Agent based model

We used agent-based stochastic simulation to compute the spatial distribution of the molecules with binding affinity for microtubules. We traced the spatial trajectory of each molecule. They diffuse in the cytoplasm when unbound. And they travel along the microtubule when bound, the velocity of which depends on the associated motor proteins, and is zero if not associated with a motor.

The microtubule-binding molecule or motor-cargo complex is simplified as a spherical particle of 10 nm in diameter. In reality, these particles can assume various shapes, and the dimensions can differ up to an order of magnitude. But the size and shape does not significantly affect the binding dynamics as long as the binding is diffusion-limited (Figure S1, increase of particle radius by an order of magnitude only doubles the effective binding rate). A free particle diffuses in the cell with a diffusion coefficient, D (1∼20 µm²/s). It binds to a microtubule when the two objects are close by within a critical distance, d₀ ( = 0.8 nm). It unbinds from the microtubule with a dissociation rate, k (∼1 s⁻¹). If the particle represents a molecular motor and/or motor-cargo complex, then the particle, when bound, will travel along the microtubule unidirectionally with a velocity, V (∼1 µm/s). The parameters of the model are taken from independent experimental measurements/estimates, with references and/or reasoning listed in Table 1.

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Table 1. Parameters used to simulate the dynein-mediated transport by microtubule spindle.

https://doi.org/10.1371/journal.pone.0034919.t001

We started with the simulation of a 3D mitotic spindle-dynein system. Located in a cell of 20 µm in diameter, the model spindle consists of microtubules originated from two opposite spindle poles distanced by 10 µm. Each spindle pole organizes 800 microtubules within the 90°-cone, which is centered along the spindle axis and based at the mid-plane. Each pole also organizes 800 astral microtubules outside the cone (Figure 1). The density of microtubules is essentially 6 times higher within the spindle cone because the cone encases ∼15% of the surface area on a spherical surface. Each microtubule is represented by a cylinder of 25 nm in diameter, extending from the spindle pole to the mid-cell plane or the cell boundary.

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Figure 1. Simulated dynein sequestration by the microtubule spindle.

(A) Time series of dynein distribution (also see Movie S1). Dyneins are initially randomly distributed throughout the cytoplasm. Dark grey lines: microtubules (only 1 out of 10 are shown to reduce clutter); blue dots: dyneins bound to microtubules; red dots: dyneins freely diffusing in the cytoplasm. Dyneins become strongly concentrated around the spindle poles within 20 s. (B) Histogram (right) of the equilibrium distribution of dyneins. More than 50% dyneins are concentrated within 200 nm around the spindle pole, and 98% within 1 µm. (C) Dyneins are initially released from one end of the cell, close to one spindle pole (big red dot) (also see Movie S2). Legends are same as in (A). Dyneins become strongly concentrated around the proximal spindle pole within 10 s, but almost none on the opposite spindle pole. Over the simulated duration of 3000 s, only 0.3% dyneins turn over to the opposite pole. Assume that the turnover is a Poisson process, then prob(wait time<t) = 1−exp(−t/τ), where τ is the characteristic turnover time, or the reciprocal of the turnover rate. Plugging in prob = 0.003 and t = 3000 s gives τ = 10⁵ s. Both simulations in (A) and (C) are carried out with 1000 non-interacting dyneins and each half-spindle consisting of 800 spindle microtubules and 800 astral microtubules. The dyneins are expelled from the spindle pole immediately as they arrive, i.e. the spindle poles are reflecting boundaries.

https://doi.org/10.1371/journal.pone.0034919.g001

Sequestration by mitotic spindle

Motivated by the observed dynein-mediated sequestration of checkpoint proteins [1], [2], and germ plasm components at the spindle poles [4], we first used our computational model to simulate the spatial regulation of dyneins by the mitotic spindle, and the conclusion naturally extends to the cargoes of the dyneins. Our simulation demonstrated that dyneins are indeed intensely sequestered by the microtubule spindle, and highly concentrated around the spindle poles (Figure 1). The results directly explain the observations on dynein localization by Lerit et al [4] mentioned in the Introduction. If dyneins are initially evenly distributed throughout the cell, within 20 s they concentrate onto the two spindle poles (Figure 1A and Movie S1). At equilibrium, 98% dyneins are located within 1 µm around the spindle poles (Figure 1B). This scenario corresponds to the experimentally observed spindles that are sufficiently immersed in the cortex layer; the germ plasm components originally in the cortex are concentrated onto the spindle poles (right of fig.6B and right of fig.3C in [4]). For the few observed spindles that only touch the cortex layer with one pole (middle left of fig.6B and lower left of fig.3C in [4]), we simulated the transport process with the dyneins initially released from one end of the cell, close to one of the spindle poles. Within 10 seconds, the dyneins are strongly concentrated on the proximal spindle pole, but almost none on the opposite pole (Figure 1C and Movie S2). Although dyneins will eventually equilibrate to an equal partition between the two spindle poles, like the equilibrium state resulting from the symmetric initial distribution, the time scale for the turnover is on the order of tens of hours (∼10⁵ s), much longer than the relevant time scale for germ cell formation, or even the life span of the syncytial embryo (∼3 hrs). Thus, the model suggests that the microtubule-mediated sequestration effect can last long enough and serve as a robust mechanism to govern the spatial localization of signaling proteins in cellular processes. In this case, the two halves of the microtubule spindle essentially partition the cytoplasm into two non-interacting regions for the dynein-associated molecules.

Intuitively, the microtubule density is critical for the spatial regulation effect; thus the change in the microtubule density over the cell cycle can affect the spatial pattern of the microtubule-binding molecules. Figure 2 demonstrates that the increase of microtubule density enhances the sequestration effect (defined as the fraction of bound particles) and the concentration effect (defined as the inverse of the mean distance of the particles to the spindle poles), and elongates the pole-to-pole turnover time. Sharp changes occur between 100 and 1000 microtubules per pole/MTOC. This is right about the range over which the microtubule density is regulated through the cell cycle. During mitosis, each spindle pole organizes around thousand microtubules, while an interphase centrosome organizes around hundred microtubules. Therefore, by changing the density of the microtubule network, the cell is able to achieve differential regulatory effects on the microtubule-binding molecules.

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Figure 2. Effects of microtubule density on the spatial regulation.

The sequestration (fraction of particles bound), concentration (inverse of mean distance to MTOC) and pole-to-pole turnover time increases, as microtubule density increases.

https://doi.org/10.1371/journal.pone.0034919.g002

Asymmetric partitioning by asymmetric spindle

Because the microtubule density is critical for the sequestration effect, an asymmetrically organized microtubule spindle would lead to asymmetric partitioning of the molecules. This can serve as a possible mechanism for the asymmetric cell division if the cell fate determinants are spatially regulated by the microtubule network. For example, the Drosophila central brain neuroblast divides asymmetrically into a neuroblast and a non-neuroblast daughter. The neuroblast daughter inherits the dominant centrosome, which, during the preceding mitosis, matures and organizes a dense microtubule array ∼1 hr earlier than the other centrosome [7]. We used the computational model to mimic the asymmetric centrosome maturation process and explore the dynein-mediated spatial regulation effect by the asymmetrically growing microtubule network (Figure 3A and C, Movie S3). During the 3000 s simulation, the number of microtubules organized by the dominant centrosome increases from 200 to 800, while the second centrosome only organizes 20 microtubules throughout the time. The second centrosome migrates from the vicinity of the dominant centrosome to the opposite side of the cell (dashed curve). As the dominant centrosome matures and grows a denser microtubule array, it imposes increasing sequestration on the dyneins. At the end of the simulation, almost all the particles are concentrated around the dominant centrosome. In the neuroblast cell, the second centrosome starts to mature at this time [7]. As we showed above, it takes ∼10⁵ s for particles to exchange between mature spindle poles (Figure 1). The huge kinetic trap prevents the particles from redistributing to the second centrosome within the time scale of mitosis. Moreover, the asymmetric distribution of the cell fate determinants requires a sufficient time delay between the maturation of the microtubule arrays organized by the two centrosomes. For a short time delay between the centrosome maturation or simply small fluctuations in the maturation process, the resultant relative differences in the microtubule density between the two arrays will gradually diminish as the number of microtubules grows. The original biased partitioning of dynein-directed particles is quenched before the microtubule density is high enough to establish the huge kinetic trap that sustains the bias (Figure 3B and D, Movie S4). This simulation showed that the microtubule network is able to achieve both asymmetric and symmetric spatial regulation, simply by controlling the time delay between the maturation of the microtubule arrays organized by each centrosome.

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Figure 3. Delayed centrosome maturation causes asymmetric particle partitioning.

(A) Time series of dynein distribution with ∼1 hr time delay between the centrosome maturation (also see Movie S3). During the simulated duration of 3000 s, the number of microtubules organized by the dominant centrosome increases from 200 to 800. The second centrosome organizes 20 microtubules throughout the time. It migrates along the trajectory depicted by the dashed curve. Dyneins are sequestered by the dominant centrosome. Dark grey lines: microtubules (only 1 out of 10 are shown to reduce clutter); blue dots: dyneins bound to the microtubules organized by the dominant centrosome; green dots: dyneins bound to the microtubules organized by the second centrosome; red dots: dyneins freely diffusing in the cytoplasm; grey circles: centrosomal area. (B) Time series of dynein distribution with ∼5 min time delay between the centrosome maturation (also see Movie S4). During the simulated duration of 3000 s, the number of microtubules organized by the first centrosome increases from 200 to 800; the number of microtubules organized by the second centrosome increases with the same rate from 150 to 750. The second centrosome migrates along the same path as in (A) (dashed curve). Dyneins ends up equally distributed around the two centrosomes. Legends are same as in (A). (C) and (D) Time trajectories of the number of dyneins bound to the microtubules associated with each centrosome. (C) corresponds to the case in (A), and (D) corresponds to (B). Blue line: the centrosome that matures earlier; green line: the centrosome that matures later (right axis in (C)). With long time delay in centrosome maturation, the number of dyneins sequestered by the second centrosome decreases to nearly zero before the centrosome matures. With short time delay, the original bias in the partitioning of dyneins diminishes as the relative difference in the numbers of microtubule organized by the two centrosomes reduces. (E) Asymmetry of particle partitioning depends on the asymmetry of microtubule numbers organized by the two centrosomes, as well as the distance between the centrosomes. The asymmetric index is defined as (N₁−N₂)/(N₁+N₂), where N₁ and N₂ are the corresponding steady state particle numbers or microtubule numbers associated with each centrosome. The simulations were run with a total number of 1000 microtubules partitioned differentially between the two centrosomes.

https://doi.org/10.1371/journal.pone.0034919.g003

We further predict that asymmetric particle partitioning not only depends on the level of asymmetry between the two microtubule arrays, but also hinges on the distance between the centrosomes (Figure 3E). At short inter-centrosome distances, achieving asymmetric particle sequestration requires a much higher level of asymmetry in the microtubule density than at large inter-centrosome distances. Our results thus suggest that the observed asymmetric sequestration effect [7], [10] could also be regulated by the inter-centrosome distance. Experiments that manipulate the microtubule density organized by each centrosome (such as by interfering the γ-tubulin recruitment level) and the inter-centrosome distance shall be a good test for these model predictions (Figure 3E).

Sequestration pattern in syncytial embryo

So far we have discussed the spatial regulation of dynein-directed particles. The concentration effect evidently relies on the minus-end motility along the microtubules. But the sequestration by microtubules also works without dynein-mediated active transport. Simple binding with microtubules is enough to generate a significant sequestration effect that affects the spatial distribution and dynamics of the molecules. In fact, such sequestration effects apply to the actin-binding molecules and the actin network, too. Therefore, the following case sheds light on the general cytoskeleton-mediated sequestration, although we adopt the parameters for the microtubule system.

The cytoskeleton, including the microtubule network and the actin network, turns the cytoplasmic space into a so-called “structured cytoplasm”, a concept proposed years ago [14], [15]. The “structured cytoplasm” can partially hold the cytoskeleton-binding components. For example, in a couple of cell cycles before the cellularization of the syncytial Drosophila embryo, Dorsal proteins are distributed in the cortex of the embryo (fig.5F of [13], also see Figure 4A, left column), where the nuclei-associated energids are semi-separated by the plasma membrane furrows and enriched with cytoskeleton [16], [17]. Fluorescence recovery after photobleaching (FRAP) and fluorescence loss in photobleaching (FLIP) experiments showed delayed exchange of Dorsal proteins between neighboring energids (fig. 5E and 5F of [13], also see Figure 4B, top row).

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Figure 4. Restricted diffusion of Dorsal in the syncytial embryo.

For the sake of computational efficiency on the much larger spatial domain, the simulation was carried out using field equations that depict the density profile of the Dorsal protein, instead of agent-based method that traces each protein particles (see Methods). The field equations were defined on a rectangular block of dimensions 35 µm W×35 µm D×14 µm H. The top area of 7 µm in height are divided into 5×5 cubic subdomains with impermeable vertical boundaries between each other. These subdomains represent the nuclei-associated energids in the syncytial embryo cortex. There are no boundaries between each subdomain and the rest of the space. (A) Comparison with FRAP experiment, top view (also see Movie S5, S6). Left column: experiment data. Middle column: computational result with microtubule-mediated sequestration and furrows. Top two panels in the right column: computational result with pure diffusion restricted by furrows only. The gray scale bar shows the normalized concentration with respect to the maximum concentration at time 0 in the computational results. Bottom panel in the right column: temporal curves of Dorsal concentration at the central point of the photobleached domain, with sequestration (blue), with pure diffusion at D = 20 µm²/s (green), or with pure diffusion at D = 3 µm²/s (red). The sequestered case thus demonstrates a phenomenological diffusion coefficient of ∼3 µm²/s. (B) Comparison with FLIP experiment, side view (also see Movie S7, S8, S9). Top: experiment data at t = 108 s. Middle: computed concentration profile with microtubule-mediated sequestration and furrows at t = 108 s. Bottom: computed concentration profile with pure diffusion restricted by furrows only at t = 108 s, with the gap width between neighboring energids of 7 µm (upper) or 1 µm (lower). The black lines mark the membrane boundaries between neighboring energids. The white box shows the area of photobleaching. The red lines show the characteristic shape of the contour lines of the concentration profiles. The color bar shows the relative Dorsal concentration. In both the FRAP and FLIP simulations, the effective binding rate k_on = 5 s⁻¹ (∼microtubule density of 3 µm⁻², or 150 microtubules within each energid, cf. Figure S1), the unbinding rate k_off = 1 s⁻¹. The diffusion coefficient of unbound protein D = 20 µm²/s except for the red temporal curve in (A) (cf. [26], the cytoplasmic diffusion coefficient of GFP is 25 µm²/s and the molecular weight of Dorsal (∼90 kD, [27]) is about 3 times that of GFP (∼30 kD, [28])). The experimental data are adapted from Delotto et al, 2007 [13].

https://doi.org/10.1371/journal.pone.0034919.g004

Here we used our computational model to show that the microtubule-mediated sequestration can help explain the observed delayed diffusion. Since the interphase microtubule network is usually extensive throughout the cell, distributed with fairly even density, we simplified our simulation, using field equations to describe the spatial concentrations of the bound and unbound Dorsal proteins. The binding was characterized by an effective binding rate derived from the agent-based simulation (Figure S1).

For the FRAP experiment, the computational results showed that pure diffusion restricted by furrows only evens out the concentration difference between the bleached and the unbleached energids in <10 s (Figure 4A and Movie S6). But with the microtubule-mediated sequestration, the FRAP takes ∼40 s as observed in the experiment (Figure 4A and Movie S5). The sequestration-hindered diffusion gives a phenomenological diffusion coefficient ∼3 µm²/s (Figure 4A, lower right panel), much lower than the actual cytoplasmic diffusion coefficient of the free particles, D = 20 µm²/s. The effective binding rate used in the simulation, k_on = 5 s⁻¹, corresponds to the microtubule density of 3 µm⁻², or 150 microtubules per energid. This is a reasonable number for interphase cells.

Our model also recapitulated the observation from the FLIP experiment, in particular, the characteristic horn-shaped contour lines from side view (Figure 4B and Movie S7). These contour lines result from the superposition of the xy-gradient (parallel to the cortex) due to the photobleaching and the z-gradient (perpendicular to the cortex) due to the sequestration. Without the sequestration effect, the membrane furrows alone can also slow down the effective diffusion (Daniels et. al., unpublished result). But these geometric obstacles alone cannot reproduce the observed z-gradient (Figure 4B and Movie S8, S9), i.e., the increasing concentration gradient from the deeper syncytium towards the embryo cortex. Instead, it leads to contour lines that resemble flipped bowls, even if the gaps between energids are very small (Figure 4B). These bowl-shaped contour lines reflect the diffusional fluxes of Dorsal through the gaps. In fact, the z-gradient exists even in the steady state before FLIP (fig.5F of [13]), which, according to our model, is a telltale sign for the sequestration by the “structured cytoplasm” of the energids.

Agent-based simulation

The scheme of the agent-based simulation is given in Figure S2. Let and denote the position and the state of the i-th particle at the discrete time point t_n. Let , a 3-by-2 matrix represent the position of the minus end and the plus end of the j-th microtubule. The minus ends are located at the spindle pole (spherical surface of radius 1 µm). The plus ends are located at the cell cortex or the cell equator, depending on which is reached first. The microtubules do not undergo dynamical changes, i.e. are time invariant. This is because the binding and unbinding largely depends on the local microtubule density, as shown by the results. Then, the equations that govern the agent-based model read as follows, where are independent random numbers drawn from the standard normal distribution N(0, 1), and are independent random numbers drawn from the uniform distribution U(0, 1).(1)

Mean field simulation

Let ρ_f be the density of free particles and ρ_b the density of bound particles. They are computed with the following equations,(2)where D_C, D_M are the diffusion coefficients of the particle in the cytoplasm and along the microtubule respectively; V is the velocity (≠0 if the particle is driven by molecular motors); is the unit vector in the direction of the microtubule, pointing toward the corresponding MTOC; k_u and k_b are the unbinding and binding rates of the particle with the microtubule. The binding rate k_b depends on the local microtubule density . The relation between the binding rate and the microtubule density is given in Figure S1. Information S2 gives more details of the derivation, as well as a compatibility check with the agent-based model.