Understanding phase behavior of plant cell cortex microtubule organization

Abstract

Plant microtubules are found to be strongly associated with the cell cortex and to experience polymerization/depolymerization processes that are responsible for the organization of microtubule cortical array. Here we propose a minimal model that incorporates the basic assembly dynamics and intermicrotubule interaction to understand the unexplored phase behavior of such a system. Through kinetic Monte Carlo simulations and theoretical calculations, we show that the self-organized patterns of plant cell cortical microtubules can be regulated by controlling single microtubule assembly dynamics. Biologically, this means that the structural reorganization can be regulated by microtubule-associated proteins via changing microtubule dynamic instability parameters, such as the microtubule plus-end growing rate, GTP-tubulin hydrolysis rate, etc. Such regulation is indirectly confirmed by various in vivo experiments. For the physical aspects, we not only construct the phase diagram that determines under what parameters ordered microtubule arrays form, but also predict that the essentially different ordered structures may appear through continuous and discontinuous transitions. The present study will play a central role in our understanding of the basic mechanism of plant cell noncentrosomal microtubule arrays.

Understanding the structural organization of the microtubule (MT) cytoskeleton is a central problem in cell biology (1–4). In plant cell, MTs are confined to a thin layer of the interior side of the plasma membrane and may organize into the so-called cortical MT array—a highly ordered structure just beneath the plasma membrane. In contrast to equilibrium ordered structures, such a MT cortical array, which is unique in the plant cell, constantly changes its appearance and even may disappear and reform during different phases of cell cycle (5–7). One open question is how the plant cell regulates the formation of cortical arrays and what dictates the self-organized patterns of MTs. From a physical point of view, it is also interesting to understand these phenomena by asking how the ordered structures are achieved through the noncentrosomal MT cortical organization and whether there are any basic mechanisms governing such dynamic order transitions.

Recent in vivo experimental studies have made great progress in revealing the possible formation of plant cell cortical arrays (5, 8). Unlike animal cells, the nucleation of new MTs is found to be widely dispersed on plasma membrane (9–11), because of the lack of centrioles in plant cells. MTs are found stably attached to the membrane, and therefore lateral and rotational diffusions of MTs are suppressed. Instead, treadmilling behavior of MTs, which is achieved by plus-end polymerization and minus-end depolymerization, was observed on the plasma membrane (9, 10), and consequent intermicrotubule collisions may frequently occur between a polymerizing MT and a preexisting one (9, 12). If two MTs encounter with a shallow angle, the MTs are highly probably zippered to form a small bundle, but encountering with a steep angle will trigger catastrophe of the stopped MTs or crossover. As shown in a previous simplified model (13), the collision between polymerizing MTs is enough to cause their alignment. Therefore, a self-organized picture may emerge under the dynamic interplay between assembly of individual MTs and their interactions. However, the regulation of the cortical array, its formation, and disappearance are still unclear. Particularly, one key issue that has not yet been studied is the phase behavior of cortical MT noncentrosomal array organization in the plant cell. In the present study, we reveal the phase behavior of interacting MTs confined in the thin shell between the vacuole and cortical membrane of plant cells by using a kinetic model on the basis of recent experimental findings (see Materials and Methods). Both kinetic Monte Carlo simulation and theoretical studies clearly demonstrate the importance of coupling between assembly dynamics and steric interaction of MTs, which widely exists in the cell cytoskeleton but normally is neglected in simulations for simplicity, to display the MT collective behavior. A full phase diagram will be helpful for further understanding how the dynamic instability parameters regulate the MT arrays, and therefore the model suggests a possible molecular mechanism that is responsible for the steady-state order-disorder transitions in plant cell cortical MTs.

Results and Discussion

The Formation of Ordered MT Arrays.

We first perform numerical simulations (see Materials and Methods) to explore the following questions: Is it possible to achieve steady-state transitions via changing the MT dynamic parameters, and how do these parameters influence the self-organization of the system? The answer to these questions is important for the further understanding of the regulation mechanisms of a biologically ordered structural formation.

Fig. 1 shows the steady-state ordering transitions of our simulations with varying the MT plus-end GTP-state polymerization rate (k_gt). In Fig. 1A, increasing k_gt without changing other MT dynamic parameters (Table 1) leads to the formation of orientational ordered MT arrays (see Movie S1 for the ordering process of MT arrays). The steady-state transition shows a discontinuous characteristic, indicating that the system can be regulated to stay in two totally different states. Such a discontinuous transition can be further displayed by the variation of the density of segments that compose the MTs, shown as a function of k_gt in Fig. 1B. Depending on different random processes, Fig. 1 C and D shows two different snapshots (i.e., disordered and ordered bistable states) of MT arrays for the same k_gt = 0.37. Fig. 1 E and F shows the angular distribution of MT lengths corresponding to Fig. 1 C and D, respectively. In the disordered state (Fig. 1E), MTs are uniformly dispersed along different orientations. In contrast, the MTs in the ordered state are concentrated on a predominant orientation, as clearly shown in Fig. 1F. Furthermore, with the variation of other dynamic parameters, our simulations show that the system undergoes similar steady-state transitions (see Figs. S1–S4), indicating that the MT arrays can be elegantly regulated by controlling the MT assembly dynamics. Such a remarkable phenomenon is called the compensation effect between the dynamic instability parameters in introducing dynamic steady-state transitions (14), which will be further discussed in Mean-Field Results.

Fig. 1.

Discontinuous isotropic-nematic-like steady-state transition observed in simulation. (A) The orientational order parameter η (see Materials and Methods) changes discontinuously with the increase of the plus-end GTP-state polymerization rate k_gt. (B) The density of segments that compose MTs changes as the steady-state transition emerges, which serves as a further indicator for the discontinuous transition. Individual simulations results are plotted (Red and Cyan). The data points with error bar are an average of the isotropic-like state with η < 0.2 or nematic-like state with η > 0.8 for the same k_gt. The red circle represents the simulation sample transforming from the isotropic state to the nematic state, which is not included for averaging. Near k_gt = 0.37, bistable disordered and ordered states can be expected. (C) Snapshot of isotropic-like state for k_gt = 0.37, which is the same as those at low k_gt, where MTs remain randomly oriented and loosely packed. (D) Snapshot of nematic-like state for k_gt = 0.37, which is the same as those in high k_gt, where MTs remain highly oriented and spontaneously densely packed. In C and D, GTP-state units are shown in red, whereas GDP-state units are green. (E) Angular distribution of MT lengths in isotropic-like state as in C, where the orientation of MTs is uniform. (F) Angular distribution of MT lengths in nematic-like state as in D. Long MTs are spontaneously distributed in a narrow orientation, the predominant angle of which is assigned to be 90°. In E and F, the color scale bar shows the local density values of the distribution of MTs over length and angle.

Table 1.

Dynamic instability parameters of the simulation

Parameter description (symbol)	Simulated values
Simulated MT segment length (a)	1 (80 nm)
Simulated time step (τ)	1 (0.2 s)
Plus-end GTP-state growing rate (kgt)	0.3 (120 nm/s)
Plus-end GTP-state shortening rate (kst)	0.005 (2 nm/s)
Plus-end GDP-state growing rate (kgd)	0.03 (12 nm/s)
Plus-end GDP-state shortening rate (ksd)	0.5 (200 nm/s)
Minus-end GDP-state shortening rate (ksm)	0.1 (40 nm/s)
Hydrolysis rate of GTP-state unit (kh)	0.05 (2.5 segments/s)
Nucleation rate (kn)	0.005 (∼4.0 μm-2 s-1)
Maximum MT length (Lm)	50 (4 μm)

When we take a = 80 nm and τ = 0.2 s, the corresponding values of parameters in the right-hand parentheses are in reasonable accordance with previously reported experimental data (9, 12). Unless otherwise indicated, all data presented in this table are used in the following simulations.

The Phase Behavior of Cortical MTs.

It has been shown experimentally that the MT nucleation mediated by γ-tubulins in the interphase plant cell happens on the cortical membrane, which is followed by severing of MT minus end (9, 10). The rates of these events are difficult to determine in experiments. Instead, a more convenient observable quantity is MT number density. Here we map out the phase diagram for the controlled MTs number, which could provide insights into further experimental observations.

In Fig. 2A, the simulated system shows three regimes of phase behavior dependent on the polymerization rate k_gt and MT number density: (i) the disordered “isotropic” (I) phase with randomly oriented MTs; (ii) the highly ordered “nematic I” (N_I) phase where long MTs are distributed in a narrow orientation; and (iii) the weakly ordered “nematic II” (N_II) phase where short MTs are distributed broadly along a preferred orientation. Fig. 2B shows the angular distribution of MT lengths near the I–N_I transition, which is similar to that of Fig. 1 E and F, signifying a possible discontinuous transition. Furthermore, the difference between N_I and N_II phases has been clearly displayed by the angular distribution of MT lengths in Fig. 2C. To identify the transition properties between different phase regimes, we analyze the typical processes marked by short lines (d)–(f) in Fig. 2A. These processes are represented by the orientational order parameter η as functions of MT number (Fig. 2D) and k_gt (Fig. 2 E and F), respectively. As shown in Fig. 2D, with the increase of MT number, the system first undergoes continuous transition from the I to N_I phases and then reenters into the disordered I phase continuously. In Fig. 2E, the I–N_I transition is discontinuous, and in Fig. 2F, the system first continuously enters the N_II phase, but with further increase of k_gt, the system undergoes discontinuous transition into the N_I phase. In the present phase diagram, a reentrant phenomenon appears: With the increase of MT number density, the system undergoes isotropic–nematic–isotropic–nematic transitions in the region 0.21 < k_gt < 0.32, because of the competition between the MT polymerization dynamics and the intermicrotubule interactions. For the first isotropic–nematic transition, polymerized long MTs are formed and organized into the nematic state with increasing MT number density. However, with further increase of MT density, steric interactions between MTs suppress the polymerization dynamics. The existing MTs become relatively short and can only be organized into the disordered isotropic phase. In the second isotropic–nematic transition, short MTs can still be reorganized into N_I or N_II states, depending on the value of k_gt. For small k_gt, the I–N_II transition is because of the competition between steric interaction and orientational entropy, which is similar to the equilibrium isotropic–nematic continuous transition. By contrast, the discontinuous I–N_I transition is because larger k_gt further elongates MTs and reinforces nematic order consequently. A deep understanding of such rich phase behavior will be provided in the framework of the next mean-field theory.

Fig. 2.

Phase behavior of the system under controlled MT number: simulations. (A) In the phase diagram, isotropic (I), nematic I (N_I), and nematic II (N_II) phases are predicted. All data points in the I (Cyan Area), N_I (Pink Area), and N_II (Buff Area) phases correspond to separate simulation runs. The disordered and ordered states are distinguished by computing the weighted order parameter η of each individual simulation: For η < 0.3 the system is considered to be in the disordered state, whereas for η > 0.3 it is in the ordered state. Horizontal and vertical short lines attached with (d), (e), and (f) display the transition processes across the phase boundaries, which are used for further analysis in D–F, respectively. (B–C) Typical angular distributions of MT lengths distinguish the transitions between I and N_I phases (B) and between N_I and N_II phases (C). (D–F) Typical transitions in phase diagram are displayed—i.e., continuous I–N_I–I transitions in D, discontinuous I–N_I transition in E, and continuous I–N_II and discontinuous N_II–N_I transitions in F (see Figs. S5–S7 for snapshots displaying their transitions).

It has been observed in vivo that a distinct difference in single MT dynamics between interphase and preprophase of plant cell cycle exists and is thought to be responsible for cortical MTs structural reforming (7). Such in vivo measurement of single MT dynamics (7, 10, 12) can be used in experiments to examine if MT polymerization is more favored in N_I phase than in I or N_II phases, which is clearly indicated by our phase map. Moreover, through fluorescent imaging of cortical MTs, it would also be possible to distinguish N_I and N_II phases as predicted in our simulations by measuring the angular distribution of MT lengths. Further roles of these phases in producing cell stress and guiding cell wall polymer deposition will be of great biological interest.

Mean-Field Results.

Now we present the mean-field model (see Materials and Methods), which gives a unified understanding of the simulation results. For the spatial homogeneous steady state, the MT number distribution can be formally solved from Eq. 2 and written as

[1]

where the effective polymerization free energy ΔG = ln[k_p0/(k_df + k_db)] depends on the plus-end polymerization rate k_p0, the plus-end depolymerization rate k_df, and the minus-end depolymerization rate k_db. The parameter A = k_n exp(-2ΔG)/(k_df + k_db) corresponds to the simulation case with dimer nucleation, and k_n represents the nucleation rate of MTs. To determine the possible phase boundaries via the dynamic instability parameters, we perform bifurcation analysis (15) around the isotropic solution f(u,l) = f_l = Ae^{ΔG·l-2f₀l〈l〉}, which shows that, at phase boundaries, , with l = 2,…,L. Here , , and . The relation between critical values A^* and ΔG^* is thus obtained, and the steady-state transition is shown in Fig. 3A by using and . Along the short green line marked in Fig. 3A, black curves in Fig. 3 B and C represent that the iterative solution begins with an isotropic condition. We find that, with increasing k_p0, the isotropic solution becomes unstable and the system self-organizes into a nematic state when k_p0 > 0.1115. On the other hand, the red curves represent that the initial state is a nematic one, showing that the transition happens when k_p0 > 0.1075. A bistable region of the system is thus expected for 0.1075 < k_p0 < 0.1115. Fig. 3 D and E shows the angular distribution of MT lengths for the disordered and ordered phases when k_p0 = 0.11. Therefore, the mean-field model reasonably captures the main features of the steady-state transitions of MTs, in accordance with the above simulation results.

Fig. 3.

Theoretically predicted steady-state phase behavior. (A) Phase transitions by varying the nucleation rate k_n with k_df + k_db = 0.1 and L = 50, which corresponds to the maximum MT length L_m of the simulation model. The short green line indicates the transition process for a fixed k_n = 0.0016 studied in B and C. (B–C) With varying polymerization rate k_p0, the density of segments that compose the MTs (B) and weighted apparent order parameter (C) change discontinuously with the isotropic initial condition (Black) and nematic initial condition (Red) in the iterative numerical solution. The system is bistable for 0.1075 < k_p0 < 0.1115. For k_p0 > 0.1115 corresponding to the intersection point of the green line and blue curve in A, the isotropic solution becomes absolutely unstable. (D–E) The angular distribution of MT length in nematic- (D) and isotropic-like (E) steady-state solution for k_p0 = 0.11.

The compensation effect is clear through our mean-field predictions. For ΔG > ΔG^∗, the system self-organizes into a nematic state, and thus the transition can be achieved by increasing the plus-end polymerization rate k_p0 or decreasing the plus-end depolymerization rate k_df and the minus-end depolymerization rate k_db according to ΔG = ln[k_p0/(k_df + k_db)]. This property means that the effects produced by increasing k_p0 can be compensated by increasing k_df or k_db. Furthermore, if other parameters may influence k_p0, k_df, and k_db, they will play similar regulation roles. In plant cells, microtubule-associated proteins (MAPs) have been identified in controlling MT assembly dynamics (16). In vivo experiments, mutation or overexpression of some MT plus-end binding proteins and MT severing proteins will cause the cortical array aberrations because of the change of MT assembly dynamics (2, 17, 18). Furthermore, through in vitro experiments, it is possible to quantitatively examine the regulation mechanism proposed here by changing the well-controlled MAPs or GTP levels.

For the controlled MT number case, by using , where ρ is the MT number density, bifurcation analysis gives the critical concentration , where . Further relations between ρ^* and are thus plotted in Fig. 4 as the solid and dotted blue curves. By introducing an effective “free-energy” functional (see Materials and Methods), it can be shown that the solid blue curves are continuous transition lines. The red curve in Fig. 4 is discontinuous first-order and is determined by equating the effective free energies of two phases. So the crossover point where the blue line at small ρ meets the red curve is called the tricritical point (TCP), whereas the point where the blue line at large ρ terminates at the red line is the so-called critical endpoint (CEP) (19).*

Fig. 4.

Phase diagram of the system under controlled MT number: theory. The isotropic (I) phase, nematic I (N_I) phase, and nematic II (N_II) phase are theoretically predicted, and the phase diagram shows the same qualitative results as in Fig. 2A. TCP and CEP are exactly determined on the phase boundary.

It is well known that the equilibrium liquid crystal I–N transition is because of a competition of translational entropy and orientational entropy. In a three-dimensional system, the I–N transition must be of first order and the phase diagram predicted here can no longer be applied. However, in two-dimensional cases, the transition can be continuous if the competition is purely because of translational and orientational entropy (15). Furthermore, as the polymerization free energy is introduced, the disorder–order transitions may become nonmonotonic. At high density ρ, the I–N_II transition is continuous because the polymerization of MTs is suppressed by steric interactions, and the competition is mainly between the translational and orientational entropies. By contrast, the discontinuous N_II → N_I transition is because of the increase of k_p0, showing that the polymerization free energy dominates the transition, and nematic order is further strengthened by elongated MTs. Finally, the discontinuous I → N_I transition is similarly driven by lowering the effective polymerization free energy, which favors long MTs, with the expense of suppressing orientational entropy. At the same time, steric interaction is reduced by forming nematic order, which is required for MT elongation. In the region between the dotted blue and solid red curves, the isotropic state is metastable (the free energy of the I phase is higher than that of the N_I phase). Dynamically, however, the system becomes bistable between the I and N_I phases near the discontinuous transition line in that region, depending on the initial conditions. Such bistability may make the cell immune to unavoidable intracellular molecular noises. Only when strong regulation signals greatly lower or rise dynamic parameter values may the change of system state be possible. This kind of robustness is absolutely important for plant cells, because MT cortical array has distinct appearances in different phases of cell cycle (5–7). These spatial arrangements are relatively stable even under the influence of intracelluar noises, which play crucial roles in guiding consequent plant cell growth and morphogenesis (1–4). Whereas previous experiments (2, 20–23) tracking the time series of MT array patterns are not conclusive on the properties of steady-state transitions, probing if there exists such bistability in cortical MTs can be useful to check the existence of discontinuous transitions in vivo.

Conclusion

We presented computational and theoretical studies that predict the emergence of steady-state transitions in plant cell noncentrosomal MT organization. The work is based on the combination of the assembly dynamics and steric interaction of MTs on a two-dimensional membrane surface. A general phase behavior is constructed, where three phase regions, including the isotropic phase, highly ordered nematic I phase, and weakly ordered nematic II phase, are identified and both continuous and discontinuous transitions are predicted. The predicted phase behavior can be reexamined in experiments by measuring the orientational order parameter and angular distribution of MT lengths. The theoretical framework that we build for these transitions will provide useful guidance as to how to explore MT dynamic parameter space to find plant cell MT arrays. In general, many cytoskeleton morphogenesis are often emerging as dynamic structures. The present combined study will provide a powerful tool to predict and analyze dynamic phase transitions of such cytoskeleton structures under various complex dynamic conditions.

Materials and Methods

Kinetic Monte Carlo Simulation.

The simulation model is designed to show the self-organization of plant cell cortex MTs and is implemented strictly at the scale of single MTs via their interactions. As has been identified in in vivo experiments, MTs are strongly associated to plant cell cortex (7, 9, 10, 24), and therefore the model can be reduced to a 2D interacting MT system that is coupled with a solution shell as the reservoir required for subunit exchange. MTs are scattered over the 2D substrate, and they will undergo nucleation, hydrolysis, polymerization, and depolymerization processes. The only intermicrotubule interaction required in simulation is steric interaction—i.e., MTs are not allowed to overlap with each other on the substrate during the simulation.

A single MT is simulated as an infinitely thin hard rod, and, in the absence of intermicrotubule encounters, its assembly dynamics is described by a mesoscopic GTP cap model (25–27) (see SI Text and Figs. S8 and S9) that contains five dynamic parameters (see Fig. 5A): k_gt[k_gd] and k_st[k_sd] describe the MT plus-end growing (g) and shortening (s) rates when the MT plus-end unit is in the GTP(t)[GDP(d)] state, and k_h describes the hydrolysis rate (i.e., the rate by converting GTP-state tubulin into a GDP-state one) of the internal MT subunits. The GTP-state subunit on the plus end does not hydrolyze in the simulation model but can depolymerize with a certain probability (i.e., the rate k_gd). Because spontaneous GTP-state tubulin hydrolysis is extremely slow, this description is appropriate in view of the docking triggered GTP-hydrolysis mechanism (27, 28). On the basis of experimental observations (9, 12), the model is further developed by introducing another parameter k_sm describing the shortening(s) rate of the MT minus end when the end unit is in the GDP state. The coexistence of minus-end depolymerization and plus-end polymerization of a single MT is responsible for its treadmilling observed in experiments (9, 12). In simulation, a maximum MT length L_m (with reflective boundary condition) is assigned to approximate the complicated regulation and confinement of MT length in cells (18, 25, 29) (see SI Text and Fig. S10).

Fig. 5.

Schematic of MT dynamics. (A) Single MT dynamics without intermicrotubule interactions. k_gt[k_gd] and k_st[k_sd] describe the MT plus-end growing (g) and shortening (s) rates when the MT plus-end unit is in the GTP(t)[GDP(d)] state, k_h describes the hydrolysis rate of the internal MT unit, and k_sm describes the shortening rate of the MT minus end when the end unit is in the GDP state. The red cross means that the hydrolysis of the MT cap is not allowed. Subunit (red, GTP; green, GDP) shapes are plotted for visual convenience. (B–C) Two successive steps obeying the evolving rules of interacting MTs: MT1 represents a newly nucleated MT undergoing plus-end polymerization and hydrolysis of the noncapped MT segment. MT2 is a long MT undergoing polymerization, hydrolysis, and depolymerization. MT3 is a MT stalled by MT2, and the polymerization process stops because of steric interaction; it also undergoes the depolymerization process. MT4 represents a two-subunit MT that is eliminated from the system in the next step as a result of depolymerization. It is also stalled by MT2. MT5 represents a MT that undergoes plus- and minus-end depolymerization. By plus-end depolymerization it loses its GTP cap on that end. MT6 is a newly nucleated MT in the later step.

Now we implement how to incorporate the steric interaction between MTs. Because of the strong cortical association, MT thermal diffusion is suppressed and negligible compared with treadmilling. Through treadmillings, MTs move along the direction of their plus end and may encounter each other. Encounter between MTs is a kind of steric interaction (i.e., when MTs are all strongly associated on the cortical membrane, they cannot penetrate each other). In a rigid MT model, as we adopted here, a direct consequence of such steric interaction is that it hinders the polymerization of the stalled MTs. That means that, when we try to add a segment on the plus end of a MT, if the added segment is found to intersect with any other preexisting MTs, then the segment should not be added on the MT because of steric interactions (see SI Text). Coupled with individual MT dynamics, the evolution rules of a 2D interacting MTs system are described as follows.

1. Nucleation: As reported in the literature (10, 11), the MT nucleating sites are scattered over the cortex, and if MTs form on the same location, they will depart at diverging angles (9). In our model, a short MT, two units both in the GTP state, forms on a 2D membrane surface with an attempt rate k_n at a random position and a random plus-end orientation. Because the steric interactions between MTs are taken into account, the attempt of randomizing nucleation is accepted only if the newly formed MT does not intersect with any other preexisting MTs. Because of experimentally reported strong and nearly continuous associations of MTs to the cortical plasma membrane (5), in our simulation, there are no translational and rotational diffusions of nucleated MTs. As shown in Fig. 5 B and C, MT1 and MT6 may represent two newly nucleated MTs in those two successive steps.

2. Polymerization: For every single MT, it can be elongated only in the plus-end directed orientation. The attempt to add a GTP-state unit onto a preexisting MT plus end depends on its plus-end state (k_gt or k_gd) and the maximum length requirement as has been stated (29). However, because the steric interactions between MTs are taken into account, the attempt of polymerization is accepted only if the updated MT does not intersect with any other preexisting MTs. In Fig. 5B, polymerization of MT3, MT4, and MT5 may be stalled, because the steric interaction requires that they cannot overlap with other MTs.

3. Depolymerization: Following single MT dynamic rules, a GTP-state or GDP-state unit on the two ends of a MT can be removed by depolymerization according to the specified depolymerization rates k_st, k_sd, and k_sm depending on the plus- or minus-end states. A MT constructed by only two subunits is eliminated from the system after depolymerization. In Fig. 5 B and C, MT4 is eliminated because depolymerization happens. In experiments, the elimination rule corresponds to the rare recovery of MTs when depolymerized to be invisible (9).

4. Hydrolysis: The hydrolysis process follows the single MT assembly dynamic rules and is irreversible during MT assembly. According to the rules presented above, we perform simulations of system size 450 × 450 in units of MT segment length (a) under a periodic boundary condition. The simulation starts from randomly oriented MT distribution by nucleation of short MTs. After that, newly nucleated MTs are added to the system, where MTs undergo polymerization/depolymerization processes and each MT unit may go through the hydrolysis process as prescribed in the simulation rules. For the controlled MT number system, randomly nucleated short MTs will be added into the system at each step to compensate MTs eliminated by depolymerization until the system reaches the required MT number.

The simulation model is an ideal version of experimental observations. Here, the mesoscopic dynamics of MTs is tightly coupled with the intermicrotubule interactions, and the hard needle-like steric interactions show similar effects as the encounter mechanism. These aspects are not captured by previous theoretical studies on the mechanism of cortical arrays formation, which are vitally important in the present self-organization model (30, 31).

Kinetic Mean-Field Theoretical Description.

In our theoretical approach, MT dynamics is coarse-grained by neglecting the internal state of the simulation unit (32), and steric interaction between MTs induced by polymerization events is taken into account on the basis of the Onsager virial theory (33). The kinetic equation can be written as follows (see SI Text and Figs. S11 and S12 for more details):

[2]

where f(r,u,l,t) is the number distribution function at position r and orientation u with length l and time t. a is the length of the MT segment. k_p is the plus-end polymerization rate, k_db is the minus-end depolymerization rate, and k_df is the plus-end depolymerization rate. The effective polymerization rate k_p would be changed via intermicrotubule interactions, and it can be written as k_p = k_p0(1 - p_r), where k_p0 is the plus-end polymerization rate without steric interactions and p_r is the rejection rate of adding a subunit because of intermicrotubule interactions. In the mean-field approximation (34), p_r can be self-consistently written as

Here, the integration kernel W(r,r^′,u,u^′,l,l^′) represents whether the added unit on the plus end of MT in the state (r,u,l) overlaps with a preexisting MT in the state (r^′,u^′,l^′) and can be written explicitly as follows:

In the present model, MTs self-organize through polymerization/depolymerization processes and MT steric interactions. The number density steady-state distribution obtained by solving Eq. 2 is a result of such self-organization processes, which are determined by dynamic parameters and intermicrotubule interactions. No other adjustment is involved to reach the final polydispersed lengths distribution. As we have shown in Figs. 1–3 through both simulation and theoretical calculation, during the ordering transition, it is often accompanied by a readjustment of MTs length-angle distribution. These features are difficult to capture for models that impose a static distribution of MT length.

Effective Free-Energy Functional.

By taking into account the contribution of polymerization force and the resulting polydispersity of MT lengths, an effective free-energy functional can be introduced to systematically determine the steady-state transitions as shown in Fig. 4, which can be written as follows:

[3]

The first term in Eq. 3, which is the entropic contribution, originates from the dispersed lengths and orientations of MTs because of the polymerization/depolymerization processes and randomly oriented nucleation. The second term accounts for the steric interaction induced by polymerization of MTs in the dynamic model. The third term, polymerization free energy, represents the energetic contribution undergoing polymerization/depolymerization force. The effective polymerization free energy ΔG represents the free energy lowered by adding one segment to the existing MTs (see SI Text). λ in the last term is a Lagrange multiplier enforcing the constraint . The minimization of Eq. 3 gives rise to the same result as in the steady-state solution Eq. 1 when A = e^-λ-1/L and ΔG = ln[k_p0/(k_df + k_db)].

Orientational Order Parameter.

Here the orientational transition is characterized by a weighted apparent order parameter η (19), which is defined as the positive eigenvalue of matrix , where n(i) is the number of segments of the ith MT and u_α(i) is the αth component of the orientational vector u(i) of the ith MT. M is the total MT number, and N is the total number of all segments that compose the MTs. Correspondingly, in the mean-field theory, is introduced for numerical analysis. The order parameter η = 0 describes a perfectly isotropic state, whereas η = 1 describes a perfectly aligned nematic state.