Morphological and dynamical properties of semiflexible filaments driven by molecular motors
Nisha Gupta Abhishek Chaudhuri Debasish Chaudhuri




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        1/2r_ij bû rnishagupta@iisermohali.ac.inabhishek@iisermohali.ac.inIndian Institute of Science Education and Research Mohali, Knowledge City, Sector 81, SAS Nagar - 140306, Punjab, India.debc@iopb.res.inInstitute of Physics, Sachivalaya Marg, Bhubaneswar 751005, India.Homi Bhaba National Institute, Anushaktigar, Mumbai 400094, India.
        We consider an explicit model of a semiflexible filament moving in two dimensions on a gliding assay of motor proteins,  which attach to and detach from  filament segments stochastically, with a detachment rate that depends on the local load experienced. Attached motor proteins move along the filament to one of its ends with a velocity that varies non-linearly with the motor protein extension. The resultant force on the filament drives it out of equilibrium. The distance from equilibrium is reflected in the end-to-end distribution, modified bending stiffness, and a transition to spiral morphology of the polymer. The local stress dependence of activity results in correlated fluctuations in the speed and direction of the center of mass leading to a series of ballistic-diffusive crossovers in its dynamics. 
        Morphological and dynamical properties of semiflexible filaments driven by molecular motors
            Debasish Chaudhuri
            December 30, 2023
        ===========================================================================================
        
        
        
        
        § INTRODUCTION
        
        The active cytoskeleton in a living cell provides its structural stability, mediates deformation and growth of the cell when necessary, and acts as transport lanes and highways for intracellular cargo <cit.>. It is made of semiflexible filaments, e.g., F-actins and microtubules, that are driven by associated motor proteins, for example myosin and kinesin family of motor proteins respectively <cit.>. Given the complexity of the cytoskeleton in  a living cell,  in vitro experiments were devised in which purified and stabilized cytoskeletal filaments and corresponding motor proteins were studied separately <cit.>. 
        Single molecule experiments on motor proteins revealed details of their dynamics, e.g., force-velocity relation, dependence of turnover on load experienced, and dependence of activity on ATP concentration <cit.>. Motion of rigid cargo under collective drive of molecular motors has been studied both experimentally and theoretically <cit.>. In a gliding assay setup, heads of molecular motors are attached to a suitably prepared cover slit irreversibly, such that the tails can actively drive the associated filaments, hydrolyzing the chemical fuel ATP. This led to observation of collective motion, e.g., formation of spiral and aster patterns in microtubules driven by kinesin <cit.> or dynein molecules <cit.>, or swirling patterns in high density F-actins floating on a myosin motility assay <cit.>.  
        
        
        
        Such patterns were explained within active hydrodynamics framework, and agent based models <cit.>. Spiral rotation and flagella like beating of individual filaments were reproduced within effective active polymer models, modeling activity as a tangential self-propulsion <cit.>, stresslets distributed over the filament contour <cit.>, or chemical activity <cit.>, in presence or absence of hydrodynamic coupling. 
        The collective dynamics in such models change from  coherently free flowing motion to frozen spiraling ones with changing activity <cit.>.   
        Generic consideration of a stiff filament in an active medium leads to the possibility of both increase or decrease of effective bending rigidity, depending on the orientation of filament segments with respect to contractile or extensile medium <cit.>.  
        It was shown that a semiflexible filament under active correlated noise transform from bending rigidity dominated to flexible polymer- like dynamics <cit.>.  Their center of mass motion showed a single crossover from a short time ballistic, to long time diffusive behavior <cit.>. In contrast, as we show in this paper, a more microscopic consideration of both the cytoskeletal filament and motor proteins allows for local stress relaxation leading to novel behavior, e.g.,  a series of ballistic- diffusive crossovers of the filament center of mass.
        
        
        
         
        Previous studies either modeled the motor proteins explicitly considering the driven object as a rigid cargo, or modeled the mechanical properties of the driven polymer explicitly, using self propulsion devoid of any underlying mechanism for relaxation.
        Thus the impact of stress dependent dynamics of motor proteins on the filament properties, despite its importance, remains elusive within such models. In this paper we set out to address this issue. 
        We perform numerical simulations, explicitly modeling the mechanical properties of the filament, and that of individual motor proteins as active harmonic springs undergoing attachment- detachment kinetics that do not obey detailed balance. The  attachment to filament is diffusion limited, and the detachment rate increases exponentially with the extension of individual motor proteins. In the attached state the  tail of a motor protein moves tangentially towards one end of the polymer in an active manner, with a velocity that depends non-linearly on the motor protein extension. 
        
        
        We characterize the non-equilibrium conformations of the polymer comparing its end-to-end distribution with that of the equilibrium filament. In theoretical studies of active systems, key concepts such as broken detailed balance and entropy production has recently been used to characterize the distance of these systems from their equilibrium counterparts <cit.>. We show that subtle changes in the local load dependence of detachment rate and active velocity of motor proteins, leads to dramatic difference in the end-to-end distribution. With increasing activity, the difference increases,  the effective bending stiffness reduces, and the polymer shows a phase coexistence between open and spiral chains.
        The most startling result is seen in the dynamics. The center of mass of the polymer shows a series of crossovers between ballistic and diffusive motion, controlled by its inertial, orientational and speed relaxation times scales.
        
        
        In Sec.<ref> we present the model and  details of the numerical simulation. All the results are discussed and analysed in Section <ref>.  Finally, we present a summary and outlook in Sec. <ref>. 
        
        
        
        
        § MODEL AND SIMULATIONS
        
        We consider an extensible semiflexible filament described as a bead-spring chain of N beads constituting (N-1) bonds of equilibrium length  such that the chain length L=(N-1), spring constant A, and finite bending rigidity $̨. This is described by  the Hamiltonian
         H = ∑_i=1^N-1A2 [  (i) - (i)  ]^2 + ∑_i=1^N-22 [  (i+1) - (i)  ]^2,
        
        with = 1/ T, the inverse temperature.
        The bond vector(i)=(i+1)-(i), where(i)denotes the position of thei-th bead. This allows one to define the local tangent(i) = [(i+1)-(i)]/b(i).
        In the limit of largeA,  instantaneous bond lengthsb(i) ≈, and the polymer maps to a worm like chain <cit.>. In addition, excluded volume interactions between the non-bonded beads of the polymer is incorporated via a Weeks-Chandler-Anderson (WCA) potential V_WCA() = 4[(/)^12 - (/)^6 + 1/4]if < 2^1/6and0otherwise.
        
        The polymer is placed on a substrate of motor protein (MP) assay (Fig. <ref>(a) ). We explicitly model MPs and their dynamics, unlike several recent studies that used effective active polymer models <cit.>. The MPs are modeled as  active elastic linkers. We assume the MP heads are attached irreversibly to the substrate at position_0^i=(x_0^i,y_0^i)placed on a two dimensional square lattice with lattice parameteradetermined by the MP density$̊. The polymer floats on this MP bed.  The  tails of MPs may bind (unbind) to (from) the nearest polymer segments stochastically. 
        The attachment process is diffusion limited. The tail of a MP attaches to a polymer segment if it lies within a capture radius r_c with an attachment rate . The extension Δ of the MP in the attached state generates an elastic load _l = -k_mΔ r. An attached MP unbinds from a polymer segment with a rate  which depends on the stress felt by the MP as 
         = ω_0exp(f_l/f_d),
         
        where ω_0 is the bare off rate, f_l = |_l| and f_d is the detachment force. The ratio : does not obey detailed balance. When attached, a MP can move on the filament towards one of its ends, depending on the MP and filament type. For example, attached Kinesin moves towards positive end of the microtubule with  active velocity v^a_t along the local tangent of the filament given by <cit.>
        v^a_t(f_t) = v_01+d_0exp(f_t/f_s),
        
        where f_t=-_l., d_0 = 0.01 and f_s is the stall force. Here v_0 denotes the velocity of MP in the absence of stress. The extension of a given MP depends on the duration and velocity with which it moves along the filament before detachment, as well as the movement of the filament segment it is attached to. This generates a stochastic and non-uniform elastic load on different MPs.
        
        We perform numerical simulations of the model to investigate structural and dynamical properties of the polymer, actively driven by MPs. The molecular dynamics of polymer is performed using the velocity-Verlet algorithm in presence of a Langevin heat bath. The bath fixes the ambient temperature T through a Gaussian white noise obeying η_i(t)  =0, and η_i(t) η_j (t')  = 2  T _̣ij(̣t-t'), where = 3 πη with η denoting viscosity of the environment. This defines the diffusivity over the bead size , D = k_BT/α. 
        The units of energy, length and time are set by T,   σ and τ = ασ^2/k_B T, respectively. 
        
        We set out to perform numerical simulations to study conformational and dynamical properties involving the longest length and time scales of the polymer, under the influence of an active MP bed pumping energy from the shortest length scales. The large separation between length and time-scales makes a fully microscopic parametrization of molecular motors prohibitively expensive in terms of simulation time. For example, the capture radius is expected to be a fraction of the size of the molecular motor, i.e., ∼ 10nm. This is three to four orders of magnitude smaller than the typical filament lengths that are used in MP assays. On the other hand the longest relaxation time of a semiflexible filament of length L varies as ∼ L^4 <cit.>. To keep the calculations tractable, we choose a capture radius r_c = 0.5, smaller than the unit of length in the model, to be qualitatively consistent with the fact that this should be the shortest length scale of the problem. 
        The active forces associated with MPs are known to be larger than that coming from thermal fluctuations, and 
        
        we use f_s = 2  T/, f_d = f_s.  
        The coarse-grained nature of the polymer segments considered allows multiple MPs to get associated with them, captured by our somewhat large MP density  in the 2d assay, ρ = 3.8 ^-2. 
        
        The large spring constant A = 100 ^-1 is chosen to keep the bond length fluctuations small (within 5%). In absence of direct measurement of effective spring constant of active MPs (we are not considering the rigor bonds), we have chosen k_m = A/ for simplicity.
        
        The attachment (detachment) of MP tails are stochastic, and performed using probabilities ṭ(ṭ). The extension in the attached state has two contributors – the MP tail is dragged along with the filament segment to which it is attached, and it can slide from one segment to another with an active velocity v_t^a. We study the influence of the active bed of MPs on the static and dynamic properties of the polymer as we vary the (a) bare processivity Ω = /( + ω_0) and (b) a dimensionless Péclet number defined as Pe = v_0σ/D. The numerical integrations are performed using ṭ = 10^-3 for Pe=1, and ṭ = 10^-4 for Pe=10,  100. Unless stated otherwise, we use ø=5/6, which corresponds to Kinesin MP property :_0 = 5:1 <cit.>. The simulations are done over 2 × 10^9 steps,  and the steady state measurements are presented over 10^6 configurations separated by 10^3 steps, discarding the first 10^9 steps. 
         
        
        
        § RESULTS
        
        At equilibrium,  mechanical and structural properties of a semiflexible filament are determined by the persistence ratio u = L/ł, where L is the contour length of the chain, and ł= 2 /̨(d-1) is the persistence length, where d=2 is the dimensionality of the embedding space <cit.>.    
        
         The active drive from processive MPs attaching (detaching) to (from) the filament generates non-equilibrium stress which have profound effect on the steady state conformational properties of the polymer. To characterize the conformational properties, we obtain probability distribution of the end-to-end distance, P(r,L), of the polymer. At equilibrium, this has the scaling form, P(r,L) = 1/L^dp(r/L,L/λ) = 1/L^dp(r̃,u) where r̃ = r/L and u = L/λ. The limits of u→ 0 and ∞ are the limits of rigid rod and flexible polymers, respectively. 
         For equilibrium worm like chain,  p(r̃,u)  shows a first-order-like transition from a single maximum at r̃ = 0 for the flexible limit of large u to a maximum at r̃ = 1 for a very rigid polymer with small u <cit.>. 
         We choose the value of u=3.33, in the regime between these two limits where semiflexibility is most strongly pronounced <cit.>, to examine the impact of active MP bed on semiflexible polymers. In  in vitro experiments, the ratio u may be tuned by controlling persistence length of the chain by, e.g., changing salt concentration in the medium thereby changing interaction, or by stabilising the chain lengths. In all our simulations, unless stated otherwise, for L=63 chains, u = 3.33 sets ł = 18.92. Two typical configurations of the MP driven polymer is shown in Fig.<ref>(i) and (ii) for Pe=100 and ø=5/6. 
         
        
        
        
        
        
         §.§ How far from equilibrium the polymer morphology is ?
        
        Under the active drive of the gliding assay of MPs, the morphology of the polymer changes. In Fig.<ref> we show how this impacts the end-to-end distribution function p(r̃, u). The conformational change with respect to the equilibrium is well captured by the logarithmic ratio,
        = ln[ p(r̃)/p_eq(r̃)].
        
        In Fig.<ref>(a) we show how the dimensionless quantity 
         changes with activity. For comparison, the equilibrium distribution p_eq is shown in  Fig.<ref>(c). 
        
        
        
        If the activity of MPs is independent of the load force acting on them, =_0 and v_t^a = v_0. This corresponds to the limit of infinitely large f_d and f_s. It is expected that the deviation  would be large for large non-equilibrium driving, quantified in terms of f_d, f_s and ø. In Fig.<ref>, we explore the impact of activity using the moderate value of Pe=1.
        
        We first consider the situation in which =_0 is kept fixed so that Ω = 5/6, and the active velocity v_t^a is varied (Fig.<ref>(a)) for three possible situations. 
        
        (i) In the absence of any directed motion of the polymer, i.e., with v_0 = 0,  shows a dip near r̃ = 0, indicating a relative bias to the  open conformations of the polymer. This indicates that a mere stochastic attachment/detachment kinetics of MPs, that does not obey detailed balance, leads to an enhancement of effective stiffness of the filament. (ii) When attached, MPs move, and if the active velocity is assumed to be  independent of the load experienced, we use v_t^a = v_0. The effect is dramatic. The filament, gliding on the attached MPs, undergoes a transition to a rotating spiral configuration (discussed further in Sec. <ref>). This gives rise to a peak in  near r̃ = 0.1. (iii) If we incorporate local stress dependence in v_t^a, the polymer is still softened but now switches between gliding and spiral states more freely. Thus in addition to the peak near r̃ = 0.1, a non-zero value at higher r̃ appears in .  The statistics, dynamics and mechanical properties of the polymer under MP drive is determined by a competition between processive active velocity of MPs and bending stiffness of the polymer.   
        
        
        
        We next consider the situation allowing the detachment rate  to be  dependent on the load force felt by individual MPs (Eq.<ref>).
        Given their similarity with the equilibrium distribution, ΔΣ≈ 0 (Fig.<ref>(c)), the corresponding non-equilibrium end-to-end distributions are shown explicitly in Fig.<ref>(b). 
         The non-equilibrium stress build-up due to activity is relaxed easily by enhanced unbinding rate of stretched MPs allowing the polymer morphology to adopt equilibrium-like conformations. The distribution is closest to equilibrium for v_t^a=0. The strongest non-equilibrium feature is observed at stress  independent activity v_t^a=v_0. At this point the distribution clearly shows a bi-modality with two maxima at r̃≈ 0, 0.8. Consideration of the stretching dependent decrease of active velocity as in Eq.<ref>, decreases the height of the flexible chain maximum at r̃≈ 0, as the polymer  switches between gliding and spiral states more easily.
        
        
        
        
        
        In most biologically relevant situations, both the turnover and active motion of individual MPs depend on their instantaneous extension.  The activity is most strongly reflected in terms of the bare velocity of MPs v_0. As was shown in Ref.<cit.>, this velocity of unloaded Kinesin MP increases from 1nm/s to finally saturate to ∼ 1 μm/s, as the ambient ATP concentration increases from 1 μM to 1mM. The change in v_0 is captured by changing Pe in our current setup.
        
        In Fig.<ref>, we show how polymer properties vary with increasing Pe when both  and v_a^t are treated as local strain dependent quantities.
        For low values of Pe, the local forces acting on the polymer backbone due to binding kinetics and motor movement is not sufficient to cause significant local curvature. As Pe is increased, due to tangential velocity of MPs and enhanced directional fluctuations, the polymer starts to coil up and rotates with a spiral configuration in the steady state (discussed further in Sec. <ref>). The impact shows up in terms of a maximum in p(r̃) near r̃=0.2 appearing for large Peclet, Pe=100 (Fig.<ref>).
        This feature is robust with respect to change in Ω (see Appendix-<ref>). 
        
        
        
        
         §.§ Competition between activity and bending stiffness
         
        For a semiflexible polymer in equilibrium, the end-to-end distribution p(r̃,u) is determined by the dimensionless ratio u=L/ł. On the other hand, in presence of motor proteins,  the statistical and mechanical properties are expected to be determined by an interplay of activity and bending rigidity.  
        
        To probe that within our model, here, we fix = ω_0, and vary the chain length L = (N-1)σ by changing N, the ratio u=L/ł, and persistence length ł of the polymer to study their impact on conformational properties. We use both stress dependent and independent v_a^t, and plot the end-to-end distributions for three different active velocities in Fig. <ref>. A comparison of Figs. <ref>(a) and (b) clearly shows that for the same u and different L, unlike in equilibrium semiflexible chains, the conformational properties of the polymer are significantly different. For example, for N = 128 and u = 3.33 (Fig. <ref>(b)), the distribution for v_0 = 0 indicates a much stiffer polymer compared to N = 64 (Fig. <ref>(a)). 
        For non-zero active velocity, the spiral states observed for N = 64 disappears for N = 128, leading to stiffer conformations devoid of spirals. If, however, we keep the value of ł fixed as we change the length of the polymer from N = 64 to N = 128 (Fig. <ref>(c)), the distributions we get compares much better with Fig. <ref>(a). This suggests that, for a given processivity ø, the conformational properties of polymers driven by MPs are determined by a competition between active velocity and bending rigidity, and not by the ratio u. 
        
        Within active polymer models with constant tangential drive, arguing that active force f_p may generate compression, a torque balance leads to a critical active force f^c_p ∼ł/L^3, beyond which straight filaments are unstable towards buckling <cit.>. 
        In the limit of stress independent activity, a simple extension of this relation to the instability of the filament under MP driving can be obtained by replacing f^c_p = ø v^c_0. This leads to a relation v^c_0 ∼ł/ø L^3. Thus buckling instabilities are expected to be controlled by the dimensionless number F = ø v_c L^3/ł. However, for polymers driven by real MPs that shows stress dependent activity and turnover, the determining factors turn out to be more subtle.
        
        
        
         §.§ Determination of effective stiffness
        
        To further characterize the steady state conformational properties of the polymer, we consider the tangent-tangent correlation function, (s) · (s^') for different Pe. 
        For an equilibrium worm like chain, one expects a single  exponential decay of the correlations, characterized by the persistence length λ as, (s)· (s^') = exp(-|s - s^'|/ł_ eff). In the long separation limit, the presence of self-avoidance leads to an effective power law correlation function determined by the Flory exponent, a behavior we ignore for relatively short length scales in the ensuing discussion. This results in a ł_ eff that is larger than ł in equilibrium simulations. The tangent-tangent correlation provides a measure for structural rigidity of the filament and can be determined from experiments by fluorescent imaging of polymer conformations.  
        In Fig. <ref>, we observe that the correlation function for small activity, Pe=1, shows a characteristic exponential decay that follows the equilibrium correlation function very closely. Fig. <ref> shows that the correlation length decreases with increase in Pe. This is indicative of a softening of the polymer with the emergence of strong bending fluctuations. Up to Pe=10 shown in the graph, the overall nature can be described by a single exponential decay, which is fitted to extract the effective persistence length λ_eff, directly. For higher values of Pe, e.g., at Pe=100, the correlations start showing oscillations, capturing emergence of spiral conformations that occur at higher activity. In such cases, the crossing of zero by the correlation function is interpreted as the persistence length. The variation of this effective persistence length with activity is listed in Table-<ref>.
        
        
        
        
         §.§ Coexistence of spiral and open chains
        
        In order to quantify the observations of the different conformational states of the polymer, we use the turning number <cit.>, ψ(s) = (1/2π) ∫_0^s ds' (ϑ/ s') where ϑ(s) is the angle subtended by the unit tangent t̂(s) with x-axis. 
        This ψ(s) is a good order parameter, clearly distinguishing  an open polymer from a spiral one and also separates clockwise and anticlockwise spiral states (Fig. <ref>(a)). The steady state probability distribution of ψ(s = L) is a Gaussian with a peak at ψ(L) = 0 for small Pe, indicating the absence of spiral states. Increasing Pe has a dramatic effect on the distribution, with symmetric peaks emerging for non-zero ψ(L) indicative of coexisting spiral states with equal probabilities of clockwise and anticlockwise winding, along with the open state characterized by ψ(L)=0. Such  phase coexistence is a characteristic feature of a non-equilibrium first order phase transition.
        Similar coexistence of spiral and open conformations were observed earlier in an active polymer model characterized by constant tangential force <cit.>. It was not  a priori clear that our current model would give rise to a similar conformational behavior, given that the activity in our model gets modified by the build up and release of local strain via load dependent activity and turnover. As we have already shown, in fact, the effect of local strain dependence reflects strongly in the end-to-end distribution functions p(r̃). Further, as we show in the following section, this implies dynamical crossovers in mean squared displacement that are unlike the active polymer model.  
        
        
        
        
        
         §.§ Anomalous dynamics of  the center of mass
        
        
        In Fig. <ref>(a) we show mean squared displacement (MSD) of the polymer center of mass as a function of time, for three different Pe values that are separated over two decades. At very short time scales the MSD shows an approximate  ballistic scaling r^2_cm∼ t^2 up to t ≈ 1 at all Pe. With increasing time, five crossovers at Pe=1 can be clearly seen, these include three ballistic-diffusive crossovers and two diffusive-ballistic crossovers. At Pe=10, numerical integration required a smaller step size restricting the results to a shorter total time t. Otherwise, all the crossovers are retained at Pe=10, with a reduction in crossover times. The qualitative behavior  changes  as the activity is increased to a larger value,  Pe=100. At this regime the first ballistic-diffusive crossover almost vanishes.  At t≈ 1 one finds a barely discernible change in the slope which quickly gets back to ballistic scaling. This is due to an effective merger of the first diffusive-ballistic crossover to the first ballistic-diffusive one. 
        
        The ballistic-diffusive crossovers discussed in this section is a recurring feature of active systems <cit.>. It is known that a persistent random walker undergoes a crossover from initial ballistic to a final diffusive motion, while directed random walkers show a crossover from short time diffusive to long time ballistic scaling <cit.>.  In the following section we present a detailed explanation of the  crossovers observed. 
         
        In Fig. <ref>(b), we show time evolution of the center of mass position of the polymer at Pe=100, indicating its various conformations associated with the trajectory. As the polymer takes a folded conformation, which is often a spiral in our system, the force generated in different segments by the gliding assay cancel each other, and the net directed force on the center of mass is negligible. As a result, the center of mass moves diffusively, getting mostly localized in a narrow region, albeit with an enhanced diffusivity. 
        When the polymer retains a more open conformation, the gliding assay indeed generates directed force on the center of mass, leading to a ballistic motion over such periods shown by long directed trails. 
        
        More quantitatively, the ballistic- diffusive crossovers are associated with changes in the evolution of the end-to-end extension r_ee, the orientation of the end-to-end vector ϕ, and the root mean squared (RMS) fluctuation of the center of mass position √( r_cm^2) along a single trajectory. In Fig. <ref>(c) we show this at Pe=100. Clearly there are time-spans over which r_ee remains close to zero, i.e., the polymer remains in a folded (spiral at Pe=100) state, e.g., between t≈ 4.5 - 5 × 10^5. 
        It should be noted that the formation of spiral happens at high Pe as was shown in Sec. <ref>. However, even at smaller Pe, the chain switches between open and non-spiral folded conformations. Non-spiral folds show a little higher value of r_ee than when spirals form. There are other time windows over which r_ee fluctuates rapidly between open and spiral states (e.g., between t≈ 0 - 4 × 10^5). 
        
         As is shown in Fig. <ref>(c), ϕ changes ballistically on a timespan over which r_ee remains close to zero in a spiral state.  In particular, between t= t_1 and t_2 the spiral rotates  clockwise ballistically reflected in a linear change in ϕ with a negative slope. During such time spans, the r_ee of spirally folded polymer remains small, and the center of mass position of the polymer does not change appreciably, as is shown by the flat segment of √( r_cm^2) in Fig. <ref>(c) in this time-window. In the window of t=t_2 and t_3 the filament opens up switching between relatively close and open conformations stochastically captured by the strong fluctuations in r_ee. In such a state the directed rotation practically stops, captured by the flat, approximately parallel to t-axis portion of the ϕ(t) curve. The polymer encounters directed drive from MPs during the time-spans over which it opens up leading to  appreciable displacement √( r_cm^2) of the centre of mass. Between t= t_3 and t_4, the polymer folds back into a spiral state again, and starts rotating in the  anti-clockwise direction this time, captured by the linear increase in ϕ, associated with characteristic flat segments of r_ee and √( r_cm^2).
        
        
         §.§ Ballistic to diffusive cross overs
         
        To analyze the crossovers of the centre of mass MSD, let us first consider the dynamics of a particle in a Langevin heat bath in absence of any active drive, m  dv/dt = -  v + η(t) where the Gaussian random noise obeys η(t) =0 and η(t) η(0)  = 2  T (̣t). The corresponding displacement fluctuation of passive origin is given by 
        
         r_p^2(t)  = 6  Tm_I^2 [ t_I - 1 + e^-t/_I],
        
        where _I = m/.
        For time scales t ≪_I this leads to a ballistic scaling of MSD, r_p^2(t) ≈ 3 v_eq t^2, with a velocity v_eq=(2  T/m)^1/2.  At longer times t ≳_I,  this crosses over to a diffusive scaling r_p^2(t)  = 6 D_eq t with D_eq =  T/.  As is shown in Fig.<ref>(a), the polymer centre of mass shows such a crossover near _I = 1 in our simulations [Given that both mass and viscous friction scales similarly with the chain length, _I = 1.].   
        
        Because of the molecular motor drive, further ballistic-diffusive crossovers beyond t_I are observed.  Following Ref. <cit.>, we identify two possible mechanisms related to activity, (i) the persistence of the direction of centre of mass velocity described by the correlation time _, and (ii) the correlated fluctuations of the speed of the centre of mass with correlation time _s.  The speed fluctuations are approximately captured by an exponential correlation C_v_s (t) = ṿ_s(t) ṿ_s(0)  / ṿ_s^2≈exp(-t/_s) where ṿ_s = v_s -  v_s (Fig. <ref> in Appendix-<ref>). Similarly, the orientational fluctuation of velocity obeys C_ (t) =  e^i [(t) - (0)] ≈exp(-t/_) where _ is the persistence time (Fig.<ref> in Appendix-<ref>). On the other hand, as we find, the velocity amplitude and orientations are only weakly correlated (Fig.<ref> Appendix-<ref>). Such correlations can be ignored to use the expression of active displacement fluctuations,
         r^2 (t)    =    r_p^2(t)  + 2  v_s ^2_^2  ( t_ -1 + e^-t/_) 
        
                 + 2 ṿ_s^2 _r^2 [ t_r -1 + e^-t/_r], 
         
        where, _r^-1 = _^-1 + _s^-1. 
        In the above expression the speed v_s and its fluctuations v_s^2 are due to activity controlled by Pe.   
        
        If ṿ_s^2 =0, the above expression would suggest a ballistic dynamics for t ≪_, crossing over to diffusion at t ≳_ as the direction of persistent motion diffuses. This is expected for structureless active Brownian particles with constant active speed. 
        
        
        However, in presence of speed fluctuations in the polymer, the other time-scale _r < _ intervenes. The total mean squared displacement of the polymer center of mass has contributions from both thermal fluctuations Eq.(<ref>) and activity Eq.(<ref>).  If the three time-scales _I ≪_r ≪_ present in the problem are well separated, they are expected to lead to three ballistic-diffusive crossovers: (a) At t ≪_I one expects a ballistic motion r^2_cm≈ 3 v_eq t^2 with a velocity v_eq = (2  T/m)^1/2. (b) At t ≳_I one crossover to diffusive regime takes place, with equilibrium diffusion constant D_ eq= T/. This is the first ballistic-diffusive crossover, and is independent of activity. (c) This regime lasts until ^∗= 3 (v^2_eq/ṿ_s^2) _I at which the chain starts to respond to the active force that drives it in a directed manner. This gives rise to the first diffusive-ballistic crossover. For ^∗ < t ≪_r, we find a ballistic behavior dictated by the active speed fluctuation ∼ṿ_s^2 t^2.  A sufficiently strong activity can enhance ṿ_s^2 to reduce ^∗ to merge this active ballistic regime to the equilibrium ballistic scaling, as is seen for Pe=100 in our simulations. (d) As t crosses _r, the scaling of r^2 (t) crosses over to another diffusive regime, the second ballistic-diffsuive crossover, with effective diffusion constant D ≈ D_ eq + 13ṿ_s^2 _r.   (e) This regime persists until ^† = 3 (v_eq^2/ v_s^2) _I + 2 (ṿ_s^2 / v_s^2 ) _r. Beyond this point the second diffusive-ballistic crossover takes place. For ^† < t ≪_, the ballistic behavior is dictated by ∼ v_s ^2 t^2. (f) For t ≳_, this ballistic regime slowly crosses over to the final diffusive behavior, the third ballistic-diffusive crossover, dictated by an effective diffusion constant   D ≈ D_ eq + 13 (ṿ_s^2 _r +  v_s ^2 _). This qualitatively explains the ballistic-diffusive crossovers obtained in Fig.<ref>(a). 
        
        Before ending this section, we note that, the fluctuations in active speed and orientation in the polymer arise essentially from the same driving mechanism due to molecular motors, and conformational relaxation of the polymer. Thus the two quantities may have similar fluctuations and significant cross-correlation. In Appendix-<ref> we show the auto-correlation functions of the center of mass speed and the active orientation, as well as the cross-correlation between the two. The auto-correlation data show longer correlation times for the orientational fluctuations. The cross-correlation function breaks time-reversal symmetry, capturing the non-equilibrium driven nature of the system, and shows  correlation even at long time gaps.       
        
        
        
        
        
        
        § DISCUSSION AND OUTLOOK
        
        Using stochastic molecular dynamics simulations we have investigated the conformational and dynamical properties of a semiflexible polymer in the presence of motor proteins, which (un)bind (from) to the polymer and perform directed active motion. Unlike in the equilibrium worm like chain, the end-to-end statistics in this case is not controlled by the ratio of persistence length and chain length, but results from a local competition between the processive active velocity and bending rigidity. As is shown in this paper, local stress dependence of turnover and active velocity provides new relaxation mechanisms giving rise to steady states unlike the active polymer models with constant tangential self-propulsion.
        The activity influences polymer morphology, mechanical properties, and dynamics in a concerted manner. With increasing activity of the motor proteins, we observed the following : (i) The end-to-end distribution characterising polymer conformation shows both stiffening and softening relative to the equilibrium morphology associated with the build up of local active stress and its relaxation. (ii) The stretching dependent active velocity and turnover of molecular motors gives rise to an interplay of three time scales, the inertial, orientational and speed relaxation times of the centre of mass, leading to a series of ballistic - diffusive crossovers in the mean squared displacement of the centre of mass. 
        
         
        These crossover time scales can be interpreted into real times using dynamics of a filament of ∼ 2 μm length. For example, considering σ = 20 nm, the 64 bead chain can be interpreted to have a length 1.84 μm. This sets f_s = f_d = 0.4 pN, slightly smaller than the pN scale in, e.g., kinesin molecules.  Assuming the viscosity of ambient fluid 100 times that of water, i.e., equivalent to that in cytoskeleton <cit.>, one obtains a viscous drag of α=0.02 pN s/μm. This sets the unit of time τ=0.002s. 
        As expected, only the non-inertial time-scales are relevant from the perspective of slow dynamics. Interpreting the predictions from Fig.<ref>(a) we find the following slow time scales, 
        the ballistic-diffusive crossover times τ_r ∼ 15s, and τ_θ∼ 15 minutes, and the diffusive-ballistic crossover time τ^†∼ 3 minutes. 
        The predictions presented here are amenable to verification in experiments on molecular motor assays.
        
        While our system reproduces some of the predictions of the standard active polymer model, some other properties that we observe are entirely due to the strain dependence of the activity and turnover of MPs. For example, the observed activity dependent reduction of effective bending stiffness, and the coexistence of spiral  and open conformations at an activity beyond a critical value are expected within the active polymer model. On the other hand, the detailed nature of end-to-end distribution functions, and the series of ballistic-diffusive crossovers observed in the center of mass dynamics are features that are unlike active polymer models <cit.>.
        
        
        
        
        
        
        
        
        § ACKNOWLEDGEMENTS
        
        A.C and D.C. acknowledge SERB, India for financial support through grant number EMR/2014/000791. DC thanks SERB, India, for financial support through grant number EMR/2016/001454, and ICTS-TIFR, Bangalore for an associateship. N.G. acknowledges UGC, India for a fellowship.
        
        
        
        
        
        
        
        
        § END-TO-END DISTRIBUTIONS FOR DIFFERENT Ω
        
        In Fig. <ref>, we show the dependence of the conformational properties of the polymer as the bare processivity Ω=/(+_0) is varied. Here we consider the scenario where both the detachment rate and the active velocity depend on the local stress. For a fixed Ω we plot the end-to-end distribution of the polymer as Pe is changed. As in Fig. <ref>(c), the distributions look similar to equilibrium distribution p(r̃) for low Pe and a peak near r̃≈ 0 appears for high Pe, indicating the emergence of spiral states. Therefore we conclude that for stress dependent , varying Ω does not affect the conformational properties significantly. Recall that a stress independent  with non-zero Pe results in coiled states of the polymer. Switching on local stress dependence in  allows the polymer to relax back to its equilibrium conformations whenever stress builds up beyond a limit, even if the processivity Ω is  high. 
        As Pe is increased, it triggers an instability towards spiral states and we see the emergence of a peak near r̃=0 in the steady state distributions. 
        
        
        
        
        § CENTRE OF MASS DYNAMICS
        
        In this section we analyse autocorrelation of the centre of mass velocity vector, focussing on the speed v_s(t), and orientation (t) separately. Here we distinguish between the direct measures of the correlation times t_s and t_ associated with multi-exponential decays of correlations, from the assumptions of single exponential decays with _s, _ used in the analysis of dynamical cross-overs in Sec. <ref>.  
        In Fig. <ref> we show the auto-correlation of speed, C_v_s (t) = ṿ_s(t) ṿ_s(0)  / ṿ_s^2. 
         A fast single exponential decay exp(-t/t_s) is observed at both Pe=1, 10, with t_s≈ 0.1, 1.0, respectively.  However, at Pe=100, we observe multiple exponential decays with time scales t_s ≈ 1,  250,  3600 (see the inset of Fig. <ref>). 
         
         The orientational correlation C_(t) =  e^i[(t)- (0)], shows multiple exponential decays at all Pe values (Fig. <ref>). The initial decay is fast with t_≈ 1. For Pe = 1, 10 we can extract the longer time scales, as shown in the log-log plot in the inset for Pe = 1, to give  t_=1500, 2000 respectively. However, for Pe = 100, in the absence of better averaging, it is difficult to extract the longest time scale. 
         
         
         Moreover, the speed and orientations remain correlated. The cross-correlation functions C_v_s,(t) =   v_s(t) (0) / [ √(ṿ_s^2)√(^2) ] calculated for Pe=1, 10, 100 are shown in 
         Fig. <ref>. All of them show significant correlation, which remarkably do not decay with increasing time-gap. The asymmetry of the data around t=0 captures the break-down of time-reversal symmetry due to the non-equilibrium molecular motor drive.    
        
        
        
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