Fig 1.
Different nucleation modes (and their hypothetical impact).
(A) Unbound nucleation (i.e., nucleation occurring from a nucleation complex dispersed in the cytoplasm) typically has a uniformly distributed orientation, whereas microtubule-based nucleation follows a specific distribution relative to the parent microtubule, with parallel, antiparallel and “branched” components. The part parallel to the parent microtubule (31% in our simulations [17]) can increase the lifetime of a microtubule bundle. Additionally, the distribution of nucleation angles is overall co-aligned with the parent microtubule, which increases the parameter regime of spontaneous microtubule alignment. Dashed lines indicate the range of zippering onto microtubules with the same orientation as the parent (). The arrow with the large arrowhead (one side only) indicates a 45° angle, i.e., neutral with respect to co-alignment with the parent. More co-aligned nucleation angles increase the aligned regime (with GDD nucleation) [17]. (B) Working hypothesis: as a consequence of the properties of microtubule-based nucleation, the persistence of bundles increases, which allows for longer retention and further communication of the information about the array stored in that bundle. This renders the array as a whole more sensitive to global cues like the size and shape (geometry) of the cell. The impact of different cues varies, as visualized by variation in font size. (C-E) Overview of model dynamics. (C) Dynamics of individual microtubules. The plus-end (+) stochastically switches between the growing and shrinking state, whereas the minus-end (-) shows steady retraction. (D) Angle dependent collision outcomes. (E) Fragment of an example simulated array. Individual microtubules are drawn with a degree of transparency, so thicker bundles appear as stronger white lines.
Fig 2.
Simulation snapshots show that the new LDD microtubule-based nucleation yields much more homogeneous arrays than GDD microtubule-based nucleation.
(A-E) Simulation snapshots of a m2 square domain taken every 6000 s (100 min). Red thick ovals represent areas of inhomogeneity, orange dashed ovals intermediate areas, and green thin ovals areas where the array became homogeneous. Note the persistence of areas of inhomogeneity in the GDD case, compared to the transient inhomogeneity in the LDD and ISO cases. (F) Local density along the axis of a cylinder (L = 40 μm; R = 6 μm) in bins of 1 μm for tightly transversely oriented arrays. Five independent simulations per nucleation mode. Time increases from purple (start of simulation) to yellow (end). In these simulations, catastrophe rate values correspond to data points marked with large symbols in Fig 3, i.e., (LDD) rc = 0.00275 s−1, (GDD) rc = 0.003 s−1, (ISO) rc = 0.002 s−1. Density values are omitted from the axes for readability. Tic marks on the vertical axis represent density increments of 3 μm−1 for LDD and ISO and 40 μm−1 for GDD. All axes start at 0. Transverse arrays were randomly selected from a larger set of simulations with the sole criterion that the array orientation angle on the cylinder mantle
([88.2°, 91.8°]).
Table 1.
Default model parameters.
Fig 3.
LDD nucleation maintains the alignment promoting effect of microtubule-based nucleation
as previously reported for GDD. Aligned regime for LDD, GDD, and ISO nucleation modes on a 4040 μm2 square periodic geometry. (A) Median over 100 independent simulation runs of the order parameter S2 as function of the control parameter G at
s (8 h, 20 min). Data points marked as larger, empty circle, square, or triangle correspond to the parameter values used for panel (B). (B) Median over 100 independent simulation runs of the order parameter S2 as a function of the simulation time for selected values of G (data points marked with large symbols in panel (A)) in the aligned regime. Error bars of (A) can be found in S1 Fig.
Fig 4.
LDD microtubule-based nucleation increases the sensitivity of array orientation to cell geometry.
Fraction of aligned arrays at s (8 h, 20 min) with a transverse (T, histograms to the right or filled symbols; when
), or longitudinal (L, histograms to the left or empty symbols; when
) orientation out of n = 2000 independent runs each, with or without directional biases: (A,B) 8% increase in catastrophe rate at the cylinder caps for the global bias; and 8% increase in maximum rescue rate on the cylinder mantle according to Eq (12) for the local bias, and (C,D) for different values of the local bias in
according to Eq (12); (C,D): in the left half of each panel, the local bias is implemented in the transverse direction, in the right half in the longitudinal direction. Note that, with these parameters, a bias of 0.04 in
corresponds to a change of 0.01 in G in the same direction. The simulation domain is a cylinder of (A,C)
m length and
m diameter, and (B,D)
m length and
m diameter. (A,B) Error bars represent 95% confidence interval according to a binomial model (computed using the binom.test function in R statistical package version 3.6.3). (C,D) Error bars have been omitted because they are approximately the same size as the symbols. In these simulations, rc = 0.00225 s−1.
Fig 5.
Orientation over time for individual simulations.
Lines depict the orientation of the array (Θ based on R2) for n = 50 simulations per nucleation mode / cylinder size. Lines are coloured by the degree of alignment, renormalized . Red lines indicate the boundaries of the “transverse” (trans) and “longitudinal” (long) categories in Fig 4. Microtubule dynamic parameters were the same everywhere in the domain, regardless of orientation. Nucleation modes: (A,B) LDD; (C,D) GDD; (E,F) ISO. Cylinder sizes: length 40 μm, diameter 12 μm (A,C,E); length 60 μm, diameter 18 μm (B,D,F).
Fig 6.
Local domain formation is much less pronounced in arrays obtained through LDD than through GDD and ISO nucleation modes.
Snapshot at s (8 h, 20 min) of the microtubule array in a periodic square simulation domain of
m2 for (A) LDD: S2 = 0.90, G = −0.19, (B) GDD: S2 = 0.90, G = −0.19, and (C) ISO: S2 = 0.84, G = −0.14. Different colours corresponds to different orientations for a microtubule or a bundle: warm colours correspond to transverse and cool colours to longitudinal orientations.
Fig 7.
Clear signs of local domain formation with isotropic nucleation on large (80x80 μm2 periodic) simulation domains.
Scatter plots of density as a function of S2, coloured by G for the three nucleation modes (rows: LDD, GDD, ISO) and three domain sizes (columns: 20 × 20, 40 × 40 and 80 × 80 μm2) as indicated in the graphs. Most points fall on a curve from low density and low S2 to high density and high S2 with increasing G. Points with relatively high G and density, but distinctly lower S2 than other points with similar G and density are in the regime of spontaneous alignment, but the occurrence of multiple aligned domains with different orientations reduces the global S2. The most cases, and most severe ones, occurred with isotropic nucleation on the largest domains. These plots are based on the same data as S4 Fig, with 100 simulations per (target) G value. Simulation time: s (8 h, 20 min; lower for the highest G values for LDD and GDD, because not all simulations terminated in time).
Fig 8.
Nucleation mode has a strong impact on sensitivity to geometry.
Orientation of n = 2000 arrays at s (8 h, 20 min) on boxes (A–C) and cylinders (D–F), all with the same total surface area. Individual points are coloured by R2 value. Note that this measure of alignment has an orientation and geometry dependent maximum value
. The basis for the surface area is a cylinder of L = 40 μm and diameter of 12 μm from Fig 4A. For cylinders (D–F), L is varied as indicated, and diameter adjusted to maintain the same total surface area. This has approximately the same surface area as a L × W × W = 17 × 17 × 17 μm3 box. For boxes (A–C), length L is varied as indicated and width W is adjusted accordingly to maintain the same total surface area. Cartoons are indicative of aspect ratios. Cartoons with green bands (ISO and GDD square box data) illustrate the array orientations belonging to the different positions in the plot. Histograms at the side of each plot show the relative distribution of transverse (top) to longitudinal (bottom) orientations. The highest peak in each histogram has a fixed height, i.e., the histograms are scaled differentially. In these simulations, rc = 0.00175 s−1. For intermediate aspect ratios, see S5 Fig.
Fig 9.
Schematic overview of the LDD nucleation algorithm.
(A) The position in the simulation domain where a nucleation complex (tentatively) appears is selected randomly, with uniform distribution. (B) Such a position, represented by the blue dot, is the centre of the nucleation area (the area within in the black circle with exploration radius R). N meta-trajectories are drawn equidistant from one another and with random orientation. Meta-trajectories are drawn up until the boundary of the nucleation area or the lattice of the first intersecting microtubule, whichever is closer. (C) Whether the resulting nucleation is microtubule-based or unbound and, in the former case, the parent microtubule and the nucleation location, is decided stochastically with probability p(di) weighted over the length of meta-trajectories, according to Eq (4). (D) In a microtubule-based nucleation, the angles follow Eq (2) [17]: a newly-nucleated microtubule initially grows along the parent microtubule with probability , in the opposite direction with probability
, or branches to either side with probability
. The branching angle is determined according to a distribution that closely matches the data in [12] (Eq (2)).