M.
Foglino
a,
E.
Locatelli
*b,
C. A.
Brackley
a,
D.
Michieletto
ac,
C. N.
Likos
b and
D.
Marenduzzo
a
aSUPA, School of Physics and Astronomy, University of Edinburgh, Edinburgh EH9 3FD, UK
bFaculty of Physics, University of Vienna, Boltzmanngasse 5, A-1090, Vienna, Austria. E-mail: emanuele.locatelli@univie.ac.at
cMRC Human Genetics Unit, Institute of Genetics and Molecular Medicine, University of Edinburgh, Edinburgh EH4 2XU, UK
First published on 11th July 2019
We present a generic coarse-grained model to describe molecular motors acting on polymer substrates, mimicking, for example, RNA polymerase on DNA or kinesin on microtubules. The polymer is modeled as a connected chain of beads; motors are represented as freely diffusing beads which, upon encountering the substrate, bind to it through a short-ranged attractive potential. When bound, motors and polymer beads experience an equal and opposite active force, directed tangential to the polymer; this leads to motion of the motors along the polymer contour. The inclusion of explicit motors differentiates our model from other recent active polymer models. We study, by means of Langevin dynamics simulations, the effect of the motor activity on both the conformational and dynamical properties of the substrate. We find that activity leads, in addition to the expected enhancement of polymer diffusion, to an effective reduction of its persistence length. We discover that this effective “softening” is a consequence of the emergence of double-folded branches, or hairpins, and that it can be tuned by changing the number of motors or the force they generate. Finally, we investigate the effect of the motors on the probability of knot formation. Counter-intuitively our simulations reveal that, even though at equilibrium a more flexible substrate would show an increased knotting probability, motor activity leads to a marked decrease in the occurrence of knotted conformations with respect to equilibrium.
Most previous theoretical and computational work on non-equilibrium forces within live cells has focused on modelling the propagation of stresses on biopolymer networks8,9 or active fields which mimic cytoskeleton dynamics.10 While molecular motors can now be readily investigated using single-molecule techniques,11–13 the emergent properties of systems where a collection of motors interact on, and through, their target polymer substrate remain poorly studied.14,15
Here we propose a coarse-grained computational model to describe the action of generic molecular motors on polymers with the aim of improving our understanding of the effects of ATP-consuming translocating machineries on the conformational and dynamic properties of their target substrates. Within our model, motors undergo free diffusion in 3-D until they encounter the polymer, at which point they become “bound” to it via an attractive interaction. While bound, a motor experiences an active force propelling it along the polymer; at the same time the polymer is subjected to an equal but opposite active force. The result is that the motor displays relative motion with respect to the substrate. The motor can unbind from the polymer, for example due to thermal motion or because it reaches the polymer's end. While our model is generic, and our aim is to understand its underlying physics, it could be viewed as a model for DNA or RNA polymerases moving along DNA or kinesin molecules stepping on microtubules.2
There has been a great deal of recent simulation work exploring active polymers, and our model differs from these in a number of ways. To our knowledge our work is the first to explicitly treat discrete motors that track along a polymer in a fully 3-D system. A number of recent studies14–20 have considered a chain of beads, where each bead experiences an active force, either directed along the chain bonds17,18,20 or in a direction independent of the chain.14,19 These works have mainly focused on regimes of different behaviour at different levels of activity (e.g., transitions from active translation to rotational motion17 or to spiral configurations18 have been observed – for a review see ref. 21). In the present work the polymer only experiences an active force at the points where the explicit motors are bound (and these vary in time), and we focus on the effect on the conformational and dynamical properties of the polymer. Another class of models have more specifically focused on motor proteins found in the cytoskeleton, like myosin and kinesin which have provided a model experimental system for active materials.22,23 These can be modelled at a field theoretic level, but there have been a number of simulation studies which resolved the motors explicitly.24–27 Unlike the current work, those studies focused on dense suspensions of motors and their substrates, and/or 2-D systems where the motors are fixed to a surface, and have often considered joined pairs of motors which can crosslink different substrate molecules to drive them into relative motion. In contrast, here we consider a single, semi-flexible polymer which is free to move in 3-D, and is subject to the action of multiple motors.
Our simulations reveal that motor activity affects the steady state conformations of the polymer substrate; we find that these are more crumpled with respect to their passive counterparts. We attribute this effective softening of the substrate to the emergence of double-folded branches, or hairpins; the effect can be tuned by changing, e.g., the strength of motor activity. We also find that, for large enough motor activity, the centre of mass of the polymer substrate can display super-diffusive behaviour at short times, with a return to diffusive behaviour at long times. Again this effect can be controlled by varying the activity or number of motors. We conclude our work by investigating the effect of motors on the propensity of the polymer to form knots. This is relevant for many biological processes involving DNA, where knotting would be detrimental (e.g., knots and tangles might hinder transcription and replication, repair of double strand breaks, or segregation of chromosomes). We find that the motor activity effectively reduces the steady state probability of finding knotted conformations. This is surprising since at equilibrium, in a good solvent, and in the absence of confinement, softer substrates show an enhanced probability of knotting.28 Thus, our findings point to an intriguing non-equilibrium effect of molecular motors that could be important in many biological systems.
The motors are represented by freely diffusing beads which interact with each other sterically (via a WCA potential) and via a Morse-like attractive potential with the polymer beads. The latter is given by
![]() | (1) |
The result of the active force is that the motors and the polymer are driven into relative motion; due to the connectivity of the chain, the polymer tends to move less compared to motors, so the latter effectively “track” along the polymer substrate. Motors can become unbound from the polymer due to thermal fluctuations (controlled by the parameter KMOT), and also active effects (e.g. tracking off the end of the polymer – this is discussed in more detail below). Since each monomer has a preferred orientation, i.e., the polymer is polar, motors will move in the parallel or anti-parallel direction depending on the sign of f. In the present work we consider only the case where all motors are the same and f ≥ 0.
For simplicity motors and polymer beads are taken to have the same diameter, σ, mass, m, and unless otherwise stated they experience the same friction ξ due to the implicit solvent (though in reality a motor such as RNA polymerase would be much bigger). We use the LAMMPs molecular dynamics software29 to perform the simulations: a Langevin equation for each polymer bead and motor is solved using a velocity-Verlet algorithm. A separate Langevin equation is solved for each monomer orientation ûi. Full details are given in Appendix A. Throughout we quote energies and times in units of kBT and respectively; how these map to physical units will depend on the system of interest, as detailed in Appendix A.
As expected, the tracking speed of the motors in general increases with f, though the relationship is complex (as noted, changing f leads to a change in 〈n〉 which can affect motor speed due to collisions). Within the range of parameters investigated, typically a motor will move from one polymer bead to the next in a few τLJ (e.g., for f = 20 and N = 200 we measure a speed ∼0.4 beads/τLJ). For an L = 500 bead polymer, motors typically spend on the order 50–100 τLJ bound to it (increasing with KMOT and decreasing with increasing f and 〈n〉).
The motor action also changes the dynamics of the polymer. At a global scale this can be quantified by measuring the decorrelation time for the polymer end-to-end vector. For an L = 500 bead polymer in the absence of motor action (f = 0) a typical decorrelation time is of the order 105τLJ. Again, there is a complicated dependence on f and N, but the motor effect leads to a decrease in this decorrelation time by around an order of magnitude. Nevertheless, the polymer relaxation time remains much longer than the motor time-scales detailed above, i.e., the positions of motors on the polymer change quickly with respect to the polymer dynamics. This implies that, after some transient time, the properties of the polymer will settle into a non-equilibrium steady-state (NESS) which is independent of initial conditions (and the system is ergodic). In the following sections we examine these steady state polymer properties by averaging measurements over time and over an ensemble of 50 independent simulations, unless otherwise stated.
To quantify the extent of polymer crumpling we measure the radius of gyration Rg defined by
![]() | (2) |
〈bi·bi+s〉 = e−s/Lp, | (3) |
As shown in Fig. 2(c and d), both Rg and Lp exhibit a marked decrease with increasing 〈n〉, in agreement with our initial qualitative observation. Furthermore, the decrease in effective persistence length Lp is found to have a power-law scaling
Lp ∼ (〈n〉f2)α, |
In order to shed more light onto the effect of the motors on the metric exponent ν relating the polymer size to its length (Rg ∼ Lν), in Fig. 3 we plot Rg a function of L for different combinations of the number of motors N and force f. We observe that passive systems (N = 0 or f = 0) closely follow the expected self-avoiding statistics (ν ≈ 0.588). More interestingly, larger values of N or f lead to polymer behaviours that are better fitted by ν ≃ 0.43. This value lies between that of an ideal chain ν = 1/2 and that of a fully collapsed conformation ν = 1/3. It is intriguing to note that a simple scaling argument can partially explain this finding using the value of α measured in Fig. 2 (see Appendix C).
To conclude this section we examine the effect of the motors on the global “shape” of the polymer by measuring two quantities: the asphericity Δ and prolateness S, which quantify the degree of spherical symmetry and the oblateness of the polymer respectively (see Appendix D for formal definitions). In Fig. 4, one can readily observe that for increasing 〈n〉 and f both shape descriptors decrease. This indicates that the coil is becoming more spherical, in agreement with our previous observations that the polymer effectively crumples under the action of the motors.
β(s) ≡ 〈bi·bi+s〉, |
A possible physical mechanism underlying hairpin formation is that the motors exert localised tangential forces on the semiflexible polymer they move on. As a result the portion of the polymer “behind” the motor (i.e., the polymer region through which the motor has just moved) is under compression, while the portion ahead is under tension – the situation is reminiscent of forced translocation through a pore.31 The conformation of the polymer could then be computed by minimising its bending energy in the presence of such a localised force. This problem may be viewed as a generalisation of the Euler buckling problem.32,33 Although a detailed calculation is outside the scope of this work, it is reasonable to expect the combination of tensile and compressive forces to result in local hairpin formation.
In Fig. 5(c) we analyse the dependence of the bond correlation function on the intrinsic stiffness of the polymer set by KBEND (recall that in the absence of motor activity, the persistence length Lp ≃ KBENDσ/kBT). To this end, we perform three sets of simulations with KBEND = 10, 20 and 30kBT, for the case of 〈n〉 = 17 and f = 20. Plotting β(s) we observe that the position of the minimum becomes progressively shifted towards higher values of s as KBEND increases, i.e., the hairpins become larger (Fig. 5(c), inset). Additionally, we find that the value of β(s) at the minimum, βmin, gets closer to zero for smaller values of KBEND.
In Fig. 5(d) we study how changing the relative impact of the active force on the motors and polymer beads affects the hairpin formation. This is achieved by changing the friction experienced by the motors (i.e., we change the viscous drag only for the motors – this changes their effective hydrodynamic size, or could reflect a difference in interactions with macromolecular crowders compared to the polymer). Increasing the friction for the motors means that they will move less as a result of the active force (which is kept at a constant magnitude). Fig. 5(d) shows that as the friction for the motors is increased relative to that of the polymer the minimum of β(s) shifts to shorter s, and gets deeper and narrower. This implies that greater motor friction leads to shorter and tighter hairpins. We can understand this by considering the opposite limit of very low motor friction; in that case the motors will move more quickly relative to the polymer. Since a motor spends less time at a given polymer bead, that bead will experience the active force for less time and we can expect the effect of the motors to be diminished. The polymer will, however, still experience the active force, so we still expect a ballistic component to its motion.
MSDCM(t) = 〈[Rcm(t0 + t) − Rcm(t0)]2〉, |
Fig. 6 shows that at short times the system is diffusive as the propulsion exerted by the motors needs a characteristic time in order to provide a displacement comparable to a Brownian displacement. Following the Rouse model34 MSDCM(t) = (6D0/L)t, where D0 = kBT/ξ is the diffusion coefficient of a single monomer. This behaviour is recovered at all times for 〈n〉 → 0.
At intermediate times, the polymer exhibits super-diffusive motion, i.e., MSDCM(t) ∼ tα with α ∼ 1.6. Finally, at large times we observed enhanced diffusion, i.e., α ≈ 1 but the diffusion constant increases with motor activity (f or 〈n〉). In Table 1 we report the measured diffusion constant D for the different cases considered. One can readily observe that the exponent α is roughly independent of 〈n〉 and f, whereas the diffusion coefficient increases with 〈n〉. Moreover, by comparing fits at short and long times for each value of 〈n〉, we find that the crossover between the super-diffusive and diffusive behaviour occurs at progressively larger times as 〈n〉 decreases. Together, this implies that the motor action leads to a persistent motion of the centre of mass of the polymer; at longer times the velocity becomes uncorrelated and the motion is again diffusive.
Intermediate times: super-diffusive | ||
---|---|---|
〈n〉 | α | μ |
17 | 1.720 ± 0.002 | 3.1 × 10−5 ± 0.5 × 10−6 |
9 | 1.690 ± 0.001 | 1.7 × 10−5 ± 0.2 × 10−6 |
5 | 1.542 ± 0.006 | 2.6 × 10−5 ± 1.3 × 10−6 |
Long times: diffusive | ||
---|---|---|
〈n〉 | α | D |
17 | 1.025 ± 0.001 | 2.7 × 10−2 ± 2.6 × 10−4 |
9 | 1.020 ± 0.002 | 2.0 × 10−2 ± 0.06 × 10−4 |
5 | 1.090 ± 0.0001 | 0.4 × 10−2 ± 0.08 × 10−4 |
These three regimes are typical of other active systems, e.g., in polymers where an active force directed along the backbone is applied to each polymer bead14,20 and in Active Brownian Particles (ABPs).35 At intermediate times, the motion of an ABP is dictated by a force which typically maintains its direction for a characteristic time, leading to super-diffusion. In our model, a correlation on the scale of the whole polymer is set by the end-to-end vector.
In order to test this hypothesis we perform simulations of a freely diffusing linear polymer 500 beads long, and measure the probability of forming physical knots as a function of lp by adopting the algorithm described in ref. 38 and 39 and publicly available at http://kymoknot.sissa.it/.40 By comparing Pknot(lp) for cases with and without motors we find that, surprisingly, the location of the maximum of Pknot is unchanged by the action of the motors, i.e., lp ≃ 5–10σ. The most apparent effect of the motor activity is a global reduction in the knotting probability across the full spectrum of bending rigidities studied (Fig. 7). In light of this finding we argue that the effect of the motors goes beyond a simple softening of the polymer backbone, and that subtler non-equilibrium effects must be at play. We also speculate that the reduced knotting probability due to motor activity may be relevant for the case of DNA under the action of motors such as polymerase. According to our model, the action of these motors would naturally have the beneficial consequence of reducing the occurrence of knots and self-entanglements in vivo.41
Our main result was that the effect of the motors is to reduce the overall size of the polymer coil by increasing the likelihood of double-folded segments, or hairpins. This “softening” of the backbone manifests as a reduction of the measured persistence length of the polymer. We found that the data collapse onto a single curve when plotted as a function of 〈n〉f2, revealing a power-law scaling Rg ∼ (〈n〉f2)−0.1 and Lp ∼ (〈n〉f2)−0.1 (insets of Fig. 2(c and d)). We also performed simulations of polymers with different lengths and obtained the explicit values of the metric exponent ν (where Rg ∼ Lν) as a function of 〈n〉 and f. These confirm that at equilibrium the polymers obey self-avoiding statistics, whereas motor activity leads to an exponent between those expected for a crumpled globule and an ideal chain (Fig. 3). A reduction in Rg and Lp has also been observed in models without explicit motors, where an active force directed tangentially to the polymer is added directly to each polymer bead.14,20 In that case the softening arises because, for an initially straight segment of the polymer (e.g., of length equal to the persistence length), small fluctuations in backbone direction will lead to components of the tangential force which are perpendicular to the end-to-end vector for the segment; this then drives the polymer into large smooth curves.14 Those studies, however, did not find hairpins (the decay in bond correlations remained exponential20), suggesting the present case involves a different mechanism.
We also found that the motors affect the dynamics of the polymer. At intermediate times the centre of mass displays super-diffusive behaviour while at longer times it follows an enhanced diffusion. Both regimes are controlled by motor activity (number of bound motors and force applied). While a super-diffusive regime was to be expected, our simulation have provided a detailed quantification of how that can be tuned through the action of the motors (Fig. 6 and Table 1). The effect of the motors on the knotting probability of the polymer was subtler. Unexpectedly, Pknot(lp) did not simply show a shift due to the change in effective persistence length, but rather there was a global reduction in the steady state knotting probability across a range of substrate rigidities (Fig. 7). We can speculate that since the motor action promotes polymer configurations with hairpin bends, but suppresses knotted ones, then these two types of configuration are incompatible. An interesting future study would be to examine knotting probabilities in other active polymer models where hairpins are not observed.
Throughout this paper we have limited ourselves to the simple case in which all of the motors travel in the same direction along the substrate. While this is the case for some systems, e.g., kinesin on microtubules, in others the motors can travel in either direction. For example, RNA polymerase follows the direction of genes, depending on the specifically oriented DNA sequences in gene promoters. This could be readily incorporated into our model in the future by either considering a mixture of two species of motors which move in different directions on the substrate, or by setting some directional binding sites. Also, we have considered the case of motors which are the same size as the polymer beads – in reality RNA polymerase is a large molecule of around 30 nm in size. However, as our results are mostly affected by the magnitude of the force, we do not expect motor size to have a qualitative affect. Nevertheless it would be interesting to examine systems with larger motors in the future.
Another interesting scenario would be to consider a mixture of passive and active polymers. Indeed, an activity-driven phase separation has been observed in another non-equilibrium polymer system – where two species of polymer were held at different temperatures.42 It would be interesting to see whether such demixing behaviour can be recovered when the system is driven away from equilibrium by a more biologically relevant mechanism, such as that presented here.
Our model naturally lends itself to the study of the response of polymer substrates to the action of molecular motors. We therefore expect that in the future is could be employed to address more specific biological questions, such as understanding the role of motors in the buckling of actin networks,8,9 or in the bending and active collective motion of microtubules43,44 by suitably tuning the rigidity of the substrate and processivity of the motors.
Finally we stress that the model presented here is among the first to explicitly account for the presence of physical motors; as a consequence, it faithfully captures the feedback between the action of the motors and the change in substrate–motor interaction due to, for instance, alterations in local polymer conformation. For this reason, we believe that our model will be pivotal in studying complex active collective phenomena where interactions between motors are mediated through the target substrate.
![]() | (4) |
〈ηα(t)〉 = 0 and 〈ηα(t)ηβ(t′)〉 = δαβδ(t − t′). | (5) |
![]() | (6) |
![]() | (7) |
![]() | (8) |
The bending rigidity of the polymer is introduced via a Kratky–Porod potential for every three adjacent DNA beads
UBEND(θ) = KBEND[1 − cos(θ)], | (9) |
![]() | (10) |
The torque on the polymer beads is due to a potential which constrains bead orientations to lie along the polymer tangent, given by
UORIENT = KORIENT[1 − cos(ϕi)], | (11) |
![]() | (12) |
The motors are represented by freely diffusing beads which interact with each other via a WCA potential and via the potential given in eqn (1). A motor is said to be “bound” to the polymer when its distance to a polymer bead is less than rmcut = σ. In addition to this interaction, when a motor is bound to a polymer bead, it experiences an active force of magnitude f which is directed along the orientation vector of the polymer bead. As a consequence of Newton's third law, the polymer bead experiences an equal and opposite force. The result is that for non-zero f the motors track along the polymer substrate while driving the motion of the polymer. In order to avoid accumulation and jamming we want motors to detach once they have reached the end of the substrate. To achieve this we impose a large force fend ≫ KMOT/σ to motors when bound to the last bead of the chain. Though large, this force does not appreciably influence the dynamics nor the conformation of the chain. Since the polymer is polar, motors can be made to move in the parallel or anti-parallel direction depending on the sign of f. In the present work we consider the case where all motors are the same and f ≥ 0.
K MOT = 10 | K MOT = 20 | K MOT = 30 | ||||||
---|---|---|---|---|---|---|---|---|
N | f | 〈n〉 | N | f | 〈n〉 | N | f | 〈n〉 |
100 | 6 | 9 | 200 | 6 | 165 | 200 | 6 | 200 |
100 | 40 | 1 | 200 | 40 | 12 | 200 | 40 | 160 |
200 | 6 | 18 | 300 | 6 | 250 | 300 | 6 | 300 |
200 | 40 | 3 | 300 | 40 | 20 | 300 | 40 | 250 |
400 | 6 | 34 | 400 | 6 | 300 | 400 | 6 | 400 |
400 | 40 | 6 | 400 | 40 | 30 | 400 | 40 | 320 |
In Fig. 8a we show how the linear density of bound motors 〈n〉/L depends on L and f, at constant N = 200. For small motor force (the thermodynamic unbinding regime) we find that 〈n〉/L increases roughly linearly with increasing L, as would be expected at equilibrium; at high f longer polymers are more populated than shorter ones. This suggests that motors moving off the end of the polymer play a more important role in the latter regime. In Fig. 8b we plot the fraction of motors which are bound, 〈n〉/N, as a function of f at fixed L = 500; data for different N collapse onto the same curve, implying that, even in the large f regime, motor crowding does not have a large effect. Interestingly, if we plot the average “used” motor force f〈n〉N (Fig. 8b, inset) we observe non-monotonic behaviour: the total force is maximal at f ∼ 15kBT/σ.
![]() | (13) |
![]() | (14) |
![]() | (15) |
Additionally we measured two polymer shape parameters, the asphericity and prolateness. These are calculated from the principal moments of the gyration tensor, which is defined as
![]() | (16) |
![]() | (17) |
![]() | (18) |
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