Timing and shape of stochastic autocatalytic burst formation.

Chemical, physical, and ecological systems passing through a saddle-node bifurcation will, momentarily, find themselves balanced at a semistable steady state. If perturbed by noise, such systems will escape from the zero-steady state, with escape time sensitive to noise. When the model is extended to include space, this leads to different points in space "escaping from zero" at different times, and uniform initial conditions nucleate into sharp peaks spreading randomly across a nearly uniform background, a phenomenon closely resembling nucleation during phase transition. We use Large Deviation Theory to determine burst shape and temporal scaling with respect to noise amplitude. These results give a prototype for a particular form of patternless symmetry breaking in the vicinity of a stability boundary and demonstrate how microscopic noise can lead to macroscopic effects in such a region.

Nonlocal mechanism of self-organization and centering of microtubule asters.

Fragments of fish melanophore cells can form and center aggregates of pigment granules by dynein-motor-driven transport along a self-organized radial array of microtubules (MTs). We present a quantitative model that describes pigment aggregation and MT-aster self-organization and the subsequent centering of both structures. The model is based on the observations that MTs are immobile and treadmill, while dynein-motor-covered granules have the ability to nucleate MTs. From assumptions based on experimental observations, we derive partial integro-differential equations describing the coupled granule-MT interaction. We use scaling arguments and perturbation theory to study the model in two limiting cases. The model analysis explains the mechanism of aster self-organization as a positive feedback loop between motor aggregation at the MT minus ends and MT nucleation by motors. Furthermore, the centering mechanism is explained by the spontaneous nucleation of MTs throughout the cytosol which acts as a volume sensing tool. Numerical simulations lend additional support to the analysis. The model sheds light on role of polymer dynamics and polymer-motor interactions in cytoskeletal organization.

Early spindle assembly in Drosophila embryos: role of a force balance involving cytoskeletal dynamics and nuclear mechanics.

Mitotic spindle morphogenesis depends upon the action of microtubules (MTs), motors and the cell cortex. Previously, we proposed that cortical- and MT-based motors acting alone can coordinate early spindle assembly in Drosophila embryos. Here, we tested this model using microscopy of living embryos to analyze spindle pole separation, cortical reorganization, and nuclear dynamics in interphase-prophase of cycles 11-13. We observe that actin caps remain flat as they expand and that furrows do not ingress. As centrosomes separate, they follow a linear trajectory, maintaining a constant pole-to-furrow distance while the nucleus progressively deforms along the elongating pole-pole axis. These observations are incorporated into a model in which outward forces generated by zones of active cortical dynein are balanced by inward forces produced by nuclear elasticity and during cycle 13, by Ncd, which localizes to interpolar MTs. Thus, the force-balance driving early spindle morphogenesis depends upon MT-based motors acting in concert with the cortex and nucleus.

Efficient chromosome capture requires a bias in the 'search-and-capture' process during mitotic-spindle assembly.

The mitotic spindle assembles into a bipolar, microtubule-based protein machine during prometaphase. One proposed mechanism for this process is "search-and-capture," in which dynamically unstable microtubules (MTs) search space to capture chromosomes. Although existing theoretical estimates suggest that dynamic instability is efficient enough to allow capture within characteristic mitotic timescales, they are limited in scope and do not address the capture times for realistic numbers of chromosomes. Here we used mathematical modeling to explore this issue. We show that without any bias toward the chromosomes, search-and-capture is not efficient enough to explain the typical observed duration of prometaphase. We further analyze search-and-capture in the presence of a spatial gradient of a stabilizing factor that biases MT dynamics toward the chromosomes. We show theoretically that such biased search-and-capture is efficient enough to account for chromosome capture. We also show that additional factors must contribute to accelerate the spindle assembly for cells with large nuclear volumes. We discuss the possibility that a RanGTP gradient introduces a spatial bias into microtubule dynamics and thus improves the efficiency of search-and-capture as a mechanism for spindle assembly.

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

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.

A force balance model of early spindle pole separation in Drosophila embryos.

The formation and function of the mitotic spindle depends upon force generation by multiple molecular motors and by the dynamics of microtubules, but how these force-generating mechanisms relate to one another is unclear. To address this issue we have modeled the separation of spindle poles as a function of time during the early stages of spindle morphogenesis in Drosophila embryos. We propose that the outward forces that drive the separation of the spindle poles depend upon forces exerted by cortical dynein and by microtubule polymerization, and that these forces are antagonized by a C-terminal kinesin, Ncd, which generates an inward force on the poles. We computed the sum of the forces generated by dynein, microtubule polymerization, and Ncd, as a function of the extent of spindle pole separation and solved an equation relating the rate of pole separation to the net force. As a result, we obtained graphs of the time course of spindle pole separation during interphase and prophase that display a reasonable fit to the experimental data for wild-type and motor-inhibited embryos. Among the novel contributions of the model are an explanation of pole separation after simultaneous loss of Ncd and dynein function, and the prediction of a large value for the effective centrosomal drag that is needed to fit the experimental data. The results demonstrate the utility of force balance models for explaining certain mitotic movements because they explain semiquantitatively how the force generators drive a rapid initial burst of pole separation when the net force is great, how pole separation slows down as the force decreases, and how a stable separation of the spindle poles characteristic of the prophase steady state is achieved when the force reaches zero.