Dynamic stability of spindles controlled by molecular motor kinetics

We analyze the role of the force-dependent kinetics of motor proteins in the stability of antiparallel arrays of polar filaments, such as those in the mitotic spindle. We determine the possible stable structures and show that there exists an instability associated to the collective behavior of motors that leads to the collapse of the structure. The agreement of our results and several experimental observations in eukaryotic cell division suggests an important role of kinesin-5 motors and microtubule bundles in the stability of the mitotic spindle. Copyright c © EPLA, 2008 Living cells display several structures that arise from the self-organization of polar filaments and motor proteins [1]. Several in vitro studies have shown that mixtures of kinesin motors and microtubules (MTs) can spontaneously develop complex spatio-temporal patterns [2]. These selforganization processes are essential for eukaryotic cell division [3]. During mitosis, motor proteins organize MTs in a bipolar structure, the mitotic spindle, which serves as a scaffold to transmit the necessary forces for chromosome segregation [4]. The mitotic spindle consists of two MT asters that overlap in the central region, with their minus ends located at the aster poles crosslinked by many different motor proteins [3,5]. One particular type of motors, the plus-ended bipolar kinesins (e.g., Eg5 or Klp61F), has been shown to be essential for the spindle stability. A decrease in their concentration below a certain threshold causes the spindle collapse [6,7], and their total inhibition prevents bipolar spindle formation [8]. In addition, Eg5 motors have been shown to drive the MT poleward flux [7] and homolog motors to induce the formation of (interpolar) MT bundles [9]. Bipolar motors are composed of two connected units, each one composed of two motor domains. Both units can move simultaneously and independently on MTs [10]. (a)Present address: Harvard University, Harvard School of Engineering and Applied Sciences 29 Oxford St., Cambridge, MA 02138, USA. These motors are able to crosslink MTs [9] and slide them with respect to each other when they are in an antiparallel configuration [10], like in the central region of the spindle (fig. 1a,c). As a result, these motors produce an outward force along the spindle axis and generate a MT flux toward the poles [7]. Typical forces involved in mitosis lay in the nanoNewton range [11]. Since individual motors cannot exert forces larger than a few picoNewtons, their collective action is required to ensure the stability of the mitotic spindle. At metaphase, this dynamic structure reaches a steady state with MTs of nearly constant length undergoing permanent treadmilling [7,12] (usually referred to as MT poleward flux), polymerizing at the + end and depolymerizing at the − end. The theoretical study of motors and MT mixtures has been recently addressed using continuum coarse-grained descriptions [13–16], which have elucidated their basic self-organizing principles. However, the coupling between force-dependent motor kinetics and local forces in selforganized structures has not been addressed. In this letter, we study the dynamic stability of antiparallel arrays of MTs under the action of longitudinal forces, in the presence of molecular motors able to collectively hold the structure by stochastically crosslinking the filaments. We analyze the effects of the motor kinetics on the stability of the structure, and show that several phenomena observed in eukaryotic cell division appear naturally in our theoretical approach. This suggests that interpolar microtubule