Microtubule dynamics and morphogenesis in Paramecium

Europ. J. Protisto!' 32, 134-144 (1996) February 23, 1996

European Journal of

PROTISTOLOGY

Microtubule Dynamics and Morphogenesis in Paramecium II. Modelling of the Conversion of a Transient Molecular Signal into a Morphogenetical Process Michel Laurent and Anne Fleury1 Service d'lmagerie Cellulaire, Universite Paris-Sud, Orsay, France 1Laboratoire de Biologie Cellulaire 4, Universite Paris-Sud, Orsay, France

SUMMARY In Paramecium, the deployment of an acetylated microtubular network is restricted to a part of the ventral cell surface, the A-paratene zone. Upon cell division, this microtubular network depolymerizes; it is reassembled only after cell separation. Convergent evidences suggest that one of the primary signals for cortical morphogenesis could be a Ca 2 + wave. Correlatively, calcium ions inhibit the enzyme which acetylates microtubules and prevent microtubule stabilization by associated proteins (MAPs). All these data have been gathered into a model which shows how a transient signal (the calcium wave) may be converted into a permanent morphological modification of the assembly state of the A-paratene-associated microtubules. In this model, biochemical switches between acetylation and depolymerization pathways acting on stabilized microtubules occur beyond the assembly/disassembly step of unstabilized microtubules. Accordingly, only the inhibitory effect of Ca2+ on the acetylation enzyme has a triggering action on the dynamics of the microtubules whereas its indirect effect on MAPs is irrelevant in this respect. This model also interprets why the A-paratene-associated microtubules resists nocodazole but not cold treatment. From this analysis, we suggest that the various control mechanisms which act at the level of each modification enzyme would make both the origin and the biological significance of the diversity of microtubules regulative.

Introduction

Paramecium is a ciliate with an extremely elaborate cortex which behaves, during division, as a mosaic of territories expressing different morphogenetic behavior. Paramecium multiplies by binary fission: the surface of the dividing cell is progressively invaded by superimposed morphogenetic waves which trigger the duplication, assembly or reorganization of each cytoskeletal cortical structure [17]. These waves are initiated near the oral groove and then propagated towards both the anterior and posterior poles of the cell. However, on the ventral surface, the morphoge0932-4739-96-0032-0134$3.50-0

netic waves do not evolve uniformly for all the cortical structures: although some of them are propagated along the whole cell, others such as the basal body duplication wave are stopped along a well-defined boundary, producing one anterior and one posterior invariant zone. Nonetheless, the invariant regions are defined morphogenetically only by the absence of both basal body duplication and segmentation of epiplasmic scales in the related areas. The fact that other cytoskeletal modifications also occur in these invariant zones at the onset of cell division [9, 17] indicates that a cortical signal is received in these invariant zones as well. Convergent experimental data [3, 18, 34] suggest that the © 1996 by Gustav Fischer Verlag, Stuttgart

Dynamics of Microtubules in Paramecium . 135

primary signal for cortical morphogenesis might be a calcium wave. We have theoretically shown [23] that the generation and the proparation of such a wave can be triggered by a simple Ca +-induced Ca 2+-release mechanism involving cortical alveoli which are calcium reservoirs in ciliates [35]. In the first part of this paper [10], we showed the presence, in the interphase cell, of a network of acetylated microtubules whose deployment is restricted to the anterior invariant zone (called the A-paratene area). These acetylated microtubules exhibit stability properties similar in behavior to microtubules observed in epithelial cells: they are cold labile but nocodazole resistant. Studies on the developmental pattern of this microtubular network has shown that it depolymerizes when the cell enters division. It reappears only after separation of the young dividers. Since the A-paratene-associated microtubules are acetylated, two complementary observations may be relevant to their dynamics during cell division: - The enzyme which ensures this post-translational modification, namely the tubulinyl acetyl transferase, is inhibited by calcium ions [25]. - Associated proteins (MAPs) which stabilize microtubuies may be phosphorylated by a Ca 2+-calmodulindependent protein kinase [1, 37] which reduces the stabilizing effect of MAPs [24, 36]. Hence, the supposed Ca 2+ wave may also be involved, directly or indirectly, in the de polymerization process of A-paratene-associated microtubules. But how are all these events correlated? Do they suffice to explain the dynamics of A-paratene-associated microtubules as observed in the course of the cell cycle? The Ca2+ wave is transient, whereas the depolymerization of this network of microtubules is irreversible and permanent (in the time scale of the cell division process). In other words, depolymerization goes on even after the passage of the Ca 2+ wave. Thus inhibition of tubulinyl acetyl transferase by Ca 2+ or an indirect effect on MAPs are not sufficient to account for the abrupt process of depolymerization. If all these events may be correlated, why does depolymerization go on even after the Ca 2+ wave has passed? Modelling of this system allows us to propose an answer to these questions: we show here that the Ca2~ wave might constitute the inductive signal which triggers a switch between two stable steady-states in a dynamical model for post-translational acetylation of A-paratene-associated microtubules. We discuss how this model can interpret experimental data and how it extends our current conception on the possible functional role of microtubular post-translational modifications.

buies persist long enough to allow enzymatic modification of tubulin [32, 38, 39] and that enzymatic activities of acetylation/deacetylation are partitioned between separate pools of protomeric and polymeric forms of tubulin, respectively. In other words, posttranslational acetylation is the consequence rather than the cause of microtubule stability (see [12] for a review). Partitioning of enzyme activities of modification/unmodification means that the enzyme which performs acetylation is active in microtubules but not in soluble tubulin [4, 25]. Stabilization results from interactions between microtubules and microtubule-associated-proteins (MAPs) and from the formation of high-order assemblies such as microtubular bundles or complexes between microtubules and organelles [12,19]. The ordered sequence: unstablilized microtubules ---. stabilized microtubules ---. post-translationally modified microtubules is now widely accepted, both for the detyrosination and acetylation mechanisms of posttranslational modifications (for a review, see (5)). Microtubules are organized in an ordered array together with other microtubules (bundling effect) and with other cytoskeletal elements (stabilization sites). This stabilizing effect [5, 12,21) may be taken into account in our model by assuming that acetylated microtubules interact with a stabilizing element to give a species which has a low rate of depolymerization.

Involvement of Ca 2+ in the Dynamics of Acetylated Microtubules In our model, the first molecular site of action of calcium ions is an enzyme that ensures acetylation of Aparatene-associated microtubules. Although the postulated mechanism of inhibition seems unusual since Ca 2+ is supposed to bind to the substrate and not to the enzyme (25), we shall consider, for the sake of simplicity, that it corresponds to the most common type of enzyme inhibition, i.e. competitive inhibition. However, another site of action may be envisaged for this ion since Ca 2+ also acts indirectly on MAPs: the extent of microtubule stabilization by the associated proteins (MAPs) depends on the affinity of these proteins for the microtubules. In fact, MAI>s are good substrates for a number of protein kinases, and their phosphorylation often suppresses the stabilizing effect on microtubules [12]. For instance, tau and MAP2 can be phosphorylated by a Ca 2+, calmodulin-dependent protein kinase [1, 37]. Phosphorylation induces a conformational change in tau protein [15] and reduces, both for tau and MAP2, their ability to stabilize microtubules [24, 36]. Tau and MAP2 proteins, originally found in brains of higher organisms, were recently shown to be present as well in ciliates (31).

Construction of the Model

Stability and Acetylation of Microtubules The increased content of acetylated tubulin in stable microtubules results from the fact that the microtu-

Modelling of the Dynamics of A-ParateneAssociated Microtubules The fact that depolymerization persists after the passage of the Ca 2 + wave strongly suggests that this system

136 . M. Laurent and A. Fleury

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behaves as a bi-stable switch. In the framework of dynamical systems theory, analogous schemes have previously been described both in a chemical [29] and in a biochemical [7, 13, 27] and cellular [22, 23] context. The stringent theoretical demand for a threshold behavior is the existence of at least one non-linear reaction. The acetylation of stabilized microtubules which is the sole enzymatic reaction in our subsystem appears to be the preferred site for such regulation. The non-linearity will be introduced by assuming that this enzyme is inhibited by an excess of its substrate. Although hypothetical in the present state of our knowledge, we may remark, however, that the in situ regulatory properties of the microtubule-acetylation enzyme are complex. Apart from the above-mentioned Ca 2 + inhibitory effect and an additional process of inhibition by coenzyme-A, Maruta et al. [25] have demonstrated the existence, in a cell body extract of Chlamydomonas, of a yet unidentified cytoplasmic inhibitor of the tubulinyl acetyl transferase activity. This inhibitory effect is distinct from that which would result from a tubulinyl deacetylase activity and might be tentatively modeled as an ante-inhihition loop. Besides this process, all individual steps are supposed to correspond to first-order kinetic rate laws (Fig. 1). Theoretical treatment of this scheme is given in the Appendix.

Results Dynamics of the System at Constant Ca 2+ Concentration Existence ofmultiple steady-states. 5teady-states are defined by constant values of the fluxes and of the concentrations of the species: for all intermediates the outflux equals the influx so that the rate of the net production (i.e. the net rate) becomes zero. As shown in the Appendix, in our system, over a wide range of input values (vo), there are three distinct MT concentration values for which the algebraic difference between production and removal processes for MT species (stabilized but unmodified microtubules) is null (Fig. 2). Hence, three steady-states (respectively named 55!> 55 2 and 55 3 ) coexist. Two of them (55 1 and 55 3 ) are stable whereas the intermediary steady-state 55 2 is unstable. We can demonstrate (see Appendix) that global fluxes essentially proceed through the depolymerization process when the system is in 55 3 and through the acetylation pathway when the system is in 55 1 , Transition between stable steady-states upon Va variations. Steady-state concentrations are different when parameter values such as the input value are modified. In order to study the dynamics of the system when fluxes are modified as the ratio between unpolymerized tubulin and assembled microtubules varies (according for instance to the dynamic instability model [26]), the input value Vo has to be considered as a variable.

Dynamics of Microtubules in Paramecium· 137

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When the system has more than one stable steadystate, it is clear that its stability must be local rather than global. If va is now considered as an extrinsic variable, the dynamical behavior of the system will be described by examining the trajectory of its steady-states (Fig. 3 and Appendix). If the system is in 55 1 (part A in Fig. 3) and is then acted on by an external influence which produces an increase in the va input value which is sufficient to bring it above the upper threshold (upper horizontal dashed line in Fig. 3) of the 55 2 region (corresponding to the negative slope in Fig. 3), the system is settled into the other stable steady-state 5S, (part B in Fig. 3) and remains there even when the external influence is removed (closed arrows in Fig. 3). The same conclusion holds for the reverse transition from 55 3 to 55 1 as va decreases (open arrows in Fig. 3), but the jump-like transition occurs for a threshold value which is lower than the previous one (lower horizontal dashed line in Fig. 3).

Switches Between Trajectories

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Ca 2+ as an inhibitor of the tubulinyl acetyl transferase catalyzed reaction. Ca 21 concentration is not governed by the fluxes which occur in the subsystem: it thus constitutes an extrinsic parameter of the model. However, since the trajectory of steady-states is determined in part by the characteristics of the kinetic process of microtubule acetylation, the global shape of this trajectory depends on the Ca 2i concentration. In the

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absence of Ca 2+ (Fig. 4, curve a) or at intermediate Ca 2 + concentration (Fig. 4, curve b), the system has two stable steady-states 55 1 and 55 3 which surround an unstable steady-state, 55 2, as previously described. On the contrary, at a high Ca 2 + concentration (Fig. 4, curve c), only one steady-state can be defined. As long as the inhibitor does not exceed a threshold concentration value, the interval between 55 1 and 55 2 steadystates on the MT concentration axis is reduced (compare curves a and b). On the contrary, the position of the stable steady-state 55 3 is unmodified on the [MT] axis. Thus, in this Ca 2i concentration range, the presence of Ca 2 + only produces a slight readjustment of the MT stationary-state value if any (i.e. if the system lies in 55 1 ), On the contrary, when the Ca 2 + concentration exceeds a threshold value, the shape of the trajectory of steady-states becomes qualitatively modified (compare curves a and c): only one single stationarystate exists and it corresponds to the high MT concentration steady-state (55 3 ), Thus, when the system initially in 55, is perturbed by a supra-threshold Ca 2 + concentration, it moves on a solution trajectory which has only a single steady-state, 55 3 , In these conditions, we observe a transition between 55 1 and 55 3 steadystates corresponding to a switch between the acetyla-

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On the Differential Effects of Cold and Nocodazole Treatments on Microtubular Stability

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Fig. 4. Influence of calcium on the steady-state properties of the system. 5witch between [Ca 2 +]-dependent trajectories resulting from the competitive inhibitory effect of Ca 2 + on the tubulinyl acetyl transferase catalyzed reaction. Net rate of MT-production was plotted as a function of MT concentration either in the absence of calcium (curve a) or for two different concentrations of calcium (curves b and c). The curves were obtained for the same set of parameter values as in Fig. 2, except for the a parameter (normalized calcium concentration) which was: 0 (curve a), 1 (curve b) or 3 (curve c). According to the Ca2+ concentration value, the graph intercepts the x-axis three times (curves a and b) or only once, i.e. the system has three or one steady-states. As discussed in Fig. 6 (Appendix), only the two extreme steady-states (on [MT] axis) are stable when three of them coexist. That means that the increase in [Ca 2 +] moves the system from a locally stable (55 1 or 55 3 ) to a globally stable steady-state (55J ). Accordingly, a supra-threshold increase of the Ca 2 + concentration will produce a jump-like transition if the system is initially in 55 1 steady-state. On the contrary, no effect will be observed if the system is in 55 3 steady-state.

tion and depolymerization pathways. On the contrary, if the system is initially in SS3' no transition occurs, whatever the range of concentration of the Ca 2+ perturbation. Ca 2 + as a cofactor in the phosphorylation process of MAPs. How can the additional indirect effect of Ca 2 + on MAPs be taken into account in our model and what are the consequences? Since the active forms (in the microtubule stabilization process) of MAPs are unphosphorylated ones and since Ca 2+ is a cofactor of phosphorylation enzymes, increase in the Ca 2+ concentration would only produce, via the indirect MAPs effect, a decrease in the value of the input rate Vo' Hence, the effect of Ca 2+ on MAPs does not modify the shape of the trajectory of steady-states of the system but acts only by producing displacements along this trajectory (see Appendix). However, the Ca 2 .,. wave occurs when the system is in the SS1 steady-state; thus, the decrease in the Vo input value (open arrows on part A of the trajectory of steady-states in Fig. 3) does not produce any transition between steady-states. Hence, this indirect effect of Ca 2 + has no consequence on the dynamics of the system.

Experimental data [10] show that the A-paratene-associated microtubules resist nocodazole but not cold treatment. Such a differential behavior is easily explained in the framework of our model. It is well known that nocodazole prevents microtubular assembly. In the scheme of Fig. 1, the effect of the drug is to diminish the Vo input value (since it occurs at the assembly/disassembly step). When this treatment is applied to interphase cells, the system is in the SS1 steady-state (part A of the trajectory of steady-states in Fig. 3). A decrease in the input value only produces a slight readjustment of the steady-state value on the part A of the solution trajectory corresponding to the acetylation without any transition between the acetylation and depolymerization pathways (open arrows on part A of the trajectory in Fig. 3). As for any other enzymatic reaction, cold treatment dramatically decreases the rate of the acetylation reaction. Such a decrease changes the shape of the trajectory of steady-states so that only the SS3 steady-state exists (the mechanism and the effect of this change are explained and illustrated above in the case of the Ca 2 + effect on the acetyl transferase reaction). Hence, cold treatment moves the system, by alteration of the trajectory of its steady-states, from a locally stable stationary state (5S 1) to a globally stable steady-state (SS3), i.e. the system switches from acetylation to depolymerization pathways. Temporal Behavior of the System

According to the dynamic properties of the model, the temporal behavior of the system is summarized in Fig. 5. Four consecutive time intervals have to be distinguished. Their properties are as follows: 1. Interphase cell. In the interphase cell, the system is on the Calow trajectory, in S5 1 steady-state (lower MT and higher MT-Ac concentrations). Hence, overall flux essentially proceeds through the microtubule acetylation pathway. Any fluctuation in MT concentration under the threshold T 1 is spontaneously eliminated, the system coming back to the 5S l steady-state. 2. Occurrence ofthe Ca2+ wave. The system switches from the Calow trajectory to the Cahigh trajectory. This new trajectory possesses only one steady-state, 55 3, which is stable. 55 3 steady-state corresponds to conditions in which overall flux essentially proceeds through the microtubule depolymerization pathway. Thus, transition from the Calow to the Cahigh trajectory goes together with a switch of the system from 55 l to 5S 3 steady-state. 3. Passage of the Ca 2 + wave. When the Ca 2+ concentration rediminishes upon passage of the Ca 2+ wave, the system switches back from the Cahigh to the Calow trajectory. However, since the latter trajectory possesses a stable steady-state corresponding to a high MT-concentration, the system stays in S5 3 steady-state, that means in a steady-state characterized by a low rate

Dynamics of Microtubules in Paramecium . 139

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Fig. 5. Temporal behavior of the system in relation to the cell cycle of Paramecium. This qualitative diagram shows how the dynamic properties of the system are temporally correlated with the interphase and division periods in the cell cycle of Paramecium and with the occurrence of the Cal+ wave (upper graph). (1): Interphase; (2): Occurrence of the Ca 2+ wave at the beginning of the division rrocess. The limits assigned to this period correspond to [Ca +] values for which the system switches first (left limit) from a locally stable (Calow trajectory) to a globally stable steady-state (Cahi&h trajectory). Upon moving of the Cal+ wave, the system switches back (right limit) to the Calow trajectory but stays in the 55 j steady-state; (3): Division period in which the system lies in SS) steady-state (depolymerization process) on the Calow trajectory. (4): Switch from 5S j to SSl steady-state upon Vo variation, on the Calow trajectory. Note that the true relative lengths of the interphase (about 5 hours) and division (30 min to 1 hour) processes are not respected, for the sake of clarity of the diagram. In addition, MTand MT-Ac concentration scales on the y-axis are arbitrary and both are not necessarily of the same order of magnitude (see text and Table 1). Dashed areas indicate the periods (and the corresponding threshold values T 1 and T 2) for which the system is susceptible to undergo a transition between stable steady-states, i.e. a switch between acetylation and depolymerization processes of A-paratene-associated microtubules. Table 1. Concentrations of MTand MT-Ac species in 5S 1 and SSj steady-states as a function of the k 3 parameter value. Steady-state concentrations were calculated by numerical integration of equations I and II (see Appendix) with the following set of parameter values: V M = 10, c = 0.001, d = 0.1, a = 0, k 2 =0.4 and k f =10 (for SSt steady-state) or 20 (for SSj steadystate)

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of microtubule acetylation. During this period, any fluctuation in MT concentration which is above the threshold T 2 is spontaneously eliminated, the system coming back to the 55 j steady-state.

Our experimental results [10] show that when the acetylated network of A-paratene-associated microtubuIes depolymerizes, it is not replaced by an equivalent unacetylated network. This means that absolute concentrations of MT in the 55) steady-state and MT-Ac (sum of MT-Acfree and MT-Acbound species) in the 55! steady-state are not of the same order of magnitude. In other words, [MT] and [MT-Ac] scales in the scheme of Fig. 5 have to be different. Although major differences between V2 and v) rates would lead to a situation in which concentrations of total microtubules ([MT] + [MT-Ac]) in 55! and 55) steady-states are not of the same order of magnitude, such an occurrence is highly questionable from a physiological point of view [22] since it seems unlikely that the depolymerization rates of unmodified and modified microtubules are very different. Experimental data [25] indicate that acetylation does not significantly affect the rate or the extent of depolymerization of microtubules. As shown in Table 1, the fact that MT-Acfree species is able to interact at a v4 rate with a stabilizing element to give [MT-Achound species explains how the switch between 55! to 55) steady-states produces a dramatic decrease in the total microtubule concentration (sum of concentrations of the MT, MT-Acfree and MT-ACbound species in Table 1), i.e. why A-paratene-associated microtubules are not replaced by an unacetylated equivalent network when they depolymerize.

Discussion In this paper, we show that the behavior of a microtubular network (A-paratene-associated microtubules) during the cell cycle in Paramecium may be interpreted as the result of biochemical switches between acetylation and depolymerization pathways acting on stabilized microtubules. The molecular signal which triggers cortical morphogenesis is supposed to be a calcium wave. Our model also interprets experimental observations on the particular stability properties of the microtubular network with respect to cold and nocodazole treatments. Both structural and parameter changes of a model are essential for the elucidation of the regulatory principles of an open system, which would be difficult to obtain without mathematical

140 . M. Laurent and A. Fleury

models since the variation of parameters or even of the structure in a real biological system is possible only to a limited extent. With this respect, our analysis demonstrates how a dynamical model allows for a coherent representation of the data: two distinct molecular sites of action of Ca 2 + might be envisaged. Our model shows that only the inhibitory effect of Ca 2+ on tubulinyl acetyl transferase may have a triggering action on the dynamics of A-paratene-associated microtubules, whereas the indirect effect of these ions on MAPs, via the well-known phosphorylation process which is calcium-dependent, is irrelevant from a dynamical point of view.

On the Triggering Action of Ca 2 I Although the existence of the Ca 2, wa ve has not yet been demonstrated experimentally, the molecular nature of the triggering signal is of little importance in our model. MT (stabilized but unmodified microtubules) has a dual action on tubuliny\ acetyl transferase: it is both substrate and inhibitor of the catalyzed reaction. This dual action constitutes the molecular basis of the feedback loop whose presence is a prerequisite for a bistable behavior in our system [11, 16]. The corollary of this observation is that it is not the Ca 2 ·- regulation which endows our model with bi-stability properties. This regulatory effect is only a biological interpretation - but the most likely interpretation with respect to the available experimental data - on the nature of an external variable of the model. The direct demonstration of the occurrence of the Ca 2+ wave is extremely difficult to give from an experimental point of view. This is a very transient phenomenon in the time scale of the cell cycle. Its observation under a microscope would suppose the injection of Ca2+ indicators in the vicinity of cortical alveoli of living but immobilized interphase cells. The injection process and the conditions of cell immobilization would have to leave the biological cycle of Paramecium undisturbed, allowing the cell not only to survive but also to divide. Nonetheless, such an experimental demonstration would allow progress in the understanding of which factors and mechanisms determine and control the spatially and temporally ordered morphological changes in this species. The simplest experiment to test this model would consist in injecting calcium in interphase cells. Our model predicts the existence of a threshold in the calcium concentration for which depolymerization would be triggered. Although some indirect and non-dynamic components might also contribute to the global action of calcium on microtubules, our analysis shows how dynamic and non-dynamic effects might be distinguished.

Dynamic and Non-dynamic effects We have seen that the complete interpretation of our experimental data consisted in considering that the A-

paratene associated microtubules are able to interact, in interphase, with a stabilizing element. Accordingly, the experimental data show that this acetylated network of microtubules originates as bundles from the posterior kinetosome of each basal body pair. The interaction which results from the bundling effect would explain why the A-paratene-associated microtubules are not replaced by an equivalent unacetylated network when it depolymerizes. Such an interaction alters the dynamics of the system but it is not per se a dynamic interaction. A dynamic process may be viewed as a process whose effects persist when its causal origin has disappeared. The effect of the transient calcium wave on the disassembly of the A-paratene-associated microtubuIes typically appears as a dynamic process; the bundling effect does not. At the same time, the A-paratene-associated microtubules appear as intracytoplasmic bundles of microtubuIes not clearly linked to any other cytoskeletal element. On the contrary, the cytospindle is a transient network which seems to be assembled from one of the microtubular ciliary rootlets, i.e. from microtubules which are already acetylated. Moreover, the cytospindIe is closely apposed to the membrane [24] and particular stabilizing interactions between cortical determinants and microtubules might occur in this case. Hence, the behavior of various acetylated microtubular networks in the course of division (the A-paratene-associated microtubules and the cytospindle, for instance), in response to a morphogenetical signal, might be quite different in a given cellular area, depending upon their relation with other cytoskeletal elements. Cellular behavior probably is the sum of dynamic and static interactions. It is probably unrealistic to expect the identification of all of the static components from the interpretation of dynamic effects.

On the Putative Role of Post-translational Modifications of Microtubules In our model, the total amount of stable microtubuies (modified and unmodified) is a function of a set of parameter values (Table I). Accordingly, the model both explains how a modified network might be replaced by another unmodified one and, as observed experimentally in the case of the A-paratene network, how post-translationally modified microtubuies may completely disappear upon transition between stable steady-states following an inductive signal. That means that in a dynamical model which exhibits multistability capacities, the regulative properties of each modification enzyme allow a double control: they regulate specifically the relative amount of each subset of distinctly modified and unmodified microtubules. This type of regulation is important if the various types of modified microtubules are functionally distinct. But, in addition, the effectors of enzymes may also regulate differentially the total amount of stable microtubules.

Dynamics of Microtubules in Paramecium· 141 Hence, a duality may exist in the biological significance of the diversity of post-translationally modified microtubules. Until now, each type of modified microtubules was believed to possess unique functional capacities. Accordingly, the primary role of the observed diversity would be to increase the functional repertoire of cells so that different effector systems might utilize differentially the array of stable microtubules [5, 14, 32]. However, little is known about what these effector systems are and how they could act. It is quite possible to imagine that the various types of modified microtubuIes are functionally equivalent but are subject to separate control mechanisms allowing differential timing, place or amount of stable microtubules. In this view, as a consequence of our model, the various control mechanisms acting at the level of each modification enzyme would modulate specifically the global amount of stable microtubules. However, this interpretation would have to be re-examined for more complex schemes allowing multiple and non-exclusive enzymatic modification pathways for microtubules (Laurent and Fleury, manuscript in preparation). Of course, different biological roles for distinct post-translational modifications of microtubules are conceivably true to different extents in different systems. Our hypothesis on a global level of regulation is reminiscent of that which supposes that the different isotypes of tubulin gene families do not have specific functions but instead specific regulatory requirements evolved to modulate the amount of tubulins in a cell and tissuespecific manner [20, 30]. It is important to note that the enzymatic mechanism of regulation that we propose allows immediate responses of the cell to physiological requirements, whereas controls at the genic level are characterized by a lag-phase (which is associated to biosynthetic processes).

Acknowledgements This work was supported by grants from the CNRS and the Universite Paris-Sud. We thank Andre Adoutte (Orsay) and Drs. Joseph Frankel and Maria Jerka-Dziadosz (Iowa City) for helpful comments and constructive criticism.

in which MT is the stabilized form of unmodified microtubuIes and MT-Acfree and MT-Acbound represent the acetylated microtubules. The kinetic processes described by V1 and v 3 correspond to the depolymerization rates of unmodified and acetylated microtubules, respectively. MT-Acfree species have an additional conversion pathway, in addition to the vrsymbolized process of depolymerization: this species is able to interact at a v4 rate with a stabilizing element to give [MT-Achound species. The latter depolymerizes according to the VI rate. Besides the v J -process, all individual steps are supposed to correspond to first-order kinetic rate laws. Acetylation of stabilized microtubules (v j-catalyzed reaction) corresponds to a kinetic process in which the enzyme is inhibited by an excess of substrate (see main text). Hence, we have presently: Vl =

kz[MT]

Vl =

k 3 [MT - Ac]

V4 = k4[MT - AC]free VI =

kdMT - AC]bound

Vj = Vr-,dMT]/([MT]l

+ c[MT] + d(1 + a))

where V M, c, d are positive constants and a is the dimensionless Ca l + concentration (i.e. molar concentration divided by the corresponding dissociation constant). The action of Ca2+ ions which is considered here solely corresponds to its inhibitory effect on the tubulinyl acetyl transferase activity. Theoretical treatment of an analogous scheme has been described elsewhere [22]. Briefly, we consider at first the input rate Vo as a constant (vo =k f) in order to describe the stability properties of the model. In a second step, Vo is introduced as a phenomenological variable with the aim to investigate the dynamics of the system. This second step will allow us to examine for instance the indirect effect of Ca2+ on MAPs since this ion modulates the rate of stabilization of microtubules Vo (which is also the input rate in our model). Insights into the dynamical behavior of the system is first obtained grapho-analytically by examining this set of equations. Temporal evolution may be described by numerical integration of differential equations I to III.

Multiplicity and Stability Properties of Steady-states Appendix According to the overall scheme of Fig. 1, dynamics of the open subsystem which is under consideration (dashed lines) is described by the following differential equations: d[MTJldt

=

Vo - VI -

Vl

d[MT - AC]free/dt = VI - V3 - V4 d[MT - AC]bound/dt = V4 - Vs

(I)

(II) (III)

Fig. 6 shows the MT-concentration dependence of kinetic processes involving stabilized microtubules MT, for the same input value Vo and in absence of Ca l +. For the purpose of analysis, these kinetic processes can be divided into an output component (sum of acetylation rate (Vj) and depolymerization rate (Vl) of stable unmodified microtubules MT) and an input component (va rate). The steady-state concentrations of MT correspond to the condition d[MT]/dt =0, i.e. MT-production rate va equals the sum v J + V1 of the rates of MT-removal processes. In other words, in any of the steady-states, the algebraic difference (which is defined as the net rate) between production and removal processes for MT species, is null. Over a wide range of va values, MT-input and -output graphs have one or three intercepts which define as many steady-states for the system.

142 . M. Laurent and A. Fleury 18 18 1<

12

" 10 ';j

IX

8

·v·;··· ..

2

o IL-~~~~~~~~~~~~~~~

o [MT]

Fig. 6. Individual rates of production (va) and removal (Vj + processes of stabilized microtubules MT. Non-enzymatic production rate (va) of MT is assumed to be constant (independent of MT concentration). V2 is assumed to correspond to a first-order rate process: V2 = k 2 [MT]. The kinetic law of disappearance of MT through the tubulinyl acetyl transferase catalyzed reaction is given by: VI =V M [MT]/( [MT]2 + c[MT] + d(l +a)), that is a reaction in which the enzyme is inhibited by an excess of its substrate. According to va value, the (Vj + V2) graph may have one or three intercepts with the (v 0) graph, i.e. the system has one or three steady-states. (5ame parameter values as in Fig. 2). V2)

In order to analyze the local stability properties of each of these steady-states, the net rate of MT-formation as a function of MT concentration has been reported on Fig. 2, when three steady-states coexist. As previously described in the case of a slightly different scheme [22], the stability properties of each stationary state 55], 55 2 and 55 3 can be investigated by determining whether infinitesimal perturbations in [MT] away from the steady-state will decay or grow with time. Briefly, let us suppose that the system initially in 55 j or 55 3 is perturbated by a slight positive fluctuation in MT concentration, 8 [MT] (right horizontal arrow in Fig. 2). In these conditions, the net rate of MT-production becomes negative, i.e. the rate of MT-consumption exceeds the rate of MT-formation. Thus, the fluctuation is spontaneously eliminated and the system comes back to its initial steady-state. If 8[MT] is negative (left horizontal arrow), the net rate of MT-production is positive: the MT-production rate exceeds the MT-consumption rate and once again the system will move back to its steady-state. Thus 55 j and 55 3 are stable steady-states. On the contrary, small fluctuations around 55 2 will be amplified (Fig. 2) and 55 2 is thus unstable.

Relation between Steady-states and Depolymerization and Acetylation Processes What is the meaning of this bi-stable behavior in terms of depolymerization and acetylation processes? In our model, the enzyme which ensures the post-translational modification of microtubules is inhibited by an excess of its substrate (v j catalyzed process in the scheme of Fig. 1). This welldocumented mechanism of enzyme inhibition [6] constitutes the feedback loop in our subsystem. The rate equation of such an enzymatic mechanism is not of the form of the MichaelisMenten equation, by virtue of a quadratic term in the substrate concentration. This term becomes significant only at high substrate concentrations: the velocity approaches the

Michaelis-Menten value when the substrate concentration is small, but approaches zero instead of Vm when the substrate concentration is large. That means that the stable steady-state 55 j obtained at the lowest MT concentration corresponds to conditions in which v j (acetylation) dominates over v 2 (depolymerization). On the contrary, Vj is almost negligible (with respect to V2) when the system is in its stable steady-state 55 3 (corresponding to highest MT concentrations). Thus, when the system is in 55 3 , global fluxes essentially proceed through the depolymerization process. Hence, switches between steady-states correspond to an alternation between acetylation and depolymerization pathways for stabilized microtubules.

Trajectory of Steady-states If va is now considered as a phenomenological variable, the dynamical behavior of the system is described by solving for [MT 55] (steady-state concentration of MT) the following equation (which is a cubic in [MT]): va -

Vj -

V2

= 0

in which k 2 , k 3, VM, c, d, and a are constant parameters. The set of points (va' [MT 55]) which are the solutions of this equation forms a curve which is called trajectory of steadystates (Fig. 3). That means that when va varies, [MT ss ] varies but the point defined by any couple of data (va, [MTss)) necessarily lies on this trajectory. However, its region of negative slope corresponds to unstable stationary states whereas the regions of positive slope are associated to stable steady-states (55 3 for the part B of the stable domain and 55 j for the part A of the stable domain). Let us suppose that the system is initially on 55 j steady-state and that a continuous increase in va input value occurs (closed arrows in Fig. 3). Until the stationary state lies on the part A of the trajectory, MT concentration is slightly readjusted, in accord with the solution trajectory. But when va exceeds the threshold value (which corresponds to the change in the sign of the slope of the solution trajectory), the system moves to the 55 3 domain of stability and a strong discontinuity (jump-like transition symbolized by the upper horizontal dashed line in Fig. 3) appears for the steady-state concentration of MT.

Moving on a Trajectory of Steady-states and Modifying the Shape of a Trajectory A trajectory of steady-states is established, in the (va' [MTss]) plane, for a given set of constant parameter values (k 2 , k 3 , V M, c, d, and a in our model). As long as these parameters are not modified, any factor which affects va value moves the system along this trajectory. On the contrary, any factor which affects the value of one or several constant parameters modifies the shape of the trajectory of steady-states. This is for instance the case for any process which alters the parameter values of the rate of acetylation (Vj process).

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Dynamics of Microtubules in Paramecium· 143

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15

16 17

18

19

20

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Key words: Paramecium - Microtubules - Acetylation - Post-translational modifications - Dynamical systems Bistability

Michel Laurent, Service d'lmagerie Cellula ire, URA 1116 CNRS, Batiment 441, Universite Paris-Sud, 91405 Orsay Cedex, France