Cell division

This final lecture of the molecular machines topic is about the machinery of mitosis, the division of one cell into two identical daughter cells (Slide 1). The lecture builds on material presented in the previous lecture (‘Cellular machinery’) and describes how the mechanisms of mitosis are studied using simulations and in vitro.


5.1 Overview of mitosis

The process of mitosis is divided into several descriptive stages, which are in general the same across different species and different cells. These are summarised in Slide 2.


There are many helpful animations showing the stages of mitosis, which include these:

http://www.cellsalive.com/mitosis.htm

http://www.youtube.com/watch?v=VlN7K1-9QB0>

Mitosis begins with prophase, where the nuclear DNA condenses from an essentially random polymer into distinct chromosomes. The DNA has been duplicated prior to cell division so there are two copies – so-called sister chromatids which are held together at a point known as the centromere. Microtubules nucleate at two centrosomes located on the outside of the nucleus. The polymerisation of overlapping microtubules between the centrosomes pushes them apart to form the mitotic spindle.

In prometaphase, the nuclear membrane disintegrates and microtubules emanating from the spindle poles attach to the centromere of each chromosome. For correct division, each chromosome requires one sister chromatid to be attached to each spindle pole. The resultant of the pulling forces from each pole causes the chromosomes to align equidistant from each spindle pole in metaphase, lying on a plane known as the metaphase plate.

Only when every chromosome has the correct attachments to each spindle pole are the protein connections between sister chromatids cleaved, and the two sets of chromosomes are pulled apart towards each spindle pole. This stage is called anaphase. When the two sets are fully separated, the cell membrane begins to contract and be pinched in where the metaphase plate was located in a process called cytokinesis. A contractile ring of actin and myosin becomes smaller and smaller until the two nuclei are pinched off into separate cells.

Each phase of mitosis is highly orchestrated, involving a host of interacting proteins, and consuming energy to drive the system irreversibly from equilibrium. However, physical principles can give insight into each stage of mitosis. Chromsome condensation is driven in part by protein complexes called condensins, which are thought to act by modifying the cross-linking and supercoiling properties of the DNA. Dynamic instability in microtubules, and the random transitions between rapidly growing and even more rapidly shrinking regimes, is an efficient method of exploring the volume within the cell and enables the chromosomes to quickly become attached to the spindle poles. Cytokinesis is thought to occur via positive feedback between acto-myosin monomers, driving a change in shape of the membrane.

The remainder of this lecture focuses on the formation and maintenance of the mitotic spindle, the structure responsible for aligning and separating the DNA into two identical sets.

5.2 The mitotic spindle

As described above, as well as maintaining cell structure and polarity during interphase, microtubules are critically important in cell division. A structure known as themitotic spindleself-assembles from nucleated microtubules. It is visualised in a time-lapse movie by modifying HeLa cells (a human cell line derived from cervical cancer cells, widely used as a model system) to produce a fusion of α-tubulin and mCherry, a fluorescent protein (Slide 3). This fluorescent tubulin remains functional and is incorporated into microtubule structures such as the mitotic spindle.


In the movie the HeLa cells also express a fusion of histone H2B and green fluorescent protein (GFP). Histones are responsible for organising the higher-order structure of DNA into chromatin fibres, and so H2B-GFP is used to visualise DNA in green. The spindle is responsible for separating the chromosomes correctly into the two daughter cells. The dynamic instability is used to explore the cellular compartment rapidly to find and capture chromosomes and bring them to the centre of the cell. Bundles of parallel microtubules from each spindle pole attach to a protein complex on each chromosome known as the kinetochore, arranging the DNA into a metaphase ‘plate’ in the centre of the cell, which is seen as a bright green line in the centre of dividing cells. The cartoon (bottom right of Slide 3) shows a cell in metaphase, with every chromosome attached to both spindle poles via bundles of microtubules. The blue tubes outside the cell represent retraction fibres, bundles of filamentous actin which form when the cell rounds up from its previously flat shape.

Slide 4 shows an enlarged region of the movie in Slide 3; both tubulin and DNA are shown in the first video, while the tubulin alone is shown underneath that, allowing the spindle to be seen.


Tubulin and DNA

Tubulin, showing spindles

Strong feedback in signalling pathways ensures that anaphase, where the chromosomes are pulled apart ready to form two new cells, does not occur until every chromosome is correctly attached to the spindle. This is known as the spindle assembly checkpoint, or spindle checkpoint, and results in the DNA lining up as a bright green line in metaphase until every chromosome is attached to both spindle poles, ready for separation into two new cells. A cascade of protein degradation triggers the transition to anaphase, which involves cleavage of the cohesin links, protein complexes which join duplicated chromosomes together, and subsequent elongation of the spindle to pull the chromosomes apart, ready for cytokinesis where two cells are formed.

The critical importance of cell division with equal sharing of chromosomes between the daughter cells means that there is strong selective pressure for complex and robust mechanisms to evolve, with checkpoints and redundant mechanisms in place to ensure correct operation. Nevertheless, the basic mechanisms of spindle formation can be studied from first principles using simulations and in vitro work (Slide 5). In this case the complexities of signalling pathways and the vast number of proteins involved in spindle formation are removed, and the minimal conditions for the creation of spindle-like structures are considered.


Adding multimeric motor proteins and ATP (as the energy source) to purified solutions of microtubules causes assembly into striking patterns, visible with dark-field microscopy (see Slide 5), where bundles of microtubules are seen as bright objects against a dark background. This system allows known concentrations of microtubules, and plus- and minus-end directed motors, to be combined and the result observed. Thus we can study the minimal conditions for spindle formation.

As described in the previous lecture (‘Cellular machinery’), microtubule motors can be classified by the direction in which they travel, either plus-end or minus-end directed. Kinesin is a plus-end directed motor, while Ncd is a mitotic motor which moves towards the minus end of microtubules. Ncd is a member of the kinesin family, as is the case for a number of genes. The name – a shortened form of non-claret disjunctional – arises from the effect, or phenotype, of removing the gene from the fruitfly Drosophila melanogaster.

Individual motors interact with only one microtubule at a time. The motors are added as multimers or complexes of a single class; having multiple motor heads on a single particle allows microtubules to interact with each other through the motor complexes. When two parallel (measured by the plus to minus direction) microtubules are bound to a motor complex, the complex will pull the microtubules together into a bundle as it moves along them. If two antiparallel microtubules are bound to the same motor complex, they will be slid apart by the action of the motors, pushed in opposite directions. The end result from adding only plus-end directed motor complexes are spherical structures with microtubules arranged radially around a central concentration of motor complexes. The microtubule plus-ends are at the centres and the minus-ends point outwards. If only minus-end directed motors are added to microtubules, inverted structures are formed, with microtubule minus-ends at the centre with the motor complexes, and plus-ends distributed radially. In vivo, these star-like structures are known as asters and are structurally similar to one pole of the mitotic spindle.

In Slide 5 the upper figure displays experimental results obtained by varying the concentration of motor proteins. The left panel (labelled d) has Ncd motors dominating kinesin activity, giving rise to the radial asters described above. Similarly, when kinesin dominates Ncd (right panel, f), aster-like structures again form, this time with kinesin motors concentrated at the centres. When the motor activities are of a similar magnitude, a new structure forms with bundles of microtubules having kinesin motors at one end and Ncd at the other (central panel, e). The structures formed in the presence of both plus- and minus-end directed motors are visually similar to the mitotic spindle, suggesting that spindle self-assembly is possible within a window of parameter space.

The results are qualitatively reproduced in silico (lower panel of Slide 5). When either minus-end motors (blue dots) or plus-end motors (green dots) are added alone, radial structures are formed (left and right figures, respectively). When both classes of motor are present, spindle-like bridges form between clusters of motor types. From a physicist’s point of view, these ‘active gels’ are fundamentally interesting to study in their own right, having properties which diverge from classical gels due to the potential mobility of the cross-links, and the consumption of energy (hydrolysis of ATP by the motor complexes) driving the system away from equilibrium. However, there are important differences between these in vitro spindles and those which assemble in living cells. The artificial spindles have a defined polarity whereby the plus-ends of the microtubules are at one pole, with a concentration of plus-end directed motor complexes, and the minus-ends and complexes are at the other pole. In contrast, functional spindles are symmetric; microtubule minus-ends nucleate at each pole, and plus-ends from opposite poles overlap at the centre of the spindle in an antiparallel fashion. An additional ingredient is required in simulations of spindle formation to stabilise the centre of the spindle against collapse.

5.3 Control of spindle length

Fully formed spindles are in a state of dynamic equilibrium (Slide 6); it is known from photobleaching experiments that there is a ‘poleward flux’ of tubulin, plus-end directed motor complexes cross-link and push apart the antiparallel microtubules in the spindle centre. Depolymerisation at the spindle poles balances the poleward flux to maintain a steady spindle length.


The following consideration of spindle length control focuses on the balance of polymerisation and depolymerisation within the spindle and the forces produced from these. The relative importance of viscous or inertial forces in a fluid depends on factors such as the viscosity and the typical lengthscale and velocity under consideration. The ratio of inertial to viscous forces is given by the Reynolds number, Re (Slide 7):

Equation 1
Equation 1


where L is the lengthscale, v is the velocity, µ is the dynamic viscosity and ρ is the fluid density. Thus if the Reynolds number is greater than 1, inertial forces are more important than viscous forces, whereas a Reynolds number less than 1 means that viscous effects are more important.


Fluid dynamics at the cellular scale are in the low Reynolds number, viscous regime, where force and velocity are linearly related; this allows sliding velocities of motors to be combined within the spindle.

Simply stated, depolymerisation at the spindle poles (where minus-ends of microtubules are nucleated) will act to decrease the spindle length, S, at a rate Vdepol, while the antiparallel overlapping microtubules at the centre of the spindle are pushed apart by motor complexes, lengthening the spindle at a rate Vsliding. This behaviour is encapsulated in Equation 2 (Slide 8):

Equation 2
Equation 2



In a similar manner, the size of the overlapping region in the centre of the spindle, L, is increased by polymerisation (at a rate Vpoly) at the plus-ends and decreased by the antiparallel sliding:

Equation 3
Equation 3


In the viscous regime, elongation (or contraction) of the spindle length at a constant rate requires the application of a constant force. Considering the forces acting in the spindle, the most prominent are the force exerted by the sliding motors at the centre of the spindle, Fsliding, the force due to chromosome attachments, Fkt (labelled ‘kt’ for the kinetochore, the complex which mediates the interaction between microtubule and chromosome), and an elastic force arising from deformations of microtubules and other structural proteins, denoted Ftension. The rate of expansion of the spindle in the viscous regime (low Reynolds number) is related to balance of forces pushing the spindle poles apart:

Equation 4
Equation 4


In Equation 4, µ is the viscous drag coefficient of moving the spindle through the cytoplasm. Each component of the force will now be described in more detail. The sliding force that can be exerted by motors in the antiparallel overlap (Slide 9) is dependent on the sliding velocity (when moving at maximal velocity the motors do not exert a force):

Equation 5
Equation 5


where α is a parameter representing kinesin activity, measured as force per unit length of overlap (L). The spindle is modelled as a simple spring with spring constant β representing the stiffness of the spindle and restoring force:

Equation 6
Equation 6



When chromosomes attach to microtubules, through the kinetochore, the microtubules exert a pulling force which brings the chromosome towards the centre of the spindle. This force is assumed to be a constant:

Equation 7
Equation 7


To reach a steady state it is necessary to couple the depolymerisation rate at the poles with the sliding force. To do this Goshima et al. (see references above) have made the assumption that the rate of depolymerisation depends on the proximity of the minus-end to the centrosome located at the spindle pole. Brownian motion of a polymer ratchet yields the depolymerisation velocity at the poles as:

Equation 8
Equation 8


In Equation 2, Vd,max is the maximal depolymerisation rate, which must be supplied from observations such as fluorescent speckle microscopy (see Slide 12). Vd,0 is the depolymerisation rate, which is observed even in the absence of a sliding force. The physical interpretation of this equation is that the microtubule minus-ends depolymerise fastest when they are close to the spindle pole (the parameter δ reflects the range of the activity of the depolymerising motors), and this distance depends probabilistically on the sliding force which pushes the microtubules towards the pole.

Then, by setting dS/dt and dL/dt to zero, equations 2, 3 and 4 can be solved straightforwardly by substitution to give expressions for the steady-state values of the spindle length and the length of antiparallel overlap (Slide 10):

Equation 9
Equation 9


Equation 10
Equation 10



These expressions give testable predictions of the effect of various proteins on spindle stability (upper image of Slide 11,). For example, the model predicts that the spindle length, S, is insensitive to α, the activity of the sliding motors. This is confirmed experimentally (lower image of Slide 11, green circles) by removing or ‘knocking out’ the existing motor gene and expressing varying levels of the motor fused to green fluorescent protein (GFP): the motor concentration (visualised as the GFP intensity) does not influence the spindle length. However, below a threshold motor level, the spindles collapse into a single spindle pole (blue line) as predicted in the theoretical graph when the level of antiparallel overlap, L, required exceeds the spindle length, S (left of the three small graphs). The central graph shows that the spindle stiffness parameter, β, affects the spindle length but not the length of the antiparallel overlap, and the right graph shows that the pulling force exerted by the kinetochores on the chromosomes would cause the spindle to collapse if the value of Fkt were too large.


For further details of the spindle length model, see Goshima et al. (reference given above).

5.4 Anaphase onset

When every chromosome is correctly bound to the spindle such that one sister chromatid is attached to each spindle pole via a bundle of microtubules, the duplicated chromosomes must be pulled apart so before the cell can divide into two, each must have a complete set of chromosomes. This stage of cell division is referred to as anaphase. Again, due to the importance of this process occurring in a robust manner, cells typically have several degenerate mechanisms for achieving this. Different organisms tend to have slightly different methods but there are some general processes.

In early anaphase, known as anaphase A, the chromosomes are pulled towards the spindle poles. After this, in anaphase B, the spindle elongates to separate the chromosomes further. Speckle microscopy uses a low concentration of fluorescent tubulin, meaning that individual points within the spindle can be followed over time.

Kymographs (Slide 12) show that in the frame of the cell the rate of tubulin moving outwards from the spindle centre remains constant between metaphase and anaphase, indicating that antiparallel sliding is unaffected by the anaphase transition. However, the poleward flux of tubulin is reduced; the movement of the speckles in the frame of the pole decreases, as depicted in the figure below the image.


The graph in Slide 13 plots measurements of poleward flux rates (y-axis) against rate of spindle elongation (x-axis) for a large number of observed cells (grey dots), and it shows that the rate of spindle elongation is negatively correlated with the speed of the poleward flux in an approximately linear way. This observation is consistent with the model that during anaphase the polymerisation and antiparallel sliding at the centre of the spindle remain constant, and spindle elongation results from a reduction in the rate of depolymerisation at the spindle poles (orange dots). The model is shown schematically in the lower figure.


The steady state of the spindle is depicted in the upper image in Slide 13, with the balance between polymerisation/sliding at the spindle centre and depolymerisation at the poles maintaining a constant length. When anaphase is triggered by degradation of the protein Cyclin B, depolymerisation at the poles is inhibited, elongating the spindle. The more effective the inhibition of depolymerisation at the poles, the faster the rate of spindle elongation.

5.5 Summary of molecular machines

As this is the final lecture in the molecular machines topic, it is instructive to summarise the main points covered.

  • Almost all processes within the cell are carried out by molecular machines. At the cellular level the behaviour is dominated by stochastic interactions and probabilistic transitions.
  • Theoretical modelling of molecular machines often describes the system in terms of a probability density in parameter space.
  • Monte Carlo simulations are commonly used to study the emergence of complex behaviour in terms of simpler rules governing the system. The physics of molecular machines often spans a range of lengthscales. To model the behaviour at mesoscopic or macroscopic lengthscales, assumptions must be made to simplify the model at the molecular scale.
  • New imaging techniques, such as TIRF, AFM and optical tweezers, allow single-molecule experiments to be performed with unprecedented precision.
  • The cell can exploit noisy behaviour – for example, the dynamic instability of microtubules to map out the cellular volume and to ‘search and capture’ chromosomes during mitosis.
  • Processes critical to correct cellular function often have two or more degenerate mechanisms to ensure that nothing goes wrong. When the cell cannot rely on probabilistic events, the large free energy change from the hydrolysis of ATP to ADP is used to drive the system from equilibrium.
  • The complexity of cell division arises from the direct evolutionary pressure to carry this out correctly. Nevertheless, consideration of physical principles such as the forces within the mitotic spindle can give insights into the mechanisms at work.