Nano Motor

7/27/2019 Nano Motor

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1 Nanomotors

In this part of the course we will study nanomotors. First we will define what we mean by nanomotor. Ananomotor is a machine that is only nanometres or 10s of nanometres across. By machine I mean somethingthat either moves in a controlled directed fashion (i.e., not randomly), or exerts forces. Here we will mainlybe considering nanomotors that move things around. These nanomotors are like cars, they are machines for

transporting things from A to B. But whereas a car is macroscopic, it is metres long, nanomotors are a billiontimes smaller. As atoms are typically a few tenths of a nm across, nanomotors are only tens or hundreds ofatoms across.

Scientists working on nanotechnology are working to make nanomotors, but so far they have only made verycrude nanomotors. However, all living organisms rely totally on a large numbers of nanomotors. Our bodiesare made of cells and each cells contain many nanomotors, a muscle cell can contain a billion nanomotors.Muscles can only exert forces because the cells of which they are composed can exert forces and the cells inturn can only exert forces because of the billion nanomotors they contain. As these nanomotors are (protein)molecules, they are also called molecular motors.

Just as cars and trucks are there to transport people and goods around, say oranges from a Tescos depotto a Tescos superstore, many molecular motors in our cells are there to transport stuff (proteins etc) around

a cell. Now, there is an obvious difference between a macroscopic object, such as an orange, and a moleculein solution. Oranges just sit there, whereas molecules diffuse around inside liquids. If a molecule is at theorigin at time t = 0, we have found that at time t, the square root of the mean of the square of the distanceof the molecule from the origin is

r2

1/2= (6D)1/2t1/2 diffusion (1)

because the molecule diffuses. Here D is the diffusion constant. So, an obvious question is: If molecules movevia diffusion anyway, why do you need molecular motors to move them? The answer is that diffusion has nodirection, you are as likely to go left as right, and so if a molecules needs to moved in a specific direction, to saya specific part of the cell, then diffusion is not adequate. Also, as we noted with diffusion in the atmosphere,the distance travelled increases only as the square root of time (as opposed to being linear in time as it is formotion in a straight line at constant speed) and so motion over large distances is very slow via diffusion.

1.1 Molecular motors and the 2nd Law of Thermodynamics

To see how the 2nd Law of Thermodynamics (=the entropy can never decrease) applies to molecular motors,consider a single motor. The motors inside cells move along railtracks inside cells, these are long (micrometreslong or more) thin ( 10 nm) filaments, that criss-cross the cell. One type of filament, called a microtubule,is illustrated in Fig. 1. Note that it is made of a regular periodic array of molecules, it is essentially aone-dimensional crystal.

Let us consider a motor moving in one dimension along one periodic row of pairs of the molecules, asshown in Fig. 1. The motor can bind to an array of positions along the row, one every 8 nm for this type

of filament1. If a filament is, for example, 8 m long this means the motor can bind to it at 1,000 differentpositions.

If we just have a filament and a motor that is not consuming any fuel then the system will go to thermo-dynamic equilibrium where the entropy is a maximum (2nd Law). We know that the entropy is a maximumwhen all possible states, 1,000 of them here, are equally likely. Then the probability of being at any positionis

p(x) =1

103= 103 (2)

1There are 13 rows in a filament and a motor can move from row to row quite easily, but for simplicity we neglect this.

Including it just multiplies the numberof positions by 13.

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The entropy S is then

S= 10

3

i=1

pi lnpi = 10

3

x=1

103 ln103 = ln103 = 6.9. (3)

where we label the binding sites from left to right as i = 1, 2, . . .. Now, molecular motors move in one direction.Let this motion be to the right. If one of the proteins is attached to a molecular motor, then it will tend to

move to the right. Ultimately, it will end up as far right as it can be, i.e., it will be at the rightmost positionfor binding. This is a single binding site and so the probability of it being there, p = 1, while the probabilityof it being anywhere else, p = 0.

S= 10

3

i=1

pi lnpi = 1 ln 1 = 0. (4)

The entropy of the protein has, due to the directed motion of the molecular motor, decreased from 6.9 to0. Of course, the total entropy must not decrease, and so the molecular motor must do work. This is justlike a fridge. This pumps heat from a cold environment (inside the fridge) to a warmer environment (thesurrounding room) and it cannot just do this as if it did so it would break the 2nd Law. Thus, all nanomotors

must in addition to moving, generate entropy by producing heat. To produce this heat they will have to burn

a chemical fuel of some sort. The molecular motors in our body burn a molecule called ATP, this powers achemical reaction which allows them to move to one place, hence reducing the entropy due to the uncertaintyin the position,

In the above example, as the motion changed the entropy due to the position of the molecule, that ofEq. (4), by 6.9, the motor can compensate by producing q = 6.9kT 2.8 1020J of heat. This is forbody temperature, for which kT 4 1021J. This amount of heat produces S = q/kT = 9.2 of entropy

Figure 1: Schematic showing two types of molecular motor, dynein and kinesin, and one type of filament,called a microtubule (i.e., one type of the railtrack inside cells that motors run along). I dont expect youto remember the names of these motors and filaments. However note that the filament is made of rows ofmolecules arranged in a helical fashion and that each row is made of a repeated (i.e., periodic) sequence of pairsof molecules. The two parts of the pair are coloured in different colours (green and blue, or in B&W photocopylight and dark grey). The microtubule filament is made of a helix 13 rows of these pairs of molecules. Theperiod is about 8 nm and so a motor can bind at a whole sequence of positions along a microtubule that are8 nm apart.

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x

x

u(x)

u(x)

diffusion

M to M*

M* to M

M

M*

l

1)

2)

3)

4)

5)

6)

7)

8)

Figure 2: Schematic of the potentials (dashed lines) as a function of x, u(x), for both states of the molecularmotor: M (top) and M (bottom). The motor itself is indicated by the black circle. The dotted lines indicatethe motor going from state M to M or vice versa, and it either diffusing, in state M, or moving to thebottom of the potential well, in state M. The progression of the motor shown is: 1) starts in state M inleftmost potential well, 2) M M, 3) diffuses (by chance to right), 4) MM, 5) moves to the bottom of

the potential that is second from the left, 6) M

M, 7) diffuses to the left, 8) moves back to the bottomof the potential that is second from the left.

and so the total entropy is then zero which is as low as it is possible to go. Any lower and the total entropychange is negative. In molecular motors this q comes from burning molecules like ATP, which produces heat.

1.2 A toy model for nanomotors: The Brownian Ratchet

Real molecular motors are complex and poorly understood. It is difficult to experimentally work out howthey function. Light microscopy, for example, is useless as the motors are much less than the wavelength oflight across. However, there is a simple model that illustrates how motors can exploit diffusion to move in a

directed way, by burning a fuel. This is the Brownian ratchet, which goes back to an idea of Feynmans inthe 1960s. It is also called a diffusive ratchet. The idea is to use a chemical reaction (molecular motors burnATP as a fuel), that takes the motor from one state, call it M, to another, call it M, and back in order torectify diffusion. By rectify I mean allow motion in one direction, we will take this to be to the right, whilepreventing it in the other direction, to the left. Our Brownian ratchet only moves in 1 dimension, which isrealistic as the molecular motors move up and down filaments inside cell. This rectification is like a rectifierin an electric circuit, which only allows a current to flow in 1 direction.

So, the motor is always restricted to move along the x axis (in practice because it is bound to a filament).However, in state M it can freely diffuse along the x axis, whereas in state M it feels a sawtooth potential

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u(x). See Fig. 2 for schematics of the 2 potentials. By freely diffuse we mean that the potential u(x) the motorfeels in state M is a constant so there are no forces on it. In state M it has a diffusion constant D. Also, itis important to note that the sawtooth potential the molecular motor feels in state M is highly asymmetric,as you go from left to right the potential gradually drops, over a distance l, then suddenly increases again. Itis this asymmetry that is going to rectify the motion of the motor.

We start with the motor in state M and at the bottom of the potential. The potential well is assumed to

be deep, i.e., deeper than the thermal energy kT. Therefore, in state M

the molecule quickly heads towardsthe minimum and stays there. Then, if we start in state M, the sequence of events that occurs is:

1. A motor in state M stays in that state only for a time , on average, before converting to state M,i.e., M M. So after a time the motor is in state M.

2. In state M the potential is flat, so it freely diffuses with diffusion constant D. Thus, t seconds after itflipped to state M, it will have diffused a distance of about (Dt)1/2 it is equally likely to be in eitherdirection.

3. The motor stays in state M for only a short period of time, on average in stays only for a time . Now,we assume that is so small that (D)1/2 l, i.e., in the time the motor is in the state M in which it

can freely diffuse it can only diffuse a distance much less than the period of the sawtooth potential instate M.

4. After a time the motor returns to state M. It has either diffused to the left or to the right, each isequally likely, i.e., each has a probability of 1/2. If it has diffused to the left it has only moved a littleway, l, to the left, and so on its return to the M state it will just slide down to the bottom of thesame well it was at the beginning: it has not gone backwards. However, if it has diffused to the right,it will have gone over the top of the sawtooth potential in the potential and so when it returns to thestate M it is in the next valley along. It will then slide down to the next minimum in the potential tothe right, and so will have moved forward a distance l to the right.

5. The motor is now in state M at the bottom of the potential well (either the same one as before or the

next one to the right). The motor is now ready to start another sequence 1) to 4).

After sufficient time has elapsed that the sequence of M to M and back to M has been repeated manytimes, the net effect is that during each MMM cycle which takes on average a time + , there isa 50% probability that the motor has moved l to the right and a 50% probability that the motor has stayedwhere it was. It will not have moved to the left: the sawtooth potential has rectified the diffusional motion.As it moves a distance l with 50% probability every + seconds the average velocity is

average velocity =1

2

l

+ . (5)

Our expression for the velocity is half the period of the potential, divided by the sums of the times in

each state, and . In cells, the motors run along two types of filament, one of these types is called amicrotubule. It has a rough surface with a periodicity of 8nm. Thus we know l, it is of order 10nm, to withinthe approximations we are currently using. Also, motor speeds have been measured and they are typicallyabout 1000 nm s1 or a little less. The final fact is that motors burn one molecule of ATP per 10 nm stepand that burning a single ATP molecule releases about 1019J of energy.

Let us see whether these observations make sense in terms of our model. A velocity of 1000 nm s1

implies that 1 10 nm step takes 102s. Thus + cannot be more that 102s. Let us consider these 2 timesseparately, first . This is the time taken to diffuse over a peak in the potential but not too far (to avoid it

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going backwards). We expect the peak to be 1 to 2 nm wide, i.e., larger than an atom but less than 10 nmof course. The diffusion constant for a protein is given approximately by the Stokes-Einstein expression

D =kT

6a(6)

where a is the radius of the protein and is the viscosity of the inside of a cell. Motors are quite big proteins

so we take a = 10nm. The viscosity of water 103Pa s. However, cells are full of proteins and othermolecules so to a protein they feel around 100 times thicker than water, so we take a viscosity = 0.1Pa s,inside a cell. Thus we have a diffusion constant D 1013m2s1 = 105nm2s1.

Also, note that the term on the bottom, 6a is the drag coefficient, i.e., if a protein molecule is draggedthrough the cell at a velocity v, the drag on the particle is

drag force = 6av 108v N. (7)

Now, we return to the time the motor takes to diffuse approximately 1 nm when it is in the freely diffusingstate M. The distance diffused is approximately (Dt)1/2 (105t)1/2nm for a protein. This distance equals1nm when t = 105s. We know that on average 1 step takes about 102s. Therefore, diffusion over the

required short distance is easily fast enough to give us the velocities measured in experiment.Now, let us consider the other time, . This is the time it takes to fall down the potential well in state

M. Going to the bottom of the well involves being pulled a distance of order l, i.e., 10 nm. This is in thethick viscous environment of the cell and so it involves doing work against friction. As we have only burnedone ATP molecule we can do no more than 1019J of work. The work done is just force times distance, andwe know the force as a function of the velocity v that we move to the bottom of the potential well. So, wehave that

work done against friction = force l = 6avl 1016v J. (8)

This increases with v of course, for a v = 1000nm s1, the work done is 1022J. This is much less than thework that can be done with a single ATP molecule. So, we have found that our model is consistent with theexperimental data. When we put experimentally measured values for the parameters such as protein size,

ATP energy etc., then we found that the maximum velocity within the model was more than the velocitymeasured in experiment.

If v were 1000 times larger then the drag would be 1000 times larger, as it increases linearly with v, andso 1019 not 1022J would be consumed. This would be the limit to how fast the motor could move. Motorsare not observed to move this fast, however, they are observed to move cargoes, pulling cargoes along willincrease the drag. The drag increases with the radius, so if a motor pulls a cargo about 100 times its size, i.e.,1000 nm, at about 1000 nm s1, then it burns 1021. This is about 10% of the maximum enegrgy in ATP.So, then it is working at about 10% efficiency, which is comparable to the efficiency of an internal combustionengine. (Of course, as the cargo is so small, its weight is not an issue.)

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