Microtubules 〚22〛 are tubular-shaped supramolecular assemblies, with inner and outer diameters of 16 and 24 nm respectively, and several microns in length. They arise from the self-assembly of a protein, tubulin, by way of reactions involving the hydrolysis of a nucleotide, guanosine triphosphate (GTP), to guanosine diphosphate (GDP). Microtubules can be formed in vitro by warming a solution containing purified tubulin and GTP from about 7 to 36 °C. A series of chemical reactions occur, tubulin assembles into microtubules, and GTP is hydrolysed to GDP. This reaction continues by processes involving microtubule disassembly and reassembly, where tubulin is added to one extremity of a microtubule and is lost from the other 〚33〛 (Fig. 1). There is hence a continual consumption or dissipation of chemical energy through the system.

Under appropriate buffer conditions, reaction–diffusion processes give rise to spontaneous macroscopic ordering 〚24–32〛. Following assembly in spectrophotometer cells, a series of periodic horizontal stripes of about 1 mm separation progressively develops over about 5 h (Fig. 2) 〚24〛. Once formed, the striped pattern remains stationary for between 48 to72 h, after which the system runs out of reactants. In each striped band the microtubules are highly oriented at either 45° or 135°, but adjacent stripes differ in having opposite orientations. The periodic variations in microtubule orientation are readily observed as birefringent stripes that, depending on their orientation, have either a yellow or blue interference colour. This orientational pattern coincides with an identical pattern of microtubule concentration, in which the microtubule concentration drops by about 25%, and then rises again, every time the microtubule orientation flips from acute to obtuse 〚30〛 Another series of stripes of about 100 μm separation occurs within the 0.5 mm stripes. These in their turn, contain another set of stripes of about 20 μm separation (Fig. 3A). At distances below this, there is another level of organisation of about 5 μm periodicity and with care it is possible to observe aligned microtubule arrays of about 1 μm separation (Fig. 3B).

Striped morphologies occur when the microtubules are prepared in upright sample containers (Fig. 2A), but a different pattern arises when they are prepared in the same containers lying flat (Fig. 2B). Once formed, the structures are stationary and independent of their orientation with respect to gravity. This behaviour is attributed to the determining role of the direction of the gravitational field during structure formation. So as to test this hypothesis, microtubules were assembled in flat horizontal containers fixed to the turntable of a record player, with the long axis of the sample container along the direction of the centrifugal field (0.14 g) 〚26〛. A striped morphology once again forms, and the direction of the stripes is perpendicular to that of the applied centrifugal field.

To establish at what moment the sample morphology depends upon the orientation with respect to gravity, the following experiment was carried out 〚27〛. Twenty samples of purified tubulin together with GTP, and at 4 °C, were placed in identical optical cells. Microtubule formation was simultaneously instigated with all the cells upright. Consecutive cells were turned from vertical to horizontal at 1 min intervals, and the samples examined 12 h later after the structures had formed. Twenty minutes after instigating microtubule formation, when the last sample was turned from vertical to horizontal, there are no obvious signs of striped structure. Since the structures forms while all the cells are flat, one might expect that they would all form the horizontal pattern. This is the case for samples inverted during the first few minutes. However, samples that were upright for 6 min or more showed striped morphologies similar to preparations that remained vertical all the time. The final morphology of the sample depends upon whether the sample container was horizontal or vertical, at a critical time 6 min after instigating assembly, at an early stage in the self-organising process. The process can be described as a bifurcation between pathways, leading to two different morphological states, and in which the direction of the sample with respect to gravity determines which morphology subsequently forms.

Preparations formed in a rocket of the European Space Agency, under conditions of very low gravity (10^–4 g) for the first 13 min of the self-organising process, showed no self-organisation 〚31〛. The experiment was carried out in the following way. Twenty-six flight samples were contained in an experimental module divided into two compartments, a ‘microgravity’ compartment and a ‘1 g on-board’ centrifuge compartment. Sample preparations were installed in the payload before lift-off and maintained at 7 °C. Once microgravity conditions were obtained, the sample temperature was raised to 36 °C, thus instigating microtubule formation, and the 1 g centrifuge switched on. Just prior to re-entry, 13 min after low gravity was achieved, the centrifuge was stopped. Hence, the only difference between the two sets of samples, is that one set was subject to weightlessness during the first 13 min. The payload was recovered, and the experimental module reinstalled in the laboratory at the launch site within 2 h from lift-off. The samples were then left for another 4 h before the module was opened and the samples examined. Some samples were filmed during these 6 h.

The samples formed in the ‘centrifuge’ part of the module formed stripes when the centrifugal field was parallel to the long axis of the cell, and the circular morphology when it was perpendicular. This result shows that the self-organising process is unaffected by payload re-entry and recovery. In contrast, the samples formed in the ‘microgravity’ part of the module show practically no self-organisation (Fig. 2C). The ‘microgravity’ preparations possess only very weak birefringence, demonstrating that the microtubules in these preparations do not have any preferred orientation. Observations down to a distance of 10 μm taken under a polarising microscope showed similarly weak birefringence. Hence, individual microtubules are relatively disordered with respect to one another. This contrasts with the ‘1 g’ preparations that show strong optical birefringence and in which many microtubules are highly oriented along the same direction. Hence in this case, gravity actually triggers the self-organising process. The gravity direction breaks the symmetry of the initially homogenous state and leads to the emergence of form and pattern.

Experimental problems frequently occur in space experiments due to air bubbles. In our case, although care was taken to prevent it, small air bubbles formed in the neck of some of the sample containers. During re-entry, when the sample is subject to high centrifugal fields, the air bubble is pushed through the sample. In one sample this process was filmed. A strongly birefringent line formed along the trajectory of the air bubble. This shows that the bubble oriented the microtubules along its trajectory. Subsequently, striped regions limited in extent, developed perpendicular to this trajectory (Fig. 2D). Hence, orienting the microtubules at an early stage in the process can also trigger self-organisation. We were expecting that self-organisation would not occur in the absence of gravity, and we found the effect of the air bubbles both intriguing and instructive.

The reaction scheme for microtubule formation is approximately the following (Fig. 4). Tubulin is incorporated into microtubules from the tubulin–GTP complex, either by addition to the growing end of a microtubule or by nucleation of a new microtubule that subsequently grows. This is associated with hydrolysis and the tubulin is incorporated into the microtubule as tubulin–GDP. Likewise tubulin is lost from the shrinking end of a microtubule as the tubulin–GDP complex. As a microtubule grows from one end, while shrinking from the other, it appears to move. The shrinking end of the microtubule leaves behind it a trail having a high local concentration of tubulin–GDP. Likewise, as it grows and incorporates tubulin–GTP from the front, it will also create a zone locally depleted in tubulin–GTP.

The tubulin–GTP complex promotes the formation of microtubules whereas the tubulin–GDP complex inhibits their formation. Once liberated, the inactive tubulin (tubulin–GDP) is progressively regenerated into the active form (tubulin–GTP) with a certain time constant, after which it is once again available to be incorporated into microtubules. During this time, the tubulin is free to diffuse. If the rate of tubulin diffusion was zero, then the tubulin would be found along exactly the same trail as the disassembled tail of the microtubule from which it came. Hence, the microtubules that either grow or reform with this tubulin will do so along exactly the same trail as before. Under these conditions, since the microtubule solution that initially forms from the tubulin solution is macroscopically homogenous, the solution will remain as before, and self-organisation will not occur. Likewise, if tubulin diffusion is very fast, then by the time the tubulin–GDP becomes tubulin–GTP, the solution is once again homogenous in tubulin–GTP. Microtubules will reform or grow in a random manner and once again there will be no self-organisation. If, however, the tubulin diffusion has an intermediate value, then self-organisation can occur. Tubulin will progressively diffuse out from the trails left by shrinking microtubules. Since rates of chemical reaction increase with increasing concentration, growing microtubules will progressively grow along the direction and in the regions where these trails exist. Likewise microtubule nucleation will be greater in a zone of high tubulin–GTP concentration. These microtubules in their turn will leave behind new trails of high tubulin–GDP concentration and so on. The growing end of microtubules will likewise create regions depleted in tubulin–GTP, and these will behave in a complementary manner. Once regions of increased microtubule concentration and of preferred orientation start to form, then the feedback mechanism outlined above will lead to a progressive reinforcement of this process. In such a way, the microtubules ‘talk to each other’ and the coupling of reaction with diffusion then progressively leads to a collective behaviour resulting in self-organisation.

When the microtubule solution first forms from the tubulin preparation, it is isotropic. However, this isotropic arrangement is unstable. Once a few microtubules have taken up a preferred orientation, then neighbouring microtubules will start to grow into the same direction. Orientational order then spreads to its neighbours and so on. Hence any small external effect that will preferentially orient just a few microtubules will trigger progressive self-organisation. As outlined above, the passage of an air bubble through the solution orients the microtubules along its path and triggers self-organisation in its wake.

Using the same argument, we would expect that magnetic fields will also trigger self-organisation. Gravity acts by way of its directional interaction with the macroscopic density fluctuations present in the solution. This will result in ‘diffusion’, which is faster in the up–down direction, thus causing a slight directional bias for the formation of microtubules that triggers self-organisation. According to Kondepudi and Prigogine 〚10, 11〛, the presence of a 1 g field in a suitable reaction–diffusion system can destabilise the equilibrium state at the bifurcation point and thus favour the formation of a macroscopic pattern.

Obviously we wanted to confirm the plausibility of this hypothesis. One approach is experimental verification. However it is difficult to devise experiments that would unambiguously demonstrate the presence and role of the chemical trails outlined above. Another approach is to numerically simulate the behaviour of a population of growing and shrinking microtubules using experimentally determined values of reaction rates and diffusion constants, and to compare this behaviour with experimental observations.

Robert et al. 〚34〛 have proposed a simple model for microtubule self-organisation based upon microtubule growth and tubulin diffusion, and involving approximately ten microtubules. However, in this model the reactive term is extremely simplified and unrealistic. In this model, the microtubules do not nucleate from the tubulin solution but were pre-positioned with a given orientation from a region simulating the centriole. We have developed a reaction–diffusion model based upon realistic reaction kinetics, and extended the simulations to a large population of microtubules.

Numerical simulations were carried out based upon the following premises. Microtubules can grow and shrink from their ends. Tubulin molecules are added as the complex tubulin–GTP at a rate dependent on the local concentration and the reaction rate. Tubulin–GDP is lost from the opposing end. The liberated tubulin–GDP diffuses into the surrounding medium with a given diffusion constant. Due to the large excess of GTP, it is converted to tubulin–GTP at a given rate. This tubulin GTP is available to be incorporated into the growing end of neighbouring microtubules, or of nucleating to form new microtubules.

The following equations represent the reaction scheme; despite their simplified appearance all three are of non-linear form.

Microtubule assembly: n (GTP·tubulin) + microtubulem → microtubule_m+n

Microtubule disassembly: microtubule_m+n → microtubulem + n (GDP·tubulin) + n Pi

Tubulin regeneration: (GDP·tubulin) + GTP → (GTP·tubulin) + GDP

The microtubule length is not fixed; microtubules are free to grow and shrink according to the reaction scheme outlined above. The simulations incorporated experimentally determined rates of reaction and diffusion taken from the literature 〚35–41〛, and they reproduced the kinetics of microtubule assembly as observed in the self-organising process. To limit calculations to a manageable size, we restricted dimensions to two, even though the experimental samples obviously have three. However, reducing the dimensions to two does not seem to cause any major changes in the calculated self-organising process.

Initially, simulations were carried out on a population containing a small number of microtubules and then extended to larger populations. Obviously, one of the things that we wanted to know was whether the chemical trails hypothesised above might be predicted by such numerical simulations. To test this hypothesis, simulations were carried out with five microtubules contained on a two-dimensional surface, 100 μm × 60 μm. Two microtubules of 10 μm length (a, b) were placed close together, parallel to one another, and with their shrinking ends pointing in the same direction. Behind these microtubules were placed the remaining microtubules (c, d, e). These microtubules were also parallel to one another, and to the first two microtubules, but were significantly further apart from each one another than were microtubules a and b (Fig. 5). The concentration of tubulin–GTP and tubulin–GDP is initially uniform. The simulation was then started. Fig. 5 shows the position of the microtubules and the concentration profiles of tubulin–GTP and tubulin–GDP, at different times. All the microtubules progressively change position, due to tubulin–GTP being added at the growing end, and tubulin–GDP being liberated at the shrinking end, at a rate of about 5 μm per minute. During this process, the microtubules form regions both at the growing end and down the sides, which are depleted in tubulin–GTP. Because of this, the two forward microtubules (a, b) that are close together grow apart. This is because it is more favourable for them to grow into regions rich in tubulin–GTP than into the regions depleted in tubulin–GTP caused by the growth of the nearby neighbour. The shrinking ends of the microtubules liberate tubulin–GDP that is progressively converted to tubulin–GTP. Hence, just behind each microtubule is a small region of high tubulin–GDP concentration. This is followed by a long diffuse trail of tubulin–GTP. The growing ends of the three backward microtubules (c, d, e) grow into the tubulin–GTP trails formed by the forward microtubules (a, b). Fig. 6 illustrates the manner by which, in the simulations, an individual microtubule directs itself towards regions of high tubulin–GTP concentration.

We then extended the simulations, to a population of about 30 000 microtubules contained on a surface of 100 μm × 100 μm. In experiments, microtubules initially form from an homogenous solution of tubulin and GTP. So as to imitate these conditions, we started the simulation with a uniform distribution of tubulin–GTP of given concentration, no tubulin GDP, and no microtubules. Nucleation of tubulin–GTP into microtubule germs was taken as being exponentially dependent on tubulin–GTP and tubulin–GDP concentrations, as proposed by Mandelkow et al. 〚41〛. The small microtubule fragments generated this way are initially homogenous and have no preferred orientation. These numerous small fragments are free, by the mechanism already described, either to grow into microtubules, or shrink and disappear. The microtubules that develop, attain an average length of about 10 μm, which is comparable to the experimentally-determined value in the self-organised preparation. Likewise, in agreement with experiment, the microtubules that first form have no preferred orientation and their distribution is uniform. Subsequently, a self-organised structure comprised of regular bands of about 5 μm separation progressively develops (Fig. 7A). This structure is comparable with the type of experimental structure that develops over a similar distance scale. In the simulation, regular variations in microtubule concentration arise because of the progressive build-up of regions that are depleted in tubulin–GTP. Self-organisation takes about 240 000 iterations and corresponds to about 2 h in real time. In experiments self-organisation takes about 5 h.

One of the features of these simulations is that the stripes always formed along the diagonal of the square reaction space. This led us to suspect that an asymmetry favouring this orientation was somehow unwittingly built into the simulation. This turned out to be the case, a very slight asymmetry occurred in the way that diffusion was digitalised as a square wave front, and this small effect sufficed to favour the formation of self-organised stripes along the diagonal. This type of asymmetry is well known in numerical simulations involving diffusion. When we eliminated it, the simulations show concentration inhomogeneities of about 5 μm, but which are uncorrelated and disorganised (Fig. 7B). When an asymmetry was once again introduced into the simulation by using slightly different rates of diffusion in the x and y directions, then, as shown in Fig. 7C, macroscopic self-organisation once again occurred.

These numerical simulations demonstrate that a relatively simple scheme, based on experimental rates of reaction and diffusion, predict the overall experimental behaviour. They hence permit a link between the microscopic and macroscopic, and between the molecular and phenomenological. The simulations confirm the hypothesis that self-organisation arises by the formation of anisotropic chemical trails produced as microtubules grow and shrink from opposing ends. Ordering arises not from static interactions but is due to the formation by reactive processes of concentration gradients in chemicals that activate and inhibit the formation of neighbouring microtubules. In this way, microtubules communicate with one another and behave as a macroscopic collective ensemble.

One of the fundamental assumptions of this reaction–diffusion model is that microtubules can change their direction of growth as a function of tubulin concentration gradients. A probable consequence is that, in some parts of the preparation, some of the microtubules will be partially curved. This is indeed the case, electron microscope images of the self-organised preparation show a significant number of partially curved microtubules. The maximum radius of curvature where these microtubules bend is approximately 0.3 μm, significantly higher than the maximum value of 3 μm permitted in the numerical algorithm.

If microtubules were unable to change their direction of growth, remaining completely straight, then self-organisation by the mechanism proposed should not occur. In agreement with this, simulations incorporating this constraint did not result in any self-organisation. This fact, taken with the experimental observation of a significant number of curved microtubules, is supplementary evidence that the reaction–diffusion scheme outlined in this article is a realistic description of the self-organising process.

The simulations also predict another major experimental observation; how self-organisation is dependent upon weak external fields. Any small effect that creates a slight preference for some microtubules to grow along one direction will trigger macroscopic self-organisation. The simulations hence provide a molecular basis for explaining how gravity, magnetic fields, and shearing can act on the system. Magnetic fields and shearing, would act by slightly favouring some microtubule orientations, whereas gravity would act by introducing an asymmetry into molecular diffusion and transport.

In this system there are at least two significant differences from the type of reaction–diffusion scheme originally proposed by Turing. In the Turing system the molecules communicate with one another across the sample by way of diffusion (fast diffusion of the inhibitor and slow diffusion of the activator). In the microtubule system, the molecules communicate by way of the reactivity of the microtubules. The fact that the microtubules are growing from one end at the rate of the order of a micron per minute whilst shrinking by approximately the same, results in a displacement of the microtubule at the same speed. At the same time, they leave behind a trail having a high concentration in inactive tubulin and create in front of them a furrow depleted in active tubulin. These chemical trails determine the pathways taken by surrounding microtubules. It is a reaction–diffusion system, since as already explained, without tubulin diffusion at the appropriate rate, self-organisation would not occur. In terms of a pure Turing system, active tubulin (tubulin–GTP) plays the role of the activator, whereas inactive tubulin (tubulin–GDP) is the inhibitor. However, in this case they both have the same diffusion constant. Self-organisation occurs because displacement of the microtubules by reaction is faster than diffusion of tubulin.

The second major difference with a normal Turing system is the reactive anisotropy of the microtubule system. In a normal reaction–diffusion scheme, the reaction has no inherent anisotropy. This is not the case with microtubules that can obviously only react at their ends. A microtubule can move in only one direction and this will equally lead to anisotropic reactive trails in active and inactive tubulin along the same direction. The system has an in in-built propensity for symmetry breaking under the effect of a weak field or some other external influence.

In this article, for reasons of presentation, we have somewhat arbitrarily separated numerical simulations from both experimental observations and the hypothetical molecular mechanism. In reality, the situation was somewhat different in that initially the proposed molecular basis of self-organisation was qualitative and rather uncertain in its details. As the simulations developed, the model progressively sharpened, leading to new insights. The combination of experiment and numerical simulation has resulted in a considerable deepening of our understanding of the phenomena involved.

Dans les organismes vivants, l’ordre et les formes apparaissent spontanément à partir d’un œuf d’apparence initialement homogène et dépourvu d’ordre. Les mécanismes biologiques par lesquels se fait cette auto-organisation sont encore mal compris. Il existe deux approches physico-chimiques possibles pouvant aboutir à de l’auto-organisation en biologie : la première est fondée sur la physique statistique et les interactions statiques, la deuxième sur les dynamiques chimiques non linéaires et les phénomènes coopératifs. Rashevsky, en 1942, puis Turing, en 1952, ont proposé que, contre toute attente, des réactions chimiques couplées à des processus de transport puissent induire des variations périodiques de concentration des réactifs. Ceci est une conséquence des systèmes chimiques maintenus éloignés de l’équilibre par une consommation permanente d’énergie. Ces systèmes sont instables et peuvent bifurquer vers plusieurs états possibles. Prigogine a prédit que l’action d’un champ externe faible à un moment précis de la réaction pouvait permettre la bifurcation du système vers un état auto-organisé. Ces mécanismes pourraient bien constituer le fondement de la morphogenèse et de l’auto-organisation dans les êtres vivants.

Nous savons que la formation in vitro de microtubules, éléments essentiels du squelette cellulaire, suit ce type de comportement. En effet, les préparations de microtubules s’auto-organisent spontanément par des processus de réaction–diffusion et les morphologies qui se forment dépendent de la présence de la gravité à un moment critique, précoce dans le processus réactif. On peut montrer que le processus d’auto-organisation microtubulaire contient effectivement une composante diffusive et une partie réactive, constituée notamment de l’assemblage et du désassemblage des microtubules.

La théorie nous a permis de comprendre le processus d’auto-organisation et sa dépendance vis-à-vis des champs externes dans sa globalité, mais il nous manquait l’approche d’un point de vue moléculaire, c’est-à-dire en termes de microtubules individuels. Ceci satisferait l’approche moléculaire des chimistes et des biologistes, et surtout permettrait d’établir un lien entre les processus biologiques macroscopiques et la foule d’événements microscopiques non linéaires qui les constituent.

Ces considérations nous ont amenés à réaliser un modèle mathématique du processus d’auto-organisation microtubulaire. Le modèle simule l’évolution d’une population de microtubules ayant chacun une cinétique réaliste propre (assemblage et désassemblage à chaque extrémité) et pouvant communiquer entre eux par l’intermédiaire de la tubuline, libre de diffuser. Les microtubules, après une phase initiale d’assemblage, sont entretenus par ajout de tubuline à l’extrémité croissante, tandis qu’ils se désassemblent à l’autre extrémité, en relâchant derrière eux une traînée concentrée de tubuline. Cette trace concentrée est « sentie » par les microtubules voisins ; ceux-ci peuvent alors croître dans la trace. Le mécanisme décrit permet de synchroniser localement les microtubules entre eux. Si quelques microtubules ont à un moment une orientation privilégiée, alors les microtubules voisins vont croître dans la même direction. Dès lors, n’importe quel petit effet qui provoque un biais directionnel chez les microtubules provoque l’auto-organisation globale de la solution. La gravité agit dans ce sens, par son interaction directionnelle sur les fluctuations de densité présentes dans la solution à un moment critique du processus.

Nous montrons lors des simulations que le comportement collectif de microtubules individuels permet d’expliquer l’auto-organisation de la solution et le rôle des champs externes dans ce processus.

Ce travail illustre bien le dialogue permanent et la complémentarité entre les expériences et les simulations, qui a été nécessaire pour la compréhension des mécanismes de l’auto-organisation en biologie.