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Chronology of motor-mediated microtubule streaming

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Cite this article as: eLife 2019;8:e39694 doi: 10.7554/eLife.39694

Abstract

We introduce a filament-based simulation model for coarse-grained, effective motor-mediated interaction between microtubule pairs to study the time-scales that compose cytoplasmic streaming. We characterise microtubule dynamics in two-dimensional systems by chronologically arranging five distinct processes of varying duration that make up streaming, from microtubule pairs to collective dynamics. The structures found were polarity sorted due to the propulsion of antialigned microtubules. This also gave rise to the formation of large polar-aligned domains, and streaming at the domain boundaries. Correlation functions, mean squared displacements, and velocity distributions reveal a cascade of processes ultimately leading to microtubule streaming and advection, spanning multiple microtubule lengths. The characteristic times for the processes extend over three orders of magnitude from fast single-microtubule processes to slow collective processes. Our approach can be used to directly test the importance of molecular components, such as motors and crosslinking proteins between microtubules, on the collective dynamics at cellular scale.

https://doi.org/10.7554/eLife.39694.001

Introduction

The vigorous motion of the intracellular fluid, known as cytoplasmic streaming, is caused by cytoskeletal filaments and molecular motors. In Drosophila oocytes this cellular-scale fluid motion, which occurs over multiple time scales, is responsible for efficient mixing of ooplasm and nurse-cell cytoplasm, for long-distance transport of intracellular material, and for proper patterning of the oocyte (Quinlan, 2016; Palacios and St Johnston, 2002; Lu et al., 2016). Although cytoplasmic streaming is known for centuries (Berthold, 1886) and kinesin-1 molecular motors and microtubules (MTs) have been identified as the components responsible for ooplasmic streaming (Quinlan, 2016; Palacios and St Johnston, 2002; Gutzeit, 1986; Serbus et al., 2005), there is considerable debate about the aetiological mechanisms for force generation. Namely, the constituent events, their order of occurrences, and their characteristic durations, which ultimately give streaming are not understood. Some studies suggest that streaming is caused by the hydrodynamic entrainment of motor-transported cargos (Monteith et al., 2016; Theurkauf et al., 1992), others that it is due to the motor-mediated sliding of adjacent MTs (Jolly et al., 2010; Lu et al., 2016; Winding et al., 2016).

Motor-mediated MT sliding occurs because molecular motors crosslink adjacent MTs and use ATP (adenosine triphosphate) molecules as fuel to 'walk’ on them unidirectionally in the direction of MT polarity (Vale and Milligan, 2000). This leads to significantly different active dynamics of MT pairs that are polar-aligned and antialigned (Gao et al., 2015; Blackwell et al., 2016; Ravichandran et al., 2017): Motors that crosslink polar-aligned MTs hold the polar-aligned MTs together, generating an effective attraction (Ravichandran et al., 2017). Active motors crosslink and slide antialigned MTs. The motors thus act as force dipoles that break nematic symmetry in MT solutions. In the absence of permanent crosslinkers, which are known to render the active network contractile (Belmonte et al., 2017), this ultimately can cause large-scale flows in the cytoskeleton (Jolly et al., 2010; Lu et al., 2016). Several approaches to analyse the collective motion, such as the displacement correlation function or the analysis of velocity distributions, are inspired by studies on collective motion of self-propelled agents (Duman et al., 2018; Needleman and Dogic, 2017; Zöttl and Stark, 2016; Bechinger et al., 2016; Elgeti et al., 2015).

In vivo, individual MTs are stationary most of the time before suddenly undergoing a burst of long-distance travel with velocities reaching 10 μm/s (Jolly et al., 2010). Also, fluorescence microscopy has shown the formation of long extended arms for an initially circular photoconverted area (Jolly et al., 2010). A possible mechanism for such a behaviour is illustrated in Figure 1: In the absence of active motor stresses, MTs in polar-aligned bundles diffuse slowly. When they encounter antialigned MTs, they can be actively and rapidly transported away from a polar-aligned bundle to another polar-aligned bundle where they again exhibit slow diffusive behaviour. These transitions between slow-bundling motion and fast-streaming bursts can give rise to Levy flight-like MT dynamics (Chen et al., 2015).

Schematic illustrating MT bundling and streaming.

Polar-aligned MTs are coloured blue, and antialigned MTs are coloured red. The grey/black MT is transported from its initial position (grey), in one polar-aligned bundle, to its final position (black), to another polar-aligned bundle, via a stream.

https://doi.org/10.7554/eLife.39694.002

In vitro and in silico, MT-motor model systems allow a systematical study of the interactions between the basic components of the cytoskeleton. The systems often contain only a small number of different components and are usually also restricted in other respects, such as a reduced dimensionality or a lack of polymerization and depolymerization of the filaments. For example, a very well studied model system contains MTs and kinesin complexes at an oil-water interface (Sanchez et al., 2012; DeCamp et al., 2015), where poly(ethylene glycol)-induced depletion interaction both keeps the MTs at the interface and induces bundle formation. In contrast, coarse-grained computer simulations in 2D are used to study both structure formation on the bundle scale (DeCamp et al., 2015) as well as single-filament dynamics, for example to quantify the effect of motor properties or of presence of passive crosslinkers (Ravichandran et al., 2017; Blackwell et al., 2016; Gao et al., 2015; Belmonte et al., 2017). Because such model systems are simpler than the cytoskeleton in biological cells, they are especially suited to study specific mechanisms in detail.

Cytoplasmic streaming is a complex multi-scale phenomenon that cannot be fully understood using antialigned filaments alone. The importance of a specific mechanism can only be studied in vivo for a specific system. This so-called ’top-down’ approach has been remarkably successful in describing streaming in the aquatic alga Chara coranilla (Woodhouse and Goldstein, 2013) and Dropsophila oocytes (Khuc Trong et al., 2015). In the former case, theoretical models have showed the importance of coupling hydrodynamic entrainment and microfilament dynamics to capture pattern formation relevant for streaming. In the latter case, simulations mimicking streaming in Drosophila oocytes have emphasised the importance of cortical MT nucleation in anteroposterior axis definition. It was shown that nucleation of MTs from the periphery is important to induce cytoplasmic flow patterns and to localise mRNAs in specific areas of the cell.

Microtubule-motor systems are intrinsically out of equilibrium, which has been shown for example by monitoring the dynamics of motors that walk along MTs (Schnitzer and Block, 1997) and by the violation of the fluctuation-dissipation theorem for tracer particles embedded into acto-myosin systems (Mizuno et al., 2007). Therefore, simulation approaches for systems of passive MTs at equilibrium have to be augmented with motor activity. Brownian dynamics simulations with MTs and explicit motors have been used to study network contractility (Belmonte et al., 2017), polarity-sorting, and stress generation at high MT densities (Gao et al., 2015), persistent motion of active vortices in confinement (Head et al., 2011a) and anomalous transport in active gels (Head et al., 2011b). Recent simulations for the defect dynamics in extensile MT systems have been performed on the coarse-grained level of MT bundles (DeCamp et al., 2015).

We employ a ’bottom-up’ approach, where we study MT streaming induced by MT sliding using a model system. In order to characterize the dynamics in the system, we use a coarse-grained model to investigate whether a purely polarity-dependent MT-MT sliding mechanism, in the absence of any hydrodynamic forces, can be sufficient to capture large-scale streaming in bulk. We identify five distinct processes that comprise streaming with their characteristic times for various MT activities and surface fractions: (1) motor-driven MT sliding, (2) polarity-inversion, (3) maximal activity, (4) collective migration, and (5) rotation. Although various experimental studies have provided high spatial resolution to describe streaming phenomena (Quinlan, 2016), MT dynamics for streaming is still poorly understood. Inspired by the biological mechanism for MT-MT sliding, we use computer simulations to provide a novel temporal perspective into streaming for a wide range of time scales, which has not been achieved so far due to limitations of experimental techniques.

In order to capture cellular-scale dynamics in computer simulations, modelling individual motors along with MTs, although done before in several studies (Mizuno et al., 2007; Hiraiwa and Salbreux, 2016; Ronceray et al., 2016; Head et al., 2014; Blackwell et al., 2016; Gao et al., 2015; Freedman et al., 2017), can prove to be unwieldy due to the wide ranges of length and time scales involved. The sizes of individual kinesin molecules that crosslink and slide MTs are three orders of magnitude smaller than that of the cells within which they bring about large-scale dynamics. Also, there is a large disparity between the residence time of a cross-linking motor (10 seconds) (Toprak et al., 2009), and the characteristic time scale of motor-induced MT streaming or pattern formation in active gels (1 hr) (Ganguly et al., 2012; Jolly et al., 2010; Lu et al., 2016; DeCamp et al., 2015). In order to capture motor-induced cellular-level phenomena, such as organelle distribution, cytoplasmic streaming, and active cytoskeleton-induced lipid bilayer fluctuations, a coarse-grained description of cytoskeletal activity seems therefore appropriate.

Coarse-grained model

Coarse-grained and continuum approaches are successfully applied to study cytoskeletal-motor systems. A well-developed model and simulation package is Cytosim that can be used to simulate flexible filaments together with further building blocks that, for example, act as nucleation sites, bind filaments together, and induce motility or severing (Nedelec and Foethke, 2007). It has been applied to study–among other processes in the cell–meiosis (Burdyniuk et al., 2018), mitosis (Lacroix et al., 2018), and centrosome centering (Letort et al., 2016). A different model that includes MT flexibility, MT polymerization and depolymerization, explicit motors, and hydrodynamics has recently been applied to study mitosis (Nazockdast et al., 2017a; Nazockdast et al., 2017b). Because the MTs are mostly radially oriented, steric interactions between MTs can be neglected and have not been taken into account. However, MT-MT repulsion is important to obtain nematic order at high MT densities, an essential ingredient for the bottom-up model systems containing suspensions of MTs and kinesins (Sanchez et al., 2012). Our model includes MT flexibility, effective-motor potentials, and excluded-volume interactions. Polymerization and depolymerization does not occur in the model system and is not taken into account. Effective-motor models in general aim to reduce the computational effort to efficiently study large systems (Swaminathan et al., 2010; Jia et al., 2008). Including hydrodynamic interactions using a particle-based approach is straightforward (Winkler and Gompper, 2018; Müller et al., 2015; Gompper et al., 2009), but beyond the scope of this paper.

Microtubules

In our two-dimensional model, MTs are modelled as impenetrable, semi-flexible filaments of length L, thickness σ, and aspect ratio L/σ. Each of the N filaments in the system is discretised into a chain of n beads with diameter σ that are connected by harmonic bonds. The configurational potential,

(1) Ui=Ubond+Uangle+Uwca.

is the sum of passive potentials, that is the spring potential Ubond between adjacent beads, the angle potential Uangle between adjacent bonds, and the volume exclusion UWCA between MTs.

The bond energy,

(2) Ubond=ks2bonds(ri,i+1-r0)2,

acts between adjacent beads of the same MT. Here, ks is the bond stiffness, r0=σ/2 is the equilibrium bond length, and ri,i+1=|𝐫i,i+1| is the distance between adjacent beads i and i+1, which make up the MT. Uangle is the bending energy, which is calculated using the position of three adjacent beads,

(3) Uangle=κr0angles(1-cosθi),

that make up the angle θi (Isele-Holder et al., 2016). It acts between all groups of three adjacent beads that make up the same MT. The bending modulus κ of the filament determines its persistence length p=κ/kBT.

MT bead pairs that are not connected by harmonic springs interact with each other via the repulsive Weeks-Chandler-Andersen (WCA) potential (Weeks et al., 1971),

(4) Uwca=4ϵ[(rijσ)12-(rijσ)6]+ϵ,

with interaction cutoff rcut=21/6σ.

Effective molecular motors

Various theoretical studies have strived to circumvent short time and length scales involved in cytoskeletal dynamics, such as diffusion and active motion of individual motors (Kruse and Jülicher, 2000; Kruse et al., 2005; Swaminathan et al., 2010; Jia et al., 2008; Aranson and Tsimring, 2005; Aranson and Tsimring, 2006; Salbreux et al., 2009; Córdoba et al., 2015). For example, in the phenomenological flux-force model the motion of MTs in one dimension occurs solely due to the orientation of neighbouring MTs (Kruse et al., 2004; Kruse et al., 2001). Many two-dimensional models, where MTs are modelled as stiff, polar rods of equal length, take motors into account using a Maxwellian model of inelastic interactions between the rods (Aranson and Tsimring, 2005; Aranson and Tsimring, 2006; Jia et al., 2008; Swaminathan et al., 2010). These probabilistic collision rules result in the alignment of rods. Although these models capture the self-organization of MT-motor mixtures into stable patterns of vortices, asters, and smectic bundles, the collision rule approaches do not reproduce the sliding of antialigned MTs described in Figure 2.

Schematic explaining the conditions that satisfy the antialigned motor potential.

The vectors, 𝐩i, 𝐩j, and 𝐦ij, represent the unit orientation vectors of MT i, MT j, and the motor vector that crosslinks the beads of adjacent MTs, respectively. The white circles represent the maximum extension of motors between the two MTs.

https://doi.org/10.7554/eLife.39694.003

Sliding of antialigned MTs due to kinesin motors has been identified as key ingredient for cytoplasmic advection in vivo (Jolly et al., 2010; Winding et al., 2016; Lu et al., 2016). Instead of modelling individual motors, in our model MT motion manifests itself as a result of a distribution of motors in an ensemble of orientations between neighbouring MT pairs. Hence, we coarse-grain MT-motor interactions using an effective motor potential that gives a contribution to Umot. A motor bond can form when the crosslinked beads are antialigned, that is the angle that a motor bond vector 𝐦ij makes with the unit orientation vector is acute on both MTs simultaneously, see Figure 2,

(5) pimij>0andpjmij<0.

Here the orientation vector assigned to bead i on an MT is 𝐩i=(𝐫i+1-𝐫i)/|𝐫i+1-𝐫i|, and the extension of a motor that crosslinks MTs i and j is 𝐦ij(si,sj)=𝐫i(si)-𝐫j(sj), with the motor heads bound at the positions si and sj along the contour of the MTs. This is similar to the activity-inducing scenario a dimeric or tetrameric motor (Ravichandran et al., 2017) encounters when it crosslinks a pair of antialigned MTs, that is the motor arms are oriented towards the + direction of either crosslinked MT.

Each effective motor is a harmonic spring of equilibrium bond length deq=σ and stiffness km that exists for one simulation time step (Although the force for each motor lasts for a single time step, this duration is not a characteristic time for the motor. Our model describes continuum propulsion forces on MTs imposed by motors. This is supported by Figure 3; Figure 3—figure supplement 1, which shows that the MT parallel velocity is proportional to the fraction of time that motors are active, independent of the duration of each active phase.). The system is inherently out of equilibrium because the motor bonds occur dependent on the relative orientation of neighboring MTs, and exist and exert forces only for short times, mimicking the ratchet model for molecular motors (Jülicher et al., 1997). The potential for a motor with extension mij=|𝐦ij| is

(6) Umot(mij)={km2(mijdeq)2mijdt0mij>dt,

and the motor binding rate is

(7) kon(mij)=paexp(-Umot(mij)kBT).

Here, pa controls the probability that an antialigned motor binds (Similarly, we can also implement motors between polar-aligned MTs). Motors bind only for extensions mij<dt=2σ, when kon/pa>exp(-1/2). This also corresponds well to the experimentally measured length of a kinesin motor (Kerssemakers et al., 2006). The motor model described here for two dimensions is analogous to the phenomenological model for one dimension described in Kruse and Jülicher (2000); these one-dimensional calculations show that the relative velocity between two antialigned MTs is a linear function of pa and km. Similarly, a kinesin-5 induced effective torque between MTs has been calculated to study forces in the mitotic spindle (Winters et al., 2018).

Results

We characterise several distinct processes comprising the phenomenology of active MT motion that arise because of the sliding of adjacent, antialigned MT pairs.

Regimes of MT dynamics

Figure 3 provides an overview of the processes that comprise MT streaming. The fundamental sliding mechanism, which is imposed through the effective motor potential, is the process which occurs at the sliding time τN,min. The three processes that occur on longer time scales are characterised by the polarity-inversion time τQ/2, the activity time τ*, and the collective-migration time τN,max. The active-rotation time τr characterises the time when an active MT reaches the end of a polar-ordered domain and changes its orientation.

Figure 3 with 10 supplements see all
Motor-driven and diffusive motion of MTs.

(a) Simulation snapshot of MTs organised by effective motors. MTs are coloured based on their orientation according to the colour legend on the right. See corresponding Video 1. (b) Trajectories of MTs within a time window of 1.2 τR separated based on the antialigned and polar-aligned categories. See corresponding Video 2. (c) Plots of the trajectory of three selected MTs coloured based on the correlation of adjacent steps in their velocity. The entire trajectory is for a time window of 300 τR is the unit vector of MT displacement. The fast-streaming and slow-diffusion modes correspond with the yellow and red parts of the trajectories respectively. The scale bar corresponds to the length of five MTs. See corresponding Video 3. (d) MSD/lag time for various levels of activity pa and MT density ϕ=0.3. The time scale of maximal activity, τ*, calculated from the time of maximal v skew is indicated by the squares on the curves. (e) Histogram of parallel velocity for various τ. The curve closest corresponding to the time scale of maximal activity, τ*, is indicated with a box marker. All figures are for ϕ=0.3. (a), (b), (c) and (e) are for pa=1.0.

https://doi.org/10.7554/eLife.39694.004

Figure 3(a) and Video 1 show multiple, small, polar-aligned MT domains with dynamic interfaces of antialigned MTs between them for MT surface fraction ϕ=0.3 and motor activity pa=1. Note that choosing different values for ϕ and pa can have a pronounced effect on the structure of the system (Figure 3—figure supplement 2). The domains are formed by polarity sorting (Wollrab et al., 2018) and are in dynamic equilibrium due to MTs that perpetually enter and leave them, see Figure 3(b) and Video 2. Tracing the individual trajectories shows that MT dynamics consists of a fast streaming mode and a slow diffusion mode, see Figure 3(c) and Video 3. Uncorrelated displacements in time correspond to slow diffusion within a polar-aligned domain of MTs, and correlated displacements correspond to fast, ballistic streaming at interfaces between domains. This leads to a highly dynamic overall MT structure illustrated by Video 4. At steady state, the length of antialigned interfaces and the size of polar-aligned domains remain constant. The polarity-inversion time τQ/2 characterises the duration that MTs stay in polar-aligned bundles or in antialigned streams.

Video 1
Steady-state dynamics of the MT-effective motor system shown in Figure 3(a) for 100 τR.

MTs are coloured based on their orientation. 

https://doi.org/10.7554/eLife.39694.016
Video 2
Streaming motion of antialigned and diffusive motion of polar-aligned MTs for 100 τR, corresponding to Figure 3(b). The scale bar corresponds to the length of 10 MTs.

Trajectories of MTs are shown within time windows of 1.2τR

https://doi.org/10.7554/eLife.39694.017
Video 3
Center-of-mass trajectories for selected MTs for 300τR, corresponding to Figure 3(c).

Fast streaming and slow diffusion is indicated by yellow and red, respectively. The scale bar corresponds to the length of five MTs.

https://doi.org/10.7554/eLife.39694.018
Video 4
Inhomogeneous dynamics over a period of 100 τR.

Fast MTs are coloured yellow, and slow MTs are coloured blue.

https://doi.org/10.7554/eLife.39694.019

Figure 3(d) shows the MT mean squared displacement

(8) MSD(τ)|di(t,τ)|2|ri(t+τ)ri(t)|2

for MT surface fraction ϕ=0.3 and various motor probabilities (We use m=1 and γ=1 in simulation units. The single MT center-of-mass dynamics is overdamped for times t/τR>2m/γτR10-2, which is shorter than the shortest time scale of interest shown in Figure 3(d).). Here, 𝐫i(t) is the center-of-mass position vector of MT i at time t, and 𝐝i(t,τ) is the displacement vector of the center-of-mass of MT i between t and t+τ. For passive MTs, pa=0.0, the ballistic regime MSDτ2 at short times due to inertia is followed by a diffusive regime MSDτ where the MT velocity is dissipated by the environment. For all simulations at finite pa, we find a superdiffusive regime 101τ/τR101 with MSDτα and α>1 with active ballistic motion. Finally, we find a diffusive regime at long times with an active diffusion coefficient that is much higher than for passive Brownian diffusion. A larger motor activity thus leads to faster filament motion. Filament dynamics is fastest for intermediate MT surface fractions ϕ=0.3 and ϕ=0.4 (Figure 3—figure supplement 3). For smaller densities the required MT-MT contacts are reduced, whereas for large densities (ϕ=0.6) excluded volume interactions lead to a larger effective friction hindering filament motion.

Figure 3(e) shows the histogram of the MT velocity that is projected on the MT orientation vector 𝐩(t)=(𝐫nb(t)-𝐫1(t))/|𝐫nb(t)-𝐫1(t)|,

(9) v(τ)=𝐝i(t,τ)𝐩(t)τ

for ϕ=0.3, pa=1.0, and various lag times τ, see Figure 3—figure supplements 1, 4 and 5 for parallel velocities at other MT surface fractions. At short lag times, the MT displacement is strongly correlated with the MT’s initial orientation vector and dominated by thermal noise, giving the largest absolute values for v. With increasing lag time, the increasing importance of the active motor force is reflected by the increasing asymmetry of the distributions of parallel velocities that are skewed towards positive velocities. At long lag times, the MTs reorient due to active forces, such that both the width of the velocity distribution and the skew again decrease. We characterise the time delay that corresponds to the maximum skew as collective-migration time τN,max. This time characterises collective motion of neighbouring MTs with similar orientations that travel in the same direction. The parallel velocity distributions also depend on lag time, filament density, and motor probability (Figure 3—figure supplements 6, 7 and 9).

Finally, the active orientational correlation time τr for MTs is denoted by τr. This time characterises the crossover between the active-ballistic and the active-diffusive regime in Figure 3(d). It therefore increases both with increasing size of polar or nematic domains as well as with decreasing rod activity at the interfaces, in agreement with the diffusion of tracer particles in Sanchez et al. (2012) (Figure 3—figure supplement 10).

Microtubule sliding

The displacement-correlation function,

(10) Cd(r,τ)=i,ij𝐝i𝐝jδ(r-|𝐫i(t)-𝐫j(t)|)tc0i,ijδ(r-|𝐫i(t)-𝐫j(t)|)t,

quantifies both spatial and temporal correlations of MT motion. Here, 𝐝i=𝐫i(t+τ)-𝐫i(t), and c0=i𝐝i2/Nτ is used for normalisation. Figure 4(a) shows displacement correlation functions for various lag times. At short times and distances, we find negative displacement correlations due to the effective motor potential, which selectively displaces neighbouring antialigned MTs. These negative correlations decay rapidly in space and do not contribute substantially for r/L=1. At intermediate lag times we find positive displacement correlations with a slower spatial decay, and at long lag times no correlations. In the limit τ0, Cd is the equal-time spatial velocity correlation function (Wysocki et al., 2014).

Figure 4 with 1 supplement see all
Displacement correlations of MTs.

(a) Spatio-temporal correlation function Cd(r,τ) for ϕ=0.3 and pa=1.0, for some selected lag times. The arrow and the colours of the curves indicate increasing lag time. The lag times are picked from a logarithmic scale. (b) Neighbour correlation function Nd(τ)=Cd(σ,τ) for ϕ=0.3 and various pa values. (c) The sliding time scale indicated by τN,min is shown for various MT surface fractions and pa values.

https://doi.org/10.7554/eLife.39694.020

The neighbour displacement correlation function Nd(τ)=Cd(σ,τ) is defined as the displacement-displacement correlation function at contact Cd(σ,τ) (Doliwa and Heuer, 2000; Wysocki et al., 2014). Figure 4(b) shows neighbour displacement correlation functions for various values of pa and ϕ=0.3. Firstly, for passive systems, Nd is positive for all MT surface fractions but considerably weaker compared to the correlations in active systems. The small positive correlation is due to steric interactions and friction because of to the roughness of MTs (made up of overlapping beads). For active systems, Figure 4(b) illustrates that the temporal dependence of Nd(τ) displays three regimes: for short times, Nd(τ) is negative and MTs slide antiparallel, for intermediate times, Nd(τ) is positive and MTs move collectively, and for long times, Nd(τ) tends to zero and there is no coordinated motion. We focus here on the first regime, whereas the other regimes will be discussed in later sections.

In the short-time regime, τ/τR10-1, the effective motor potential propels neighbouring antialigned MTs away from each other and Nd is negative. This is aided by higher pa but hindered by higher ϕ, which opposes active motion sterically (Figure 4—figure supplement 1). The times τN,min at which the minima occur represent maximal MTs propulsion because of effective motor interactions, due to presence of the antialigned neighbours. At this time MTs move a small fraction of their length. In Figure 4(c) the sliding times are collected for different MT surface fractions showing that the sliding is strongly enhanced by activity, where τN,min decreases approximately exponentially with pa and increases with surface fraction.

Polarity inversion of local MT environment

In order to characterise an MT’s neighbourhood, we define a pairwise motor partition function (Gao et al., 2015; Ravichandran et al., 2017),

(11) qij=ρ2i=1nbj=1nbe-Umot(mij)/(kBT),

where ρ=nb/L is the linear density of binding sites on a single MT, and mij is the extension of the motor bound at positions si and sj on MTs i and j, respectively (Ravichandran et al., 2017). Local polar order thus weighs pairwise interactions of MTs on the basis of motor binding site availability. It is a function of relative orientation and distance between the beads that are used to model the MTs. Because of the Boltzmann weight, qij is significant only for pairs of MTs in close proximity. When two MTs are perfectly overlapping each other, qij=1. When two MTs are sufficiently far away, , qij=0 because the MTs are outside the motor cut-off range. Since the motor energy Umot(mij) increases quadratically with increasing motor extension, the partition function qij decays rapidly for increasing distance between the binding sites on the MTs.

The polarity of an MTs environment is quantified by the local polar order parameter ψ(i). MTs within motor cut-off range are defined to be antialigned if (pipj)<0 and polar-aligned if (𝐩i𝐩j)0. By taking the sum of all interacting MTs ji with MT i (Gao et al., 2015; Ravichandran et al., 2017), we ensure that the local polar order parameter

(12) ψ(i)=ji(𝐩i𝐩j)qijjiqij

depends on the polarity of the neighbourhood of MT i. Here qij is given by Equation 11. The environment of the MT can now be classified into polar (subscript-P, 0.5<ψ(τ)<1), antipolar (subscript-A, 1<ψ(τ)<0.5), and mixed (subscript-M, 0.5<ψ(τ)<0.5), see Figure 3—figure supplement 8 and Figure 5—figure supplement 1. When a single MT’s environment changes from predominantly antipolar to polar its active motion is stopped and it only moves diffusively.

By tracking changes in ψi for single MTs, see Video 2, we measure the time that MTs spend in antialigned or polar-aligned environments for various values of pa and ϕ. The change in local polar order of MT i can be written as

(13) Δψi(τ)=ψi,0-ψi(τ),

where ψi,0=ψi(τ=0). Figure 5(a) shows ψi,A(τ) and ψi,P(τ) for pa=1 and pa=0. In both cases, we find that ψi,A(τ) increases with time, indicating antialigned MTs leaving their antialigned environments, and that ψi,P(τ) decreases with time, indicating polar-aligned MTs leaving their polar-aligned environments. At long times, ψi,A and ψi,P converge to the long-time mean ψi,=0 for passive systems, and to ψi,(τ)>0 for active systems. The time scale for relaxing ψi to the equilibrium value is, as expected, shorter for the active than for the passive system.

Figure 5 with 3 supplements see all
Local polar order of MTs.

(a) Mean local polar order ψi(τ) for pa=0.0 and pa=1.0 at ϕ=0.3, for MTs starting from antialigned (dotted line) and aligned (solid line) environments at τ=0. (b) Deviation of local polar order Q(τ) for ϕ=0.3 for various pa for antialigned MTs. (c) Relaxation time for the polar order parameter, τQ/2 for various pa and ϕ, estimated by the time for Q to decrease to half its initial value.

https://doi.org/10.7554/eLife.39694.023

In order to quantify the change in ψi, we construct the deviation of local polar order,

(14) Q(τ)=1-Δψi(τ)ψi,0-ψi,.

Q(τ) for the antialigned MTs is shown in Figure 5(b); the lag time for that Q(τ) reaches half its initial value is τQ/2,A. While MTs that stay within a polar-ordered domain determine the offset for ϕi,A at long times, only MTs entering polar aligned domains determine τQ/2,A. Figure 5(c) shows that τQ/2,A decreases almost exponentially with pa and increases with ϕ. In stationary state, the time scales for the inversion of local polar order of initially antialigned and initially polar-aligned MTs are equal, τQ/2,A=τQ/2,P, compare Figure 5(b) and Figure 5—figure supplement 2. The dependence of ψi, ϕ and pa is shown in Figure 5—figure supplement 3.

Maximal activity

The mean squared displacements of MTs are ballistic, diffusive, or superdiffusive depending on the lag time, see Figure 3(d). This is reflected in the distributions of the parallel velocity v, see Equation 9 and Figure 3(e). The v distributions become increasingly asymmetric with increasing lag time when active propulsion dominates over Brownian motion for antialigned MTs–and again less asymmetric when the lag time is further increased and orientational memory is lost. Because of the high number of parallel MTs in our simulations the position of the main peak is at v=0 as expected for passive MTs. A skew of the distribution can then be understood as a superposition of a high peak of non-propelled polar-aligned MTs and a small peak that is shifted to positive values of v for antialigned MTs, see also Videos 2 and 4.

In Figure 6, we plot skews of v distributions (Figure 3(e)) as function of lag time for various pa values for ϕ=0.3. Both maximal skew and maximal lag time for that we detect finite skews increase with increasing ϕ, Figure 6—figure supplement 1. The lag time at which the skew of the v distribution is maximal is defined as the activity time τ*, where the ratio of the displacements due to active forces to the thermal displacements is largest. This activity time τ* falls into the regime where the MSD is most superdiffusive, see Figure 3(d). As shown in Figure 6(b), τ* exponentially decreases with increasing pa and increases with ϕ. Increasing motor concentration, and thus a higher amount of active forces in the system, is akin to exponentially shifting the activity time to shorter values.

Figure 6 with 3 supplements see all
MT parallel velocity distributions.

(a) Skew of parallel velocity (v) distribution computed as function of lag times for different pa for ϕ=0.3. The probability distributions that correspond to the maximal skew are shown in Figure 3—figure supplement 6 together with distributions for few other lag times. (b) Lag time at which maximal skew is observed in the v(τ) distribution (compare Figure 1). The ordinate is log-scaled to show that τ* is exponentially decreasing with pa.

https://doi.org/10.7554/eLife.39694.028

The proportion of aligned (passive) and antialigned (propelled) MTs depends strongly on the area fraction, where the number of antialigned MTs decreases with surface fraction ϕ. This corresponds to larger domain sizes and less interfaces with increasing ϕ, see Figure 3—figure supplement 2 and Figure 6—figure supplement 2. Further, we notice that for all surface fractions ϕ increasing pa widens the v(τ) distributions, see Figure 3—figure supplement 6. The widening of the distribution becomes less pronounced with increasing ϕ. The shifting of the negative part of the v distribution (v(τ)<0) to more negative values with increasing pa is because τ* decreases simultaneously.

Collective migration

The observables discussed so far characterise the motion of individual MTs. They only take collective effects into consideration indirectly, for example via the asymmetry of the v distribution for polar-aligned MTs. For a direct discussion of the time scale of collective effects, we return to Figure 4(b). For times around τ/τR=10, in the intermediate time regime, we observe a positive neighbour displacement correlation. This behaviour is altogether absent at low surface fractions, ϕ=0.2, but for larger ϕ the positive correlations increase with increasing ϕ. This suggests that neighbouring MTs in a particular stream (likely polar-aligned) travel in the same direction. These polar-aligned MTs will collectively migrate in the same direction because they are in a similarly antialigned environment, that is at the same interface with another domain. Correlations in their motion can only manifest at longer lag times, since at short lag times the correlation contribution will be dominated by fast-moving antialigned MTs. We denote the lag time for the maximum of Nd(τ), when collective migration occurs, τN,max.

In order to explicitly show that positive neighbour displacement correlations observed in the intermediate time regime are due to collective migration of similarly oriented MTs, we can predict the results of photobleaching or photoactivation experiments (Gao et al., 2015; Mitchison, 1989; Hush et al., 1994). Experimentally, in a photobleaching experiment a high-intensity laser beam can be used to inactivate fluorescent molecules in a circular region (Axelrod et al., 1976). The time evolution of the distribution of the light-inactivated regions gives clues about the underlying mechanisms which mediate this motion. Figure 7(a) illustrates that we expect little or no MT sliding to occur in a polar-aligned region, and the photobleached area maintains its shape. In Figure 7(b), however, the photobleached area is antialigned and we expect bundles of antialigned domains to slide away, causing the photobleached spot to separate into two elongated regions. In our simulations, we perform a similar measurement, where instead of inactivating regions to inhibit fluorescence, we selectively label MT beads within a certain region. We then track their locations for t=τN,max and investigate their displacements. MTs move in response to the effective motor potential and form streams. In Figure 7(c), we visualise a four-MT length radius circular area for ϕ=0.4 and pa=1.0. The structures become more diffuse for ϕ=0.3 and more compact for ϕ=0.5 (Figure 7—figure supplement 1).

Figure 7 with 1 supplement see all
Collective motion of MTs.

Schematic of expected evolution of photobleached regions in (a) polar-aligned and (b) antialigned regions. (c) Selectively visualised MTs in a circular region within the simulation box, and their evolution after a time of τN,max, for ϕ=0.4 and pa=1.0. The black backgrounds are predictions of FRAP results.

https://doi.org/10.7554/eLife.39694.033

Active rotation

The longest relevant time scale for the MT dynamics is that of active rotational motion characterised by the orientational correlation function

(15) 𝐩(t)𝐩(t+τ)=e-τ/τr.

By fitting Equation 15 to the simulation data (Figure 8(a)), we obtain the transition time to long-time active diffusive behaviour, τr. Figure 8(b) shows τr for various pa. For passive systems (pa=0), τr increases with MT surface density ϕ. For active systems, τr decreases with increasing pa and with decreasing ϕ. In the nematic state, MTs are no longer able to rotate freely as in the isotropic case. The decrease of τr with increasing pa is more pronounced at higher MT surface fractions. Smaller values of τr correspond to smaller domain sizes. In larger domains, the streams appear at interfaces between polar-ordered domains and antialigned MTs. The streams extend in the same direction over larger lengths, for longer times, and MTs do not rotate away from their initial orientation as quickly. Also, MTs that are trapped in aligned MT bundles are less likely to exit their environments and their rotational diffusion is smaller for higher ϕ and lower pa. Only for τ>τr the MTs show again diffusive motion with an active diffusion coefficient DAv||2τr, see Figure 3(d).

MT orientational correlation and active diffusion. 

(a) Orientational correlation function for ϕ=0.3 for various antialigned motor probabilities pa. (b) Inverse of rotational diffusion, τr for various antialigned motor probabilities pa and surface fractions ϕ (c) Active diffusion coefficient DA for pa=1.

https://doi.org/10.7554/eLife.39694.035

Discussion

In our two-dimensional simulation model, dipolar effective motor forces that drive antialigned MT pairs are sufficient to bring about MT streams which are perpetually created and annihilated, akin to MT streaming in biology. Processes that occur on several characteristic times characterise streaming in our MT-motor mixtures: the characteristic time τN,min corresponds to the strongest anti-aligned motion of neighbouring MTs, the time τQ/2 that an MT stays within a stream, the time τ* that corresponds to maximal skew of the MT velocity distribution, the collective migration time τN,max that characterises maximal directed active motion, and the active rotation time τr that corresponds to single rods traveling the distance of a polar-aligned domain when they loose their orientational memory. Figure 9 and Table 1 summarise our findings. (We do not show ϕ=0.2, because there is no evidence of streaming in these systems, and the chronology of events is not consistent with those observed for higher surface fractions.)

Chronology of MT streaming. Events from antialigned MT propulsion to MT rotation (left to right) which make up the streaming process, for various antialigned motor probabilities pa and surface fractions ϕ=0.3, ϕ=0.4, and (c) ϕ=0.5 as indicated.
https://doi.org/10.7554/eLife.39694.037
Table 1
Table of time scales involved in MT dynamics.

The time scales reported are approximate values for various antialigned motor probabilities pa and surface fractions ϕ.

https://doi.org/10.7554/eLife.39694.039
SymbolsTime scale (τR)Derivation
Passive
diffusion
τDτ<10-1Slope of MSD 1
Antialigned
propulsion
τN,minτ10-1Minimum of Nd
StreamingτQ/210-1<τ<100ψi decay time
Maximal
skew
τ*100<τ<101Maximum skew of 𝐩0𝐝
Collective
migration
τN,maxτ101Maximum of Nd
Active
rotation
τrτ>101Orientational correlation
time

All characteristic times increase with decreasing motor attachment probability pa. We expect the sliding time τN,min and the active rotation time τr to diverge with vanishing motor attachment probability, while the collective migration time τN,max and the polarity-inversion time τQ/2 attain finite values due to thermal motion and steric interaction between the MTs. The activity time τ* is not defined for passive systems. Overall, the characteristic times increase with increasing MT surface density, because the lifetime and the coherence of the streams increases. However, our simulations also reveal details of the multi-scale process of streaming in MT-motor mixtures. For example, the time for an MT to transition from a polar-aligned to an antialigned environment is similar to the sliding time for low motor forces, where thermal motion dominates, and to the activity time for high motor forces when the streams are more stable.

The closest ’bottom-up’ experimental system to our simulation model is the in vitro model system of microtubule bundles, kinesin complexes, and depletants at the oil-water interface, investigated in Sanchez et al. (2012). A detailed quantitative comparison is currently not possible because the characteristic length scales in simulations and experiment are quite different. The depletion-induced MT bundle formation in the experiments leads to a characteristic length scale of the order of 10μm, whereas the MTs in the simulations have lengths below 1μm. However, on a more qualitative level interesting correspondences are revealed. The transition between diffusive and ballistic MSDs in the simulations has also been reported for the in vitro model system (Sanchez et al., 2012). This allows the comparison of the active diffusive regime for times τ>τr and for lengths longer than a typical domain size. Whereas the motion in suspensions of passive MTs is diffusive, a ballistic regime at large lag times is found for increasing concentrations of active motors (simulations) and for increasing ATP concentration (experiment).

Some of our results can be used to interpret experimental results in vivo. For example, using Particle Image Velocimetry (PIV) in Drosophila cells, fluid velocity distributions have been measured for wild-type oocytes and those lacking pat1, a protein required for kinesin heavy chain to maximise its motility (Ganguly et al., 2012). The main peak is close to a velocity of 10 nm/s, which hints that the majority of the MTs are propelled. As in our simulations, heavy tails in the velocity distribution have been reported in the experiments. A compariston of the experimental data for wild-type and pat1-deficient systems showed that for the wild-type system the mean speed was slower and the velocity distribution had heavier tails. This qualitatively agrees with our findings for varying pa. It was suggested in Ganguly et al. (2012) that the heavy tails in the velocity distribution of the cytosol reflect a combination of an underlying distribution of motor speeds, and a complex MT network geometry. From our simulations, we conclude that neither a complex three-dimensional cytoskeletal geometry nor a combination of different motor speeds are required to reproduce cytoskeletal velocity distributions with heavy tails.

We have studied the characteristic times of MT-motor dynamics relevant for the cytoskeleton using a coarse-grained motor model and Langevin Dynamics simulations in the overdamped regime. This allows us to access both the single-MT level as well as the collective-MT level. In previous studies that use a similar coarse-graining technique for the motor activity, the focus has been on understanding and capturing biologically relevant cytoskeletal structures (Aranson and Tsimring, 2005; Jia et al., 2008). Here, for the first time, we have decomposed the time scales of activity from single MTs to system-scale ordering and streaming.

MT advection has also been analysed using photoconversion in interphase Drosophila S2 cells, where MTs were observed to buckle and loop (Jolly et al., 2010). MT motion was visualised by photoconverting a circular region within the cell. These MTs were observed over a 7 minute period, during which 36% of the MTs were determined to be motile. It was observed that MTs spent most of the time not moving, but underwent abrupt long-distance streaming. They were found to achieve velocities up to 13 μm/min, during these bursts of active motion. These observations are very similar to those in our simulations, where MTs spend most of their times in stable polar-aligned bundles, but when in contact with an antialigned MTs coherently stream over large distances. We find similar fractions of motile MTs between 30% and 40% also for ϕ=0.3 in our simulations. Our study provides the basis for a more detailed quantitative comparison with experiments because the model can be easily extended to include further relevant aspects, such as a 3D cytoskeletal network, crosslinking proteins, and cellular confinement.

Our simulations show collective migration of MTs that is maximal at τN,max. Using a FRAP-like visualisation of our data, we find elongated MT stream patterns similar to those observed in experiments (Jolly et al., 2010). This confirms that similarly oriented MTs move colletively in the same stream. Based on the polarity-sorting mechanism of MTs, qualitatively similar FRAP results have previously been predicted using computer simulations (Gao et al., 2015). Experimental studies of a system on various length and time scales should allow testing the chronology that we predict. For example, systems with different fractions of fluorescent MTs with fixed lengths should give access to both collective as well as single-MT dynamics, for example using FRAP/photoactivation for systems with many labeled MTs to quantify collective dynamics and confocal microscopy for systems with few labeled MTs to investigate correlations in single-MT motion.

We have studied collective motion in active gels based on single MTs. Our spatio-temporal displacement correlation functions show that antialigned MTs slide away from each other in opposite directions for short time windows, while in agreement with experiments positive correlations occur for long time windows (Ganguly et al., 2012). Our two-dimensional simulations resemble systems close to an interface that have been used to experimentally study hierarchically assembled active matter (Sanchez et al., 2012). They also lay the foundations for future studies of 3D systems and have allowed us to test parameter regimes using less computationally expensive, two-dimensional systems. Furthermore, although we observe streaming without hydrodynamics, hydrodynamic interactions may still be an important player for motor-MT systems, which can be investigated in future studies.

To summarize, our results provide a direct handle to fully characterise MT streaming over a wide range of time and length scales. Future experimental studies using modern microscopy techniques may allow testing our predictions. Future theoretical and computer simulation studies may provide further insights, such as the importance of the aspect ratio of the MTs, the presence of motors between polar-aligned MTs, and the effect of crosslinkers important for buckling and looping of flexible MTs.

Materials and methods

Langevin dynamics

We simulate the MT-motor systems in two dimensions using periodic boundary conditions. The motion of the beads is described by the Langevin equation,

(16) md2ridt2=Ui+Fmotγdridt+ξi(t),

where 𝐫i is the position of bead i, m is the mass of a bead, γ is the friction coefficient of the solvent for bead motion, Fmot=-Umot is the active motor force and ξi is the Gaussian-distributed thermal force. The friction coefficient can be estimated using the Stokes friction γ=6πηR for a spherical particle with radius R in a solvent with viscosity η. The thermal forces ξi have ξi=0 and, from the fluctuation-dissipation theorem,

(17) ξα(t)ξβ(t)=2γkBTδαβδ(tt),

where kB is the Boltzmann constant, T is the temperature, and ξα(t) is the α-th component of the vector ξi(t).

Langevin dynamics simulations allow the use of larger time steps compared with Brownian dynamics simulations without a particle mass. The friction constant γ and bead mass m are chosen such that the center-of-mass motion of passive MTs at the same density is diffusive at length scales larger than a fraction of the MT length and at time scales τ/τR0.01 (see SI), such that passive MTs only move ballistically at times shorter than the relevant times.

The simulation package LAMMPS has been employed to perform the simulations (Plimpton, 1995), see Source code file 1.

System parameters

Each simulation consists of nf=1250 semiflexible filaments with aspect ratio 10, each made up of nb=21 overlapping beads, which reduces the friction of the otherwise corrugated MTs (Abkenar et al., 2013; Isele-Holder et al., 2015). The MT surface fraction ϕ=nfLσ/Lb2 is controlled by adjusting the box size Lb. Our effective-motor model is a coarse-grained model and individual effective motors in the simulations may not represent individual motors in experiments. However, the system parameters are based on those of biological systems, see Table 2.

Table 2
Parameter values used in the simulations.
https://doi.org/10.7554/eLife.39694.040
ParameterSymbolValueNotes/Biological Values
Thermal energykBT4.11pNnmroom temperature
MT lengthL0.625μm2.5±1.4μm (Howard et al., 1989)
MT diameterσ25 nm(Chrétien and Wade, 1991)
MT bond angle constantκ2.055×104pNnm2rigid MTs
MT bond spring constantks13.15pN/nmpreserves MT length (Isele-Holder et al., 2015)
Dynamic viscosityη1 Pa sviscosity of cytoplasm (Wirtz, 2009)
Characteristic energyof WCA potentialϵ4.11 pN nm(Bates and Frenkel, 2000;
Bolhuis and Frenkel, 1997; McGrother et al., 1996)
Motor spring constantkm6.6×10-3 pN/nm0.33pN/nm per kinesin (Coppin et al., 1995), high number of effective motors
Equilibrium motor lengthdeq25 nmMT-MT distance at contact
Motor dwell timeδt4.16×10-4 s

The WCA potential is used with the interaction cutoff at 21/6σ, such that the potential between MTs is purely repulsive. The bond stiffness is large, such that the contour length of the MTs remains approximately constant throughout a simulation run. The angle potential is chosen such that MTs are rigid; the persistence length is p=200L. We use a time step of duration δt=5.31×10-6τR. Each run for a particular parameter set consists of 3 107 time steps.

We nondimensionalise the key parameters using the MT diameter σ or length L, thermal energy kBT, and the single-MT rotational diffusion time τR, see Table 3.

Table 3
Dimensionless parameters and ranges of the values used in the simulations.
https://doi.org/10.7554/eLife.39694.041
ParameterSymbolValue
MT surface fractionϕ0.2-0.5
MT aspect ratioL/σ10
Reduced MT bond angle stiffnessκσ/kBT200
Reduced MT persistence lengthp/L200
MT bond spring constantksσ2/kBT2000
Reduced motor spring constantkmσ2/kBT1
Reduced motor equilibrium lengthdeq/σ1
Antialigned motor probabilitypa0-1.0
Reduced single-bead frictionγ/(kmδt)171.6
Reduced system sizeLb/L16-25

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Decision letter

  1. Raymond E Goldstein
    Reviewing Editor; University of Cambridge, United Kingdom
  2. Anna Akhmanova
    Senior Editor; Utrecht University, Netherlands

In the interests of transparency, eLife includes the editorial decision letter and accompanying author responses. A lightly edited version of the letter sent to the authors after peer review is shown, indicating the most substantive concerns; minor comments are not usually included.

Thank you for sending your article entitled "Chronology of motor-mediated microtubule streaming" for peer review at eLife. Your article is being evaluated by three peer reviewers, one of whom is a member of our Board of Reviewing Editors, and the evaluation is being overseen by Anna Akhmanova as the Senior Editor.

Given the list of essential revisions, including new experiments, the editors and reviewers invite you to respond within the next two weeks with an action plan and timetable for the completion of the additional work. We plan to share your responses with the reviewers and then issue a binding recommendation.

Summary:

This article describes a computational study of a model for cytoplasmic streaming, motivated by the phenomenology found in Drosophila oocytes. The authors introduce a coarse-grained model in which details of the molecular motors' activity that can lead to sliding between adjacent microtubules are subsumed into an effective orientation-dependent potential. The microtubules are modelled as linked spheres, and the whole setup is described by a Langevin equation. The results include many aspects of the correlation functions between the microtubules, with different regimes characterized by different types of alignments. Probability distribution functions of speeds are computed, of the kind that can be measured experimentally. The authors make some contact with experimental studies of streaming using PIV and advance the hypothesis that heavy-tailed velocity distributions arise without the previously conjectured need for varying motor speeds or complex cytoskeletal geometry.

Essential revisions:

1) Whereas the mechanism postulated by authors does produces streaming in the in silico system studied here, it is not guaranteed that the same mechanism occurs inside the cell or in vitro. Streaming occurs in a vast range of systems, from plants upwards, and in many of these systems the filaments are both organized and unchanging in their conformations. For sure in others the situation is different, but the present manuscript appears to suggest that cytoplasmic streaming is always associated with interfilament sliding, whereas that is not the case. This needs to be clarified.

The manuscript contains various measurements and predictions, but it is rather unclear which of them would unambiguously demonstrate the postulated mechanism if recovered in experiments. This point is crucial and should come across the manuscript with no ambiguity. In other words, the authors should clearly explain which one of their predictions an experimentalist should recover in order to verify that the mechanism behind MTs streaming is indeed that proposed in this paper.

There is a significant literature on the streaming problem in Drosophila that has not been cited. Examples include Woodhouse, et al., 2013 and Khuc Trong, et al. 2015, which discuss in detail self-organization processes and the role of cytoskeletal architecture on streaming patterns. Of particular interest in the case of Drosophila is the nucleation of microtubules from the periphery of the oocyte, leading to anchoring there. This is not accounted for in the present paper.

2) The authors do not explain what promotes short-ranged polar alignment in the absence of directed motion and cross-linking. In other words, why the polar bundles form in the first place? Can the authors exclude that the interlocking of the beads forming an individual MT has nothing to do with it?

3) While reading the manuscript, it is very tempting to think about the active suspensions of MT bundles and kinesin pioneered in the Dogic Lab. The authors do refer to some of this work and, toward the end of the manuscript, explicitly say that the collective motion found here resembles that observed in those experiments. Yet, it is rather puzzling that they avoid making a direct comparison. If the present numerical approach could serve as a particle model of these active suspensions, this should be clearly said and motivated (with an eye to the rich theoretical literature around the topic). If not, it would be useful to know where are the fundamental differences and how both these model systems compare to cytoplasmatic streaming in vivo.

4) It is somewhat unclear how the model outlined in Sec. II. reconciles with the non-equilibrium nature of kinesin-based propulsion. MT-kinesin interactions are modeled through conservative forces, that can be expressed as derivatives of of potential energy U_mot. Furthermore, the motor binding rate follows the Boltzmann distribution. This raises the question of whether the authors are attempting to describe cytoskeletal activity as an equilibrium process. Most models of cytoskeletal fluids (both discrete and continuous) are based on the assumption that kinesin moves at constant speed from the minus to the plus end (or vice versa for specific types of kinesin). This manifestly violates detailed balance and is consistent with experimental observations (see e.g. Schnitzer and Block, Nature 1997). One can debate on whether kinesin is in fact delivering a constant power, as opposed to move at constant speed, but both scenarios appear to lie outside of the scope of the present model.

5) The reviewers raised questions about the degree of novelty of your methodology. While recognizing that the specific results in the paper concerning sliding motility are new, they pointed to recent work published by the Shelley group (Nazockdast et al., 2017) which has introduced a 3D computational framework that accounts for polymerization and depolymerization kinetics of fibers, their interactions with molecular motors and other objects, their flexibility, and hydrodynamic coupling. Their model has been applied in (Nazockdast et al., 2017).

The present authors should clearly compare and contrast their work with these recent papers.

6) Regarding the simulations, the reviewers are unclear why a Langevin equation with a mass term was used in what is clearly an overdamped problem. It is also unclear why it is possible a priori to neglect hydrodynamic interactions between filaments, and the significance of working in only two dimensions. All of

these issues need clarification.

Reviewer #1:

This article describes a computational study of a model for cytoplasmic streaming, motivated by the phenomenology found in Drosophila oocytes. The authors introduce a coarse-grained model in which details of the molecular motors' activity that can lead to sliding between adjacent MTs are subsumed into an effective orientation-dependent potential. The MTs are modelled as linked spheres, and the whole setup is described by a Langevin equation. The results include many aspects of the correlation functions between the microtubules, with different regimes characterized by different types of alignments. Probability distribution functions of speeds are computed, of the kind that can be measured experimentally. The authors make some contact with experimental studies of streaming using PIV and advance the hypothesis that heavy-tailed velocity distributions arise without the previously conjectured need for varying motor speeds or complex cytoskeletal geometry.

The subject matter of this paper is certainly appropriate for eLife, and as a computational study it is reasonably well done. Less clear to me is the significance of the results. In part this is due to what appears to be a superficial understanding of the literature on streaming. Streaming occurs in a vast range of systems, from plants upwards, and in many of these systems the filaments are both organized and unchanging in their conformations. For sure in others the situation is different, but the present manuscript appears to suggest that cytoplasmic streaming is always associated with interfilament sliding.

Second, there is a significant literature on the streaming problem in Drosophila that has not been cited. Examples include Woodhouse et al., 2013 and Khuc Trong, et al. 2015, which discuss in detail self-organization processes and the role of cytoskeletal architecture on streaming patterns. Of particular interest in the case of Drosophila is the nucleation of microtubules from the periphery of the oocyte, leading to anchoring there. This is not accounted for in the present paper.

Regarding the simulations, I am unclear why the authors would solve a Langevin equation with a mass term in what is clearly an overdamped problem. It is also unclear to me why it is possible a priori to neglect hydrodynamic interactions between filaments, and the significance of working in two dimensions.

Overall, I think the contributions of this paper are interesting, but the lack of proper biological context is a significant weakness.

Reviewer #2:

The manuscript by Ravichandran et al. introduces a computational framework for studying microtubule (MT) dynamics with focus on motor-driven sliding motility. MTs are modeled as spring chains with standard stretching and bending energy contributions, and steric MT-MT interactions are described by WCA repulsion. Motor activity is modeled by cross-linker springs between MTs that bind with exponential rates depending on relative orientation between motors and MT pairs. The model neglects hydrodynamics and does not account for MT nucleation and de/polymerization, although both effects could likely be added in future extensions of this framework. Simulations are restricted to 2D systems. Simulated MT numbers are O(1000) and the authors make a commendable effort to provide biologically relevant values for all model parameters.

The paper is clearly written and the numerical study has been performed carefully.

My main concern regarding suitability for publication in eLife is novelty.

Recent work published by the Shelley group, see Nazockdast et al., 2017, has introduced a 3D computational framework that accounts for polymerization and depolymerization kinetics of fibers, their interactions with molecular motors and other objects, their flexibility, and hydrodynamic coupling. Their model has been applied in Nazockdast et al., 2017.

In view of this previously published work, I believe that the present manuscript does not constitute the type of major conceptual or computational advance typically expected for publication in eLife. That said, it seems to me that the specific results in the paper concerning sliding motility are new and certainly deserve publication in some other form.

Reviewer #3:

Ravichandran and coworkers report a comprehensive computational study of microtubules (MT) streaming. This process, observed both in vivo and in vitro, is generally ascribed to the sliding motion promoted by kinesin molecules, but the microscopic mechanism behind the kinesin-mediated MT-MT interactions is still debated. Numerical simulations suggest that streaming results from the interaction between bundles of polar-aligned MTs and is particularly sensitive to the time scale associated with the reorientation of individual MTs. The paper appears technically sound, clearly written and nicely illustrated. Unfortunately, there are various points where the authors have been too vague.

1) Whereas the mechanism postulated by authors does produces streaming in the in silico system studied here, it is not guaranteed that the same mechanism occurs inside the cell or in vitro. The manuscript contains various measurements and predictions, but it is rather unclear which of them would unambiguously demonstrate the postulated mechanism if recovered in experiments. This point is crucial in my opinion and should come across the manuscript with no ambiguity. In other words, the authors should clearly explain which one of their predictions should an experimentalist recover in order to verify that the mechanism behind MTs streaming is indeed that proposed in this paper.

2) The authors do not explain what promotes short-ranged polar alignment in the absence of directed motion and cross-linking. In other words, why the polar bundles form in the first place? Can the authors exclude that the interlocking of the beads forming an individual MT has nothing to do with it?

3) While reading the manuscript, it is very tempting to think about the active suspensions of MT bundles and kinesin pioneered in the Lab of Zvonimir Dogic and now investigated by various other groups around the world. The authors do refer to some paper by the Dogic Lab and, toward the end of the manuscript, explicitly say that the collective motion found here resembles that observed in those experiments. Yet, they avoid making a direct comparison. I find this puzzling. If the present numerical approach could serve as a particle model of Dogic's active suspensions, this should be clearly said and motivated (with an eye to the rich theoretical literature around the topic). If not, it would be useful to know where are the fundamental differences and how both these model systems compare to cytoplasmatic streaming in vivo.

4) It is somewhat unclear how the model outlined in Sec. II. reconciles with the non-equilibrium nature of kinesin-based propulsion. MT-kinesin interactions are modeled through conservative forces, that can be expressed as derivatives of of potential energy U_mot. Furthermore, the motor binding rate follows the Boltzmann distribution. This raises question on whether the authors are attempting to describe cytoskeletal activity as an equilibrium process. Most of models of cytoskeletal fluids (both discrete and continuous) are based on the assumption that kinesin moves at constant speed from the minus to the plus end (or vice versa for specific types of kinesin). This manifestly violates detailed balance and is consistent with experimental observations (see e.g. Schnitzer and Block, Nature 1997). One can debate on whether kinesin is in fact delivering a constant power, as opposed to move at constant speed, but both scenarios appear to lie outside of the scope of the present model.

In summary, whereas the authors have been quite meticulous in calibrating the parameters to experimental values, it is unclear to me whether what they present are indeed properties of MTs and kinesin or simply properties of their model. Therefore, I am unable to recommend this paper for publication in eLife in the present form.

https://doi.org/10.7554/eLife.39694.046

Author response

[Editors' note: the authors’ plan for revisions was approved and the authors made a formal revised submission.]

Essential revisions:

1) Whereas the mechanism postulated by authors does produces streaming in the in silico system studied here, it is not guaranteed that the same mechanism occurs inside the cell or in vitro. Streaming occurs in a vast range of systems, from plants upwards, and in many of these systems the filaments are both organized and unchanging in their conformations. For sure in others the situation is different, but the present manuscript appears to suggest that cytoplasmic streaming is always associated with interfilament sliding, whereas that is not the case. This needs to be clarified.The manuscript contains various measurements and predictions, but it is rather unclear which of them would unambiguously demonstrate the postulated mechanism if recovered in experiments. This point is crucial and should come across the manuscript with no ambiguity. In other words, the authors should clearly explain which one of their predictions an experimentalist should recover in order to verify that the mechanism behind MTs streaming is indeed that proposed in this paper.There is a significant literature on the streaming problem in Drosophila that has not been cited. Examples include Woodhouse, et al., 2013, and Khuc Trong, et al. 2015, which discuss in detail self-organization processes and the role of cytoskeletal architecture on streaming patterns. Of particular interest in the case of Drosophila is the nucleation of microtubules from the periphery of the oocyte, leading to anchoring there. This is not accounted for in the present paper.

We acknowledge that cytoskeletal streaming has been reported across multiple biological systems and that various mechanisms can account for this motion. Our goal is to isolate and address anti-aligned filament sliding as a relevant mechanism of activity, and to study whether it alone could be responsible for streaming. Through our findings, we wish to emphasize that this mechanism can singly bring about some of the hallmarks of MT streaming observed in experiments; however, it is largely underappreciated in both biological and soft-matter literature so far. We thus propose that further theoretical and experimental studies of filament sliding and MT streaming are required to exactly determine their role in long-range MT organization and transport in various contexts (e.g. neurons, Drosophila oocytes). We certainly do not want to claim that filament sliding is only and always responsible for MT streaming.

We agree with the reviewers that this point is not clearly discussed in our manuscript. We now state this more clearly in the Introduction by discussing other “bottom-up” in vitro and in silico approaches and contrasting them with “top-down” approaches. For the latter, we discuss other streaming mechanisms, such as the cargo transport reported for Chara corallina by Woodhouse et al., 2013, and nucleation of MTs from the periphery reported for Drosophila oocytes by Khuc Trong et al. 2015.

An important point is that we focus in our study on the motion on several scales far from boundaries. We have extended a paragraph in the Discussion section on how our model can be tested by experimentalists. We think here mainly about fluorescence experiments that compare both, collective and single-filament dynamics. Addition of small fractions of fluorescent filaments should allow the analysis of single-filament motion (Levy flight-like motion, velocity distributions), while photobleaching or photoactivation experiments can reveal collective motion.

2) The authors do not explain what promotes short-ranged polar alignment in the absence of directed motion and cross-linking. In other words, why the polar bundles form in the first place? Can the authors exclude that the interlocking of the beads forming an individual MT has nothing to do with it?

We believe that this is a misunderstanding regarding the non-equilibrium nature of our simulations, see also point 4) of the essential revisions. In absence of directed motion (pa=0, thermal equilibrium) polar alignment cannot be obtained in our simulations. For suspensions of passive filaments, neighboring filaments do not experience sliding forces and therefore nematic and polar configurations are indistinguishable. For suspensions of active filaments, a filament in an oppositely-oriented region is propelled and transported after several simulation steps to an aligned region, in which case the active propulsion ceases.

Independent of polar alignment, potential interlocking of rough filaments is indeed an important issue to be considered carefully in the simulations. In previous work reported by Abkenar et al., 2013, Isele-Holder et al., 2015, Abaurrea Velasco et al. (Soft Matter 13, 5865 (2017)), and Duman et al., 2018, we have studied the effect of filament discretization on the dynamics of active systems. We found that by using overlapping beads within filaments, where the bond-length is half the cut-off of the excluded volume potential, the friction is strongly reduced. In particular the filament model in Isele-Holder et al. is exactly the same as the model used in this manuscript, which we now state in section Materials and methods, System parameters.

3) While reading the manuscript, it is very tempting to think about the active suspensions of MT bundles and kinesin pioneered in the Dogic Lab. The authors do refer to some of this work and, toward the end of the manuscript, explicitly say that the collective motion found here resembles that observed in those experiments. Yet, it is rather puzzling that they avoid making a direct comparison. If the present numerical approach could serve as a particle model of these active suspensions, this should be clearly said and motivated (with an eye to the rich theoretical literature around the topic). If not, it would be useful to know where are the fundamental differences and how both these model systems compare to cytoplasmatic streaming in vivo.

We thank the reviewers for raising the issue of the connection between our work and the active-suspension model systems consisting of MT bundles and kinesin motors pioneered in the Dogic lab. The work of Dogic et al. has indeed been a very important motivation for us, and we completely agree that a more detailed comparison is desirable. Dogic’s studies of 2D model systems -- with MTs and kinesins at an oil-water interface – do not mimic or reproduce any specific biological cell in vivo but instead aim at a general understanding of self-organization and dynamics of MTs by the activity generated by molecular motors.

In the previous version of the manuscript, we referred to Dogic‘s active suspensions only for very specific results in the Introduction section. In the Discussion section, we referred to the hierarchical nature of the 2D systems. The main difference between both studies is a mismatch in system size between the experimental model system and our theoretical / numerical model. While the single-filament resolution of our simulations is not achieved in the experiments, the simulations deal with much smaller systems than the experiments. Furthermore, the two model systems are different regarding the filament aspect ratio (6:1) and the filament length to system size ratio. Filament and motor concentrations at the interface are unknown in the experiments and the filaments bundle. Therefore, a comparison of results for the two systems is at this stage only possible on a qualitative level.

We have added a new paragraph on in vitro and in silico approaches to the Introduction, where we introduce the active suspensions pioneered in the Dogic lab as well as computer simulations and where we highlight the single-filament vs. bundle nature. We contrast these “bottom-up” model systems with “top-down” studies for cytoplasmic streaming in vivoin the following paragraph. See also our reply to point 1).

We have also added a new paragraph to the Discussion section and Figure 3—figure supplement 10 that compares our results with Dogic’s results. Here, we focus on mean squared displacements.

In Figure 3—figure supplement 10, we compare the MSDs from our simulations to those measured experimentally by Dogic et al., in 2012. The MSDs have been measured using micron sized tracer particles immersed in the active nematic in Dogic’s work and using individual filaments in our work. The different system parameters make it difficult to find a common normalization of the lag time as well as of the MSDs. Following the concept of our manuscript, we have normalized time by the single-filament rotation time and the MSD by the filament length squared. Despite the difficulties to obtain a quantitative comparison, the qualitative features of the MSDs are very similar. For passive systems, the MSDs represent diffusive motion. When the motor probability pa in our model or the ATP concentration in the experiments is increased, the filaments become more active and the active velocity leads in both cases to an active ballistic motion. Our simulations, however, show in addition also very clearly the active diffusive regime at long times (the plateau). Here the filaments lose their orientational correlation. In contrast, in Dogic’s systems the filaments move mostly on straight trajectories for all studied lag times. The (active) rotational diffusion in our simulations is enhanced because of a lower filament density and a smaller MT aspect ratio compared with the experiments. The shorter active rotation time is also the reason why the active ballistic regime (MSD~(vt)2) is not fully developed in our simulations, but affected by crossover regimes to diffusive motion.

4) It is somewhat unclear how the model outlined in Sec. II. reconciles with the non-equilibrium nature of kinesin-based propulsion. MT-kinesin interactions are modeled through conservative forces, that can be expressed as derivatives of of potential energy U_mot. Furthermore, the motor binding rate follows the Boltzmann distribution. This raises the question of whether the authors are attempting to describe cytoskeletal activity as an equilibrium process. Most models of cytoskeletal fluids (both discrete and continuous) are based on the assumption that kinesin moves at constant speed from the minus to the plus end (or vice versa for specific types of kinesin). This manifestly violates detailed balance and is consistent with experimental observations (see e.g. Schnitzer and Block, Nature 1997). One can debate on whether kinesin is in fact delivering a constant power, as opposed to move at constant speed, but both scenarios appear to lie outside of the scope of the present model.

We thank the reviewers for raising the issue that the current presentation of the model may be confusing regarding the implementation of the active motor force. The nonequilibrium activity is a key aspect of our model, and it fully captures the non-equilibrium nature of the MT-kinesin systems. The activity enters via a periodic switching on and off of harmonic potentials that are generated by the extension of the molecular motors – very similar to a ratchet model. Here, the requirement of acute angles between the stalks of the motors and the polar filaments generates net propulsion forces. To better highlight the non-equilibrium nature, we have added a new paragraph on the nonequilibrium nature of the system to the general part of the Introduction. In addition, we have added a short statement before Equation 6, which emphasizes that because of the temporary and orientation-dependent “motor bonds” the system is inherently out of equilibrium.

5) The reviewers raised questions about the degree of novelty of your methodology. While recognizing that the specific results in the paper concerning sliding motility are new, they pointed to recent work published by the Shelley group (Nazockdast, et al., 2017) which has introduced a 3D computational framework that accounts for polymerization and depolymerization kinetics of fibers, their interactions with molecular motors and other objects, their flexibility, and hydrodynamic coupling. Their model has been applied in Nazockdast, et al., 2017. The present authors should clearly compare and contrast their work with these recent papers.

The model of the Shelley group features several important aspects of cytoskeletal systems including filament flexibility, filament polymerization and depolymerization, and hydrodynamic interactions. Steric interactions between filaments are not included. Consequently, the authors have studied a system where steric interactions are not expected to play a major role, the mitotic positioning where the microtubules are oriented radially. Our model features filament flexibility, an effective-motor potential between neighboring antiparallel filaments that coarse-grains walking of the motors, and steric interactions between the filaments. Hydrodynamic interactions are not included. We study dense systems, where the screening of hydrodynamic interactions may decrease their importance and at the same time steric interactions are essential. Both simulation frameworks use parallelized codes to allow for investigation of large systems. Our code builds on the freely available LAMMPS simulation package. Together with the source code and the input file provided with this submission, the simulations can easily be implemented by other researchers.

We have added a new paragraph to the section Introduction, Coarse-grained model, where we compare and contrast our work with the model the Shelley group (New York) and the Cytosim simulation package of the Nedelec group (Heidelberg).

In particular, we also highlight that our filament-based simulations are dissimilar from calculations and simulations that strive to capture microscopic, biological details. We coarse-grain individual motor interactions into an anti-aligned motor potential, and we show that our model still captures hallmarks of MT streaming. A novel aspect is that we demonstrate that microscopic details of motor interaction might be unnecessary when trying to capture long length- and time-scale phenomena. Moreover, we distill processes occurring in various time scales into observables which can be identified and verified through other simulations and experiments.

6) Regarding the simulations, the reviewers are unclear why a Langevin equation with a mass term was used in what is clearly an overdamped problem. It is also unclear why it is possible a priori to neglect hydrodynamic interactions between filaments, and the significance of working in only two dimensions. All of these issues need clarification.

The referees are correct that the dynamics in the experimental system is overdamped. We use Langevin Dynamics (LD) simulations with a mass term instead of a Brownian Dynamics (BD) simulation because of the numerical stability of the Langevin approach. When the mass term is included, up to 100 times longer time steps can be used in the simulations. However, we operate the code with parameters that ensure that the systems are in the overdamped regime for the experimentally relevant time scales.

Author response image 1
Mean squared displacements of filaments obtained from Langevin Dynamics (LD) and Brownian Dynamics (BD), for single filaments (left) and filaments in suspensions with packing fraction 𝜙 = 0.3 (right).

The simulation parameters are the same as in the manuscript.

https://doi.org/10.7554/eLife.39694.045

Author response image 1 shows a comparison of LD and BD simulations for (a) single filaments and (b) dense systems. The results demonstrate that our choice of parameters is justified because the crossover time from (inertial) ballistic to diffusive motion is at much shorter times than the time for a filament to diffuse over its own length. We have added a short paragraph to the section Materials and methods, Langevin Dynamics.

Hydrodynamic interactions between filaments may very well play an important role for cytoplasmic streaming. Nevertheless, there are several reasons why it makes sense to study a model in which hydrodynamic interactions are not taken into account:

1) Hydrodynamic interactions are long-ranged, and therefore induce coupling of motion not only of neighboring but also of distant filaments. Furthermore, they require numerically a significantly larger effort, as well as large system sizes. Simulations that include hydrodynamic interactions are therefore usually restricted to smaller number of particles.

2) In order to elucidate mechanisms, it is often helpful to start from simple systems, and to add additional features subsequently.

3) Boundary effect can strongly affect hydrodynamic flows, and hydrodynamic interactions are screened near surfaces and membranes.

4) In fact, our model for the filament-motor systems can be combined rather easily with particle-based mesoscale hydrodynamic approaches, which have been developed in recent years, and which are ideally suited for this purpose. Two examples for previous studies of filament hydrodynamics in a different context are:

"Semidilute polymer solutions at equilibrium and under shear flow", C.-C. Huang, R.G. Winkler, G. Sutmann, and G. Gompper, Macromolecules 43, 10107 (2010);

“Migration of semiflexible polymers in microcapillary flow", R. Chelakkot, R.G. Winkler, and G. Gompper, EPL 91, 14001 (2010).

5) It will certainly be very interesting to investigate the effect of hydrodynamic interactions in our model, which we plan to do in the near future. However, this goes far beyond the current study.

We have studied 2D systems motivated by Dogic’s model systems for active suspensions. For these experimental model systems, the dynamics of the entire systems, i.e., of all filaments, can be followed using light microscopy. Furthermore, the 2D geometry might be relevant for filament-motor systems close to boundaries, such as for cytoskeletal filaments next to plasma membranes of cells. We now motivate the reduced dimensionality in the paragraph on in vitro and in silico models in the Introduction, where we highlight that such less complex bottom-up studies are especially suited to study specific mechanisms in more detail.

https://doi.org/10.7554/eLife.39694.047

Article and author information

Author details

  1. Arvind Ravichandran

    Theoretical Soft Matter and Biophysics, Institute of Complex Systems and Institute for Advanced Simulation, Forschungszentrum Jülich, Jülich, Germany
    Contribution
    Investigation, Visualization, Methodology, Writing—original draft, Writing—review and editing, Code development-simulation, Code development-data analysis, Code development-visualization
    Competing interests
    No competing interests declared
    ORCID icon "This ORCID iD identifies the author of this article:" 0000-0001-6247-5229
  2. Özer Duman

    Theoretical Soft Matter and Biophysics, Institute of Complex Systems and Institute for Advanced Simulation, Forschungszentrum Jülich, Jülich, Germany
    Contribution
    Investigation, Methodology, Code development-data analysis, Code development-visualization
    Competing interests
    No competing interests declared
  3. Masoud Hoore

    Theoretical Soft Matter and Biophysics, Institute of Complex Systems and Institute for Advanced Simulation, Forschungszentrum Jülich, Jülich, Germany
    Contribution
    Methodology, Writing—original draft, Code development-simulation
    Competing interests
    No competing interests declared
    ORCID icon "This ORCID iD identifies the author of this article:" 0000-0003-1442-4739
  4. Guglielmo Saggiorato

    Theoretical Soft Matter and Biophysics, Institute of Complex Systems and Institute for Advanced Simulation, Forschungszentrum Jülich, Jülich, Germany
    Present address
    LPTMS, CNRS, Université Paris-Sud, Université Paris-Saclay, Orsay, France
    Contribution
    Conceptualization, Methodology
    Competing interests
    No competing interests declared
  5. Gerard A Vliegenthart

    Theoretical Soft Matter and Biophysics, Institute of Complex Systems and Institute for Advanced Simulation, Forschungszentrum Jülich, Jülich, Germany
    Contribution
    Conceptualization, Investigation, Methodology, Writing—original draft, Writing—review and editing
    For correspondence
    g.vliegenthart@fz-juelich.de
    Competing interests
    No competing interests declared
    ORCID icon "This ORCID iD identifies the author of this article:" 0000-0003-2459-8652
  6. Thorsten Auth

    Theoretical Soft Matter and Biophysics, Institute of Complex Systems and Institute for Advanced Simulation, Forschungszentrum Jülich, Jülich, Germany
    Contribution
    Conceptualization, Investigation, Methodology, Writing—original draft, Writing—review and editing
    For correspondence
    t.auth@fz-juelich.de
    Competing interests
    No competing interests declared
    ORCID icon "This ORCID iD identifies the author of this article:" 0000-0002-6618-2316
  7. Gerhard Gompper

    Theoretical Soft Matter and Biophysics, Institute of Complex Systems and Institute for Advanced Simulation, Forschungszentrum Jülich, Jülich, Germany
    Contribution
    Conceptualization, Resources, Investigation, Methodology, Writing—review and editing
    For correspondence
    g.gompper@fz-juelich.de
    Competing interests
    No competing interests declared
    ORCID icon "This ORCID iD identifies the author of this article:" 0000-0002-8904-0986

Funding

International Helmholtz Research School of Biophysics and Soft Matter (Graduate Student Fellowship)

  • Özer Duman
  • Guglielmo Saggiorato

The authors declare that there was no funding for this work.

Acknowledgements

OD and GS acknowledge support by the International Helmholtz Research School of Biophysics and Soft Matter (IHRS BioSoft). CPU time allowance from the Jülich Supercomputing Centre (JSC) is gratefully acknowledged.

Senior Editor

  1. Anna Akhmanova, Utrecht University, Netherlands

Reviewing Editor

  1. Raymond E Goldstein, University of Cambridge, United Kingdom

Publication history

  1. Received: June 29, 2018
  2. Accepted: December 28, 2018
  3. Accepted Manuscript published: January 2, 2019 (version 1)
  4. Version of Record published: January 18, 2019 (version 2)

Copyright

© 2019, Ravichandran et al.

This article is distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use and redistribution provided that the original author and source are credited.

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