**Cellular Machinery**

Processes at the molecular length-scale are dominated by stochasticity – the locations of individual protein molecules, and even their number, are heavily influenced by randomness and are inherently unpredictable. Nevertheless, for the cell to function correctly the localisation of many proteins must be controlled at some level, both spatially and temporally. In order to react to chemical signals and stimuli, and to move accordingly, the cell must distinguish front from back. For one cell to divide into two functioning cells, the DNA must be accurately separated so that each daughter is given a full complement of chromosomes. This lecture is concerned with microtubules and their varied functions, from trafficking proteins and other cargo to where they are needed in the cell, to finding and pulling apart duplicated chromosomes into two new cells in mitosis. To do this, individual molecular subunits self-assemble into vast networks and highly ordered structures, spanning the whole cell. The manner in which the cell carries this out gives a fascinating insight into how stochastic noise is suppressed or exploited in biological systems.

Important functions carried out in cells include trafficking (the directional transport of cellular components and fuel to where they are needed within the cell) and mitosis (the fundamental process by which cells divide and replicate). The mechanics of these various functions will be discussed in this lecture (Slide 1).

Slide 1 The structure and motion of cells can be fruitfully explored through physical experimentation and modelling.

### 4.1 Introducing microtubules

Microtubules are biological polymers, the monomeric subunit being a heterodimer consisting of a-tubulin and b-tubulin molecules. Each microtubule *in vivo* is made up of typically 13 microfilaments arranged into a tube (although microtubules consisting of 14 microfilaments have been formed *in vitro*), and each filament is built by adding the dimers in an end-to-end manner. Microtubules play a key role in the structure and function of cells, as cells exploit a phenomenon known as dynamicinstability, where microtubules randomly switch between rapidly growing and rapidly shrinking regimes, as well as specific binding of motor proteins to carry out a varied range of functions.

The dimeric nature of the subunit gives rise to a polarity in the microtubule, with the ‘plus’ end and ‘minus’ end having different polymerisation and depolymerisation kinetics. The video below shows an animation showing the composition of a microtubule.

Slide 2 Polarity of a microtubule

Microtubules are also discussed in the topic ‘Cell structure’ in the lectures ‘Basics of cellular structures and functions’ and ‘Cellular mechanics’.

#### 4.1.1 Dynamic instability

Microtubules exhibit dynamic instability (see video below) whereby filaments can switch between rapidly growing and rapidly shrinking phases. The polymerisation is driven through the hydrolysis of a guanosine triphosphate (GTP**) **molecule associated with each dimer. GTP performs a similar role in the cell to ATP in driving non-equilibrium processes. The conversion of GTP to GDP (guanosine diphosphate) is thought to cause a shape change of the dimer into a bent conformation, which does not allow subsequent polymerisation. When this occurs with a dimer away from the end of a filament, the rigidity of the microtubule prevents the shape change and the microtubule remains stable. However, if the GTP is lost at the end of a filament, the conformation change can cause rapid depolymerisation. This is known as a catastrophe. The microtubule can be ‘rescued’ if the depolymerisation reaches a GTP-rich region whose conformation is stable, and polymerisation can subsequently begin again.

Dynamic instability in a microtubule

The random transitions between growing and shrinking phases has been quantitatively reproduced using a kinetic model, where the rate of adding or removing a tubulin dimer is calculated from the free energy change, ∆*G*. A cartoon of a microtubule growing end is shown in the upper left of Slide 4. By considering the interaction between the neighbours in the microtubule lattice and whether they are tubulin-GTP or -GDP, the probability of adding or removing a subunit can be calculated. The resulting prediction of the behaviour is shown in the right figure for different lattice arrangements The same qualitative behaviour is obtained for all microtubule lattices, with transition rates and velocities depending on whether there are 13 (curves a and c) or 14 (curve b) protofilaments in the lattice, and the relative alignment of dimers between adjacent protofilaments (curves a and c).

The behaviour of microtubules covers a range of lengthscales, creating some problems for modelling these processes. For example, when considering the formation of the mitotic spindle, the micron-scale structure which ensures that chromosomes are equally divided during cell division (lower image on Slide 4), one does not want to have to model the behaviour of individual tubulin dimers; instead, some level of coarse-graining is required. Typically, microtubules or bundles of microtubules are modelled as structures which undergo random transitions between growing and shrinking phases, and the consequences for the cell can be modelled (Slide 5) without having to consider the molecular details of the mechanism of the transitions.

The net rate of change of microtubule length, via addition or subtraction of tubulin dimers, is calculated from the probabilities (*P*_{g} and *P*_{s} respectively) of finding the microtubule in a growing or shrinking phase:

Equation 1a

Equation 1b

where* f*_{res} and *f*_{cat} are the frequencies of rescues and catastrophes, respectively, measured as events per unit time. Rescues and catastrophes are modelled as Poisson processes, with the duration between events being exponentially distributed. This assumes that there is a single rate-limiting step for each transition. The exponential distribution arises from the fact that there is a constant probability per unit time of the transition occurring.

If *V*_{g} is the rate of dimer addition to the end of the microtubule (i.e. the rate of elongation) during the growing phase, and *V*_{s} is the rate of dimer loss from the microtubule end during the shrinking phase, the average rate of change of microtubule length can be expressed as flux, *J*:

Equation 2

hence

Equation 3

If the flux is greater than zero, microtubules will grow more than they shrink and, availability of components and energy notwithstanding, they will grow to an infinite length; however, when *J* < 0, the distribution of lengths will reach a steady state. Overall the microtubules will shrink more than they grow; however, at any given time there will be some microtubules that have grown from zero length and have not yet shrunk back to zero.

The time dependence of the microtubule lengths has been calculated by considering the rate of change of the probability distribution of growing ends, *p _{g}*(

*L,t*), and shrinking ends,

*p*(

_{s}*L,t*) (Slide 6):

Equation 4a

Equation 4b

These equations have been solved for the steady state in the case *J* < 0. However, the result can be derived using physical arguments as follows. Periods of uninterrupted growth will be exponentially distributed with mean period 1/*f*_{cat} and therefore the length added is also exponentially distributed, with mean *V*_{g}*/f*_{cat}. Similarly, the length decreased during a period of shrinkage is exponentially distributed with mean *V*_{s}/*f _{r}*

_{es}. The steady-state distribution of lengths can be obtained by subtracting the distribution of shrinkages from the distribution of growths. For the special case of combining two exponential distributions, another exponential distribution is obtained:

Equation 4b

which is written more compactly as

Equation 5b

where

Equation 6

Thus the distribution of lengths is exponentially distributed and the length, *L*, is the average length of a microtubule.

Dynamic instability is exploited by the cell to exert localised forces to move objects rapidly within the cell, or drive large-scale changes in cell behaviour – for example, during mitosis.

### 4.2 Trafficking and transport

Slide 7 and the related video summarise the key areas covered in this section.

A number of molecular motors specifically recognise and bind to microtubules and move in a directional manner. The polarised nature of microtubules – having a plus end and a minus end, as described in Section 4.1, Slide 2 – and the specificity of recognition means that different motors move in different directions along the microtubule. The kinesins are a family of plus-end-directed motors, while the dynein complex moves towards the minus end. Kinesins and dyneins are recognised by different cargo, allowing for the localisation of proteins which is critical for correct cellular function to occur. From a physical viewpoint, the mechanism of kinesin or dynein stepping along the microtubule is similar to the movement of myosin along actin filaments, with the free energy change from the hydrolysis of an ATP molecule at each step driving the directed motion in a non-equilibrium manner.

The previous lecture in this topic, Molecular Machines Lecture 3 ‘Single molecule methods’, includes a movie of kinesin ‘walking’ along a microtubule.

The movement of kinesin along a microtubule has been visualised by labelling with fluorescent markers. Individual quantum dots bound to kinesins have been observed to move along Texas-Red-labelled microtubules (Slide 8, left image) using the high-contrast technique of total internal reflection fluorescence (TIRF) microscopy (described previously in the lecture ‘Single-molecule methods’). In TIRF microscopy, instead of exciting a fluorescent sample with perpendicular illumination, the angle of incidence of the light source is such that the beam is totally internally reflected with an evanescent wave propagating only tens of nanometres into the sample. Thus only fluorophores very close to the lower coverslip are excited, giving very high contrast and the potential for single molecule detection.

By imaging only nanometres from the coverslip, binding and unbinding events are clearly visible. To illustrate the motion of the quantum dot, the intensity is measured along the length of the microtubule and displayed vertically. The left figure in Slide 8 shows individual fluorescence images every 0.5s. To visualise the movement and velocity of the motor, the intensity of the quantum dot (green channel) is measured along the length of the microtubule and displayed as a single column of pixels. The time frames (captured every 50–200 ms) are then stacked together with time increasing from left to right (central figure). This representation is called a kymograph and it allows trends in the motion to be quickly illustrated.

A standard use of a kymograph is to illustrate linear motion, seen as straight diagonal lines, as shown in the central image of Slide 8, and to measure velocities. The kymograph (centre) also shows stalling events (horizontal line) and unbinding events (lines terminating), and the velocity of the kinesin can be measured from the gradient. By mutating specific amino acids within the kinesin, its ability to move processively along can be impaired. The graphs on the right show that adding an excess concentration of mutated kinesins (*x*-axis) causes a blockage to the normal kinesin, reducing the velocity but not the run length. In the two graphs the black squares show the values calculated by fitting the velocity and run length distributions with models (Gaussian distribution for velocity, exponential distribution for run length), while the red circles show the directly calculated mean values. Both of these measures show a decrease in velocity as the concentration of mutant kinesin increases (x-axis), while the run length remains relatively unaffected. This implies that the kinesin stalls in a strongly bound state when there is a queue ahead of it.

#### 4.2.1 Active diffusion

The binding of a molecular motor and its cargo to a microtubule is not irreversible. Rather, the motor binds for short runs along the microtubule before unbinding, and freely diffusing until another binding event occurs, as shown in the cartoon at the left of Slide 9.

This cycle of binding, directed motion, unbinding and diffusion can be modelled as a biased random walk on a potential gradient. When the microtubules have a defined polarity with respect to the geometry of the cell – for example, if the minus-ends are nucleated at the centre of the cell and the plus-ends extend towards the cell periphery – the concentration of plus-end directed motors is enhanced at the plus-ends of the microtubules (left figure in Slide 9). For parallel arrays of microtubules, cargo will be directed preferentially towards one end of the cell (left figure). Thus concentration gradients of any cargo will be set up within the cell.

This has been demonstrated experimentally by growing microtubule asters (arrays emanating radially from a centre) between two coverslips and observing the behavior of motors within the array (right figure in Slide 9). The kinesin motors aggregate at the centre of the asters where there is a concentration of plus-ends. As the number of microtubules in the aster is varied (increasing left to right), the localisation of the kinesin becomes more pronounced at the centre.

In contrast to the directed trafficking from cell centre to cell periphery, the case of non-polarised lattices of microtubules has also been considered theoretically, where the polarity of individual bundles of microtubules alternates along the lattice (see Slide 9). In this case there will be no concentration gradient created since the lattices are isotropic. However, the motors and any cargo experience enhances diffusion across the lattice.

Thermal fluctuations cause particles to move around randomly; this Brownian motion is a result of collisions with the many molecules surrounding the particle. The statistics of Brownian motion have been studied in detail, with the mean squared displacement (MSD) increasing linearly with time. In the case of thermally driven fluctuations, the diffusion coefficient, extracted from the gradient of the MSD, is dependent on the size of the particle, the viscosity of the medium and the temperature. If, however, the particle motion is driven by some other energy source, the ‘diffusion’ constant must be interpreted differently. In passive diffusion, the direction of motion of a particle is randomised by collisions with other particles at high frequency. In contrast, in active diffusion, particles are driven to make steps of larger distance before their direction is randomised. In the case of motors binding to lattices of microtubules, the step distance before randomisation is defined by how long the motors remain bound to the microtubule before detaching and how fast the motors process along the microtubule.

A particle undergoing Brownian motion moves in a three-dimentional random walk (Slide 10) where the mean squared displacement (msd) after *N* steps, (*r _{N}r*

^{2}) is directly proportional to the time,

*t*, allowed:

Equation 7

where *D* is the diffusion coefficient (Brownian motion and diffusion are discussed in more detail in the thermodynamics topic: Lecture 2 ‘Brownian motion’).

Measurement of the concentration gradient allows the diffusion constant to be calculated. One-dimensional simulations of motors moving through the isotropic lattice, using the binding (directed motion) unbinding (diffusion) model reveal two regimes in the MSD (left figure of Slide 9). The lowest curve indicates the diffusion of unbound motors, with a power law of unity and one-dimensional diffusion constant *D*_{ub}. The remaining curves show motion of motors which bind to the lattice with decreasing unbinding rates (10^{–2}, 10^{–3}, 10^{–4} from lower to upper). For short times the motion is superdiffusive (MSD power law >1) as the motors bind to the lattice and move in a directed manner for short lengths of time.

At times much longer than the typical run duration before unbinding, the diffusive behaviour is recovered but with a greatly increased diffusion constant. The crossover time between diffusive and superdiffusive motion increases as the average run duration increases, as expected. A number of geometries of microtubule arrays have been studied *in silico*,and classes perform one-dimensional (A1, A2) or two-dimensional transport (B1,B2). The active diffusion coefficient depends on the precise system architecture. In A2 and B2, each stripe contains filaments of one polarity (orientation of plus-end to minus-end). This means that motors which unbind are more likely to rebind and continue moving in the same direction as before, giving a larger active diffusion coefficient than the architectures A1 and B1, where each stripe contains filaments of both orientations.

Thus networks of randomly oriented microtubules can distribute cargo, such as organelles and vesicles, more quickly than random diffusion would allow, overcoming the viscosity of the intracellular environment. The effective step distance of the random walk is determined by the velocity of the motor and the probability of remaining bound each time the motor advances. Since *D* ~ *r*^{2}/*t*, the active diffusion constant is proportional to the square of the motor velocity rather than the inverse of the particle size, so large vesicles may be distributed in this manner.

### 4.3 Multiple motors

For important or large cargo, multiple motors bind to a microtubule or a bundle of microtubules to ensure that the cargo remains bound to the fibre. Slide 11 describes an experiment where kinesin motors are bound to polystyrene beads to measure the dynamics of the motors moving along microtubules grown *in vitro*. By varying the concentration of kinesin, the number of active motors on a single bead can be controlled.

The upper figure in Slide 11 shows the run length distributions for nine kinesin concentrations. At the lowest concentration the run length is measured to have an approximately exponential distribution (top left, concentration 0.1 μg ml^{–1}_{ }of kinesin), implying that only a single motor is binding the bead to the microtubule at a time, and unbinding from the microtubule is governed by a single rate-limiting step. As the number of motors attached to the cargo increases (increasing kinesin concentration), the run length distribution is less well fit by an exponential distribution; motors which unbind remain close to the microtubule due to the remaining bound motors, and thus are more likely to rebind, extending the run length. The increase in average run length with increasing kinesin concentration is summarised in the central lower figure (blue dots). In contrast, the motor concentration does not affect the velocity of the beads along the microtubule (figure bottom-left, black squares), which is determined by the rate of stepping of the kinesin motor.

As described above (Slide 10), increasing the run length greatly increases the active diffusion constant. A similar effect on the run length was found for the quantum-dot-labelled kinesin experiments described earlier (Slide 8) as the number of kinesins per quantum dot was increased. The effect is shown in the upper figure in Slide 12, in qualitative agreement with the experiment of Beeg et al.

Behaviour of greater complexity is observed when the cargo is bound to motors with opposing directionalities (lower figure in Slide 12). In this case the cargo can move backwards and forwards via a tug-of-war mechanism. Since unbinding rates of motors are dependent on the force exerted, motors of the opposite directionality to the current direction of motion are less likely to remain bound, stabilising runs in a given direction (plus or minus). Thus there is positive feedback maintaining binding of cooperative motors, while fluctuations in the number of bound motors means that the direction of motion will switch at random.