Figure 6.

Testable parameters of a piecewise continuous spring. (A) The most common form of a spring is given by a Hookean spring equation F = −k_spring (L_spring − L_rest), in which k is the spring constant, and L_rest is the spring rest length. Simulations of a linear spring fail to account for behavior of the spindle and the pericentric chromatin upon experimental depletion of pericentric cohesin or condensin (see Table 1). (B) A nonlinear spring with a threshold length (L_threshold) recapitulates increase in spindle length and fluctuations, asymmetric chromatin stretching, and kinetochore declustering. The threshold represents the length/force at which a compact loop transitions to a stretched loop. Cohesin and condensin increase the threshold of the chromatin loops maintaining compaction (equilibrium arrows shifted toward loops). Perturbation of the chromatin spring through depletion of pericentric cohesin or condensin decreases the length/force the loops can resist, causing the loops to stretch freely (equal amounts of compact and stretched loops; Fig. 2, A and C). The L_threshold variable is an alternative way to modulate the native linear spring constant (k) and rest length (L_rest). (C) Experimentally observed stretching of two chromosomes could be simulated through the addition of a cross-linking spring between neighboring chromosomes (k_cross-link; Fig. 5). Cross-linked chromatin springs can distribute tension, thereby increasing the ability of a single chromatin spring to resist reaching L_threshold or extreme stretching (equilibrium arrows shifted toward the looped state). Simulation and experimental data suggest condensin and cohesin modulate L_threshold and k_cross-link, respectively.