Eric N. Cytrynbaum

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.

Mechanisms of Self-Organization of Cortical Microtubules in Plants Revealed by Computational Simulations

Plant cortical microtubules self-organize into parallel yet dispersed arrays transverse to the elongation axis of elongating plant cells. Computational modeling yields several novel predictions and shows that plus-end entrainment can give rise to order.

Force generation by a dynamic Z-ring in Escherichia coli cell division.

FtsZ, a bacterial homologue of tubulin, plays a central role in bacterial cell division. It is the first of many proteins recruited to the division site to form the Z-ring, a dynamic structure that recycles on the time scale of seconds and is required for division to proceed. FtsZ has been recently shown to form rings inside tubular liposomes and to constrict the liposome membrane without the presence of other proteins, particularly molecular motors that appear to be absent from the bacterial proteome. Here, we propose a mathematical model for the dynamic turnover of the Z-ring and for its ability to generate a constriction force. Force generation is assumed to derive from GTP hydrolysis, which is known to induce curvature in FtsZ filaments. We find that this transition to a curved state is capable of generating a sufficient force to drive cell-wall invagination in vivo and can also explain the constriction seen in the in vitro liposome experiments. Our observations resolve the question of how FtsZ might accomplish cell division despite the highly dynamic nature of the Z-ring and the lack of molecular motors.

Stability conditions for the traveling pulse: Modifying the restitution hypothesis

As a simple model of reentry, we use a general FitzHugh-Nagumo model on a ring (in the singular limit) to build an understanding of the scope of the restitution hypothesis. It has already been shown that for a traveling pulse solution with a phase wave back, the restitution hypothesis gives the correct stability condition. We generalize this analysis to include the possibility of a pulse with a triggered wave back. Calculating the linear stability condition for such a system, we find that the restitution hypothesis, which depends only on action potential duration restitution, can be extended to a more general condition that includes dependence on conduction velocity restitution as well as two other parameters. This extension amounts to unfolding the original bifurcation described in the phase wave back case which was originally understood to be a degenerate bifurcation. In addition, we demonstrate that dependence of stability on the slope of the restitution curve can be significantly modified by the sensitivity to other parameters (including conduction velocity restitution). We provide an example in which the traveling pulse is stable despite a steep restitution curve. (c) 2002 American Institute of Physics.

Centering of a radial microtubule array by translocation along microtubules spontaneously nucleated in the cytoplasm

Positioning of a radial array of microtubules (MTs) in the cell centre is crucial for cytoplasmic organization, but the mechanisms of such centering are difficult to study in intact cells that have pre-formed radial arrays. Here, we use cytoplasmic fragments of melanophores, and cytoplasts of BS-C-1 cells to study MT centering mechanisms. Using live imaging and computer modelling, we show that the MT aster finds a central location in the cytoplasm by moving along spontaneously nucleated non-astral MTs towards a point at which MT nucleation events occur equally on all sides. We hypothesize that similar mechanisms, in the presence of the centrosome, contribute to this centering mechanism and ensure the robustness of cytoplasmic organization.

A mechanochemical model explains interactions between cortical microtubules in plants.

Microtubules anchored to the two-dimensional cortex of plant cells collide through plus-end polymerization. Collisions can result in rapid depolymerization, directional plus-end entrainment, or crossover. These interactions are believed to give rise to cellwide self-organization of plant cortical microtubules arrays, which is required for proper cell wall growth. Although the cell-wide self-organization has been well studied, less emphasis has been placed on explaining the interactions mechanistically from the molecular scale. Here we present a model for microtubule-cortex anchoring and collision-based interactions between microtubules, based on a competition between cross-linker bonding, microtubule bending, and microtubule polymerization. Our model predicts a higher probability of entrainment at smaller collision angles and at longer unanchored lengths of plus-ends. This model addresses observed differences between collision resolutions in various cell types, including Arabidopsis cells and Tobacco cells.

The effect of spatial scale of resistive inhomogeneity on defibrillation of cardiac tissue

Defibrillation of cardiac tissue can be viewed in the context of dynamical systems theory as the attempt to move a dynamical system from the basin of attraction of one attractor (fibrillation) to another (the uniform rest state) by applying a stimulus whose form is physically constrained. Here we give an introduction to the physical mechanism of cardiac defibrillation from this dynamical perspective and examine the role of resistive inhomogeneity on defibrillation efficacy. Using numerical simulations with rotating waves on a one-dimensional periodic ring, we study the role of the spatial scale of resistive inhomogeneity on defibrillation. For a rotating wave on a periodic ring there are three stable attractors, namely the uniform rest state, a wave traveling clockwise and a wave traveling counterclockwise. As a result, the application of a stimulus has the potential for three different outcomes, namely elimination of the wave, phase resetting of the wave, and reversal of the wave. The results presented here show that with resistive inhomogeneities of large spatial scale, all three of these transitions are possible with large amplitude shocks, so that the probability of defibrillation is bounded well below one, independent of stimulus amplitude. On the other hand, resistive inhomogeneities of small spatial scale produce a defibrillation threshold that is qualitatively consistent with that found experimentally, namely the probability of defibrillation success is an increasing function that approaches one for large enough stimulus amplitude. Extending these results to higher dimensions, we describe conditions for successful defibrillation of functional reentry with large scale spatial inhomogeneity, but find that elimination of anatomical reentry is quite difficult. With small spatial scale inhomogeneity, there are no similar restrictions.

Predictions from a stochastic polymer model for the MinDE protein dynamics in Escherichia coli.

The spatiotemporal oscillations of the Min proteins in the bacterium Escherichia coli play an important role in cell division. A number of different models have been proposed to explain the dynamics from the underlying biochemistry. Here, we extend a previously described discrete polymer model from a deterministic to a stochastic formulation. We express the stochastic evolution of the oscillatory system as a map from the probability distribution of maximum polymer length in one period of the oscillation to the probability distribution of maximum polymer length half a period later and solve for the fixed point of the map with a combined analytical and numerical technique. This solution gives a theoretical prediction of the distributions of both lengths of the polar MinD zones and periods of oscillations--both of which are experimentally measurable. The model provides an interesting example of a stochastic hybrid system that is, in some limits, analytically tractable.

Sequential Action of MalE and Maltose Allows Coupling ATP Hydrolysis to Translocation in the MalFGK2 Transporter

Background: MalE and maltose together stimulate the ATPase activity of MalFGK2. Results: Binding of open-state MalE triggers the cleavage of ATP; maltose triggers the release of Pi. Conclusion: Open-state MalE stabilizes the transporter in the outward-facing conformation; maltose triggers return to the inward-facing state. Significance: The complementary action of MalE and maltose allows coupling ATP consumption to transport. ATP-binding cassette (ABC) transporters have evolved an ATP-dependent alternating-access mechanism to transport substrates across membranes. Despite important progress, especially in their structural analysis, it is still unknown how the substrate stimulates ATP hydrolysis, the hallmark of ABC transporters. In this study, we measure the ATP turnover cycle of MalFGK2 in steady and pre-steady state conditions. We show that (i) the basal ATPase activity of MalFGK2 is very low because the cleavage of ATP is rate-limiting, (ii) the binding of open-state MalE to the transporter induces ATP cleavage but leaves release of Pi limiting, and (iii) the additional presence of maltose stimulates release of Pi, and therefore increases the overall ATP turnover cycle. We conclude that open-state MalE stabilizes MalFGK2 in the outward-facing conformation until maltose triggers return to the inward-facing state for substrate and Pi release. This concerted action explains why ATPase activity of MalFGK2 depends on maltose, and why MalE is essential for transport.

Feasibility and effectiveness of group exams in mathematics courses

Abstract Active learning techniques, such as peer instruction and group work, have been gaining a lot of traction in universities. Taking a natural next step in re-evaluating current practices, many institutions recently started experimenting student-centred group exams. In order to assess the feasibility and effectiveness of collaborative assessments, we implemented a study that focuses on lower-level and intermediate-level mathematics courses with large student populations. We collected and analyzed data relating to student material retention, their perceptions of the format, as well as the effects of the group composition (in terms of the strength of its individual members) and question types (multiple choice versus short answer) on the exam success. Although we establish that long-term learning is not strongly influenced by the group exams, students find them useful and enjoyable, with the majority of students preferring group exams to individual ones. Examining group composition and question type results, we conclude that multiple choice questions are more suitable for group exams. We also comment on the format in view of limited people and time resources available for test creation, administration, and marking. Finally, we consider instructor perspectives on the process.