3.4. Molecular motors ~ 10 mm ~ mm ~ 10 nm ~ 100 nm

Consider the energetics and size of an engine

3.4.1 Brownian motors Motors are devices which convert stored energy (chemical, electrical, thermal, solar….) - sometimes called fuel - into mechanical (kinetic) energy of motion (Translation, rotation, oscillation…..) Macroscopic motors obey the laws of thermodynamics (in particular Carnot, no perpetuum mobile ….). The “operating parameters” of the motor (P,T, fuel density…) may vary in time because of the operation principle (e.g. cyclic machines) but the random fluctuations of those parameters are negligible.

Motors are systems (a bit) away from thermodynamic equilibrium (eg temperature gradient and heat flow in the Carnot machine), but they may operate in a steady state (no change of parameters). This requires a steady in-flow of fuel. Question: What happens if the motor gets smaller and smaller? Viscosity dominates - less power (dissipation rate) Therml fluctuations become dominant Totally new modes of operation

Let's look at typical forces in a molecular motor E/a = f Smallest measurable forces: Langevin forces responsible for the Brownian motion of bacteria, pollen grains, and other small objects in water at room temperature. The average force buffeting a bacterium every second is comparable to its weight, about 10-14 N = 10 fN.

Typical forces of molecular motors (convert chemical energy from adenosine triphosphate (ATP) into mechanical work). ATP is the common source of stored chemical energy in all life. Hydrolysis of one ATP molecule yields an energy of ~14 kBT (at room temperature ~ 6 × 10-20 J). Thus, at the molecular dimensions ~ 10 nm, the characteristic forces of such motors are ~10-11 N (10 pN).

Cohesion forces associated with hydrophobic interactions and cooperative hydrogen bonding. Such interactions are essential for the stability of biomolecules and their native folded configurations. These forces are of order 10-10 N, the typical force required to break a noncovalent bond and denature a protein. The strongest forces at the molecular level are those required to break covalent bonds with dimensions ~ 0.1 nm and typical binding energies of 1 eV, giving 10-9 N.

Nature has invented many molecular motors (nano-motors) Nature has invented many molecular motors (nano-motors). Here are a few (!) examples: 3 2 exercise 4 5 cf 3.5

Remember the Feynman ratchet

Put this into an abstract model

The basic phenomenon is the separation of densities amplitude a frequency f J. Eggers, PRL 83, 5322 (1999).

How does the separation happen? Let's suppose we have an ideal Gas: Then we have a barometric density with height. n(z) = g<N>/T exp(-gz/T) Where the Temperature T is given by: T = <v2>/3 = 2m(af)2/D<N> The diffusion constant D basically depends on the restitution coefficient e

Consider the flux through a boundary With a barometric density the flux through a hole at height h will be: F ~ n(h) v (h) ~ <N>3/2 exp(-4B<N>) And for an ideal system of smooth, round particles with radius r, the diffusion constant can be calculated, such that: B = 4p r2 (1-e)2 gh / (af)2

Steady state flux Now in the steady state, the fluxes between the compartments have to be equal, while the total number of particles is conserved: F(n1) = F(n2) , n2 = 1 – n1 n13/2 exp(-4Bn1) = (1 – n1)3/2 exp(-4B(1-n1)) exp(-4B(1/2-n1)) n1 = 1 + exp(-4B(1/2-n1)) With the solution n1 = 1/2, but also n1 = 1,0 for large B

Bifurcation point Let's look at the stability of the solution n1 = 1/2, so set n1 = 1/2 + e exp(4Be) 1/2 + e = 1 + exp(4Be) Which after a little algebra gives 2e = tanh (2eB) Such that the solution is unstable for B >1and we have a pitchfork bifurcation.

Compares well to experiment Remember: B = 4p r2 (1-e)2 gh / (af)2 K. Van der Weele et al., EPL 53, 328 (2001).

How can these motors work despite of thermal noise? Diffusing ratched model for single headed kinesin

A diffusing particle in thermodynamic equilibrium in a symmetric potential does not give directed motion Proba. distrib. because diffusion Activated hopping rate ~ e-EB/kT EB A diffusing particle in thermodynamic equilibrium in a asymmetric potential does not give directed motion either Activated hopping One needs a switching of the potential from high to low (which consumes energy in a dissipative system) in order to get directed motion (system out of equilibrium). More less few confined diffusion particles

3.4.2 Actin-myosin motor How muscles contract

The motions of muscles are driven by molecular motors that move unidirectionally along protein polymers (actin or microtubules). Myosin and kinesin both convert chemical energy into motion Each of the 350 heads of myosin form about 5 cross-bridges per second

A closer look

How muscle contraction works

The rebinding of myosin is strongly Ca dependent

Another view of the binding process

Force measurements on singe actin-myosin motor Force amplitude typically 4 pN discrete force strokes - load-independent force - average step size of about 11 nm very efficient motor directed motion out of random diffusion of ATP

Influence the binding by forces…

Gives the free energy landscape of the process

Cross-bridge Kinetics In order for the myosin head to bind to the actin filament, the myosin must act like a spring The elastic energy required to extend or compress the spring is supplied by thermal activation

Thermodynamics of the Molecular Motor ∆G’0 is the increase in the standard free energy when each of the components has 1 M concentration ATP is present at higher concentrations than ADP in muscle The free energy is negative in these conditions, making the cross-bridge formation spontaneous

Steady-state Tension and Stretch Activation Tension increases as filament displacement increases up to a mechanical maximum at 20 nm (ratchet slips) Hydrolysis rate of ATP increases as filament displacement increases up to a maximum of 7 nm

Filament Sliding: The Motor as a Ratchet To model skeletal muscle, we must consider multiple attachment sites due to the helical structures of actin and myosin Lower phosphate concentration exists during the stretching stages, so the tension is higher When the tension is completely removed from the system the filament quickly returns to its original length, but when tension is still applied the velocity is slower because it must work against the tensile force

How does myosin walk?

Fluorescence imaging of labelled cargo on myosin allows a step determination

Distance indicates hand-over-hand movement

Levering motion of the arms

Keeping one of the arms fixed

Controlled motion of actin filaments on myosin carpets (using ATP as fuel)

Movement of actin filaments along the PTFE ridges coated with skeletal muscle myosin S1

The patterns shown here biased actin filament movement confining it to be unidirectional.

3.4.3 Kinesin - microtubule system The kinesin-microtubule system is responsible for the movement of proteins and vesicles within cells.

Kinesin transporting a colloid

Trapping with an optical tweezer to apply a force

Measure the power consumption one kinesin step of 8 nm consumed one ATP

Force dependence of the walking speed

Walking mechanism

Again from fluorescence data

On a closer look, kinesin seems to be limping

Mechanics of the kinesin-microtubule system Kinesin walks towards the plus-end of microtubules (right side of picture). The motor domains attach to successive beta-Tubulin subunits spaced 8 nm apart, and each head advances 16 nm at a time. The two heads must move in a non-equivalent fashion in order to avoid twisting the stalk. The upper (C-terminal) part of the coiled-coil neck is shown to be permanently connected since it has a high coiled-coil potential, similar to a leucine zipper. The lower part of the neck is shown to open and close reversibly in order to allow the heads to detach, move, and reattach. This is coupled to a reorientation of the linker region (yellow) between the neck helix and the motor domain.

Evidence from FRET for waiting times

The two-headed motor is more efficient than the one-headed

3.4.4 The F0 - F1- ATP synthase complex Sketch of processes involved in the synthesis of ATP A rotating machine H+ e- H+

A closer look: F1 ATPase F0 This machinery works a bit like a hydroelectric generator: The proton flow through the F0 subunit embedded in the membrane rotates a shaft in the stator-like F1 subunit to synthesize ATP. Conversely, ATP hydrolysis in F1 causes a reverse rotation of the shaft and a reverse flow of protons.

First real-time study of the F1-ATPase system. - On addition of ATP, 120° step-wise rotation of the attached actin filament Only a single ATP molecule is hydrolized per step. 90° and 30° substeps are associated, respectively, with the protein binding to ATP and the release of the ATP hydrolysis products Those results, together with an estimate of the energy dissipated by the drag on the rotated actin, imply that the efficiency of the F1 subunit is nearly 100%.

An animation showing why the F1 motor is rotating counter clockwise and how the chemical reaction and the rotation are coordinated by two switches: switch 1 (red) controls the ATP binding; switch 2 (blue) controls the phosphate release

3.4.5 Swimming bacteria N.B. Swimming at low Reynolds numbers Re does not work like our swimming (which is at high Re) The friction coefficient of the moving arm must be different in the forward and backward motion, the motion must not be reciprocal (symmetric with respect to time reversal)

The rotatory motion of a helix provides translational motion, because it is not reciprocal N.B. Bodies with anisotropic friction coefficients (eg rods) experience a force which is not parallel to the dragging velocity field

A rotatory motor for swimming bacteria

Technical specifications Driving Force: Proton or sodium electrochemical gradient Number of Protons per revolution ~ 1000 Energy per proton ~ 2.5 x 10-20 J  (6kT) Maximum rotation rate 300 Hz (protons)  1700 Hz (sodium) Torque at stall ~ 4 x 10-18 Nm (= 4 nN nm) Maximum power output ~ 10-15 W Efficiency 50-100% (stall)  ~ 5% (swimming cell) Number of steps per revolution ~ 50 per torque generator

Recap Sec. 3.4 Small motors are driven by Brownian motion Outside of equilibrium, this can lead to directed motion Muscles contract by the walking of myosin on actin filaments Fluorescence experiments show how these molecules work mechanically and determine the energy landscape Other molecular motors are used in vesicle transport, ATP synthase and locomotion