentration diﬀerence. The motor regime is on the rightside and the pump regime on the left one. The shaded zone
corresponds
to the gradient
gap where powerare2.
The efficienciesof the nanoscale
motorand
andeﬃciency
pump vanish
null. The deterministic power and eﬃciency (dashed line) areatthe boundaries of this gray zone.
always greater than the stochastic ones (solid line) especiallynear the gradient gap. FE = 10 pN, the rest of the fixedparameters in the figures are the same as in fig. 2.

On the contrary, the pump power is the variation in time of the piston so no mechanical work can be extracted, andPerez-Carrasco, J. M. Sancho, Europhys. Lett. 100, 40001 (2012)f the gain in free energy in the reservoirs. This is the free the flux isR.stillin the gradient direction so there is no active
nergy of a particle passing between reservoirs times the pumping of particles. Therefore, during this leakage gap

What becomes diﬀerent at nanoscales?
ρ1ρ2
A. The piston will be subject to thermal fluctuations that aﬀect its motion.
B. The number of particles transported per a step may be small and stochastic
motions of individual particles have to be resolved.
FE

B

The Brownian motion of particles i = 1,2,...,N that interact with the piston
was numerically simulated, together with the stochastic motion of the piston itselfγdX= − ∑Vp′(x i − X )− FE + ξ (t )
dt
i

γp

dx i
dt

= Vp′(x i − X )+ ξ P (t )where the interaction potential was⎡ ⎛ x − X ⎞2⎤⎥
Vp (x i − X ) = H exp ⎢ − ⎜ i⎟⎢ ⎝ a ⎠ ⎥⎣⎦The numerical study was performed for a more efficient, two-piston turbine.

R. Perez-Carrasco, J. M. Sancho, Phys. Rev. E 88, 042705 (2013)

AR TURBINE

Comparison of analytical predictions
numerical
simulation
PHYSICALand
REVIEW
E 88, 042705(2013) data
for the eﬃciencies of the turbine

umerical (symbols) and(5) and the flux of thebine, which is a goodo. Actually, even for
ternal assisting force
slightly slower thanmatch the values preultiple-vane scenario

0 .6

ρ1 = 80 mM, ρ2 = 20 mM,
A = 150 nm2 , D = 4 nm

η

0 .4

0 .2

0 .0

0

20

40

FE (pN)

60

80

Dashed and solid curve are analytical predictions (for one and two
FIG. 10. (Color online) Efficiency of the turbine computed
pistons) accounting only for thermal fluctuations for the pistons, symbols
numerically(symbols),
analyticallyshow numericalsimulation
results using the two-vane turbine (solidline) through (20) and the one-vane turbine (dashed line) (7).

deviations from both the deterministic and the one-piston
system. Additionally, these results also reassert the phenomenology described in the ideal one-piston turbine model,
R. Perez-Carrasco,
J. M. Sancho,
this time enhanced by finding a larger
leakage zone
and aPhys. Rev. E 88, 042705 (2013)
larger deviation of the expected efficiency maxima that can