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First published online 2 March 2004
doi: 10.1242/jcs.00919

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Computational model of dynein-dependent self-organization of microtubule asters

¹ Laboratory of Cell and Computational Biology, Department of Mathematics and Center for Genetics and Development, University of California, Davis, CA 95616, USA
² Department of Physiology and Center for Biomedical Imaging Technology, University of Connecticut Health Center, Farmington, CT 06032-1507, USA

View larger version (87K): [in a new window]   Fig. 1. Self-organization in the melanophore cell fragment. Live fluorescent images of MTs are shown. Pigment granules can be seen as black speckles. (Left) A random MT network and uniformly dispersed pigment granules before adrenalin treatment (dynein stimulation). (Right) After dynein is stimulated, pigment granules aggregate and MTs re-organize into a polar aster.

View larger version (50K): [in a new window]   Fig. 2. Pathway of the self-organization phenomenon. Phase contrast images at the left show pigment distribution in two fragments of different size and shape. A schematic model of MT/pigment re-organization is on the right. (A) Initially, the pigment granules are distributed homogeneously across the fragments and MTs are positioned and oriented randomly. (B) Within a few minutes (for fragments with a characteristic dimension of 20-50 µm), multiple local pigment aggregates emerge, presumably near the minus ends of pre-existing MTs. (C) In the next few minutes, local aggregates nucleate new MTs, some of which pass through other aggregates. (D) Dynein-mediated granule transport along these latter MTs eventually lead to a merging of local aggregates into a single focus and ultimately, to nucleation of a single polar MT aster. Bars, 10 µm

View larger version (17K): [in a new window]   Fig. 3. (A) MT treadmilling: Most MTs ( 80%) in the discoid fragment (radius 20 µm) with a pigment aggregate at the center have their minus ends in the aggregate, and plus ends at the periphery. MTs are in this state for 40 minutes. About 10% of the MTs have their minus ends in the aggregate with their plus ends growing toward the fragment boundary. The plus ends of another 10% of MTs are stalled at the boundary, while the corresponding minus ends shorten. MTs are in each of these states for 5 minutes. Rare MTs (1%) treadmill. (B) The evolution of MTs throughout aggregation and aster formation (stimulus at 4 time units). (Top) Computed total length of MT polymer before and after stimulation of dynein motors. The total length of MT polymer increases by a factor of 3 during aggregation and MT aster formation (from before stimulus until steady state). (Bottom) Percentages of MTs, with minus ends anchored in the pigment aggregate and plus ends at the periphery (solid), with minus ends anchored and plus ends growing (dashed), with minus ends shortening and plus ends at the periphery (dot-dashed), and treadmilling (dotted), agree well with the corresponding observed percentages.

View larger version (14K): [in a new window]   Fig. 4. (Top) The possible density dependencies of the MT nucleation rate are an increasing function with (1) neither saturation nor threshold behavior (solid curve), (2) without threshold but with saturation (dashed curve), and (3) with both threshold and saturation (dotted curve). (Bottom) The possible density dependencies of the MT minus-end disassembly rate are a decreasing function without threshold behavior (solid curve) and with threshold behavior (dashed curve). The granule density, nucleation rate and disassembly rate are plotted in the non-dimensional units described in the text [g=1 in the untreated fragment, n(1)=1, v(0)=1].

View larger version (8K): [in a new window]   Fig. 5. 1D implementation of the model in a long narrow fragment of length 2L. Two dynamic sub-populations of MTs with opposite orientations are characterized by the densities of the plus (pr,l) and minus (mr,l) ends and the polymer densities Nr,l. Three pigment sub-populations are described by the densities of the granules gliding to the right (gr) and left (gl) with speed vg, and static granules (gs) dissociated from the MTs.

View larger version (76K): [in a new window]   Fig. 6. Illustration of the numerical implementation of the 2D computational model. Each MT generates an effective velocity field (arrows) in the rectangular domain of influence (see isolated MTs at the periphery). The corresponding velocities are minus-end-directed and decrease away from the MT. The velocity fields of individual MTs add locally and geometrically. Note that the velocity field in the center is not parallel to any individual MT but results from the vector sum of individual contributions. The shading shows the granule density generated fast by the shown effective velocity field.

View larger version (19K): [in a new window]   Fig. 7. (A) Analytical solutions of the 1D model initial pseudo steady state equations. In the fragment, the MT minus- and plus-end densities are linear functions of the spatial coordinate. Upon dynein stimulation, the granules aggregate loosely to the center of the fragment within tens of seconds. (B) Numerical solutions of the 1D model steady state equations. In several minutes, granules aggregate tightly in the center with most MT minus ends focused in the pigment aggregate. Note that the plus ends at the fragment boundary are not shown. The total numbers of the plus and minus ends are equal.

View larger version (92K): [in a new window]   Fig. 8. Computer simulations of the 2D model (compare with Figs 1 and 2). MTs are shown as segments. The granule density is illustrated by shading with darker areas corresponding to higher densities. (A) Initially, the MTs are distributed randomly and the granule density is uniform. (B) The granules quickly aggregate into a few local foci before MTs can re-organize. (C) The local aggregates coalesce into a single loose aggregate over a few minutes simultaneously with the re-organization of the MTs. (D) The pigment aggregate tightens over the next few minutes and the MT aster is clearly seen.

View larger version (94K): [in a new window]   Fig. 9. MTs and granule density are illustrated as in Fig. 8. Level-curves of granule density are added for emphasis. Both images show the state of a fragment soon after dynein stimulation. (A) At low and moderate nucleation rates, a few local aggregates evolve initially. (B) At high nucleation rates, the granules initially coalesce into a single loose aggregate.

View larger version (15K): [in a new window]   Fig. 12. The time course of aggregation in a 1D fragment. (A) Phase contrast images of a 1D fragment (the width is much smaller than the typical MT length). (B) Line scans of intensity through the midline of the fragment. (C) Simulation results from the 1D model with an inhomogeneous MT array (see Movie 2, http://jcs.biologists.org/supplemental/). The top row shows the initial state of the pigment distribution (roughly uniform). During the aggregation process, local aggregates form, as seen in the second row. Finally, a single tight aggregate forms as the system approaches steady state.

View larger version (73K): [in a new window]   Fig. 10. Self-organization in the bi-lobed fragment. (Left) Phase contrast images show pigment distribution. (Right) Results of a computer simulation in the bi-lobed fragment. The experimental images are obtained before the adrenalin treatment, and 5 and 10 minutes after it, respectively. Bars, 10 µm.

View larger version (11K): [in a new window]   Fig. 11. The observed relationship between space (fragment size) and time (aggregation time) constants for the 20 fragments, as explained in the text. A straight-line fit to the data (least squares sense) is superimposed with dashed lines representing one standard deviation above and below.