Open Access


BMC Biophysics20114:18

DOI: 10.1186/2046-1682-4-18

Received: 4 August 2011

Accepted: 16 November 2011

Published: 16 November 2011








1 Background

Filamentous proteins are prevalent within eukaryotic cells and perform a variety of crucial tasks relating to cellular integrity, locomotion, transport and division [1, 2]. 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, 5]. 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 [79]. 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].


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 [1214]. 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 [15, 16]. Accompanying simulations of semiflexible filaments [16] and subsequent hydrodynamic theories [17, 18] appeared to reproduce the observed structures. However, as discussed in Ref. [19], it is unlikely that the simulations of Surrey et al. [16] and the theories and simulations of Ref. [17, 18] 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, 21]. 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, 22] and at high concentration exhibit vortex-like 'swirls' [23, 24], 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 [2628], but with differing microscopic mechanisms.



2 Methods

We consider a system of N semiflexible polar filaments, which can be connected by motor proteins. Each filament consists of M = 30 monomers separated by a bond length b with Hookean bond potentials with a spring constant 100 kBT/b2. Self-avoidance of filaments is introduced by repulsive Lennard-Jones potentials with diameter σ and energy parameter ε = 5 kBT. A natural choice is σ = b. Semi-flexibility is described by curvature elasticity with bending rigidity κ = 200 bkBT such that the persistence length ℓ p = κ/k B T is ℓ p = 20L/3 with L = Mb the filament length.

Only motors simultaneously connected to two different filaments are explicitly represented. The concentration of free motors in solution is assumed to be spatiotemporally uniform, which is a valid assumption for rapidly-diffusing free motors when the ratio of attached to free motors is small. This concentration is renormalized into a constant rate of attachment as discussed below. Motors are modeled as two-headed Hookean springs with a spring constant kBT/b2 and dynamics defined by four rates as shown in Figure 1(a): (i) The attachment rate kA for a motor to attach to two monomers within a predefined range, here taken to be the excluded volume radius 21/6σ (so kA is the product of a molecular attachment rate and the free motor concentration); (ii) the detachment rate kD of each head independently from its filament (detachment of either head results in removal of the whole motor from the system); (iii) the movement rate kM of each head independently towards the filament's [+]-end, and (iv) the detachment rate kE for motor heads already at a [+] end. The movement rate is attenuated by an exponential factor e - Δ E k B T with ΔE the change in motor spring energy for the trial move. Except where otherwise stated, kE = kD below.
Figure 1





3 Results

Results are presented here in terms of the normalised attachment rate kA/kD, the normalised motor rate kMτ b where τ b = Lbγ/4kBT is the approximate time for a filament to freely diffuse one monomer distance; the normalised pressure P/P0 with P0 = ε/b3 with ε = 5 kBT the Lennard-Jones repulsion energy, and, where relevant, the scaled end-detach rate kE/kD.

3.1 Stationary states

Figure 2


from which can be defined the mode amplitudes Q m ,


To determine the corresponding state, the measured Qm up to m = 3 are compared to known values for ideal states, and that with the closest Euclidean distance is taken to be the state. The values for pure asters and nematic phases are easy to derive; for an aster p ^ ( θ ) = ( - cos θ , - sin θ ) , (Q0, Q1, Q2, Q3) = (0,1,0,0), whereas for the nematic state where p ^ ( θ ) = ( 0 , 0 ) , (Q0, Q1, Q2, Q3) = (0,0,0,0). Note that while an isotropic state would give the same Qm as for the nematic, such states only arise for kA and P well below the considered ranges. For spindles and semi-aster states there is a degree of choice in how the target Qm are calculated, so we choose simple forms that permit exact evaluation of the Q m . For the spindle, p ^ ( θ ) = ( - cos θ , sin θ ) for θ (-π/4, π/4) or θ (3π/4, 5π/4) and zero otherwise, for which (Q0, Q1, Q2, Q3) = (0, 2/π2 + 1/2, 0, 2/π2). For the semi-aster, p ^ ( θ ) = ( - cos [ θ 3 ] , - sin [ θ 3 ] ) for θ (-3π/4, 3π/4) and zero otherwise, for which (Q0,Q1,Q2,Q3) = (9/π2, 9/4π2, 333/1715π2, 9/100π2). Variations in these forms have been tested and although the boundaries between states shift slightly, the underlying trends remain the same.

The occurrence of the four steady-states, plus a fifth 'vortex' state to be discussed below, with motor density and speed are presented in Figure 3 for two different pressures. The observed configurations correlate with the density and distribution of motors along the filaments. The mean number of motors per filament nmot/N is plotted in Figure 4 and shows an expected increase with the attachment rate kA as well as the pressure. The increase with pressure can be understood as due to the closer packing of the filaments, increasing the number of potential attachment points for motors and hence nmot. The approximate scaling n mot ~ k A 3 2 is faster than the linear relationship measured for constant volume, two-dimensional simulations [19], presumably due to similar reasons: As kA increases so does the motor density which, in this constant-pressure ensemble, allows the system to contract, presenting more potential attachment points between monomers and hence further increasing the motor density. A derivation of the value 3/2 of the exponent is not available so far.
Figure 3

Figure 4


Figure 5


3.2 Dynamics and vortices

Figure 6

Snapshot of a vortex. Snapshot of a vortex rotating in the anti-clockwise direction as presented, for parameters P/P0 = 0.024, kA/kD = 35 and kMτ b = 7.5 × 10-3 taken at a time t/τb ≈ 5.3 × 103. The colour code is the same as in Figure 2.

Figure 7




Figure 8

Vorticity and contraction times. Contraction time to a semi-aster tcont/τ b (left axis; solid circles) and vorticity V (right axis; open squares) versus pressure P/P0 for kA/kD = 35 and kMτ b = 7.5 × 10-3. The contraction time was determined by the fit of the radius to a hyperbolic tangent, or assigned the maximum value of tcont = 1.6 × 104τ b if no contraction had occurred within this time. Each point represents 5 independent runs.
Figure 9




Figure 10

Mean-squared angular changes. Mean-squared changes in angle 〈(Δθ)2〉 versus lag time Δt for the pressures given in the legend, kA/kD = 35 and kMτb = 7.5 × 10-3. The short thick line segments have the slope given.

3.3 Enhanced detachment from ends

In the simulations that accompanied the microtubule experiments, it was claimed that the residence time at the microtubule [+]-ends played a crucial role in determining the vortex stability, with an enhanced end-detachment rate required to form vortices [16]. By contrast, for our model it is the fraction of motors at filament [+]-ends that determines vortex stability relative to an aster. The vorticity, motor density and fraction of motors at [+]-ends are plotted in Figure 11 against end-detachment rates kEkD for two sets of kA, kM and P. It is clear that increasing kE can both destroy a vortex that existed when kE = kD, and create a vortex when kE = kD gave an aster. In order of increasing kE, the sequence aster → vortex → semi-aster (where any vortex is either absent or too short lived to be discerned) is typically observed, although we do not claim this sequence is followed by all points in parameter space. There is a slight increase in motor density in the semi-aster state as evident from the figure, resulting from an increase in potential attachment points due to the increased density.
Figure 11


Thus residence time at [+]-ends, which is k E - 1 , is not the determining factor with regards vortex stability here. Rather, vortices coincide with around 25% of motors at [+]-ends as highlighted in the figure. There is no critical dependency on the motor density, although we speculate that below some minimum value spindles or nematic states would be observed instead. The critical fraction 25% will likely depend on parameters that were not varied in this work, such as filament length M and the motor spring stiffness. A systematic survey of these parameters, or of kEkD for all kA, kM and P, is however beyond the scope of this work.

3.4 Controlled volume


Figure 12


Although the magnitude of the rotational velocity remains fixed (note that the characteristic velocity v is the same for both runs and constant in time), the direction aperiodically reverses as evident in the changes of sign in the figure. The statistics of time intervals between direction switching suggest that the underlying mechanism may be the same as for the contraction to the semi-aster state in the constant pressure case. Specifically, the mean switching time Δtswitch/τ b = 21.6 × 104 ± 5.8 × 104 is consistent with the mean contraction time ≈ 1.7 × 104 measured earlier, and again is consistent with an exponential distribution (significance level P ≈ 0.2 from n = 7 values using the Anderson-Darling test [31, 32]). This suggests that the same spontaneous fluctuation that permits contraction under constant pressure instead promotes rotational reversal under constant volume.

4 Discussion





In very recent experiments on actin (without myosin) in confined geometries [34], pattern formation strikingly similar to our nematic state in Figure 2(c) was observed. Since we find such states for a low density of slow motors, this is entirely consistent with the approach to the passive systems investigated in these experiments.

5 Conclusions





Authors’ Affiliations

Computational Biophysics, University of Twente
School of Computing, Leeds University


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