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Biol Cell (1990) 68, 177-181 177 ~t) Elsevier, Paris Original article Self organization of the microtubule network. A diffusion based model Claudine Robert ~, Mohamed Bouchiba 2, Raoul Robert 3, Robert L Margolis 4, Didier Job 5 / Facultd de Mddecine, Ddpartement de Statistiques et lnformatique, Domaine de la Merci, 38700 La Tronche, France; 2Facultd des Sciences de Tunis, Ddpartement de Mathdmatiques, Campus universitaire 1060 Tunis, Tunisia; 3Laboratoire d'Analyse Numdrique, Universitd de Lyon I, 43 Bd du 11 novembre 1918, 69622 Villeurbanne Cedex, France; 4Fred Hutchinson Cancer Research Center, 1124 Columbia Street, Seattle, WA 98104, USA; SlNSERM, U 244, D~parternent de Recherche Fondamentale, Centre d't~tudes Nucldaires 85 X, 38041 Grenoble Cedex, France (Received 9 November 1989; accepted 17 April 1990) Summary - Microtubules form organized polymer networks in cells. Experimental evidence indicates that their mechanical proper- ties do not play a significant role in the generation of such regular patterns. This spatial organization seems closely related to their dynamic behavior. Here we use mathematical modeling to define conditions under which microtubular dynamics result in self organiza- tion. We demonstrate that random diffusional processes can generate regular microtubule organizations under specified kinetic con- ditions which are found to be compatible with the known properties of tubulin polymers. The organizing forces are the tubulin concentration gradients which are generated by the polymer growth. The present analysis has been restricted to the phase of establish- ment of the microtubule network. The same conceptual framework, applied to steady state pattern maintenance suggests that the kinetic requirements for self organization might ultimately be responsible for such extraordinary in vivo microtubule dynamics, as the rapid turnover and "dynamic instability" of the interphase network. microlubules / self organization / diffusion / model Introduction It is currently considered that microtubules are the prin- cipal organizers of intracellular space [1]. Consequently, the mechanisms that might produce and stabilize a specific arrangement of microtubules in cells have recently been subjected to intensive experimental [3, 11, 13, 15, 16] and theoretical [13] scrutiny. In general, the microtubule net- work originates from centrosomes. It has been proposed that the specific orientation of the polymer network and the changes in cell morphology that might result are regulated by centrosome orientation [12, 13, 22] and the action of various extrinsic [11, 15] and intrinsic [13, 16] factors which might trigger localized stability transitions in polymers [11, 13]. An underlying but untested assump- tion in this regard is that polymer growth is a self organiz- ing process .which is capable of reacting in an ordered manner to specific signals. Microtubules were long considered as rigid entities because they were often observed to follow relatively straight cellular pathways. This view is rapidly changing in the light of new microscopic methods which show that they are dynamic moieties [26] with a remarkably rapid in vivo subunit turnover [19-21]. Furthermore, there is now substantial evidence that the configuration of microtubules is not primarily governed by their intrinsic rigidity, but is related to their dynamics. It has been firmly established that in ceils, stable inac- tive microtubules have a highly tortuous configuration [3, 9, 21, 25]. These polymers make all sorts of U turns knots and sharp angle turns. These observations are important because 1), they show that microtubules are flexible structures ; and 2), that they most probably are normally organized by dynamic pro- cesses. This is the most straigtforward interpretation of the anarchic aspect of the subsets of non dynamic polymers. It has been shown that self organization can arise from nonlinear thermodynamics leading to structures which are energy dissipative [17]. Microtubules could very well form dissipative structures [14] because they continuously con- sume energy to maintain their structure and under certain conditions, behave as nonlinear oscillating systems [4, 14, 18, 24]. The present paper does not consider this aspect of microtubule self organization. It is restricted to the relatively simple case of the establishement of the microtubular network. During this phase of rapid polymer growth, the kinetics of the system are most probably linear, as has been extensively demonstrated in vitro [5]. In the present paper we show that elementary laws of diffusion suffice to confer self organizing properties on rapidly growing microtubules. Methods In the absence of other strong organizing forces, the vector of microtubule growth can only be governed by diffusion laws. We have chosen mathematical modeling to elucidate the polymer distribution mechanism, since direct experimental observ- ations of growing or steady state polymers in an undisturbed

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Biol Cell (1990) 68, 177-181 177 ~t) Elsevier, Paris

Original article

Self organization of the microtubule network. A diffusion based model

Claudine Robert ~, Mohamed Bouchiba 2, Raoul Robert 3, Robert L Margolis 4, Didier Job 5

/ Facultd de Mddecine, Ddpartement de Statistiques et lnformatique, Domaine de la Merci, 38700 La Tronche, France; 2 Facultd des Sciences de Tunis, Ddpartement de Mathdmatiques, Campus universitaire 1060 Tunis, Tunisia; 3 Laboratoire d'Analyse Numdrique, Universitd de Lyon I, 43 Bd du 11 novembre 1918, 69622 Villeurbanne Cedex, France; 4Fred Hutchinson Cancer Research Center, 1124 Columbia Street, Seattle, WA 98104, USA; S lNSERM, U 244, D~parternent

de Recherche Fondamentale, Centre d't~tudes Nucldaires 85 X, 38041 Grenoble Cedex, France

(Received 9 November 1989; accepted 17 April 1990)

Summary - Microtubules form organized polymer networks in cells. Experimental evidence indicates that their mechanical proper- ties do not play a significant role in the generation of such regular patterns. This spatial organization seems closely related to their dynamic behavior. Here we use mathematical modeling to define conditions under which microtubular dynamics result in self organiza- tion. We demonstrate that random diffusional processes can generate regular microtubule organizations under specified kinetic con- ditions which are found to be compatible with the known properties of tubulin polymers. The organizing forces are the tubulin concentration gradients which are generated by the polymer growth. The present analysis has been restricted to the phase of establish- ment of the microtubule network. The same conceptual framework, applied to steady state pattern maintenance suggests that the kinetic requirements for self organization might ultimately be responsible for such extraordinary in vivo microtubule dynamics, as the rapid turnover and "dynamic instability" of the interphase network.

microlubules / self organization / diffusion / model

I n t r o d u c t i o n

It is currently considered that microtubules are the prin- cipal organizers of intracellular space [1]. Consequently, the mechanisms that might produce and stabilize a specific arrangement of microtubules in cells have recently been subjected to intensive experimental [3, 11, 13, 15, 16] and theoretical [13] scrutiny. In general, the microtubule net- work originates from centrosomes. It has been proposed that the specific orientation of the polymer network and the changes in cell morphology that might result are regulated by centrosome orientation [12, 13, 22] and the action of various extrinsic [11, 15] and intrinsic [13, 16] factors which might trigger localized stability transitions in polymers [11, 13]. An underlying but untested assump- tion in this regard is that polymer growth is a self organiz- ing process .which is capable of reacting in an ordered manner to specific signals.

Microtubules were long considered as rigid entities because they were often observed to follow relatively straight cellular pathways. This view is rapidly changing in the light of new microscopic methods which show that they are dynamic moieties [26] with a remarkably rapid in vivo subunit turnover [19-21].

Furthermore, there is now substantial evidence that the configuration of microtubules is not primarily governed by their intrinsic rigidity, but is related to their dynamics.

It has been firmly established that in ceils, stable inac- tive microtubules have a highly tortuous configuration [3, 9, 21, 25]. These polymers make all sorts o f U turns knots and sharp angle turns.

These observations are important because 1), they show that microtubules are flexible structures ; and 2), that they most probably are normally organized by dynamic pro- cesses. This is the most straigtforward interpretation of the anarchic aspect of the subsets of non dynamic polymers.

It has been shown that self organization can arise from nonlinear thermodynamics leading to structures which are energy dissipative [17]. Microtubules could very well form dissipative structures [14] because they continuously con- sume energy to maintain their structure and under certain conditions, behave as nonlinear oscillating systems [4, 14, 18, 24]. The present paper does not consider this aspect of microtubule self organization. It is restricted to the relatively simple case of the establishement of the microtubular network. During this phase of rapid polymer growth, the kinetics of the system are most probably linear, as has been extensively demonstrated in vitro [5].

In the present paper we show that elementary laws of diffusion suffice to confer self organizing properties on rapidly growing microtubules.

M e t h o d s

In the absence of other strong organizing forces, the vector of microtubule growth can only be governed by diffusion laws.

We have chosen mathematical modeling to elucidate the polymer distribution mechanism, since direct experimental observ- ations of growing or steady state polymers in an undisturbed

178 C Robert et al

medium would be quite difficult to obtain. Mathematical models and computer simulations are on the contrary well suited to this particular case. The laws of diffusion that underlie the model- ing are well established, and the generated patterns are qualitat- ively independent of wide variance in numerical values attributed to the relevant paramaters.

We have modeled the behavior of polymers growing in a finite medium (a circle, representing an idealized cell) and originating from a small region (representing the centrosome). At time zero the concentration of building blocks (tubulin) is considered to be homogeneous. As polymer growth proceeds, the resulting local tubulin concentrations have been calculated using the classical law of Fick. The rate of growth of the polymer is constrained to be proportional to the local tubulin concentration : a hypothesis which is substantially supported by experimental data [5].

Other physical assumptions are that the fluid medium is at steady state so that convection movements and polymer diffu- sion are negligible. The only dynamic processes involved are tubulin diffusion and microtubule growth. The plausibility of these assumptions and the underlying kinetic requirements are discussed below.

We denote x, a vector of the 3-dimensional space ; t, the time variable ; and O an open bounded domain which represents the part of the cell filled by the tubulin solution. We denote 0(t,x) the concentration of tubulin molecules in solution at time t and position x and X(t) the position at time t of a growing microtubule end.

But for the influence of microtubule growth, the tubulin con- centration follows the Fick's law of diffusion, and so 0 satisfies the standard diffusion equation:

80 (I) 8-] - Coa0 = 0,

where c o is a diffusion constant and A stands for the Laplacian operator.

80 At the boundary ~D o l d we fix Neumann's condition (~n = 0)

which states that there is no flux of tubulin through the cell membrane.

The subsequent evolution of the tubulin concentration is governed by the microtubule dynamics.

Every microtubule end X(t) is supposed to satisfy the com- monly used equation:

dX ~0 - c~0

(2) d t - I V0 [

which states that the speed of growth of the microtubule is pro- portional to the concentration O(t,X(t)), its direction being given by the gradient ~70 of 0.

The fact that microtubule growth depletes the neighbouring fluid from tubulin molecules has also to be accounted for. This is easily done by introducing a negative source term in equation (l) :

80 (3) ~ - CoA0 = - cl0~(x- X(t)),

where cl is another constant and the function ~o(x) is a smooth approximation of the Dirac mass at 0.

Finally choosing a value ao(x) for the initial tubulin concen- tration and the starting positions of the microtubules, we ob- tain a complete set of equations (SN) which modelizes the simultaneous growth of N microtubules:

n 80 - CoZa0 = - c10 ~ ~ ( x - Xi(t)), ~n = 0 on 8a,

i = l

dX i V0 (SN) d t - ca0 I V0[ (t, Xi(t)), i = I . . .N,

O(O,x) = 9o(xL Xi(O) = Xio, i = I... N.

It can be shown [2] that (SN) leads to a well posed mathematical problem: for any initial concentration 0o(x) and starting positions Xio ..... XNo in Q, there is a unique solution of (S n) where Xl(t )... Xn(t ) are smooth functions of t for all t.

Roughly speaking (SN) iS a dynamic system with memory, the memory effect being given by the diffusion process. The existence of global (in time) solutions of (S N) does not depend on the values of the positive constants c o, ct, c 2, whose dimensions are respectively LZT - J, L3T - i, L4T - ~M- I.

We have used the model (S N) to perform numerical simula- tion. A detailed analysis of the computational process will be published elsewhere [2]. Briefly, the simulation is bidimensional, we rescale the variables, take for 12 a disk of radius one and 00 as identical to 1. The qualitative features of the patterns generated by the simulations do not depend on the particular values assigned to the new non-dimensional constants Co, C~, C 2 (see the details of the dimensional analysis in the discussion below). We must nevertheless choose values that fit with time and spatial discretiz- ation steps. For small N (N < 10), we perform the computations with CoZC t z0.5 and C2= 1, with a time step dt within the in- terval 0.1_>dt>0.01 and a spatial step of about 0.05.

R e s u l l s

Graphic results o f s imulat ions are presented in figures 1 - 5 . Figure 1 shows the pa t te rn generated by po lymers grow-

ing by distal subuni t add i t i on f rom nuclea t ion sites which are regularly and radial ly a r ranged a round the centrosome. The first obse rva t ion is tha t the po lymers grow l inear ly . This is not due to intr insic mechanica l r igidi ty , but solely to the fact that the po lymer i za t i on process modi f ies the local tubul in concen t ra t ion in such a way that it is maxi- mal jus t ahead o f the p o l y m e r ' s g rowing end. The overal l s y m m e t r y and s impl ic i ty o f the f igure look decept ively na tura l . The final pa t t e rn is ac tua l ly cond i t i oned by the initial o r i en ta t ion o f seeding sites, and by subs tan t ia l "c ross - ta lk ing" among polymers . The lat ter case is created by mod i f i ca t ions in free tubul in d i s t r ibu t ion genera ted by the growth o f neighbouring polymers . These influences are dep ic ted in f igures 2 and 3.

In f igure 2 the seeding sites have been or ien ted in a biased manne r t oward the left. The result shows there is a complex in terp lay between cen t ro soma i ins t ruct ions and the spon t aneous t endency o f f i laments to a r range themselves in an o rde red and symmet r ica l manner .

A n ext reme example o f these conf l ic t ing inf luences is shown in figure 3, where we have mode led the init ial g rowth as inward f rom rad ia l ly , d ispersed sites o f or igin. Microtubules begin to grow as specified, but rapidly escape and again genera te a symmet r i ca l pa t t e rn which rad ia tes ou tward .

Figure 4 shows the influence o f an obstacle to diffusion. Here , g rowth ini t iates at the cen t ro some and the ini t ial ly radia l d i rec t ion o f g rowth has been r a n d o m l y pe r tu rbed . The growing po lymers are nevertheless regular ly spaced and skirt the obstacle. Similarly, they begin to bend as they a p p r o a c h the cell m e m b r a n e .

F ina l ly , f igure 5 shows the effect o f a synchron i sm on the g rowth o f mic ro tubu les . The po lymers s tar t g rowing one af ter another with a shift t ime dt = 0.01 which is abou t 3% o f the to ta l g rowing t ime.

Microtubule network organization 179

We see that while figure 5 is slightly different from figure 1 it displays the same global tendancies to grow out- ward and fill all cellular space.

Discussion-Conclusion

The main conclusion of these studies is that, in theory, an organized polymer network can arise from diffusion effects alone, under the influence of gradients of freely diffusible effectors. Such effectors may be tubulin subunits as above, or MAPs or other factors whose effects would mimic a local modification of the tubulin concentration. The polymers in turn would integrate, in a 3-dimensional pattern, such informational input from all parts of the cell.

The validity of this conclusion in real systems depends on the relatives rates of tubulin diffusion and microtubule growth. They should be of the same order of magnitude. A value of Co" 6.10 -8 cm ' / s has been determined for the diffusion coefficent of tubulin in vivo [19]. The half-

diffusion time of tubulin, tl/2, in a sphere of radius R = 20 ~tm can then be derived from the classical equation: tl/2z R' /4c o. One finds tl/2 = 16 s. For the rate of growth, values as high as 50/~m/min have recently been observed in vivo [8]. It is true that this value is extreme, but we are not aware of any other experimental determination in vivo. No one really knows how fast microtubules grow during the phase of establishement of a network because this period of the cell life is too transient to have ever been directly observed. It is certainly realistic to consider that microtubule formation is much more rapid at this time, when all tubulin is in dimers, than at any other period of the cell cycle. Another condition, though, is that the microtubule growth has a perceptible effect on the in- tracellular tubulin gradients• In other words, accurately rescaled, the values of c o, c~ and c 2 should be comparable to the set of values used in the computer simulation.

From the value of the microtubule growth rate used above and a mean value for the tubulin concentration of 2 mg/ml, one can derive an estimation of c 2 : c2= 0.4 10- t

Fig 1

/ \

\ .

Fig 2 Fig 3

/ /

~'~.~ (_.t~ j

Fig 5 Fig 4

Fig 1. The 8 starting positions of the microtubules are equally distributed on a circle of radius 0. I. The initial direction of growth is radial, dt = 0.04. Ten iterations (computed elementary spatial steps) have been performed. Fig 2. Same as 1, except that the eight starting positions are equally distributed on a quarter of the circle. Fig 3. The 8 starting positions are equally distributed on a circle of radius 0.3. The initial direction of growth is inward, toward the circle's center; dt =0.03. Fifteen iterations have been performed. Fig 4. Sixteen starting position have been equally distributed on a circle of radius 0.1. The initially radial direction of growth has been randomly perturbed. The value for dt is 0.03. Twenty iterations have been performed. The flux of tubulin through the boundary of the obstacle is supposed to be nil. Fig 5. The eight starting positions are as an 1. The microtubules start growing one after another with a shift of time dt = 0.01. We begin by the one indicated by the arrow and go on clockwise. The step of time is dt=0.01 and we perform thirty iterations.

180 c Robert et al

cm4/g/s. To obtain c t we just have to multiply c 2 by the linear mass density of the polymer, which is known to be about 27.10 -13 g /cm. Doing so we obtain cl= 1.10 10 -13 cm3/s. Dimensional analysis with characteristic time T, length L and concentration O, yields the fol lowing re la t ionsh ips : C 0 = c o T / L 2, C ] = c ] T / L 3, C2=c2GT/L ; with T = 10 s, L = 10~m, 0 = 2 mg/ml, one finds C0z 0.6, Ci= 10 -3, C2z 0.8. Roughly speaking, C I has to be multiplied by the number of microtubules in the cell to account for the overall tubulin consumption due to the polymerization process. Taking a number of microtubules of about 500, we find final values in agree- ment with those used for computer simulation. Thus the real system can potentially fulfill the basic requirements of our model. However, we should stress the fact that the mathematical analysis in [2] shows that the qualitative behavior of the system does not depend on the different values of the parameters; there is no bifurcation in the model. Introducing much faster tubulin diffusion coef- ficients and slower rates for microtubules growth would need more spatial and time points for the computat ions and would progressively lead to a prohibitive compu- tational cost.

Finally, we present a quantitative argument to support the assumption that convection movements are negligible. If we perform the same change of scale as the one which was made for the diffusion equation (giving a normalized diffusion constant of 0.6), a fluid of kinematic viscosity v get a normalized viscosity v n = v T / L 2 = 10 7 V ( T = 10 S, L = 10 -3 cm). Taking for granted that the cytoplasm viscosity is greater than that of water (v = 0.01 cm-'/s) we see that the normalized viscosity is greater than 105 . This shows that at the scale of length and time where we work in our model the viscosity of the fluid appears so high that free convection cannot exist and we can only possibly observe some form of highly strained viscous motion.

In conclusion, a diffusion based model for the establish- ment of the microtubule network, is compatible with known data. It does not by itself solve problem of the generation of stable patterns at steady state. Several re- quirements have to be fulfilled to avoid the degeneration of an established network : it should be dynamic, and its kinetics have to be more rapide than those of polymer dif- fusion. Furthermore, the steady stage state microtubule dynamics should be such as to generate favourable tubulin gradients.

As a wlqole, these considerations suggest a rationale for the rapid turnover and the dynamic instability [13, 16] of the microtubules in interphasic cells: according to a dif- fusion based model, stable microtubules which escape the morphogenetic tubulin concentration gradients will degenerate into disorganized strands. As outlined above several studies have actually shown that the subset of polymers in interphase cells that remain indefinitely stable are distinguishable as randomly curled and disorganized [3, 9, 21, 251.

In our model, we have deliberately neglected the spon- taneous tendency of polymers in solution to line up [6, 7]. In the case of microtubule in vitro, this phenomenon ap- pears within a 5 to 15 min delay [10], whereas we can realistically assume that the phase of establishment of the microtubule network does not last more than about one minute.

It is also clear that a number of additional influences should be considered after the initial establishment of the network. Thus intracellular membrane systems and microtubules seem to reciprocally stabilize their spatial ar-

rangement [23]. Also, selective microtubule stabilization through end binding to specialized structures most pro- bably plays an important role [13]. Finally, nonlinear kinetic properties of microtubules could results in very strong forces for organization. Work in this direction is now in progress.

Acknowledgments

We are grateful to J Tabony and J Haiech for useful discussions during this work.

References

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Microtubule network organization 181

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