- Email: [email protected]
Geometric features of microtubule dynamics
Physica A 257 (1998) 156–164
Geometric features of microtubule dynamics Silvina Ponce-Dawson a; ∗ , John E. Pearson b , William N. Reynolds c a Departamento
de Fsica and I.A.F.E., Facultad de Ciencias Exactas y Naturales, U.B.A., Ciudad Universitaria, Pabellon I, (1428) Buenos Aires, Argentina b XCM MS F645, Computational Science Methods, Applied Theoretical Physics Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA c Bios Group L.P., 317 Paseo de Peralta, Santa Fe, NM 87501, USA
Abstract Microtubules are long and sti polymers that form the cytoskeleton of eucaryotic cells. They perform a series of tasks, such as determining the cell shape and providing a network of “rails” along which molecular motors transport organelles to di erent parts of the cell. They are particularly important during the process of cell division, since they provide the forces by which replicated chromosomes are segregated into what will be the two daughter cells. Microtubules are formed from a protein called tubulin and undergo a process called dynamic instability. In this paper we study, via numerical simulations of some simpli ed models, how the interaction c 1998 between microtubules and the di usion of free tubulin a ects their spatial organization. Elsevier Science B.V. All rights reserved. Keywords: Microtubules; Dynamic instability; Spatial organization
1. Introduction All eucaryotic cells (i.e., cells that have a nucleus) have a network of protein laments which is called the cytoskeleton [1]. This network is highly dynamic and allows the cell to change its shape and to carry out coordinated and directed movements. Three types of protein laments form the cytoskeleton, among them, the microtubules. Microtubules are long, sti polymers that are made out of a protein called tubulin. They are thought to be the primary organizers of the cytoskeleton. They extend throughout the cytoplasm and govern the location of organelles and other cell components. Microtubules are polar structures, with a plus end capable of rapid growth and a minus end that, in most cells, is stabilized by embedding in a structure called the centrosome. Microtubules are highly dynamic: after growing outward for many minutes by adding tubulin dimers to their plus ends, they may suddenly start to lose subunits, ∗
Corresponding author. E-mail: [email protected].
c 1998 Elsevier Science B.V. All rights reserved. 0378-4371/98/$19.00 Copyright PII: S 0 3 7 8 - 4 3 7 1 ( 9 8 ) 0 0 1 3 8 - 1
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shrinking very fast and, in many cases, even disappearing. While this happens, new microtubules are formed, that emanate from the centrosome. These rapid shortenings are called catastrophes. The process by which they occur is called dynamic instability [2] and it is thought to be related to the hydrolization of GTP, which binds to the tubulin subunits. Microtubules are involved in cell division, intracellular transport, positioning of cell membranes and the determination of cellular shape. For example, after chromosome replication, the nuclear envelope breaks down and the duplicated chromosomes are distributed more or less randomly throughout the cell, which now has two centrosomes. Before the cell can divide, the duplicated chromosomes must be segregated. During mitosis, (the segregation process) microtubule laments emanating from both centrosomes (also called poles) form a bipolar spindle. The microtubules nucleate at the poles and then randomly grow and shrink until they attach to a pair of replicated chromosomes. The microtubules in the spindle serve as rails along which the chromosomes can move. The chromosomes rst migrate to the cell equator, then separate and migrate to the poles, all via microtubule mediated forces [4]. The stage is set for the cell to divide only after this segregation process is complete. During mitosis the microtubule array is much more dynamic than during interphase (the period before mitosis) [1]. On the other hand, while in interphase the centrosome nucleates long microtubules that extend throughout the cytoplasm, at mitosis there is a larger number of shorter and less stable microtubules. The rate at which dynamic instability occurs increases dramatically during mitosis. For this reason, the half-life of an average microtubule goes from about 5 min in interphase to 15 s in mitosis. It is thought that this is due to a steep increase in the probability that a typical growing microtubule will convert to a shrinking one. At the same time there is, apparently, a change in the centrosome itself that allows the nucleation of a larger number of microtubules. All these changes help the microtubules perform their task during mitosis, which is the capture of the replicated chromosomes. As in mitosis, dynamic instability is a necessary part of many of the other microtubule functions. In this paper we are interested in studying what kind of spatial organization the interaction between the polymerization=depolymerization of the microtubules and the di usion of tubulin can give rise to. In order to study this, we have developed two simpli ed models. These models have many similarities with those studied in [3], but while the studies reported in [3] mostly focused on the properties of the microtubule distribution in the direction of growth, in this paper we will also discuss the structure perpendicular to this direction. In both models the dynamics occurs on a twodimensional rectangular domain in which microtubules can grow only from one side of the domain in a direction perpendicular to it. The results of our simulations show that, due to a competition between neighboring microtubules, the spatial distribution in the direction perpendicular to that of growth is highly non-uniform. As reported in [3] we also nd that, depending on the parameters, two “populations” form: one of many short tubes and another one of rapidly growing longer ones. The time at which this population of rapidly growing tubes is formed is
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highly a ected by the distance between tubes perpendicular to the direction of growth. In particular, we could observe that upon increasing the separation between nucleating sites the formation of this population of “escaping” tubes is delayed so that the distinction between two “classes” of tubes cannot be done. We also observe that there is always an optimal distance between nucleating sites at which more “escaping” tubes are formed. This gives a typical length-scale whose change with the various parameters we have studied. In particular, we have found that for time large enough this length-scale is inversely proportional to the square root of the velocity of growth. We have also observed that the shorter tubes are not characterized by a nite length-scale: the closer the nucleating sites, the larger is the number of tubes that can be formed.
2. The models The rst model we have developed is a lattice gas in which tubulin particles perform a random walk, in order to simulate di usion, and can attach with a certain probability to the growing plus ends of the microtubules. On the other hand, these plus ends, with a certain probability can also lose tubulin subunits which are then incorporated into the population of tubulin that is able to di use. The process of losing tubulin units can stop with a certain probability (i.e. free tubulin can also attach to shrinking plus ends in which case the disassembly process is stopped: a rescue). Free tubulin can also attach to the wall in order to form a new microtubule. Performing a sequence of numerical simulations of this model on a 256 × 1024 domain we found that the spatial distribution of microtubules was far from uniform. As shown in Fig. 1, we observe that there are groups of long tubes surrounded by shorter ones and that there seems to be a typical separation between the di erent groups. In order to characterize this distance we calculated an auto-correlation function in the following way: yZmax
dy L(y)L(y + y) ;
C(y) =
(1)
ymin
where L(y) is the length of the microtubule that grows o the site x = 0, y (all tubes grow along the x direction from sites with x = 0). We observe that, besides the maximum at y = 0 there is a local maximum at a nite value, y ≈ 50, the length that characterizes the separation between the groups of microtubules. In order to study this structure with greater detail we developed a second model, which can be called a mean- eld or lattice-Boltzmann model. In this case, instead of following particles, we work with distribution functions of microtubules and free tubulin. Also in this case the tubes grow from the wall at x = 0 (i.e. all minus ends are at x = 0). As done in [3], we separate the microtubules in two groups: shrinking and growing ones. We call S(x; y; t)xy and G(x; y; t)xy the number of (plus) tips of shrinking and growing tubes, respectively, and, T (x; y; t)xy, the number of (free) tubulin dimers, all of them in [x; x + x] × [y; y + y] at time t. We update the various
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Fig. 1. Snapshot of the distributions of tubes from a lattice-gas simulation. Each horizontal line represents one tube that grows from the left wall.
quantities di erently at the the wall and in the bulk. We also split the time step in two. Let us write down the equations in the bulk. We rst let tubulin di use: T˜ (x; y) = P0 T (x; y; t) + Pup T (x; y − y; t) + Pdown T (x; y + y; t) +Pleft T (x + x; y; t) + Pright T (x − x; y; t) ;
(2)
then we let the plus ends interact with the free tubulin: S(x; y; t + t) = P+− G(x; y; t) + P−0 S(x; y; t) + P−− S(x + x; y; t) ;
(3)
G(x; y; t + t) = P−+ S(x; y; t) + P+0 G(x; y; t) + P++ G(x − x; y; t) ;
(4)
T (x; y; t + t) = T˜ (x; y) + xP−− S(x + x; y; t) − xP++ G(x − x; y; t) : (5) In Eqs. (2)–(5) the various quantities are de ned as follows. Pright , Pleft , Pup , Pdown and P0 , are the probabilities that a free dimer move to the right, to the left, up, down, or does not move, respectively. We consider Pright = Pleft = Pup = Pdown , P0 6= 0, all of them independent of x, y and t, and Pright + Pleft + Pup + Pdown + P0 = 1. P++ (x; y; t) is the probability that the plus tip of a growing tube located at (ˆx; y) ˆ ∈ [x; x+x] × [y; y+y] at time t move to the right (move to xˆ + x) at time t + t, P+− (x; y; t), the probability that a growing tube with tip in [x; x + x] × [y; y + y] at time t transform into a shrinking tube (if this happens, the tip stays at the same place, only changes “state”), and P+0 the probability that nothing happens during that time step. Analogously, given a shrinking tube, P−− (x; y; t) is the probability that the tip moves to the left, P−+ (x; y; t) the probablity that it becomes growing (also in this case the tip does not move) and P−0 (x; y; t) the probability that nothing happens (it stays at the same place without
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changing the state) during t. In principle, all these probabilities depend on the density of tubulin, and, thus, on (x; y) and t. As done in [3] we will consider P−+ = aT˜ , P+− = b, P++ = cT˜ , and P−− = d, with a, b, c and d constant. In any case, at all times and positions, the sum of the probabilities satisfy P++ + P+− + P+0 = 1 = P−− + P−+ + P−0 . In order that a growing tip move to the right, from xˆ to xˆ + x, it uses up x dimers of tubulin, where is the number of dimers per unit length in a tube. Analogously, when a shrinking tube shrinks (going from xˆ + x to xˆ ), it frees x dimers of tubulin. At the wall the equation for the di usion of density is similar to Eq. (2) with the only di erence that we do not let tubulin di use outside of the domain. On the other hand, dimers of tubulin can bind with probability Pb to form a new tube. All other processes at the left wall involving tubes remain the same as in the bulk, with the following exceptions: no growing tubes tips can grow from the left into the wall; shrinking tubes with tips at x = 0 that “decide” to shrink free dimers of tubulin and the tube disappears. In this way Eqs. (3) – (5) are replaced by S(0; y; t + t) = P+− G(0; y; t) + P−0 S(0; y; t) + P−− S(x; y; t) ;
(6)
G(0; y; t + t) = P−+ S(0; y; t) + P+0 G(0; y; t) + Pb T˜ (0; y) ;
(7)
T (0; y; t + t) = T˜ (0; y) − Pb T˜ (0; y) + xP−− S(x; y; t) + P−− S(0; y; t) : (8) We can take the continuum limit of these equations, assuming that the various P±± =t remain nite. In the bulk, we get @ @T = f−− S − f++ G + (v− S + v+ G) + D∇2 T ; @t @x
(9)
@ @S = f+− G − f−+ S + (v− S) ; @t @x
(10)
@ @G = f−+ S − f+− G − (v+ G) ; @t @x
(11)
where we have de ned f±± ≡ P±± =t, v+ ≡ P++ x=t, v− ≡ P−− x=t and D ≡ (Pright + Pup )x2 =t, the di usion coecient of tubulin. Actually, writing the continuum limit in this way corresponds to keeping the dimers size nite [3]. If instead of doing this we treat the dimers as points and consider that P−+ =t, P+− =t, P−− x=t and P++ x=t remain nite, then the terms f−− S and f++ G in Eq. (9) should be replaced by -functions sources and sinks. Both the lattice-gas and the lattice-Boltzmann models involve a series of simpli cations. Perhaps, the crudest is the assumption that all free tubulin can polymerize into microtubules. Actually, only tubulin that has a GTP molecule bound to it can polymerize. Most of the tubulin that is freed up during the rapid depolymerizations is bound to GDP rather than to GTP. So, a chemical reaction should occur so that the GDP transforms into GTP before that tubulin can be incorporated into a new microtubule. Neglecting this e ect is equivalent to assuming that the timescale for recharging the
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GDP into GTP is shorter than the one associated with polymerization. Other simpli cations involve the dependences we assumed for the various probabilities as functions of the tubulin density. Other more realistic dependences, like the ones described in [5] will be studied in the future.
3. Results We performed a series of numerical simulations with the lattice-Boltzmann model for D = 0:375, d = 0:05, e = 0:75, a = 1, a = 0:15 and various values of b and c. By starting with no tubes and a noisy distribution of tubulin density with mean value hT i = 0:05, we could also observe the formation of clumps as shown in Fig. 1. In order to study this clump formation in more detail we performed a series of simulations in which we did not allow the tubes to grow from all sites on the walls, but just from some of them, separated by a given spacing. By varying the spacing we could analyze how the competition between ngers a ected their growth. As reported in [3] we also observed that for bd=ac¡1 two “populations” of tubes are formed, which is re ected in the existence of a local maximum, at x = xmax (t), for the function hGi(x; t) ≡
X
G(x; y; t) ;
(12)
y
as shown in Fig. 2, which corresponds to a simulation with a = 0:15 and b = 0:0005. There we can observe the local maximum for three values of the spacing (2, 8 and 12) and two values of c (1 and 1.749). Since c is proportional to the velocity of growth, the tubes with larger c have advanced further than those with c = 1. Also, the formation of the local maximum is delayed when c is smaller. We can also observe that the local maximum is more or less smeared in the simulations with large value of the spacing. We have observed the same e ect in other simulations, namely, that increasing the spacing makes the separation between these two populations less visible and also delays the formation of the local maximum in those cases in which we could observe it.
P
Fig. 2. G(x; y; t) as a function of x for t = 1200. The gure on the left corresponds to c = 1 and the y one on the right to c = 1:7490. All other parameters are the same, as described in the text. The solid line corresponds to a spacing between growing sites equal to 8, the dashed line to 2 and the dashed-dotted line to 12.
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P
P
Fig. 3. G(x = xmax ; y), with xmax the position at which the local maximum of G(x; y; t) occurs, as y y a function of the spacing between growing sites. The value of the spacing at which this curve reaches its maximum provides a length-scale.
Fig. 4. Same as Fig. 2 but at x = 0 (the wall).
We also observe that the number of tubes per growing site in the population that escapes (the one with tips around the local maximum of hGi) increases with the spacing. Thus, the total number of tubes increases with the spacing up to a certain value, after which it decreases. This is re ected in Fig. 3, where we have plotted P y G(x = xmax ; y) as a function of the spacing for a simulation with a = 1, b = 0:0005 and c = 1:5. The value of the spacing at which this curve reaches its maximum provides the spacing at which the formation of the escaping population is somehow “optimal”. The existence of such a length-scale is not observed at the wall, i.e., for the population of short tubes that do not escape, as can be observed in Fig. 4. This agrees with what can be observed in Fig. 1, since the separation between tubes occurs for the long ones but not for the short ones. On the other hand, this supports the idea that the separation
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Fig. 5. Length-scale (as obtained from Fig. 2) as a function of c. Since in all simulations the amount of tubulin is the same, this is equivalent to plotting the length-scale as a function of P++ , f++ or v++ . The straigth line we have superimposed has a − 12 slope.
between tubes observed in Fig. 1 is due to a competion between tubes to use up the available free tubulin. Finally, we have studied how the length-scale changes with the various parameters. The dependence on most of them is very weak, with the exception of c, the parameter proportional to f++ , and, thus, to the velocity of growth. For b = 0 there are three length-scales that can be constructed with the various parameters [6]: ‘v+ ≡ D=v+ , which corresponds to the size of the correlated volume seen by the advancing growing p tip, ‘f ≡ D=f++ , which corresponds to the volume over which tubulin can di use during the timescale of growth, and ‘T ≡ f++ =(DT ) with T , which corresponds to the distance over which the tubulin density is perturbed by consumption. For b 6= 0 there is a release of tubulin each time a tube depolymerizes and we did not know what e ect this could have on the scaling. In anycase, the dependence of the lengthscale with c follows more or less a c−1=2 law, as shown in Fig. 5. Since f++ = cT=t this corresponds to a length-scale similar to ‘f . Thus, we conclude that, basically, the escaping tubes advance as if there were no shrinking tubes (the b = 0 case). 4. Conclusions Microtubules are polymers made out of tubulin which are continually polymerizing and depolymerizing. Using lattice-gas and lattice-Boltzmann models we observed that, as reported in [3], under certain circumstances two populations of microtubules can form. A group of fewer long and stable ones and another one of shorter and more dynamic ones. The distribution of escaping tubes is characterized by a typical lengthscale: due to a competition for the available tubulin, not too many tubes can grow if their separation is not big enough. This length-scale is absent in the population of
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shorter tubes: the rapid depolymerizations replenish the amount of tubulin available and, for this reason, this competing e ect cannot be observed. By increasing the rate of change between growing and shrinking tubes, as occurs in mitosis, the separation in two populations does not occur, and, as observed, all tubes are short. In this case, there is no typical separation. This is a desirable feature for the tubes to probe the space in their search for chromosomes during mitosis. The separation in two populations is also a ected by the spacing between nucleating sites. If this spacing is too big, even when, according to [3], two populations should form, the formation of the escaping population is so much delayed that is almost unobservable. Thus, the change from a distribution characterized by the formation of long tubes to another one with short and more dynamic ones can also be achieved by a change in the spatial distribution of nucleating sites at the centrosome. There is the idea that “templates” of -tubulin in the centrosome could provide the nucleating sites for the tubes [7]. Therefore, a rearrangement of such sites could also be responsible for the changes observed in mitosis which, as described before, are also associated to changes in the centrosome [1]. Acknowledgements This work was supported in part by Fundacion Antorchas, Universidad de Buenos Aires and CONICET (Argentina) and the Department of Energy at LANL (USA). We are indebted to the organizers of LAWNP’97 for their support and hospitality. References [1] B. Alberts, D. Bray, J. Lewis, M. Ra , K. Roberts, J.D. Watson, Molecular Biology of the Cell, Garland, New York, 1994. [2] T.J. Mitchison, M.W. Kirschner, Cell Byophys. 11 (1987) 35. [3] M. Dogterom, S. Leibler, Phys. Rev. Lett. 70 (1993) 1347; Europhys. Lett. 24 (1993) 245. [4] R.B. Nicklas, Science 275 (1997) 632; R.B. Nicklas, S.C. Ward, J. Cell Biol. 126 (1994) 1241; Ann. Rev. Biophys. Biophys. Chem. 17 (1988) 431. [5] S.R. Martin, M.J. Schilstra, P.M. Bayley, Biophys. J. 65 (1993) 578; P.M. Bayley, M.J. Schilstra, S.R. Martin, J. Cell Sci. 95 (1990) 33. [6] D.K. Fygenson, Ph.D. Thesis, Princeton University, 1995. [7] M. Moritz, M.B. Braunfeld, J.W. Sedat, B. Alberts, D.A. Agard, Nature 378 (1995) 638.
Geometric features of microtubule dynamics Silvina Ponce-Dawson a; ∗ , John E. Pearson b , William N. Reynolds c a Departamento
de Fsica and I.A.F.E., Facultad de Ciencias Exactas y Naturales, U.B.A., Ciudad Universitaria, Pabellon I, (1428) Buenos Aires, Argentina b XCM MS F645, Computational Science Methods, Applied Theoretical Physics Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA c Bios Group L.P., 317 Paseo de Peralta, Santa Fe, NM 87501, USA
Abstract Microtubules are long and sti polymers that form the cytoskeleton of eucaryotic cells. They perform a series of tasks, such as determining the cell shape and providing a network of “rails” along which molecular motors transport organelles to di erent parts of the cell. They are particularly important during the process of cell division, since they provide the forces by which replicated chromosomes are segregated into what will be the two daughter cells. Microtubules are formed from a protein called tubulin and undergo a process called dynamic instability. In this paper we study, via numerical simulations of some simpli ed models, how the interaction c 1998 between microtubules and the di usion of free tubulin a ects their spatial organization. Elsevier Science B.V. All rights reserved. Keywords: Microtubules; Dynamic instability; Spatial organization
1. Introduction All eucaryotic cells (i.e., cells that have a nucleus) have a network of protein laments which is called the cytoskeleton [1]. This network is highly dynamic and allows the cell to change its shape and to carry out coordinated and directed movements. Three types of protein laments form the cytoskeleton, among them, the microtubules. Microtubules are long, sti polymers that are made out of a protein called tubulin. They are thought to be the primary organizers of the cytoskeleton. They extend throughout the cytoplasm and govern the location of organelles and other cell components. Microtubules are polar structures, with a plus end capable of rapid growth and a minus end that, in most cells, is stabilized by embedding in a structure called the centrosome. Microtubules are highly dynamic: after growing outward for many minutes by adding tubulin dimers to their plus ends, they may suddenly start to lose subunits, ∗
Corresponding author. E-mail: [email protected].
c 1998 Elsevier Science B.V. All rights reserved. 0378-4371/98/$19.00 Copyright PII: S 0 3 7 8 - 4 3 7 1 ( 9 8 ) 0 0 1 3 8 - 1
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shrinking very fast and, in many cases, even disappearing. While this happens, new microtubules are formed, that emanate from the centrosome. These rapid shortenings are called catastrophes. The process by which they occur is called dynamic instability [2] and it is thought to be related to the hydrolization of GTP, which binds to the tubulin subunits. Microtubules are involved in cell division, intracellular transport, positioning of cell membranes and the determination of cellular shape. For example, after chromosome replication, the nuclear envelope breaks down and the duplicated chromosomes are distributed more or less randomly throughout the cell, which now has two centrosomes. Before the cell can divide, the duplicated chromosomes must be segregated. During mitosis, (the segregation process) microtubule laments emanating from both centrosomes (also called poles) form a bipolar spindle. The microtubules nucleate at the poles and then randomly grow and shrink until they attach to a pair of replicated chromosomes. The microtubules in the spindle serve as rails along which the chromosomes can move. The chromosomes rst migrate to the cell equator, then separate and migrate to the poles, all via microtubule mediated forces [4]. The stage is set for the cell to divide only after this segregation process is complete. During mitosis the microtubule array is much more dynamic than during interphase (the period before mitosis) [1]. On the other hand, while in interphase the centrosome nucleates long microtubules that extend throughout the cytoplasm, at mitosis there is a larger number of shorter and less stable microtubules. The rate at which dynamic instability occurs increases dramatically during mitosis. For this reason, the half-life of an average microtubule goes from about 5 min in interphase to 15 s in mitosis. It is thought that this is due to a steep increase in the probability that a typical growing microtubule will convert to a shrinking one. At the same time there is, apparently, a change in the centrosome itself that allows the nucleation of a larger number of microtubules. All these changes help the microtubules perform their task during mitosis, which is the capture of the replicated chromosomes. As in mitosis, dynamic instability is a necessary part of many of the other microtubule functions. In this paper we are interested in studying what kind of spatial organization the interaction between the polymerization=depolymerization of the microtubules and the di usion of tubulin can give rise to. In order to study this, we have developed two simpli ed models. These models have many similarities with those studied in [3], but while the studies reported in [3] mostly focused on the properties of the microtubule distribution in the direction of growth, in this paper we will also discuss the structure perpendicular to this direction. In both models the dynamics occurs on a twodimensional rectangular domain in which microtubules can grow only from one side of the domain in a direction perpendicular to it. The results of our simulations show that, due to a competition between neighboring microtubules, the spatial distribution in the direction perpendicular to that of growth is highly non-uniform. As reported in [3] we also nd that, depending on the parameters, two “populations” form: one of many short tubes and another one of rapidly growing longer ones. The time at which this population of rapidly growing tubes is formed is
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highly a ected by the distance between tubes perpendicular to the direction of growth. In particular, we could observe that upon increasing the separation between nucleating sites the formation of this population of “escaping” tubes is delayed so that the distinction between two “classes” of tubes cannot be done. We also observe that there is always an optimal distance between nucleating sites at which more “escaping” tubes are formed. This gives a typical length-scale whose change with the various parameters we have studied. In particular, we have found that for time large enough this length-scale is inversely proportional to the square root of the velocity of growth. We have also observed that the shorter tubes are not characterized by a nite length-scale: the closer the nucleating sites, the larger is the number of tubes that can be formed.
2. The models The rst model we have developed is a lattice gas in which tubulin particles perform a random walk, in order to simulate di usion, and can attach with a certain probability to the growing plus ends of the microtubules. On the other hand, these plus ends, with a certain probability can also lose tubulin subunits which are then incorporated into the population of tubulin that is able to di use. The process of losing tubulin units can stop with a certain probability (i.e. free tubulin can also attach to shrinking plus ends in which case the disassembly process is stopped: a rescue). Free tubulin can also attach to the wall in order to form a new microtubule. Performing a sequence of numerical simulations of this model on a 256 × 1024 domain we found that the spatial distribution of microtubules was far from uniform. As shown in Fig. 1, we observe that there are groups of long tubes surrounded by shorter ones and that there seems to be a typical separation between the di erent groups. In order to characterize this distance we calculated an auto-correlation function in the following way: yZmax
dy L(y)L(y + y) ;
C(y) =
(1)
ymin
where L(y) is the length of the microtubule that grows o the site x = 0, y (all tubes grow along the x direction from sites with x = 0). We observe that, besides the maximum at y = 0 there is a local maximum at a nite value, y ≈ 50, the length that characterizes the separation between the groups of microtubules. In order to study this structure with greater detail we developed a second model, which can be called a mean- eld or lattice-Boltzmann model. In this case, instead of following particles, we work with distribution functions of microtubules and free tubulin. Also in this case the tubes grow from the wall at x = 0 (i.e. all minus ends are at x = 0). As done in [3], we separate the microtubules in two groups: shrinking and growing ones. We call S(x; y; t)xy and G(x; y; t)xy the number of (plus) tips of shrinking and growing tubes, respectively, and, T (x; y; t)xy, the number of (free) tubulin dimers, all of them in [x; x + x] × [y; y + y] at time t. We update the various
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Fig. 1. Snapshot of the distributions of tubes from a lattice-gas simulation. Each horizontal line represents one tube that grows from the left wall.
quantities di erently at the the wall and in the bulk. We also split the time step in two. Let us write down the equations in the bulk. We rst let tubulin di use: T˜ (x; y) = P0 T (x; y; t) + Pup T (x; y − y; t) + Pdown T (x; y + y; t) +Pleft T (x + x; y; t) + Pright T (x − x; y; t) ;
(2)
then we let the plus ends interact with the free tubulin: S(x; y; t + t) = P+− G(x; y; t) + P−0 S(x; y; t) + P−− S(x + x; y; t) ;
(3)
G(x; y; t + t) = P−+ S(x; y; t) + P+0 G(x; y; t) + P++ G(x − x; y; t) ;
(4)
T (x; y; t + t) = T˜ (x; y) + xP−− S(x + x; y; t) − xP++ G(x − x; y; t) : (5) In Eqs. (2)–(5) the various quantities are de ned as follows. Pright , Pleft , Pup , Pdown and P0 , are the probabilities that a free dimer move to the right, to the left, up, down, or does not move, respectively. We consider Pright = Pleft = Pup = Pdown , P0 6= 0, all of them independent of x, y and t, and Pright + Pleft + Pup + Pdown + P0 = 1. P++ (x; y; t) is the probability that the plus tip of a growing tube located at (ˆx; y) ˆ ∈ [x; x+x] × [y; y+y] at time t move to the right (move to xˆ + x) at time t + t, P+− (x; y; t), the probability that a growing tube with tip in [x; x + x] × [y; y + y] at time t transform into a shrinking tube (if this happens, the tip stays at the same place, only changes “state”), and P+0 the probability that nothing happens during that time step. Analogously, given a shrinking tube, P−− (x; y; t) is the probability that the tip moves to the left, P−+ (x; y; t) the probablity that it becomes growing (also in this case the tip does not move) and P−0 (x; y; t) the probability that nothing happens (it stays at the same place without
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changing the state) during t. In principle, all these probabilities depend on the density of tubulin, and, thus, on (x; y) and t. As done in [3] we will consider P−+ = aT˜ , P+− = b, P++ = cT˜ , and P−− = d, with a, b, c and d constant. In any case, at all times and positions, the sum of the probabilities satisfy P++ + P+− + P+0 = 1 = P−− + P−+ + P−0 . In order that a growing tip move to the right, from xˆ to xˆ + x, it uses up x dimers of tubulin, where is the number of dimers per unit length in a tube. Analogously, when a shrinking tube shrinks (going from xˆ + x to xˆ ), it frees x dimers of tubulin. At the wall the equation for the di usion of density is similar to Eq. (2) with the only di erence that we do not let tubulin di use outside of the domain. On the other hand, dimers of tubulin can bind with probability Pb to form a new tube. All other processes at the left wall involving tubes remain the same as in the bulk, with the following exceptions: no growing tubes tips can grow from the left into the wall; shrinking tubes with tips at x = 0 that “decide” to shrink free dimers of tubulin and the tube disappears. In this way Eqs. (3) – (5) are replaced by S(0; y; t + t) = P+− G(0; y; t) + P−0 S(0; y; t) + P−− S(x; y; t) ;
(6)
G(0; y; t + t) = P−+ S(0; y; t) + P+0 G(0; y; t) + Pb T˜ (0; y) ;
(7)
T (0; y; t + t) = T˜ (0; y) − Pb T˜ (0; y) + xP−− S(x; y; t) + P−− S(0; y; t) : (8) We can take the continuum limit of these equations, assuming that the various P±± =t remain nite. In the bulk, we get @ @T = f−− S − f++ G + (v− S + v+ G) + D∇2 T ; @t @x
(9)
@ @S = f+− G − f−+ S + (v− S) ; @t @x
(10)
@ @G = f−+ S − f+− G − (v+ G) ; @t @x
(11)
where we have de ned f±± ≡ P±± =t, v+ ≡ P++ x=t, v− ≡ P−− x=t and D ≡ (Pright + Pup )x2 =t, the di usion coecient of tubulin. Actually, writing the continuum limit in this way corresponds to keeping the dimers size nite [3]. If instead of doing this we treat the dimers as points and consider that P−+ =t, P+− =t, P−− x=t and P++ x=t remain nite, then the terms f−− S and f++ G in Eq. (9) should be replaced by -functions sources and sinks. Both the lattice-gas and the lattice-Boltzmann models involve a series of simpli cations. Perhaps, the crudest is the assumption that all free tubulin can polymerize into microtubules. Actually, only tubulin that has a GTP molecule bound to it can polymerize. Most of the tubulin that is freed up during the rapid depolymerizations is bound to GDP rather than to GTP. So, a chemical reaction should occur so that the GDP transforms into GTP before that tubulin can be incorporated into a new microtubule. Neglecting this e ect is equivalent to assuming that the timescale for recharging the
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GDP into GTP is shorter than the one associated with polymerization. Other simpli cations involve the dependences we assumed for the various probabilities as functions of the tubulin density. Other more realistic dependences, like the ones described in [5] will be studied in the future.
3. Results We performed a series of numerical simulations with the lattice-Boltzmann model for D = 0:375, d = 0:05, e = 0:75, a = 1, a = 0:15 and various values of b and c. By starting with no tubes and a noisy distribution of tubulin density with mean value hT i = 0:05, we could also observe the formation of clumps as shown in Fig. 1. In order to study this clump formation in more detail we performed a series of simulations in which we did not allow the tubes to grow from all sites on the walls, but just from some of them, separated by a given spacing. By varying the spacing we could analyze how the competition between ngers a ected their growth. As reported in [3] we also observed that for bd=ac¡1 two “populations” of tubes are formed, which is re ected in the existence of a local maximum, at x = xmax (t), for the function hGi(x; t) ≡
X
G(x; y; t) ;
(12)
y
as shown in Fig. 2, which corresponds to a simulation with a = 0:15 and b = 0:0005. There we can observe the local maximum for three values of the spacing (2, 8 and 12) and two values of c (1 and 1.749). Since c is proportional to the velocity of growth, the tubes with larger c have advanced further than those with c = 1. Also, the formation of the local maximum is delayed when c is smaller. We can also observe that the local maximum is more or less smeared in the simulations with large value of the spacing. We have observed the same e ect in other simulations, namely, that increasing the spacing makes the separation between these two populations less visible and also delays the formation of the local maximum in those cases in which we could observe it.
P
Fig. 2. G(x; y; t) as a function of x for t = 1200. The gure on the left corresponds to c = 1 and the y one on the right to c = 1:7490. All other parameters are the same, as described in the text. The solid line corresponds to a spacing between growing sites equal to 8, the dashed line to 2 and the dashed-dotted line to 12.
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P
P
Fig. 3. G(x = xmax ; y), with xmax the position at which the local maximum of G(x; y; t) occurs, as y y a function of the spacing between growing sites. The value of the spacing at which this curve reaches its maximum provides a length-scale.
Fig. 4. Same as Fig. 2 but at x = 0 (the wall).
We also observe that the number of tubes per growing site in the population that escapes (the one with tips around the local maximum of hGi) increases with the spacing. Thus, the total number of tubes increases with the spacing up to a certain value, after which it decreases. This is re ected in Fig. 3, where we have plotted P y G(x = xmax ; y) as a function of the spacing for a simulation with a = 1, b = 0:0005 and c = 1:5. The value of the spacing at which this curve reaches its maximum provides the spacing at which the formation of the escaping population is somehow “optimal”. The existence of such a length-scale is not observed at the wall, i.e., for the population of short tubes that do not escape, as can be observed in Fig. 4. This agrees with what can be observed in Fig. 1, since the separation between tubes occurs for the long ones but not for the short ones. On the other hand, this supports the idea that the separation
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Fig. 5. Length-scale (as obtained from Fig. 2) as a function of c. Since in all simulations the amount of tubulin is the same, this is equivalent to plotting the length-scale as a function of P++ , f++ or v++ . The straigth line we have superimposed has a − 12 slope.
between tubes observed in Fig. 1 is due to a competion between tubes to use up the available free tubulin. Finally, we have studied how the length-scale changes with the various parameters. The dependence on most of them is very weak, with the exception of c, the parameter proportional to f++ , and, thus, to the velocity of growth. For b = 0 there are three length-scales that can be constructed with the various parameters [6]: ‘v+ ≡ D=v+ , which corresponds to the size of the correlated volume seen by the advancing growing p tip, ‘f ≡ D=f++ , which corresponds to the volume over which tubulin can di use during the timescale of growth, and ‘T ≡ f++ =(DT ) with T , which corresponds to the distance over which the tubulin density is perturbed by consumption. For b 6= 0 there is a release of tubulin each time a tube depolymerizes and we did not know what e ect this could have on the scaling. In anycase, the dependence of the lengthscale with c follows more or less a c−1=2 law, as shown in Fig. 5. Since f++ = cT=t this corresponds to a length-scale similar to ‘f . Thus, we conclude that, basically, the escaping tubes advance as if there were no shrinking tubes (the b = 0 case). 4. Conclusions Microtubules are polymers made out of tubulin which are continually polymerizing and depolymerizing. Using lattice-gas and lattice-Boltzmann models we observed that, as reported in [3], under certain circumstances two populations of microtubules can form. A group of fewer long and stable ones and another one of shorter and more dynamic ones. The distribution of escaping tubes is characterized by a typical lengthscale: due to a competition for the available tubulin, not too many tubes can grow if their separation is not big enough. This length-scale is absent in the population of
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shorter tubes: the rapid depolymerizations replenish the amount of tubulin available and, for this reason, this competing e ect cannot be observed. By increasing the rate of change between growing and shrinking tubes, as occurs in mitosis, the separation in two populations does not occur, and, as observed, all tubes are short. In this case, there is no typical separation. This is a desirable feature for the tubes to probe the space in their search for chromosomes during mitosis. The separation in two populations is also a ected by the spacing between nucleating sites. If this spacing is too big, even when, according to [3], two populations should form, the formation of the escaping population is so much delayed that is almost unobservable. Thus, the change from a distribution characterized by the formation of long tubes to another one with short and more dynamic ones can also be achieved by a change in the spatial distribution of nucleating sites at the centrosome. There is the idea that “templates” of -tubulin in the centrosome could provide the nucleating sites for the tubes [7]. Therefore, a rearrangement of such sites could also be responsible for the changes observed in mitosis which, as described before, are also associated to changes in the centrosome [1]. Acknowledgements This work was supported in part by Fundacion Antorchas, Universidad de Buenos Aires and CONICET (Argentina) and the Department of Energy at LANL (USA). We are indebted to the organizers of LAWNP’97 for their support and hospitality. References [1] B. Alberts, D. Bray, J. Lewis, M. Ra , K. Roberts, J.D. Watson, Molecular Biology of the Cell, Garland, New York, 1994. [2] T.J. Mitchison, M.W. Kirschner, Cell Byophys. 11 (1987) 35. [3] M. Dogterom, S. Leibler, Phys. Rev. Lett. 70 (1993) 1347; Europhys. Lett. 24 (1993) 245. [4] R.B. Nicklas, Science 275 (1997) 632; R.B. Nicklas, S.C. Ward, J. Cell Biol. 126 (1994) 1241; Ann. Rev. Biophys. Biophys. Chem. 17 (1988) 431. [5] S.R. Martin, M.J. Schilstra, P.M. Bayley, Biophys. J. 65 (1993) 578; P.M. Bayley, M.J. Schilstra, S.R. Martin, J. Cell Sci. 95 (1990) 33. [6] D.K. Fygenson, Ph.D. Thesis, Princeton University, 1995. [7] M. Moritz, M.B. Braunfeld, J.W. Sedat, B. Alberts, D.A. Agard, Nature 378 (1995) 638.