Computational model of dynein-dependent self-organization of microtubule asters
Computational model of dynein-dependent self-organization of microtubule asters (PDF)
E. N. Cytrynbaum, V. Rodionov and A. Mogilner Journal of Cell Science 117, 1381-1397 (2004)
Polar arrays of microtubules play many important roles in the cell. Normally, such arrays are organized by a centrosome anchoring the minus ends of the microtubules, while the plus ends extend to the cell periphery. However, ensembles of molecular motors and microtubules also demonstrate the ability to self-organize into polar arrays. We use quantitative modeling to analyze the self-organization of microtubule asters and the aggregation of motor-driven pigment granules in fragments of fish melanophore cells. The model is based on the observation that microtubules are immobile and treadmilling, and on the experimental evidence that cytoplasmic dynein motors associated with granules have the ability to nucleate MTs and attenuate their minus-end dynamics. The model explains the observed sequence of events as follows. Initially, pigment granules driven by cytoplasmic dynein motors aggregate to local clusters of microtubule minus ends. The pigment aggregates then nucleate microtubules with plus ends growing toward the fragment boundary, while the minus ends stay transiently in the aggregates. Microtubules emerging from one aggregate compete with any aggregates they encounter leading to the gradual formation of a single aggregate. Simultaneously, a positive feedback mechanism drives the formation of a single MT aster – a single loose aggregate leads to focused MT nucleation and hence a tighter aggregate which stabilizes MT minus ends more effectively leading to aster formation. We translate the model assumptions based on experimental measurements into mathematical equations. The model analysis and computer simulations successfully reproduce the observed pathways of pigment aggregation and microtubule aster self-organization. We test the model predictions by observing the self-organization in fragments of various sizes and in bi-lobed fragments. The model provides stringent constraints on rates and concentrations describing microtubule and motor dynamics, and sheds light on the role of polymer dynamics and polymer-motor interactions in cytoskeletal organization.
Key words: Cytoskeleton, Self-organization, Mathematical model, Molecular motors, Microtubules, Mitosis
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.
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.
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.
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.
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.
E. N. Cytrynbaum, V. Rodionov and A. Mogilner Journal of Cell Science 117, 1381-1397 (2004)
Polar arrays of microtubules play many important roles in the cell. Normally, such arrays are organized by a centrosome anchoring the minus ends of the microtubules, while the plus ends extend to the cell periphery. However, ensembles of molecular motors and microtubules also demonstrate the ability to self-organize into polar arrays. We use quantitative modeling to analyze the self-organization of microtubule asters and the aggregation of motor-driven pigment granules in fragments of fish melanophore cells. The model is based on the observation that microtubules are immobile and treadmilling, and on the experimental evidence that cytoplasmic dynein motors associated with granules have the ability to nucleate MTs and attenuate their minus-end dynamics. The model explains the observed sequence of events as follows. Initially, pigment granules driven by cytoplasmic dynein motors aggregate to local clusters of microtubule minus ends. The pigment aggregates then nucleate microtubules with plus ends growing toward the fragment boundary, while the minus ends stay transiently in the aggregates. Microtubules emerging from one aggregate compete with any aggregates they encounter leading to the gradual formation of a single aggregate. Simultaneously, a positive feedback mechanism drives the formation of a single MT aster – a single loose aggregate leads to focused MT nucleation and hence a tighter aggregate which stabilizes MT minus ends more effectively leading to aster formation. We translate the model assumptions based on experimental measurements into mathematical equations. The model analysis and computer simulations successfully reproduce the observed pathways of pigment aggregation and microtubule aster self-organization. We test the model predictions by observing the self-organization in fragments of various sizes and in bi-lobed fragments. The model provides stringent constraints on rates and concentrations describing microtubule and motor dynamics, and sheds light on the role of polymer dynamics and polymer-motor interactions in cytoskeletal organization.
Key words: Cytoskeleton, Self-organization, Mathematical model, Molecular motors, Microtubules, Mitosis
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.
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.
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.
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.
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.
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