The mechanism of anaphase spindle elongation: uncoupling of tubulin incorporation and microtubule sliding during in vitro spindle reactivation.

To study tubulin polymerization and microtubule sliding during spindle elongation in vitro, we developed a method of uncoupling the two processes. When isolated diatom spindles were incubated with biotinylated tubulin (biot-tb) without ATP, biot-tb was incorporated into two regions flanking the zone of microtubule overlap, but the spindles did not elongate. After biot-tb was removed, spindle elongation was initiated by addition of ATP. The incorporated biot-tb was found in the midzone between the original half-spindles. The extent and rate of elongation were increased by preincubation in biot-tb. Serial section reconstruction of spindles elongating in tubulin and ATP showed that the average length of half-spindle microtubules increased due to growth of microtubules from the ends of native microtubules. The characteristic packing pattern between antiparallel microtubules was retained even in the new overlap region. Our results suggest that the forces required for spindle elongation are generated by enzymes in the overlap zone that mediate the sliding apart of antiparallel microtubules, and that tubulin polymerization does not contribute to force generation. Changes in the extent of microtubule overlap during spindle elongation were affected by tubulin and ATP concentration in the incubation medium. Spindles continued to elongate even after the overlap zone was composed entirely of newly polymerized microtubules, suggesting that the enzyme responsible for microtubule translocation either is bound to a matrix in the spindle midzone, or else can move on one microtubule toward the spindle midzone and push another microtubule of opposite polarity toward the pole

Diversity and critical behavior in prisoner's dilemma game

The prisoner's dilemma (PD) game is a simple model for understanding cooperative patterns in complex systems consisting of selfish individuals. Here, we study a PD game problem in scale-free networks containing hierarchically organized modules and controllable shortcuts connecting separated hubs. We find that cooperator clusters exhibit a percolation transition in the parameter space (p,b), where p is the occupation probability of shortcuts and b is the temptation payoff in the PD game. The cluster size distribution follows a power law at the transition point. Such a critical behavior, resulting from the combined effect of stochastic processes in the PD game and the heterogeneous structure of complex networks, illustrates the diversity of social relationships and the self-organization of cooperator communities in real-world systems

On a q-difference Painlev\'e III equation: II. Rational solutions

Rational solutions for a $q$-difference analogue of the Painlev\'e III equation are considered. A Determinant formula of Jacobi-Trudi type for the solutions is constructed.Comment: Archive version is already official. Published by JNMP at http://www.sm.luth.se/math/JNMP