Bidirectional motion of filaments: Role of motor proteins and passive cross linkers

TL;DR

This study models the interplay between motor proteins and passive cross linkers in microtubule overlaps, revealing dynamic phase transitions and oscillatory behaviors relevant to cell division stability.

Contribution

The paper introduces a novel model analyzing how active motor proteins and passive cross linkers jointly influence microtubule overlap dynamics, including phase transitions and oscillations.

Findings

Dynamic phase transitions with changing activity levels.

Emergence of stable limit cycle oscillations via Hopf bifurcation.

Overlap stability depends on whether microtubules are movable or fixed.

Abstract

In eukaryotic cells, motor proteins (MP) bind to cytoskeletal filaments and move along them in a directed manner generating active stresses. During cell division a spindle structure of overlapping antiparallel microtubules (MT) form whose stability and dynamics under the influence of MPs has been studied extensively. Although passive cross linkers (PCL) were known to provide structural stability to filamentous network, consequences of the interplay between ATP dependent active forces of MPs and passive entropic forces of PCLs on MT overlap remains largely unexplored. Here, we formulate and characterize a model to study this, using linear stability analysis and numerical integration. In presence of PCLs, we find dynamic phase transitions with changing activity exhibiting regimes of stable partial overlap with or without oscillations, instability towards complete overlap, and stable limit cycle oscillations that emerge via a supercritical Hopf bifurcation characterized by an oscillation frequency determined by the MP and PCL parameters. We show that the overlap dynamics and stability depend crucially on whether both the MTs of overlapping pair are movable or one is immobilized, having potential implications for in vivo and in vitro studies.

Figures

Tables

Table 1. Table 1: Parameters: Two values of v 0 subscript 𝑣 0 v_{0} and f d subscript 𝑓 𝑑 f_{d} correspond to ATP concentrations of 5 ​ μ 5 𝜇 5\,\mu M and 2 mM respectively.

active velocity	$v_0$	0.03, 0.8$\mu$m/s Schnitzer2000 ; Block2003
stall force	$f_s$	7.5 pN Carter2005 ; Block2003
back velocity	$v_{back}$	0.02$\mu$m/s Carter2005
detachment force	$f_d$	1.8, 2.4 pN
		(Appendix-B)
attachment rate	${\omega }_a$	20/s Schnitzer2000 ; Chaudhuri2016
detachment rate	${\omega }_d$	1/s Block2003
motor stiffness	$k_m$	1.7 pN/nm comment1
ambient viscous friction	$\gamma$	893 $k_B$T-s/$\mu$m2 Lansky2015
friction parameter	$\zeta$	0.22 (dimensionless)

Equations

$$\frac{d{n}_m}{dt}={\omega }_a(N-n_m)-{\omega }_d{n}_m\exp \left[\frac{|f_l|}{f_d}\right],$$

$$v_{m}(f_{l})=\left\{\begin{array}[]{ll}v_{0}&\mbox{for}\;\;\;f_{l}\leq 0\\ v_{0}\left(1-\frac{f_{l}}{f_{s}}\right)&\mbox{for}\;\;\;0<f_{l}\leq f_{b},\,f_{b}>f_{s}\\ -v_{back}&\mbox{for}\;\;\;f_{l}>f_{b},\\ \end{array}\right.$$

$$\frac{d{y}^i}{dt}=v_m(k_m{y}^i)+\frac{dx}{dt}.$$

$${\gamma }_f\frac{dx}{dt}=-\sum _{i=1}^{n_m}{k}_m{y}^i+F_p(n_p),$$

$$\frac{d\tilde{x}}{d\tau }$$

$$\frac{d\tilde{y}}{d\tau }$$

$$\frac{d{\tilde{n}}_m}{d\tau }$$

$$\tilde{v}_{m}=\left\{\begin{array}[]{ll}\tilde{v}_{0}&\mbox{for}\;\;\;\tilde{y}\leq 0\\ \tilde{v}_{0}\left(1-\frac{\tilde{k}_{m}\tilde{y}}{\tilde{f}_{s}}\right)&\mbox{for}\;\;\;0<\tilde{k}_{m}\tilde{y}\leq\tilde{f}_{b}\\ -\tilde{v}_{back}&\mbox{otherwise},\end{array}\right.$$

$${\tilde{y}}_0$$

$${\tilde{n}}_m^0$$

$${\tilde{x}}_0$$

$$a_{11}=-(N{\tilde{n}}_m^0{\tilde{f}}_s{)}^2/n_p\exp (\zeta n_p),a_{12}=-{\tilde{k}}_{m}N{\tilde{n}}_m^0/\exp (\zeta n_p),$$

$$a_{13}=-{\tilde{f}}_{s}N/\exp (\zeta n_p),a_{21}=a_{11},$$

$$a_{22}=-\left(\frac{{\tilde{v}}_0{\tilde{k}}_m}{{\tilde{f}}_s}+\frac{{\tilde{k}}_{m}N{\tilde{n}}_m^0}{\exp (\zeta n_p)}\right),a_{23}=a_{13},a_{31}=0,$$

$$a_{32}=-\frac{{\tilde{k}}_m{\tilde{n}}_m^0}{{\tilde{f}}_d}\exp \left[{\tilde{f}}_s/{\tilde{f}}_d\right],a_{33}=-\left(\tilde{\omega }+\exp \left[{\tilde{f}}_s/{\tilde{f}}_d\right]\right).$$

$$F(\lambda )={\lambda }^3+A{\lambda }^2+B\lambda +C$$

$$A$$

$$B$$

$$C$$

$$C=\bigg{[}\frac{1}{3}A+\frac{2}{3}\sqrt{A^{2}-3B}\bigg{]}\bigg{[}-\frac{1}{3}A+\frac{1}{3}\sqrt{A^{2}-3B}\bigg{]}^{2},$$

$$d_{\tau }{\tilde{x}}_1$$

$$d_{\tau }{\tilde{y}}_1$$

$$d_{\tau }{\tilde{n}}_{m_1}$$

$$d_{\tau }{\tilde{x}}_2$$

$$d_{\tau }{\tilde{y}}_2$$

$$d_{\tau }{\tilde{n}}_{m_2}$$

$$d_{\tau }\left(\begin{array}{c}\delta {\tilde{x}}_1\\ \delta {\tilde{y}}_1\\ \delta {\tilde{n}}_{m_1}\\ \delta {\tilde{x}}_2\\ \delta {\tilde{y}}_2\\ \delta {\tilde{n}}_{m_2}\end{array}\right)=\left(\begin{array}{cccccc}{b}_{11}& b_{12}& b_{13}& -b_{11}& 0& 0\\ b_{11}& b_{22}& b_{13}& -b_{11}& 0& 0\\ 0& b_{32}& b_{33}& 0& 0& 0\\ -b_{11}& 0& 0& b_{11}& b_{12}& -b_{13}\\ -b_{11}& 0& 0& b_{11}& b_{22}& -b_{13}\\ 0& 0& 0& 0& -b_{32}& b_{33}\end{array}\right)\left(\begin{array}{c}\delta {\tilde{x}}_1\\ \delta {\tilde{y}}_1\\ \delta {\tilde{n}}_{m_1}\\ \delta {\tilde{x}}_2\\ \delta {\tilde{y}}_2\\ \delta {\tilde{n}}_{m_2}\end{array}\right)$$

$$b_{11}=-(N{\tilde{n}}_{m_1}^0{\tilde{f}}_s{)}^2/n_p\exp (\zeta n_p),b_{12}=-N{\tilde{n}}_{m_1}^0{\tilde{k}}_m/\exp (\zeta n_p),$$

$$b_{13}=-\frac{N{\tilde{f}}_s}{\exp (\zeta n_p)},a_{22}=-\left[\frac{{\tilde{v}}_o{\tilde{k}}_m}{{\tilde{f}}_s}+\frac{N{\tilde{n}}_{m_1}^0{\tilde{k}}_m}{\exp (\zeta n_p)}\right],$$

$$b_{32}=-{\tilde{n}}_{m_1}^0{\tilde{k}}_m/{\tilde{f}}_d\exp ({\tilde{f}}_s/{\tilde{f}}_d),b_{33}=-\tilde{\omega }-\exp ({\tilde{f}}_s/{\tilde{f}}_d).$$

$$d_{\tau }\left(\begin{array}{c}\delta x_r\\ \delta \bar{y}\\ \delta {\bar{n}}_m\end{array}\right)=\left(\begin{array}{ccc}2b_{11}& b_{12}& b_{13}\\ 2b_{11}& b_{22}& b_{13}\\ 0& b_{32}& b_{33}\end{array}\right)\left(\begin{array}{c}\delta x_r\\ \delta \bar{y}\\ \delta {\bar{n}}_m\end{array}\right).$$

$$d_{\tau }\left(\begin{array}{c}\delta y_r\\ \delta n_m^r\end{array}\right)=\left(\begin{array}{cc}{b}_{22}& b_{13}\\ b_{32}& b_{33}\end{array}\right)\left(\begin{array}{c}\delta y_r\\ \delta n_m^r\end{array}\right).$$

$$d_{\tau }\bar{x}=b_{12}{y}_r+b_{13}{n}_m^r.$$

$${\lambda }^3+A^{\prime }{\lambda }^2+B^{\prime }\lambda +C^{\prime }$$

$$A^{\prime }$$

$$B^{\prime }$$

$$C^{\prime }$$

$$A$$

$$B$$

$$C$$

Full text

I Introduction

The cytoskeleton in eukaryotic cells is a complex network of interlinking filaments associated with motor proteins (MP) and passive cross linkers (PCL). It plays a central role in controlling the cell shape and motility apart from being also involved in intra-cellular transport. For example, cytoskeleton controls mechanical processes underlying the essential mechanism of cell division. In fact, the process of chromosome segregation to daughter cells in eukaryotes is achieved by the formation of spindle structure comprising of overlapping microtubules (MTs) Alberts2007 ; Wittmann2001 ; Walczak2008 . In a spindle, interpolar MTs from the two microtubule originating centers (MTOC) grow antiparallel to each other to interdigitate at the cell midzone. On the other hand, astral MTs grow towards the cell membrane, where active force generators like Dynein capture these MTs and pull them towards the cell cortex Grill2003 . The Kinesin-8 family of motor proteins (MPs) present at the interdigitating midzone control MT length Fukuda2014 , and generate sliding force reducing MT overlap Grill2003 ; Grill2005 ; Pecreaux2006 ; Su2013 . The deatails of the resultant dynamics has been thoroughly studied during the last decade Gross2002 ; Kural2005 ; Levi2006 ; Leduc2010 ; Pavin2012 ; Chaudhuri2016 ; McLaughlin2016 .

A large body of experimental and theoretical work focussed on active MP-filament systems Holzbaur2010 , e.g., actin- myosin complex that forms the cell cortex Tolic-Norrelykke2004 ; Brugues2012 , and MP driven active self assembly of filamentous proteins Lam2014 ; Lam2016 . Theoretical and experimental studies of spindle dynamics, and spindle oscillation, mainly focused on two aspects – active force generation due to MPs and dynamic instability of the microtubules Grill2005 ; Muhuri2011 ; Malgaretti2016 ; Ma2014 ; Pavin2012 ; Leduc2010 ; Scharrel2014 . These studies have clarified the subtle and rich interplay between MTs and MPs in the emergence and stability of the cytoskeleton, as well as qualitative differences of active complexes with respect to their equilibrium counterparts. Although, presence of PCLs in the cytoskeleton has been known for a long time, relatively less attention has been paid to their role in determining the various structural properties of cytoskeletal network. Only recently it has become clear that they may also play important role in the spindle formation during mitosis. For example, it has been shown that PCLs can increase sliding viscous friction between overlapping MTs Tawada1991 , and provide stability to the overlap Braun2011 . In fact, it has been predicted that PCLs can stabilize the overlap of antiparallel MTs in the mitotic spindle, competing against MP pull Johann2015 ; Johann2016 . Recent studies on in vitro motility assays have identified the entropic force induced by PCLs as a relevant physical mechanism contributing to the stability of overlapping MT parallel arrays Walcott2010 ; Lansky2015 . Nonetheless, the dynamic interplay between PCLs that promote MT overlap and MPs that pull them away from each other, and their impact in the self-assembly and stability of MP and PCL complexes remain largely unexplored.

In this paper, we present a comprehensive study of the non-equilibrium phases that develop as a result of the competition between the active forces generated by MPs and the passive forces due to PCLs. We consider a simplified geometry, where two anti-parallel, length-stabilized MTs are pushed inwards to larger overlap by PCLs, while MPs pull them apart. This geometry, reminiscent of the MT arrangement in the mitotic spindle, allows us to identify the fundamental principles that determine the impact that the dynamic interaction between MPs and PCLs has on the MT morphology. Specifically, we will focus on two complementary arrangements and will compare the behavior of two movable MTs with the situation in which one of the two MTs is surface immobilized. Our study shows the existence of a variety of non-equilibrium phases, comprising of stable and unstable overlapping arrays, as well as the emergence of persistent characteristic oscillations. The development of such oscillations can be understood in terms of a stable limit cycle arising via a supercritical Hopf bifurcation. The general framework identifies the relevant parameters that compete and control the relative stability of these different phases. For example, we show that the arrangements can be stabilized by tuning the MP activity, which itself depends on the ATP concentration in the medium. Further, the stabilization is strongly dependent on whether one partner MT is immobilized, which could emerge in the cell via formation of strong bonds to the cytoskeletal matrix. The theory also provides an illustration of how the cell can control its dynamic arrangements by regulating the ATP concentration and the amount of passive cross linking. Our detailed predictions may be verified in proposed in vitro experiments, while many other qualitative aspects may be tested across in vivo studies.

II Model

We consider two simple geometric arrangements comprising of a pair of antiparallel, overlapping MTs in presence of MPs and PCLs. In the first arrangement, we consider a situation in which one of the two overlapping MTs is made immobilized by fixing it to a substrate. The other MT can move freely under the entropic force of PCLs and active force due to MPs, one end of which are attached irreversibly to the substrate while the other end move unidirectionally on the MT as depicted in Fig.1. In the second arrangement, we take up a situation where both the overlapping MTs are free to move and are subjected to active and passive forces due to MPs and PCLs, respectively, as depicted in Fig.5. The active forces exerted by MPs are controlled and characterized by the ambient ATP concentration, and the concentration of MPs, $\phi_{MP}$. The stability of overlapping morphology is determined by the MP dynamics and its competition with the entropic forces exerted by $n_{p}$ number of PCLs diffusing within the overlap region.

Building on well-established approaches to the theoretical study of filament- motor protein complexes Grill2005 , MPs are modeled as active stretchable springs undergoing active attachment- detachment dynamics, governed by energy consumption via ATP hydrolysis, and do not obey detailed balance. On an average, the maximum number of motor proteins that can attach to a MT is $N=L\,\phi_{MP}$, where $L$ denotes the length of the MT. Within mean field approximation, the kinetics of $n_{m}$ attached MPs can be expressed as

[TABLE]

where $\omega_{a}$ and $\omega_{d}$ denote the attachment and bare detachment rates respectively. The actual detachment rate of MPs increases exponentially with the load they are subjected to, $|f_{l}|$, irrespective of whether the load is extensile or compressive, and the detachment force $f_{d}$ sets its force scale. Modeling the MP as a linear spring of spring constant $k_{m}$, its extension $y^{i}$ leads to the load force felt by it $f_{l}=k_{m}y^{i}$.

The ATP dependent activity allows attached MPs to walk along a filament, with a speed that depends on the force they are subjected to, $v_{m}(f_{l})$, toward one of the filament ends. According to the experimental evidence for a variety of MPs, including kinesin Carter2005 , we consider a piecewise linear force-velocity relation

[TABLE]

where, $f_{s}$ denotes the stall force and $v_{0}$ stands for the intrinsic MP velocity. For a load force beyond stall, $f_{l}\geq f_{b}>f_{s}$, the velocity saturates to an extremely small negative value $v_{back}=v_{0}\,(f_{b}-f_{s})/f_{s}$ Leduc2010 ; Carter2005 , while supportive loads do not affect the intrinsic MP motion. All the parameters $v_{0}$, $\omega_{a}$, $\omega_{d}$, $f_{s}$, and $f_{d}$ characterizing MPs are potentially functions of the ATP concentration in the ambient fluid. An assumption of Michaelis-Menten kinetics of ATP hydrolysis successfully describes the ATP dependence of $v_{0}$ for kinesin, where $v_{0}$ increases with ATP concentrations to eventually saturate Schnitzer2000 . This large variability of $v_{0}$ leads to various interesting dynamical regimes. We return to this point later in the paper.

Neglecting fluctuations, the mean extension of $i$-th MP $y_{i}$ attached to a MT is determined by its active velocity $v_{m}$ and the variation of the MT- overlap $x$,

[TABLE]

Finally, the dynamics of the overlap length, $x$, between the two antiparallel MTs is determined by the mechanical balance of the forces they are subjected to,

[TABLE]

where, the left hand side corresponds to the viscous friction force associated with the relative motion between the two MTs, quantified by the viscous friction $\gamma_{f}$. This force is balanced by the total restoring force of the MPs as they distort when attached to a MT due to their inherent rectification, and the force exerted by PCLs, $F_{p}$, that depends on the number of PCLs, $n_{p}$. Here, $\sum_{i=1}^{n_{m}}k_{m}y^{i}$ denotes the total force exerted by the attached MPs. We stick to the mean field assumption that the total reaction force is uniformly shared by all the attached MPs, hence $f_{l}=-\frac{1}{n_{m}}\sum_{i=1}^{n_{m}}k_{m}y^{i}$, an assumption valid for a rigid rod like MT Badoual2002 ; Vilfan1998 .

In the presence of $n_{p}$ number of diffusing PCLs cross-linking two MTs in the overlap region, an entropic force $F_{p}=k_{B}Tn_{p}/x$ develops. This one dimensional ideal gas pressure pushes the system to increase its overlap region Lansky2015 . Moreover, PCLs increase the viscous friction associated with the relative motion of the MTs to $\gamma_{f}=\gamma\exp(\zeta n_{p})$, see Appendix-A.

In the mitotic spindle, both the force due to cortical dynein transmitted from astral MTs via the spindle pole body Grill2003 ; Grill2005 , and force due to bipolar kinesins potentially present at the interdigitating interpolar MTs Sharp1999 , slide MTs actively to reduce their overlap. On the other hand passive cross linkers like Ase1 generate entropic force that has the opposite effect, and tends to increase the overlap between the MTs. We analyze in depth the interplay between these two mechanisms and the impact they have in the stability of overlapping MTs. We select two simplified, although experimentally realizable in vitro assays consisting of kinesins, MTs and PCLs, where active forces due to kinesins compete with entropic force due to passive PCLs. We are able to identify the relevant morphological structures for the corresponding overlapping MTs and describe a thorough study that provides the relevant phase diagrams for these systems.

III Microtubule conformations stabilized by PCLs

III.1 One movable MT

Let us consider first a setup where one MT is attached irreversibly to a cover slip. This MT is associated with a second, moving MT via PCLs, such as Ase1 Lansky2015 , as depicted in Fig.1. One end of MPs, e.g., kinesins are irreversibly attached to the cover slip, while the other end stochastically bind and generate force on the movable MT. In the MT overlap region, PCLs freely diffuse, generating an entropic force $F_{p}=k_{B}Tn_{p}/x$ tending to increase the overlap length, $x$, and increasing the effective viscous friction felt by the moving MT, $\gamma_{f}=\gamma\exp(\zeta n_{p})$ Lansky2015 . The system evolves according to Eqs.(1)–(3), where we assume that after detachment the $i$ -th MP immediately relaxes to equilibrium with mean extension $y_{i}=0$. The average extension associated to $n_{m}$ MPs is expressed as $y=\sum_{i}^{n_{m}}y^{i}/n_{m}$.

Using the characteristic thermal energy $k_{B}T$, the time scale associated to MP detachment rate $\omega_{d}^{-1}$, and the typical length scale coming from force balance $l_{0}=\sqrt{k_{B}T/\gamma\omega_{d}}$, the dynamic equations, Eqs. (1)-(3) governing the motion of the overlapping MTs can be expressed in dimensionless form

[TABLE]

where, $\tau=t\omega_{d}$, $\tilde{x}={x}/{l_{0}}$, $\tilde{y}={y}/{l_{0}}$, $\tilde{n}_{m}={n_{m}}/{N}$ the fraction of attached MPs, $\tilde{k}_{m}={k_{m}l_{0}}/{f}$, $\tilde{\omega}={\omega_{a}}/{{\omega_{d}}}$, $\tilde{f}_{s}={f_{s}}/{f}$, $\tilde{f}_{d}={f_{d}}/{f}$, where $f=\sqrt{k_{B}T\gamma\omega_{d}}$ 111In dimensionless units, the MP velocity under a load reads

\tilde{v}_{m}=\left\{\begin{array}[]{ll}\tilde{v}_{0}&\mbox{for}\;\;\;\tilde{y}\leq 0\\ \tilde{v}_{0}\left(1-\frac{\tilde{k}_{m}\tilde{y}}{\tilde{f}_{s}}\right)&\mbox{for}\;\;\;0<\tilde{k}_{m}\tilde{y}\leq\tilde{f}_{b}\\ -\tilde{v}_{back}&\mbox{otherwise},\end{array}\right.

where $\tilde{v}_{0}=v_{0}/v$, $\tilde{v}_{back}=v_{back}/v$ with respect to the characteristic velocity $v=l_{0}/\omega_{d}$. .

Table-1 lists the used parameter values, corresponding to kinesin at room temperature. Under these conditions, the units of length, force and velocity are set by $l_{0}=33\,$nm, $f=0.125\,$pN, and $v=l_{0}\omega_{d}=33\,$nm/s, respectively.

III.1.1 Linear stability of overlapping morphologies

The overlapping MTs steady configurations are obtained from the fixed points of Eqs.(4), which read as,

[TABLE]

and express the fact that the steady state emerges from force balance, $n_{m}k_{m}y=n_{p}k_{B}T/x$. We can identify the stability of these overlapping morphologies analyzing the time evolution of small perturbations around the fixed points of Eq.(4), in terms of the MT overlap, MP extension and number of attached motors, $|\psi\rangle\equiv(\delta\tilde{x},\delta\tilde{y},\delta\tilde{n}_{m})^{T}$. 222Note that the expansion up to linear order around the fixed point, where load force $k_{m}y=f_{s}$, requires that the force-velocity relation is continuous and differentiable at least once at this point. The piece-wise linear force-velocity relation used in this paper obeys this condition, with the discontinuities being at load force $=0$, and at $f_{b}$ where $f_{b}>f_{s}$. The fact that kinesin has a small but finite back velocity has been observed experimentally Carter2005 and allows $v_{m}(k_{m}y)$ to be analytic at $k_{m}y=f_{s}$.

To that end, we identify all the elements of the $3\times 3$ stability matrix $\utilde{a}$ , that quantifies the evolution of these deviations, $d|\psi\rangle/d\tau=$ $\utilde{a}$ $|\psi\rangle$,

[TABLE]

The steady configuration is stable when all the eigenvalues, $\lambda$ of $\utilde{a}$ have a real negative component. The eigenvalues are obtained solving the characteristic equation

[TABLE]

where, $A=-{\rm Tr}$( $\utilde{a}$ ), $B=\frac{1}{2}(a_{ii}a_{jj}-a_{ij}a_{ji})$ with $a_{ij}$ denoting elements of the matrix $\utilde{a}$, and $C=-$det( $\utilde{a}$ ). 333The full expressions for the coefficients of $F(\lambda)$ in terms of the elements of the stability matrix $\utilde{a}$ read

$\displaystyle A$ $\displaystyle=$ $\displaystyle\frac{(N\tilde{n}_{m}^{0}\tilde{f}_{s})^{2}}{n_{p}\exp(\zeta n_{p})}+\frac{\tilde{v}_{0}\tilde{k}_{m}}{\tilde{f_{s}}}+\frac{N\tilde{n}_{m}^{0}\tilde{k}_{m}}{\exp(\zeta n_{p})}+\tilde{\omega}+\exp\left[\frac{\tilde{f}_{s}}{\tilde{f}_{d}}\right]$

$\displaystyle B$ $\displaystyle=$ $\displaystyle\frac{(N\tilde{n}_{m}^{0}\tilde{f}_{s})^{2}}{n_{p}\exp(\zeta n_{p})}\left[\frac{\tilde{v}_{0}\tilde{k}_{m}}{\tilde{f}_{s}}+\tilde{\omega}+\exp\left[\frac{\tilde{f}_{s}}{\tilde{f}_{d}}\right]\right]$

(8)

$\displaystyle+\frac{\tilde{v}_{0}\tilde{k}_{m}}{\tilde{f}_{s}}\left[\tilde{\omega}+\exp\left[\frac{\tilde{f}_{s}}{\tilde{f}_{d}}\right]\right]$

$\displaystyle+\frac{N\tilde{n}_{m}^{0}\tilde{k}_{m}}{\exp(\zeta n_{p})}\left[\tilde{\omega}+\left(1-\frac{\tilde{f}_{s}}{\tilde{f}_{d}}\right)\exp\left[\frac{\tilde{f}_{s}}{\tilde{f}_{d}}\right]\right]$

$\displaystyle C$ $\displaystyle=$ $\displaystyle\left(\frac{(N\tilde{n}_{m}^{0}\tilde{f}_{s})^{2}}{n_{p}\exp(\zeta n_{p})}\right)\left(\frac{\tilde{v}_{0}\tilde{k}_{m}}{\tilde{f_{s}}}\right)\left(\tilde{\omega}+\exp\left[\frac{\tilde{f}_{s}}{\tilde{f}_{d}}\right]\right).$

(9)

Since the active velocity $v_{0}\geq 0$, one has $A>0$, $C\geq 0$, while B may change its sign with changing $f_{d}$ or $f_{s}$. As a result, there are only four possible combinations of roots of Eq.(6), with one of them, $\lambda_{1}$, remaining always real and negative, as is shown in the following subsection. Thus the long time dynamics is characterized essentially by the two other roots. This gives rise to four possible dynamical phases: (i) linearly stable ($s$): both these roots are real and negative $\lambda_{2,3}<0$ corresponding to a stable node, (ii) stable spiral ($ss$): $\lambda_{2,3}=\alpha\pm i\beta$ with $\alpha<0$, (iii) unstable spiral ($us$): $\lambda_{2,3}=\alpha\pm i\beta$ with $\alpha>0$, and (iv) linearly unstable ($u$): both the other roots $\lambda_{2},\lambda_{3}$ are real and positive giving rise to unstable nodes. In general it is possible to have one positive and one negative real root corresponding to a saddle point, however, it is discarded by the form of coefficients in the polynomial $F(\lambda)$. A final possibility is when $\lambda_{2,3}=\pm i\beta$ that occurs as $\alpha$ changes sign and reaches $\alpha=0$ denoting a center. As we will see later, this captures the onset of stable oscillations.

All these phases characterize different types of overlap dynamics. Stability occurs when opposing forces due to active MPs and PCLs balance each other. In our analysis, instability is mediated by detachment of all the MPs from the movable MT leading to its complete overlap with the immobile MT, driven entirely by the PCLs. Complex eignvalues denote development of oscillatory dynamics. We assume a constant number of PCLs in the overlap region, a reasonable choice given the long association time of PCLs to MTs at the overlap. At small overlap $x$, the entropic extensile force $k_{B}Tn_{p}/x$ dominates not allowing MT- overlap to vanish. Changing ATP concentration in the ambient medium changes activity parameters of MPs, as a result changing the effect of their competition with PCLs. We study the phase diagram in the plane of two such important active parameters, $\tilde{v}_{0}$ and $\tilde{f}_{d}$.

III.1.2 The phase diagram

The change in behavior of the eigenvalues allows us to build a phase diagram that determines the expected MT configurations and dynamics. We use the fact that the characteristic polynomial in Eq.(6) behaves asymptotically as $F(\lambda)\sim-|\lambda|^{3}$ for large negative $\lambda$ ensuring that it has to cross the $F(\lambda)=0$ line at some negative $\lambda$ value. Thus one of the roots always remains real negative, $\lambda_{1}<0$. The phase diagram that identifies all possible configurations can be built once we identify the three relevant boundaires delimiting the change of stability of the different regimes described in the previous subsection. Specifically,

a) Boundary between linearly (un)stable and (un)stable spiral phases: The cubic polynomial in Eq.(6) has a minimum at $\lambda_{m}=-\frac{A}{3}+\frac{1}{3}\sqrt{A^{2}-3B}$. For $B>0$ ($B<0$) the minimum occurs at negative (positive) $\lambda$. At $F(\lambda_{m})=0$, two degenerate negative (positive) real roots emerge, $\lambda_{3}=\lambda_{2}=\lambda_{m}$ making the dynamics linearly stable (unstable). On the other hand $F(\lambda_{m})>0$ implies complex conjugate eigenvalues. The corresponding phase boundaries are given by (See Appendix-C)

[TABLE]

where $B>0$ ($B<0$) denotes the boundary between (un)stable spiral and linearly (un)stable phase.

*b) Boundary between stable spiral and unstable spiral phases: * This phase boundary is characterized by the change in sign of the real part of complex conjugate roots which occurs at $\alpha=0$. Hence, the cubic function in Eq.(6) should read as $F(\lambda)=(\lambda+|\lambda_{1}|)(\lambda+i\beta)(\lambda-i\beta)$. Comparing coefficients of $\lambda$ with Eq.(6) we get $C=AB$ as the equation for this phase boundary, with a necessary condition that $B>0$. At this boundary, the stable spiral becomes unstable spiral, non-linear stabilization of whose oscillation- amplitude may give rise to a stable limit cycle.

Fig.2 displays the phase diagram in the $\tilde{v}_{0}$-$\tilde{f}_{d}$ plane obtained both from the linear stability analysis and numerical solution of the model (a further detailed view of the phase diagram at a different $n_{p},\,N$ value is depicted in Fig. 8($a$) in Appendix-D). Continuous lines denote phase boundaries derived from linear stability analysis, while the points correspond to observed non-equilibrium morphologies obtained from numerical integration of the non-linear dynamics, Eq. (4). The non-linear dynamics shows existence of various steady states characterized by (i) a stable phase with stable partial overlap of MTs, (ii) a stable spiral phase, where the system shows decaying oscillation to a final stable state of partial overlap, and (iii) a stable limit cycle phase with persistent oscillation of overlap.

III.1.3 Stability, instability and limit cycle

We use Fig. 3 to analyze the overlap dynamics in detail. Deep inside the stable spiral phase denoted by $\tilde{f}_{d}=16$, $\tilde{v}_{0}=3$, one obtains an inward spiral in the phase plot in fraction of bound MPs $\tilde{n}_{m}$ and overlap $\tilde{x}$ finally reaching a fixed point. This characterizes relaxation to a final stable state. The time scale of the decay of oscillation amplitude gets longer with decreasing $\tilde{v}_{0}$, and finally the dynamics changes to small amplitude stable oscillation reached via a super-critical Hopf bifurcation Strogatz2014 . This is characterized by a stable limit cycle surrounding an unstable fixed point in the $\tilde{n}_{m}-\tilde{x}$ plane, as shown in Fig. 3($b$). Initial conditions inside or outside the limit cycle, flow to the limit cycle via a spiral path in the phase space. The amplitude of this limit cycle oscillation grows as one moves away from the phase boundary denoting the onset of a super-critical Hopf bifurcation. The values of $\tilde{f}_{d}$, $\tilde{v}_{0}$ used in Fig. 3($a$) and ($b$) can be attained for wild type kinesins by changing ATP concentration, as is shown in Sec. III.1.6.

At large enough $\tilde{v}_{0}$, the dynamics of the average MP extension $\tilde{y}$ is essentially controlled by the force-velocity relation of MPs. A small increase of $\tilde{y}$ from the steady state is compensated by a negative response. As a result $\tilde{n}_{m}$ and $\tilde{x}$ remain stable.

At intermediate values of $\tilde{v}_{0}$, a limit cycle oscillation develops due to non-linearity, cooperativity, and phase lag. The mechanism is elucidated in Fig.s 3($c$)-$(e)$. Consider a state with the largest possible overlap $\tilde{x}$. The corresponding entropic force due to PCLs $\sim n_{p}/\tilde{x}$ is weak, thus MPs slide the movable MT to a smaller $\tilde{x}$. As a result, the mean extension $\tilde{y}$ relaxes to smaller values. This relies on the cooperativity of the MPs via load sharing. The force-velocity relation of MPs contain a finite $\tilde{v}_{0}$, that generate a phase lag between $\tilde{x}$ and $\tilde{y}$. This phase lag can easily be noticed comparing Fig.s 3($c$) and $(d)$. The decrease in $\tilde{y}$ reduces stress on the bound MPs leading to lesser detachment, and as a result an overall increase in the number of bound motors $\tilde{n}_{m}$. This mechanism keeps on reducing $\tilde{y}$ and increasing $\tilde{n}_{m}$ until the latter reaches its maximum. Due to the phase lag, even at this point overlap $\tilde{x}$ keeps on decreasing until reaching its own minimum. At this point $\tilde{y}$ starts to increase due to a finite $\tilde{v}_{0}$. This starts to decrease $\tilde{n}_{m}$ leading to an increase in $\tilde{x}$ that in turn increases $\tilde{y}$. This positive feedback leads to enhanced detachment of the bound motors. However, beyond a point the attachment rate $\omega_{a}$ wins over the overall detachment rate as $\tilde{n}_{m}$ becomes too small. This is the point of minimum $\tilde{n}_{m}$ where $\tilde{y}$ reaches its largest value and overlap nears its peak. Then the cycle repeats.

For vanishing $\tilde{v}_{0}$, the mean motor extension $\tilde{y}$ follows $\tilde{x}$, with negligible phase lag. Starting from a steady state, a slight increase in the extension $\tilde{y}$ decreases the number of attached MPs, $\tilde{n}_{m}$. This in turn increases the overlap $\tilde{x}$, which leads to further decrease in $\tilde{n}_{m}$. This mechanism underpins a run away instability, where the movable MT increases indefinitely the extent of its overlap with the immobile MT.

It is interesting to note that the time scales of growth or decay of perturbation near the phase boundaries show diverging behavior, reminiscent of equilibrium phase transitions (see the Appendix-E).

III.1.4 Stable limit cycles: Oscillation amplitude and frequency

At the boundary between the stable spiral and unstable spiral phases, predicted by the linear stability analysis, two modes are purely oscillatory, denoted by $\lambda_{2,3}=\pm i\beta$ (and $\lambda_{1}<0$). Inserting this functional form in the secular determinant, Eq.(6) gives $\beta=\sqrt{B}$. Thus, the frequency of oscillation characterizing the Hopf bifurcation boundary reads $f_{\omega}=\sqrt{B}/2\pi$, where $B$ is given by Eq.(8), which is sensitive to activity of MPs via $\tilde{f}_{s}$, $\tilde{v}_{0}$, $\tilde{f}_{d}$, $\tilde{\omega}$, as well as the impact of PCLs through its number $n_{p}$ and modification of viscous friction $\gamma_{f}$. Fig.4($a$) shows a good agreement between the expression $f_{\omega}$ with the numerically calculated limit cycle oscillation frequency along the super critical Hopf bifurcation boundary. Along this line, $f_{\omega}$ varies between 0.2 to 2 Hz, a range easily observable in experimental assays.

As we move away from the Hopf bifurcation boundary, the nature and properties of the frequency and amplitude of the stable limit cycle change substantially. For example, if we move in the phase diagram Fig.(2) along a constant $\tilde{v}_{0}$ line from one Hopf bifurcation boundary to the other by varying $\tilde{f}_{d}$, the frequency of oscillation $f_{\omega}$ follows $\sqrt{B}/2\pi$ as one crosses the phase boundary from the stable spiral phase. However, inside the stable limit cycle $f_{\omega}$ drops to a relatively small value and remains almost independent of $\tilde{f}_{d}$ (Fig.4($b$)). The amplitude of oscillation vanishes towards the phase boundary (Fig.4($c$)), capturing a characteristic of supercritical Hopf bifurcation.

On the other hand, if we move along a constant $\tilde{f}_{d}$ line reducing $\tilde{v}_{0}$ starting from a stable spiral phase, at the supercritical Hopf bifurcation again the frequency gets the form $f_{\omega}=\sqrt{B}/2\pi$, however, it changes its functional dependence drastically as we enter the stable limit cycle phase where we observe a sharp decrease in the oscillation frequency associated with similar increase in amplitude (Fig.4($d$), ($e$)).

III.1.5 Re-entrant transitions:

Increasing $\tilde{f}_{d}$ at a fixed value of $\tilde{v}_{0}$ it is possible to obtain re-entrant transitions from stable spiral - stable limit cycle - stable spiral phases, as shown in Fig. 2. Staying within the stable limit cycle phase, one can go from the lower to the upper Hopf bifurcation boundary, varying $\tilde{f}_{d}$. The frequency and amplitude of oscillation show non-monotonic variation in general, with the frequency reaching a minimum and the amplitude a maximum in the middle of the limit cycle region. The phases below and above the limit cycle along the $\tilde{f}_{d}$ axis are both stable spiral. The small $\tilde{f}_{d}$ stable spiral region reaches a final stability with large overlap $\tilde{x}$, and as a result with little stored free energy in PCLs. On the other hand, the large $\tilde{f}_{d}$ stable spiral region maintains stability with a big fraction of attached MPs pulling the movable MT into a very small overlap $\tilde{x}$, a stability with a large amount of stored free energy, i.e., low entropy. In between these two limits, for half a period MPs win the tug-of-war and in the other half PCLs win, leading to oscillation.

III.1.6 Experimental realizations

Experiments on wild type kinesins have provided evidence that MP- activity depends on the ambient ATP concentration. The intrinsic kinesin velocity, $v_{0}$, varies by more than one order of magnitude from $\sim 30\,$nm/s to $\sim 800\,$nm/s with change in ATP concentration from 5$\,\mu$M to 2 mM Schnitzer2000 . With this change, the detachment force $f_{d}$ increases weakly from 1.8 to 2.4 pN. This estimate is obtained from the measurement of run lengths of kinesins (see Appendix-B). In terms of dimensionless parameters, changes in $\tilde{f}_{d}$ from 14.4 to 19.2, and $\tilde{v}_{0}$ from $\sim 1$ to 25, are accessible varying the ATP concentration from $\mu$M to mM range. In in vitro experiments, the length of MTs may be stabilized using taxol. Fig. 2 suggests that carrying out such experiments will allow one to observe all the non-equilibrium phases discussed above. Further, different types of kinesins, and kinesin mutants may allow to probe a wider range of the phase space in the $\tilde{v}_{0}$-$\tilde{f}_{d}$ plane.

III.2 Both MTs movable

The insight gained from the above analysis allows us to extend it to a situation where both MTs are movable, as depicted in Fig. 5. This geometry is closely related to the mitotic spindle, where at the midzone antiparallel MTs are pulled apart by MPs and stabilized by PCLs. However, for simplicity and specificity, we still consider an in vitro experiment with two movable MTs on an kinesin MP assay, in the presence of Ase1 PCLs. The activity of MPs can be tuned by changing ATP concentration.

Let $x_{1}$ and $x_{2}$ denote two tips of the antiparallel overlapping MTs. Extension of MPs associated with these two MTs are denoted by $y^{(i)}_{1}$ and $y^{(i)}_{2}$, such that the mean extensions are $y_{1}=\sum_{i=1}^{n_{m_{1}}}y^{(i)}_{1}/n_{m_{1}}$ and $y_{2}=\sum_{i=1}^{n_{m_{2}}}y^{(i)}_{2}/n_{m_{2}}$ where $n_{m_{1}}$ and $n_{m_{2}}$ are the number of MPs attached to the left and right MTs respectively (Fig.5). $N$ denotes the total number of MPs available to both the left and right MTs. The overlap length is given by $x=x_{1}-x_{2}$. Within mean field approximation, the equations of motion in terms of dimensionless variables can be expressed as,

[TABLE]

where, $d_{\tau}\equiv d/d\tau$, $\tilde{n}_{m_{1,2}}=n_{m_{1,2}}/N$, $\tilde{y}_{1,2}=y_{1,2}/l_{0}$, $\tilde{x}_{1,2}=x_{1,2}/l_{0}$, with $\{1,2\}$ denoting the two MTs and MPs associated with them, respectively. The kinesin MPs that are attached to the right MT, move towards left, leading to a self propelled velocity $-\tilde{v}_{0}$ and average extension $\tilde{y}_{2}<0$. This leads to the difference between the form of the equations for $\tilde{y}_{2}$ with respect to that of $\tilde{y}_{1}$. Note that the detachment rates enhance with the extension of MPs, irrespective of the sign of extension.

III.2.1 Linear stability and the phase diagram

For a given value of parameters, the fixed point is given by $(\tilde{x}_{1}^{0}-\tilde{x}_{2}^{0})={n_{p}}/{\tilde{n}_{m_{1}}^{0}N\tilde{f}_{s}}$, $\tilde{y}_{1}^{0}={\tilde{f_{s}}}/{\tilde{k}_{m}}$, $\tilde{y}_{2}^{0}=-{\tilde{f_{s}}}/{\tilde{k}_{m}}$, $\tilde{n}_{m_{1}}^{0}=\tilde{n}_{m_{2}}^{0}={\tilde{\omega}}/[{\tilde{\omega}+\exp({\tilde{f}_{s}}/{\tilde{f}_{d}})}]$. Linearized time evolution of small perturbations around this fixed point is given in the form of the matrix equation

[TABLE]

where,

[TABLE]

Let us define a set of new variables using the following linear transformations $\bar{x}=(\tilde{x}_{1}+\tilde{x}_{2}),x_{r}=(\tilde{x}_{1}-\tilde{x}_{2}),\bar{y}=(\tilde{y}_{1}-\tilde{y}_{2}),y_{r}=(\tilde{y}_{1}+\tilde{y}_{2}),\bar{n}_{m}=(\tilde{n}_{m_{1}}+\tilde{n}_{m_{2}}),n_{m}^{r}=(\tilde{n}_{m_{1}}-\tilde{n}_{m_{2}})$. In these definitions, some of the variables has clear physical meaning and interest, e.g., $x_{r}$ is the overlap length of the MTs, $\bar{y}$ denotes the total mean extension of the MPs generating pulling force on the overlap, and $\bar{n}_{m}$ denotes the total number of motors generating the pull. The center of mass of the MT pair is denoted by $\bar{x}/2$. In terms of these new variables, Eq.(12) breaks down into a block diagonal form with each block being independent of the other.

The dynamics of the overlap length $x_{r}$ depends on the force due to MPs via the mean MP extension $\bar{y}$ and the total number of MPs in the attached state $\bar{n}_{m}$, and is governed by,

[TABLE]

The stability matrix in the above equation, $\utilde{b}$ , has exactly the same structure as $\utilde{a}$ in the previous section, only difference being that in $\utilde{b}$ the element $2b_{11}$ has replaced the element $a_{11}$ of the matrix $\utilde{a}$.

The relative variables $y_{r}$ and $n_{m}^{r}$, gets decoupled and can be expressed in terms of the stability matrix $\utilde{b}^{\prime}$ such that,

[TABLE]

Finally, while the center of mass of the overlapping MT pair does not impact any other variable, it itself evolves as

[TABLE]

Eq.(13) is completely independent of the dynamics of $\bar{x}$, $y_{r}$ and $n_{m}^{r}$. Thus stability of Eq.(14) and Eq.(15) does not affect the stability of Eq.(13) involving the overlap $x_{r}$. Consequently, diagonalizing the $3\times 3$ stability matrix $\utilde{b}$ we get the characteristic equation

[TABLE]

where, $A^{\prime}=-{\rm Tr}$( $\utilde{b}$ ), $B^{\prime}=\frac{1}{2}(b_{ii}b_{jj}-b_{ij}b_{ji})$ and $C^{\prime}=-$det( $\utilde{b}$ ) are equivalent to the coefficients $A,\,B,\,C$ in the previous section 444 The full expressions of the coefficients in terms of the elements of the stability matrix $\utilde{b}$ read

$\displaystyle A^{\prime}$ $\displaystyle=$ $\displaystyle\frac{2(N\tilde{n}_{m_{1}}^{0}\tilde{f}_{s})^{2}}{n_{p}\exp(\zeta n_{p})}+\left[\frac{\tilde{v}_{o}\tilde{k}_{m}}{\tilde{f}_{s}}+\frac{N\tilde{n}_{m_{1}}^{0}\tilde{k}_{m}}{\exp(\zeta n_{p})}\right]+\tilde{\omega}+\exp\left[\frac{\tilde{f}_{s}}{\tilde{f}_{d}}\right],$

$\displaystyle B^{\prime}$ $\displaystyle=$ $\displaystyle\frac{2(N\tilde{n}_{m_{1}}^{0}\tilde{f}_{s})^{2}}{n_{p}\exp(\zeta n_{p})}\left[\frac{\tilde{v}_{o}\tilde{k}_{m}}{\tilde{f}_{s}}+\tilde{\omega}+\exp\left[\frac{\tilde{f}_{s}}{\tilde{f}_{d}}\right]\right]$

$\displaystyle+\frac{\tilde{v}_{o}\tilde{k}_{m}}{\tilde{f}_{s}}\left[\tilde{\omega}+\exp\left[\frac{\tilde{f}_{s}}{\tilde{f}_{d}}\right]\right]$

$\displaystyle+\frac{N\tilde{n}_{m_{1}}^{0}\tilde{k}_{m}}{\exp(\zeta n_{p})}\left[\tilde{\omega}+\left(1-\frac{\tilde{f}_{s}}{\tilde{f}_{d}}\right)\exp\left[\frac{\tilde{f}_{s}}{\tilde{f}_{d}}\right]\right],$

$\displaystyle C^{\prime}$ $\displaystyle=$ $\displaystyle\left(\frac{2(N\tilde{n}_{m_{1}}^{0}\tilde{f}_{s})^{2}}{n_{p}\exp(\zeta n_{p})}\right)\left(\frac{\tilde{v}_{o}\tilde{k}_{m}}{\tilde{f}_{s}}\right)\left(\tilde{\omega}+\exp\left[\frac{\tilde{f}_{s}}{\tilde{f}_{d}}\right]\right).$

.

Analyzing properties of $\lambda$ in the same manner as in Sec. III.1, we obtain the phase boundaries presented by lines in Fig.(6). As expected, this leads to a prediction of the same qualitative features of overlap as in Fig.(2), showing stable node, stable spiral, unstable node and unstable spiral phases.

The relative quantities $y_{r},\,{n}^{r}_{m}$ may be excited using an asymmetric perturbation in $y_{1,2}$ and $n_{m_{1,2}}$. Their decoupled dynamics given in Eq. (14) can be analyzed by diagonalizing the stability matrix $\utilde{b}^{\prime}$ to obtain the characteristic equation $\lambda^{2}-p\lambda+q=0,$ where, $p=$Tr.( $\utilde{b}^{\prime}$ ) $=b_{22}+b_{33}$, and $q=$ det( $\utilde{b}^{\prime}$ ) $=b_{22}b_{33}-b_{13}b_{32}$. The eigenvalues $\lambda_{1,2}=(p\pm\sqrt{p^{2}-4q})/2$, with $p<0$ while $q$ may change sign. With change in parameters, if $q$ becomes negative, $y_{r}$ and $n_{m}^{r}$ gets unstable. Thus $q=0$ denotes the phase boundary of instability, which is shown in Fig.(6) by the dashed black line.

One may find oscillations in this asymmetric perturbation if the discriminant $\Delta=p^{2}-4q$ is negative, however, $\Delta=(b_{22}-b_{33})^{2}+4b_{13}b_{32}>0$, discarding any possibility of oscillation in this mode.

A general perturbation contains all possible components, symmetric and asymmetric. Although these components are linearly decoupled, as shown above, instability in one can destabilize all other quantities. Numerical integration of Eq.(11) shows non-equilibrium phases characterized by stability, stable spiral behavior, and instability in the MT overlap. The linear instability in asymmetric fluctuations characterized by $y_{r}$, $n_{m}^{r}$ destabilzes any possibility of limit cycle oscillations in overlap $x_{r}$ within the region enclosed by the dashed black line in Fig. 6. This behavior is unlike that in Fig. 2, overriding the similarities of ($x_{r}$, $\bar{y}$, $\bar{n}_{m}$) mode with one movable MT case. The phase diagram, Fig. 6, shows that the overlap between MTs has a stable region, either linear or oscillatory, denoted by a stable node or spiral respectively. The MTs may also show linear instability towards complete overlap. In stark contrast to Fig. 2, the stable limit cycle oscillation phase is completely ruled out. The fact that when both the MTs in an overlapping pair are movable, the competition between MPs and PCLs lead to stability of partial overlap or instability towards complete overlap, is consistent with the predictions of Ref. Johann2015 . In Fig. 8($b$) of Appendix-D, we show a phase diagram for another set of values of $n_{p}$ and $N$, adding to the understanding of overall structure of the phase boundaries.

IV Discussion

We considered two related systems, analyzing competition between active and passive forces on the overlap of filaments cross-linked by PCLs and driven by MPs. PCLs impact this dynamics via entropic force generation and enhancement of viscous friction between the overlapping pair. Within a mean field approximation, the motion of MTs and associated MPs in the presence of PCLs is described by a set of coupled non-linear differential equations. The resultant dynamics was analyzed within linear stability and using numerical integration. This showed non-equilibrium phases characterized by stability, via either stable node or stable spiral, instability towards complete overlap, and stable limit cycle oscillations. For a given number of PCLs, the transitions between these non-equilibrium phases can be controlled by MP activity that depends on ambient ATP concentration. A dramatic change in the dynamics occurs depending on whether one of the partner MT of an overlapping pair is held fixed, or both the MTs are movable.

If one MT is immobilized, the overlap dynamics can give rise to a stable limit cycle oscillatory phase that occurs via a supercritical Hopf bifurcation, where the oscillations are maintained by the inherent non-linearity, cooperativity through load sharing, and a phase lag between the mean MP extension and the evolution of overlap. Earlier studies mainly focussed on the impact of active mechanism due to MPs on mitotic spindle. In contrast, we have shown that the PCLs present at the overlap play a crucial role in controlling the non-equilibrium phases. In particular, PCLs control phase boundaries and the oscillation frequency at the onset of limit cycle. Within the stable limit cycle phase, the oscillation amplitude of overlap stays within tens of nm and the frequency varies by one order of magnitude between 0.1 to 1 Hz with the change in activity parameters. This frequency range is within that of various other cellular processes, and may potentially interfere with them.

The possibility of having a stable limit cycle oscillation phase disappears as the fixed partner of an anti-parallel MT pair is allowed to move. Any asymmetric perturbation between the two MTs grows to destabilize the possibility of stable oscillations. The cell may control the switching between a stable limit cycle oscillatory phase and non-oscillatory phase, by fixing or releasing one of the partner MTs of the overlapping pair to or from the cytoskeletal matrix using molecular bonds. The non-equilibrium phases might be tuned by the cell also by controlling the local ATP concentration, and changing the number of PCLs present in the overlap. Our work is thus directly relevant for understanding the impact of PCLs in the stability of mitotic spindle structure. We used parameterization considering MTs under the influence of kinesin MPs and Ase1 PCLs. This makes our predictions amenable to quantitative comparison with in vitro experiments.

We performed all the analysis within a mean field approximation ignoring stochasticity. It remains to be seen how consideration of passive and active noise impacts the mean field phase diagram. It will be interesting to study the influence of PCL turnover on the overlap dynamics.

Acknowledgements.

DC thanks Amitabha Nandi and Chaitra Hegde for discussions, and in particular Pieter Rein ten Wolde for his useful comments and suggestions. D.C. acknowledges SERB, India for financial support through grant numbers EMR/2016/0001454, and EMR/2014/000791. I.P. acknowledges MINECO and DURSI for financial support under projects FIS2015-67837- P and 2014SGR-922, respectively.

Appendix A Dependence of viscous friction on PCLs

The mechanism of generation of viscous friction due to the PCLs may be described as follows. In presence of PCLs, a MT must overcome an energy barrier in order to move with respect to the other MT. Assume that the PCLs are extensible springs with spring constant $k_{p}$ with the two ends of a PCL attached to two MTs. Let the bottom end of one PCL move by a distance $\delta$ to the right, keeping the other end fixed. This will pull the top MT to the right, while rest of the $n_{p}-1$ PCLs will pull it to the left bringing the system to a new force balance, with increased potential energy. The relative motion of MTs is mediated by hopping of PCLs. The energy reaches its maximum when equal number of PCLs, $n_{p}/2$, pull to the right and left. This transition state energy is $2\times\,\frac{1}{2}k_{p}\delta^{2}\times(n_{p}/2)$. The top MT will slide to right with respect to the bottom MT, once this energy barrier is crossed Lansky2015 . The rate of this barrier crossing $\sim\exp(-k_{p}\delta^{2}n_{p}/2k_{B}T)$ gives the MT diffusivity $D_{MT}=k_{B}T/\gamma_{f}$, leading to an exponential increase of viscous friction with the number of PCLs $\gamma_{f}=\gamma\exp(\zeta n_{p})$ where $\zeta=k_{p}\delta^{2}/2k_{B}T$. A comparison with experiments gives $\gamma=8.93\times 10^{2}\,k_{B}$T-s/$\mu$m2 and $\zeta=0.22$ Lansky2015 , assuming fluorescent intensity from Ase1 as equivalent to $n_{p}$. Within this picture, we ignore the viscous drag due to the ambient fluid as it turns out to be orders of magnitude smaller than that typical magnitudes of the rest of the forces involved. For example, the viscous friction felt by a MT of $10\,\mu$m length in water (of viscosity 0.001 Pa-s) is $\sim 10^{-3}\,k_{B}$T-s/$\mu{\rm m}^{2}$, five orders of magnitude smaller than $\gamma_{f}$.

Appendix B Dependence of $\tilde{f}_{d}$ on ATP concentration

A free MP, when attached to a conjugate filament, moves with velocity $v_{0}$ dependent on ATP concentration. The time-scale over which a MP remains associated to a MT and moves actively is proportional to the duty ratio $\Omega=\omega_{a}/[\omega_{a}+\omega_{d}\exp(f_{l}/f_{d})]$. In the presence of resistive load $f_{l}$, thus, the run length of a kinesin molecule over MT is given by $\ell_{r}=\bar{\kappa}v_{m}(f_{l})\Omega$, where, $\bar{\kappa}$ is a proportionality constant having the dimension of time. Measurements of such run-lengths for kinesins at two different ATP concentrations were presented in Ref. Schnitzer2000 . We use this data to extract the detachment force $f_{d}$ (see Fig. 7). Fitting the data to $\ell_{r}=\bar{\kappa}v_{m}(f_{l})\Omega$ within the linear force-velocity regime, shows that $f_{d}$ does not vary much, although values of $v_{0}$ strongly depend on the ATP concentration. Fixing values of $v_{0}$ from independent measurements of Ref. Schnitzer2000 , we find that $f_{d}=2.4\,$pN for 2mM ATP concentration, and $f_{d}=1.8\,$pN for 5$\mu$M ATP concentration (Fig 7).

Appendix C Calculating phase boundary

At the phase boundary, with two degenerate roots at $\lambda_{m}=-\frac{A}{3}+\frac{1}{3}\sqrt{A^{2}-3B}$, the characteristic equation is $(\lambda+\lambda_{1})(\lambda-\lambda_{m})^{2}=0$. Comparing it with Eq.(6) one gets

[TABLE]

Eq.s (16) and (17) are satisfied by $\lambda_{m}$ and

[TABLE]

As a result Eq.(18) gives the equation of phase boundaries

[TABLE]

with $B\textgreater 0$ denoting the boundary between $ss$ and $s$ phases while $B\textless 0$ gives the boundary between $us$ and $u$ phase.

Appendix D Phase diagrams

In Fig. 8($a$) and ($b$), we show the phase diagrams corresponding to the two models discussed in Sec. III.1 and Sec. III.2 for different number of PCLs $n_{p}=20$ and total motor proteins $N=50$. The overall structure of the phase diagrams remain unaltered, however, the appearance of particular phases does depend on the number of PCLs, $n_{p}$. The upper branch of the linear stability line between the stable ($s$) and the stable spiral ($ss$) phases are clearly visible in these diagrams.

Appendix E Time scales near phase boundary

The linear stability analysis gives an estimate of the time-scale $\bar{\tau}$ of growth or decay of the perturbation as $\exp(t/\bar{\tau})$ near the onset of instability, be it linear or oscillatory. Here $\bar{\tau}>0$ denotes time required for instability to grow, and $\bar{\tau}<0$ denotes time required for a perturbation to die out within a stable phase. Fig.9 shows these estimates around phase transitions in Fig.(2). Magnitude of both the time scales are small deep inside a given phase, and shows signature of divergence at the phase boundaries.