Filamentous proteins are prevalent within eukaryotic cells and perform a variety of crucial tasks relating to cellular integrity, locomotion, transport and division [1
]. Such tasks are often active
in that they can only proceed in concert with energy-consuming mechanisms, including directed filament growth and motor protein-generated tension, placing such processes outside the realm of equilibrium thermodynamics [3
]. Self-organisation of motor protein-filament mixtures will be selected for when it robustly reproduces static or dynamic structures beneficial to the cell's viability. An example is the mitotic spindle that forms during division of fission yeast
cells. It has been shown that this bipolar structure, consisting of microtubules emanating from spindle pole bodies towards an overlapping midplane region, exists and functions essentially as normal even in cells with no nucleus-associated microtubule organizing center [4
]. The plausible conclusion is that the interaction between filaments and motor proteins in the confined cell geometry controls the location of the pole bodies. For budding yeast
this self-organisation scenario has been reinforced by the evolution of more sophisticated regulatory mechanisms [6
]. Also, egg cell extracts from the amphibious genus Xenopus
can generate a well-formed spindle apparatus despite entirely lacking cell walls [7
]. Nonetheless an understanding of the principles underlying self-organization of bioflaments driven by motor proteins in confined spaces is of direct relevance to many organisms [10
Given the complexity of real cells it is often advantageous to consider simplified model systems, and this approach has been adopted to investigate the role of confinement in filament-motor mixtures. Experiments on growing microtubules confined to spherical emulsion droplets revealed a droplet-size dependency on the observed structure [11
]: Droplets larger than ≈ 29 μm
in diameter contained asters
with the polar microtubules pointing towards the centre, controlled by the motor protein dynein, whereas smaller droplets were found to contain semi-asters
with the aster's focus near the interface. These findings demonstrate that the degree of confinement can partly determine structure formation, but as motor density and speed were not control variables in these experiments their influence could not be assayed.
A strikingly non-equilibrium property of filament-motor mixtures is their ability to spon-taneously generate flows due to their active components, even in the absence of boundary driving forces [12
]. Assays of microtubule-oligomeric kinesin mixtures in a quasi-two dimensional geometry with flat, parallel confining walls found a dynamic rotating structure denoted a vortex
]. Accompanying simulations of semiflexible filaments [16
] and subsequent hydrodynamic theories [17
] appeared to reproduce the observed structures. However, as discussed in Ref. [19
], it is unlikely that the simulations of Surrey et al.
] and the theories and simulations of Ref. [17
] describe the same type of vortex, because the hydrodynamic theories are based on a nematic order-parameter description, while simulations of semi-flexible filaments in Ref. [16
] neglect self-avoidance (and thus nematic order). Mesoscopic models based on the Smoluchowski equations have not resolved this issue [20
]. Simulations of self-avoiding filaments strictly in two dimensions showed no evidence of a vortex state [19
]. The microscopic picture underlying vortex formation thus remains unknown. Gliding assays of filaments along motor beds permit quantitative comparison to models [9
] and at high concentration exhibit vortex-like 'swirls' [23
], although in this situation the active forces are unbalanced monopoles, unlike dipoles generated by motors connecting two filaments in the bulk [25
]. Vortex-like motion is often observed in self-propelled systems such as bacterial swimmers [26
], but with differing microscopic mechanisms.
It is apparent that the combined influence of confinement and activity on structure formation and spontaneous flows in filament-motor mixtures is presently not well understood. Our aim here is to acquire a deeper understanding of this problem in a broad sense, not restricted to any one biological realisation, i.e.
microtubule-dynein or actin-myosin. It is therefore desirable to study model systems in which all parameters can be freely varied. The application of continuum equations, which are coarsegrained over lengths much larger than the filament length L
, to structures of only a few L
in spatial extent is not guaranteed to be successful. We therefore adopt a discrete numerical model in which motors and filament segments are explicitly represented, and all physical mechanisms that are potentially relevant (steric hinderance, thermal fluctuations etc
.) are incorporated. This model is an extension of one previously employed in two dimensions [19
], where it was found to produce some signatures of active gels such as super-diffusion and anomalous small wavelength density fluctuations, but not vortices.
We consider arrays of filaments confined to a quasi- two dimensional cylinder, with a height of a few filament diameters which permits filament overlap, and an external pressure at the curved walls. We then systematically vary the motor density, speed and applied pressure. Four steady-state configurations arise within the covered parameter space, including an aster and semi-aster as observed in confined emulsion droplets [11
], and also a spindle-like state that spontaneously emerges from the motor-filament interaction in the confined geometry, possibly reproducing the fission yeast observations [4
]. These states are described in Sec. 3.1 along with a fourth nematic state that links to known equilibrium phases. We also find a fifth, vortex state associated with a definite rotation of filaments about a fixed center that appears to be always transient. The existence and properties of these vortices are characterised in Sec. 3.2. To highlight the important role played by motors at filament plus-ends, we independently vary the detachment rate of motors from plus-ends in Sec. 3.3 and show that vorticity is associated with a critical fraction of plus-ended motors. The observation of vortices in fixed volume systems described in Sec. 3.4 confirm that they are driven at least partly by motor motion and not boundary fluctuations, and in Sec. 4 we discuss possible future directions.