Theoretical Soft Matter and Biophysics, Institute of Complex Systems, Forschungszentrum Jülich, 52425 Jülich, Germany

Computational Biophysics, University of Twente, 7500 AE Enschede, The Netherlands

School of Computing, Leeds University, Leeds LS2 9JT, UK

Abstract

Background

Robust self-organization of subcellular structures is a key principle governing the dynamics and evolution of cellular life. In

Results

Here we present results of numerical simulations of a discrete filament-motor protein model confined to a pressurised cylindrical box. Stable spindles, nematic configurations, asters and high-density semi-asters spontaneously emerge, the latter pair having also been observed in cytosol confined within emulsion droplets. State diagrams are presented delineating each stationary state as the pressure, motor speed and motor density are varied. We further highlight a parameter regime where vortices form exhibiting collective rotation of all filaments, but have a finite life-time before contracting to a semi-aster. Quantifying the distribution of life-times suggests this contraction is a Poisson process. Equivalent systems with fixed volume exhibit persistent vortices with stochastic switching in the direction of rotation, with switching times obeying similar statistics to contraction times in pressurised systems. Furthermore, we show that increasing the detachment rate of motors from filament plus-ends can both destroy vortices and turn some asters into vortices.

Conclusions

We have shown that discrete filament-motor protein models provide new insights into the stationary and dynamical behavior of active gels and subcellular structures, because many phenomena occur on the length-scale of single filaments. Based on our findings, we argue the need for a deeper understanding of the microscopic activities underpinning macroscopic self-organization in active gels and urge further experiments to help bridge these lengths.

1 Background

Filamentous proteins are prevalent within eukaryotic cells and perform a variety of crucial tasks relating to cellular integrity, locomotion, transport and division

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

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

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,

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

2 Methods

We consider a system of _{B}
^{2}. Self-avoidance of filaments is introduced by repulsive Lennard-Jones potentials with diameter _{B}
_{B}
_{
p
}= _{
B
}T is ℓ_{
p
}= 20

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 _{B}
^{2 }and dynamics defined by four rates as shown in Figure _{A }for a motor to attach to two monomers within a predefined range, here taken to be the excluded volume radius 2^{1/6}
_{A }is the product of a molecular attachment rate and the free motor concentration); (ii) the detachment rate _{D }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 _{M }of each head independently towards the filament's [+]-end, and (iv) the detachment rate _{E }for motor heads already at a [+] end. The movement rate is attenuated by an exponential factor _{E }= _{D }below.

Model definition

**Model definition**. (a) Summary of key model parameters including the rates of motor attachment _{A }and detachment _{D}, and the bare stepping rate _{M}. See text for details. (b) Plan view showing the filaments oriented with their light-shaded [+]-ends towards the center. The arrows denote the external pressure acting on the circular elastic wall. (c) Side view of the same, showing the confining walls perpendicular to the

Simulating 3D filament gels in a spherical or cubic box at physiologically-relevant densities is computationally prohibitive when excluded volume interactions are included. To reduce computational demands while still permitting filament overlap, we therefore adopt a quasi-2D simulation cell with parallel confining walls normal to the z-axis spaced 5

All walls repel the monomers with the same Lennard-Jones non-bonding potential as for filaments. _{0 }to ℓ_{0 }+ _{0 }to _{0 }+ ^{3}
_{B}
_{0}
^{2 }per triplet. The chosen coefficients ensure an approximately circular wall shape throughout the deformation without significantly countering the external pressure for typical filament densities, as confirmed by the far smaller final wall radii measured when the filaments were absent.

The filaments, motors and elastic wall are all updated stochastically. The filaments obey Brownian dynamics _{
i
}the angle between filament

3 Results

Results are presented here in terms of the normalised attachment rate _{A}/_{D}, the normalised motor rate _{M}
_{
b
}where _{
b
}= _{B}
_{0 }with _{0 }= ^{3 }with _{B}T the Lennard-Jones repulsion energy, and, where relevant, the scaled end-detach rate _{E}/_{D}.

3.1 Stationary states

For the parameter space considered, we observe four classes of steady-state configuration as shown in Figure

Snapshots of steady-states

**Snapshots of steady-states**. Snapshots of steady-states for a low motor attachment rate _{A}/_{D }= 1 (left) and a higher rate _{A}/_{D }= 30 (right). Conversely, the top line is for fast motors _{M}_{b }= 3.75 × 10^{-2 }and the bottom line for motors 10 times slower. Filaments are shaded light (dark) towards their plus (minus) ends, respectively. These states are referred to as (a) spindle, (b) aster, (c) nematic and (d) semi-aster. The other parameters are _{0 }= 0.03 and _{E }= _{D}. Movies of the same parameter values are available from the supplementary information.

**Movie showing the transition from the initial conditions to a spindle steady state. System parameters are identical to Figure **

Click here for file

**Movie showing the transition from the initial conditions to an aster steady state**. System parameters are identical to Figure

Click here for file

**Movie showing the transition from the initial conditions to a nematic steady state**. System parameters are identical to Figure

Click here for file

**Movie showing the transition from the initial conditions to a semi-aster steady state**. System parameters are identical to Figure

Click here for file

To quantify to which state a system belongs, each filament's polarity vector is projected onto the **a**
_{
m
}and **b**
_{
m
},

from which can be defined the mode amplitudes _{
m
},

The _{
m
}are invariant under global rotations of the whole box.

To determine the corresponding state, the measured _{m }up to _{0}, _{1}, _{2}, _{3}) = (0,1,0,0), whereas for the nematic state where _{0}, _{1}, Q_{2}, _{3}) = (0,0,0,0). Note that while an isotropic state would give the same _{m }as for the nematic, such states only arise for _{A }and _{m }are calculated, so we choose simple forms that permit exact evaluation of the _{
m
}. For the spindle, _{0}, _{1}, _{2}, _{3}) = (0, 2/^{2 }+ 1/2, 0, 2/^{2}). For the semi-aster, _{0},_{1},_{2},_{3}) = (9/^{2}, 9/4^{2}, 333/1715^{2}, 9/100^{2}). 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 _{mot}/_{A }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 _{mot}. The approximate scaling _{A }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.

State diagrams

**State diagrams**. Variation of steady-state with motor density and speed for (a) _{0 }= 0.01 and (b) _{0 }= 0.024. Markers denote actual states determined as described in the text and the boundaries are equidistant between pairs of data points. The vortex region in (b) is a transient configuration that is explained in section 3.2 and is delineated as those states with a vorticity

Motor density

**Motor density**. No. of motors per filament _{A}/_{D }for the motor speeds and external pressure denoted in the legend. The thick black dashed line has a slope of 3/2. Where data for the required _{mot }was interpolated from runs with

Motors move to the [+]-end and dwell there until detaching, thus a greater fraction are expected to occupy filament [+]-ends, potentially resulting in tight binding mediated _{A}, which corresponds to the crossover to the aster state with a high degree of [+]-end binding. Further indication of the importance of [+]-end binding is presented in section 3.3 where enhanced end-unbinding rates _{E }> _{
D
}are considered.

Plus-end motors

**Plus-end motors**. Fraction of motors with at least one head at a filament's [+]-end versus _{A}/_{D }for the same data as in Figure 4, so the top two lines correspond to fast motors and the bottom to slow motors.

3.2 Dynamics and vortices

The stationary states described above admit no spontaneous non-equilibrium flows, despite the motor motion generating a positive energy flux: The increase in the stored motor elastic energy due to motor motion and thermal drift of the connected filaments is balanced by the loss due to detachment, with no observed net translocation or rotation of the filaments in steady-state. Collective rotation of all filaments about a fixed center arises for one region of the considered parameter ranges, but appears to be a transient flow that irreversibly contracts to a non-rotating semi-aster configuration. These states are referred to here as

Snapshot of a vortex

**Snapshot of a vortex**. Snapshot of a vortex rotating in the anti-clockwise direction as presented, for parameters _{0 }= 0.024, _{A}/_{D }= 35 and _{M}_{b }= 7.5 × 10^{-3 }taken at a time _{b }≈ 5.3 × 10^{3}. The colour code is the same as in Figure 2.

**Movie showing the transition from the initial conditions to a dynamic vortex, and subsequent contraction to a semi-aster**. System parameters are identical to Figure 6.

Click here for file

**Identical run to additional file **
**, with one filament highlighted and the remaining translucent**.

Click here for file

Collective rotation of the whole system can be quantified by the mean angular velocity of filament centre-of-mass vectors r relative to the system center, or alternatively by the net transverse velocity of each filament's centre-of-mass relative to its polarity, _{min }+ Δ^{cont})/Δ^{cont }found from this fit. In all cases, ^{cont }coincides with the rapid decay of

Example of vorticity

**Example of vorticity**. Examples of the vorticity _{A}/_{D }= 35, _{M}_{b }= 7.5 × 10^{-3 }and _{0 }= 0.024. For clarity error bars are only given for a single run. The thick horizontal line segments denote the mean value up to the time when the vortex contracts to a semi-aster, and are plotted up to this time.

It is now possible to define a vorticity order parameter for each point in parameter space. For each run α, the mean _{
a
}and standard deviation _{
α
}of

This is then averaged over all runs with the same parameters to give the mean vorticity _{E }supports the existence of a critical fraction for vortex formation, as discussed in Sec. 3.3.

The reciprocal relationship between vorticity and contraction time is clearly evident when both quantities are plotted together; see Figure

Vorticity and contraction times

**Vorticity and contraction times**. Contraction time to a semi-aster ^{cont}/_{b }(left axis; solid circles) and vorticity _{0 }for _{A}/_{D }= 35 and _{M}_{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 ^{cont }= 1.6 × 10^{4}_{b }if no contraction had occurred within this time. Each point represents 5 independent runs.

Contraction time and vorticity statistics

**Contraction time and vorticity statistics**. (a) Probability distribution of contraction times on log-linear axes for _{0 }= 0.024, _{A}/_{D }= 35 and _{M}_{b }= 7.5 × 10^{-3}. The thick line gives the best fit to an exponential distribution which has a mean ≈ 1.7 × 10^{4}, corresponding to ≈ 0.75 full rotations (the longest vortex survived for ≈ 2.6 rotations). Data corresponds to 20 independent runs. (b) Normalised probability histogram of signed vorticity for _{0 }= 0.020 (white bars in the background; 5 runs) and _{0 }= 0.024 (shaded bars in the foreground; 20 runs).

Assuming the true distribution is exponential, this would suggest that contraction is triggered by spontaneous fluctuations that occur at a constant rate in time. From observation of movies of filament arrangements, a likely candidate is the transient void formation frequently observed near the outer wall, where nearby filaments are attached purely by motors at their [+]-ends and not along their length. Such voids, when large enough, lead to a 'hinge'-like mechanism in which the void expands and one section of the polarity field inverts, leading to the semi-aster.

The onset of vorticity is also evident in the histogram of the

Independent confirmation of vorticity can be inferred from the mean-squared angular deviation 〈(Δ^{2}〉 already defined in Sec. 2. This is plotted in Figure ^{cont}. There is a crossover from linear behavior (Δ^{2 }~ Δ^{2 }~ (Δ^{2 }for pressures well into the vortex regime. Since this quantity is the angular analogue of the mean squared displacement for translation degrees of freedom, these two limits can be regarded as

Mean-squared angular changes

**Mean-squared angular changes**. Mean-squared changes in angle 〈(Δ^{2}〉 versus lag time Δ_{A}/_{D }= 35 and _{M}_{b }= 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 _{E }≥ _{D }for two sets of _{A}, _{M }and _{E }can both destroy a vortex that existed when _{E }= _{D}, and create a vortex when _{E }= _{D }gave an aster. In order of increasing _{E}, 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.

Varying end-detach rates

**Varying end-detach rates**. Variation of vorticity, motor density and fraction of [+]-end motors with _{E }for _{A}/_{D }= 35 and _{m}_{b }= 7.5 × 10^{-3 }(solid lines, filled squares) and _{A}/_{D }= 60 and _{M}_{b }= 37.5 × 10^{-3 }(dashed lines, open diamonds). _{0 }= 0.024 in both cases. The thick dashed line in the lower plot corresponds to 25%. Quantities were measured just prior to contraction, or in steady-state if there was no contraction or it happened too rapidly to discern.

Thus residence time at [+]-ends, which is ∝ _{E }≠ _{D }for all _{A}, _{M }and

3.4 Controlled volume

One message from the previous sections is that the observed steady-state is predominately determined by the density of motors and the fraction at [+]-ends. It may appear that the primary role of motor motion, which would be the source of any non-equilibrium effects in this model, is merely to select the fraction of [+]-ended motors, faster motors giving a higher fraction. It might even be speculated that even the transient vortex state is driven, not by motor motion, but rather as a protracted buckling event powered by the pressurised walls.

It is straightforward to show that motor motion can drive vortex motion, however. Plotted in Figure _{
b
}≈ 1.6 × 10^{4}, the system switches to a rotating vortex state that appears to be long-lived; the total time window in this figure is an order of magnitude longer than the longest vortex described in Sec. 3.2 (which has the same parameters). Since there is no energy input from the walls, the only possible cause for this rotation is the motor motion. Thus the pressure ensemble is important to let the system adjust its density to the vortex state; however, the same pressure also destabilizes the vortex state, because it favors further contraction into the semi-aster.

Fixed volume vortices

**Fixed volume vortices**. (a) Filament rotation for two independent runs at fixed volume. The imposed radius

Although the magnitude of the rotational velocity remains fixed (note that the characteristic velocity ^{switch}/_{
b
}= 21.6 × 10^{4 }± 5.8 × 10^{4 }is consistent with the mean contraction time ≈ 1.7 × 10^{4 }measured earlier, and again is consistent with an exponential distribution (significance level

4 Discussion

It has been demonstrated that the vortices described here involve the collective rotation of the filaments about a fixed centre. This was predicted by the nematodynamics theory of Kruse

The variation of steady-state structure with motor speed and density shown in Figure _{E }confirms that a strong binding at [+]-ends can stabilise an aster relative to a vortex or semi-aster. The motor speed _{M }plays a role in selecting the distribution of motors along the filament, but also contributes to the rotation of vortices as inferred from the fixed volume system in Sec. 3.4. Therefore we claim that the observed vortex is a genuine non-equilibrium state powered at least partially by the unidirectional motion of energy-consuming motor heads along the filaments, although at constant pressure they appear to be transient. It is not clear if varying some other parameters may produce stable vortices.

Systematically quantifying the role of all of the model parameters is clearly challenging for such a high-dimensional parameter space, and here we have adopted the pragmatic approach of holding most parameters fixed while varying those deemed most likely to be critical. Eventually the impact of all parameters on structure and dynamics will need to be quantified if a broad description of active gels is to be attained. Here, we highlight two parameters likely to reveal novel or interesting behaviour. First, the filament length

While it was always our intention to model filament-motor systems as generally as possible, it is nonetheless insightful to consider the corresponding parameters for an actual system. Taking the filament diameter to be _{B}

In very recent experiments on actin (without myosin) in confined geometries

5 Conclusions

We have systematically varied motor speed and density in filaments confined to a pressurised cylindrical cell, and have uncovered four qualitatively different types of steady state, namely aster, semi-aster, spindle and nematic. The corresponding regions of parameter space for each state were delineated by modal analysis of the filament polarities. Furthermore, in one region of parameter space we found a vortex state in which filaments rotated about the system centre for a finite time before buckling to a semi-aster. Quantitative analysis of rotation speed and mean-squared angular displacement provided unambiguous evidence of coherent filament rotation in this state. The vortex state persisted for far longer times with fixed walls, albeit with stochastic changes of direction, demonstrating that motors and not pressure alone are necessary for the observed vortex rotation.

Competing interests

The authors declare that they have no competing interests.

Authors' contributions

DAH, WJB, and GG designed research; DAH performed simulations and analyzed data; DAH, WJB, and GG wrote the paper. All authors read and approved the final manuscript.

Acknowledgements

Financial support of this project by the European Network of Excellence "SoftComp" through a joint postdoctoral fellowship for DAH is gratefully acknowledged.