Nanomotors: Nanoscale Machines

Nanomotors: Nanoscale machines

October 31, 2016

1 Introduction to nanomotors

In this part of the course we will study nanomotors. First we will deﬁne what we mean by nanomotor. A motor (of any size) is a type of machine for moving in a controlled direction (to transport something) and/or to exerting forces. Examples of motors are trucks that can transport objects such as oranges from a warehouse to a branch of Tescos, and winches for pulling a broken-down car onto the back of a breakdown truck. Moving in a controlled direction and exerting a force go together as generally you need to exert forces to move. Trucks, winches etc, are macroscale, i.e., large machines. Nanomotors have the same function but are microsocopic, they are only nanometres or tens of nanometres across. So nanomotors too transport (tiny) cargos and pull with (tiny) forces. As nanomotors are typically

Figure 1: Announcement of 2016 Nobel Prize in Chemistry, for work on designing and making man- made molecular machines/nanomotors. Although impressive these man-made molecular motors are much less sophisticated than the natural molecular motors that are in our cells.

1 molecules, they are also called molecular machines. Nanomotors consume energy to function, as do motors of all sizes. Note that the 2nd Law of Thermodynamics forbids doing work without consuming energy. However, as we will see here, as they are so small, how they perform their functions of transport and exerting forces, can be very diﬀerent. Scientists working on nanotechnology are working to make man-made nanomotors. So far they have only made very simple nanomotors, although Fraser Stoddart, Jean-Pierre Sauvage and Ben Feringa shared the 2016 Nobel Prize in Chemistry, for developing nanomotors, see Figure 1. However, all living organisms use and rely on a large numbers of biological nanomotors. Our bodies are made of cells and a typical one of our cells contains millions or more of nanomotors, and a muscle cell can contain billions of nanomotors. Muscles can only exert forces because the muscle cells of which they are composed can exert forces, and muscle cells in turn can only exert forces because of the billions of nanomotors they contain. As these nanomotors are (protein) molecules, they are also called molecular motors.

Figure 2: 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 don’t expect you to remember the names of these motors and filaments. However note that the filament is made of rows of molecules arranged in a helical fashion and that each row is made of a repeated (i.e., periodic) sequence of pairs of molecules. The two parts of the pair are coloured in different colours (green and blue, or in B&W photocopy light and dark grey). The microtubule filament is made of a helix of 13 rows of these pairs of molecules. The period is about 8 nm and so a motor can bind at a whole sequence of positions along a microtubule that are 8 nm apart. The Brownian Ratchet model is a generic model, not a detailed model of a specific nanomotor such as dynein or kinesin, but roughly speaking state M corresponds to a motor bound to a filament by only one its two “legs”, which allows motion of the other leg. These two legs are labelled “head domain” above. State M ∗ then corresponds to the motor having both legs bound to the filament.

2 Figure 3: Fluorescence image of a cell, in which actin filaments have been tagged to fluoresce red, and microtubules have been tagged to fluoresce green. When printed in black and white they both look grey, the microtubules are the thinner more crinkly ones. Black and white cannot do justice to multicolour fluorescence images, so please take a look at images online, e.g., Google ‘fluorescence images cells actin’. The cell is a few tens of micrometres across. The cell is full of proteins, RNA, DNA, etc but in a fluorescence image such as this one, you will see molecules that have fluorescent tags on them. Only the actin filaments, the microtubules, and DNA have these fluorescent tags. DNA has a blue tag, which is why there is a blue oval in the image. This is the nucleus, which is where almost all the DNA is. Image is from Wikimedia, by James Faust and David Capco.

2 Like trains need railway tracks, our nanomotors need ﬁla- ments to move along

The motors inside cells move along ‘railtracks’ inside cells, these are long (micrometres long or more) thin (∼ 10 nm) filaments, that criss-cross the cell. One type of filament, called a microtubule, is illustrated in Fig. 2. Note that the filament is made of a regular periodic array of molecules, it is essentially a one-dimensional crystal. Also shown in Fig. 2, are the two types of motors, called dynein and kinesin, that move along filaments of the type shown. I don’t expect you to remember these names but there are three types of motors: dynein, kinesin and actin, that move along two types of filament: microtubules and actin filaments. Dynein and kinesion move along microtubules and perform many functions, including transporting molecules along our very long nerve cells, and moving our chromosomes (our DNA) during cell division. Myosin motors on actin filaments are what generate forces inside our muscles. Just as cars and trucks are there to transport people and goods around, say oranges from a Tesco’s depot to a Tesco’s 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 or a car, and a molecule in solution. Oranges just sit there, as do cars if their engine is off.

3 But molecules diﬀuse around inside liquids. If x(t) is the displacement of a diﬀusing molecule at time t, then RMS (root-mean-square) of x(t) increases with the square root of time, i.e.,

hx2(t)i1/2 ' D1/2t1/2 diffusion (1) because the molecule diffuses. Here D is the diffusion constant. As molecules diffuse, an obvious question is: If molecules move via diffusion anyway, why do you need molecular motors to move them? The answer is that diffusion has no direction, you are as likely to go left as right, and so if a molecules needs to moved in a specific direction, to say a specific part of the cell, then diffusion is not adequate. Also, as we noted earlier, the distance travelled increases only as the square root of time (as opposed to being linear in time as it is for motion in a straight line at constant speed) and so motion over large distances is very slow via diffusion.

2.1 A simple model for nanomotors: The Brownian Ratchet model Real biological molecular motors are complex and poorly understood. They are poorly understood for the simple reason that they are so small, it is difficult for us to study them as they work. Light microscopy, for example, is useless as the motors are much less than the wavelength of light across. However, there is a simple model that illustrates how molecular motors can actually exploit the inevitable diffusion of molecules, to move in a directed way. This is the Brownian ratchet model, which goes back to an idea of Feynman’s in the 1960s. It is also called the diffusive ratchet model. Real motors almost certainly work a little differently but this simple model does illustrate the basic ideas behind motors at the nanoscale. Directed motion can only be done by burning a fuel, so all motors, including molecular motors, burn fuel. Thermodynamics does not allow anything to do work without consuming a fuel, as that would violate the second law of thermodyamics.

2.1.1 Assumptions of the simple Brownian Ratchet model The Brownian Ratchet model is a model of rectified Brownian motion. Brownian motion is another name for diffusion, because one of the first to observe the diffusion of particles in water was the botanist Robert Brown. Rectify means to allow motion in only one direction, we will take this to be to the right, while preventing it in the other direction, to the left. You may have come across rectification in electric circuits, which means to allow the current to only flow in one direction. Here, nanomotors burn fuel (the molecule ATP), in order to rectify diffusion and move in a directed way. Our Brownian ratchet only moves in 1 dimension, which is realistic as the molecular motors move along filaments inside cells, see Fig. 2. Our simple Brownian ratchet model follows from a set of assumptions: (A) The motor molecule can exist in two states: M and M ∗. Burning one ATP molecule is needed to take the molecule once round the cycle, i.e., M → M ∗ → M. So, burning ATP molecules as they go, the molecule can cycle M → M ∗ → M → M ∗ → M → · · ·. (B) The motor can only move in one dimension (which we make the x axis). This is reasonable as real motors move along filaments. (C) In state M, the motor can freely diffuse along the x axis, with a diffusion constant D. By freely diffuse we mean that the potential energy of the motor in state M is a constant so there are no forces on it.

4 Figure 4: Schematic of the potentials (dashed lines) as a function of x, u(x), for both states of the molecular motor: M (top) and M ∗ (bottom). The motor itself is indicated by the black circle. The vertical dotted lines indicate the motor going from state M to M ∗ or vice versa, the horizontal dotted lines indicate diffusion in state M, and the diagonal dotted lines indicate moving to the bottom of the potential well in state M ∗. The progression of the motor shown is: 1) starts in state M ∗ in leftmost potential well, 2) M ∗ → M, 3) diffuses (by chance to the right), 4) M → M ∗, 5) moves to the bottom of the potential that is second from the left, 6) M ∗ → M, 7) diffuses (by chance to the left), 8) moves back to the bottom of the potential that is second from the left.

(D) In state M ∗ it feels the sawtooth potential u(x). See Fig. 4 for schematics of the potential. The potential well is assumed to be deep, i.e., much deeper than the thermal energy kT . Therefore, in state M ∗ the molecule quickly heads towards the minimum and stays there. It is 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. However, on going from right to left there is gradual increase, then a sudden decrease. It is this asymmetry between left-to-right and right-to-left movements, that is going to rectify the motion of the motor.

(E) It takes a time τ + τ ∗ seconds, on average to go round the cycle once. Here τ is the average time the motor spends in state M before converting to M ∗, and τ ∗ is the average time the motor spends in state M ∗ before converting to state M. We assume that the time τ is so small that (Dτ)1/2 l, i.e., that in the time τ it is in state M, it can only diﬀuse a distance much less than the period of the sawtooth potential in state M ∗. Here D is the diﬀusion constant for the molecule in state M.

5 Having deﬁned the model, we will outline how the motor works by going through one cycle, one step at a time.

2.1.2 One cycle in the Brownian Ratchet model We start with the motor in state M ∗ and at the bottom of the potential. Then, the sequence of events that occurs is: 1. A motor in state M ∗ stays in that state only for a time τ ∗, on average, before converting to state M, i.e., M ∗ → M. So after a time τ ∗ the motor is in state M. 2. In state M the potential is flat, so it freely diffuses with diffusion constant D. The motor stays in state M for only a short period of time, on average it stays only for a time τ. Thus, τ seconds after it flipped to state M, it will have diffused a distance of about (Dτ)1/2 — it is equally likely to have diffused to the left or to the right. 3. After a time τ the motor returns to state M ∗. It has either diffused to the left or to the right, each is equally likely, i.e., each has a probability of 1/2. If it has diffused to the left it has only moved a little way, l, to the left. Recall that we assumed that τ is so small that (Dτ)1/2 l. i.e., that in the time τ it can only diffuse a distance much less than the period of the sawtooth potential in state M ∗. So, after a time τ it has moved only l to the left, and therefore on its return to the M ∗ state it will just slide down to the bottom of the same 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 the state M ∗ it is in the next valley along. It will then slide down to the next minimum in the potential to the right, and so will have moved forward a distance l to the right. 4. 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 many times, the net effect is that in half the M ∗ → M → M ∗ cycles the motor has moved l to the right, and in the other half the motor has stayed where 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 τ ∗ + τ seconds the average velocity is 1 l average velocity = . (2) 2 τ ∗ + τ 2.2 Applying the Brownian Ratchet model to understanding nanomotors in cells In cells, the motors run along two types of filament, one of these types is called a microtubule. A microtubule has a crystal-like surface with a periodicity of 8 nm, which sets the length l = 8 nm. Motor speeds have been measured and they are typically a few hundred nm/s. For comparison, for a free protein, i.e., one not bound to a filament, a typical diffusion constant of a protein inside a cell is around D = 10−12m2s−1. The final fact we need is that motors burn one molecule of ATP per 8 nm step and that burning a single ATP molecule releases about 10−19J of energy. Using these numbers we can consider both how motors transport stuff around inside cells, and how they exert forces. Let’s start with transport.

6 2.2.1 Transport in a cell Our longest cells are the nerve cells we use to control our legs. They run from the base of our neck all the way down to our feet. So they are over 1 m long. Consider the transport of a protein molecule down the full length of one of these cells. By diﬀusion this would take a time ' (1 m)2/D = 1/10−12 = 1012s. This is over 10,000 years. This is obviously far too slow. If the protein is instead transported by a motor moving at 100 nm/s, then it takes a time ' (1 m)/10−7) = 107s. This is a few months, which is slow but not unreasonable.

2.2.2 Motor exerting forces In one step a motor burns 1 ATP molecule. This releases 10−19J of energy, and so a motor can do no more than 10−19J of work in a step. As this step is about 10 nm long, the maximum force a single motor can exert is around 10−19J/10 nm= 10−11N. So one motor can exert a force of at most 10 pN. To exert larger forces, multiple motors in parallel are required. For example, it is believed that the jaws of an adult T. rex could bite its prey with a force of around 50, 000 N. This force would require of order 1015 molecular motors pulling in parallel. Motors pulling in parallel add up their forces (like electric currents in parallel). To move distances more than 10 nm, motors in series are needed — motors in series increase the distance moved. So as the jaws of T. rex move tens of cms, it uses many more motors than this to bite down on its prey. Large animals exert large forces using huge numbers of molecular motors, each of which only exerts a tiny force.

Figure 5: Fossilised skull of T. rex in the American Museum of Natural History. Image from Wikimedia.