Towards synthetic molecular motors: a model elastic-network study

The proposed model motor consists of a cyclically operating machine that interacts with the filament by employing a ratchet mechanism which transforms cyclic machine movements into the translational motion of the filament. First, we explain how the machine is designed. Its interactions with the filament are described in the next section.

The EN machine which we use has been proposed by Togashi and Mikhailov [23] and has previously been employed in other applications [49–51]. Nonentheless, only a brief description of the machine has so far been provided [23]. It has been designed [23] by using evolutionary optimization methods, in such a way that its conformational dynamics and ligand-induced internal motions closely resemble those of actual protein machines. Particularly, its equilibrium states, with and without the ligand, have large attraction basins, so that even after large perturbations the system returns back to them. Moreover, relaxation proceeds along a well-defined path in the conformational space.

The model machine is made up of two relatively rigid domains connected by a flexible hinge region (figure 1). A substrate ligand can bind in a binding pocket situated in the hinge region and induce a closing motion of the domains about the hinge. Furthermore, the substrate ligand can get converted to the product and get released, thus bringing about the reverse opening hinge motion. The machine has 64 identical particles connected by a set of identical elastic links.

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The elastic energy of the machine is given by

$$\begin{eqnarray}&&{ \mathcal E }({\bf{R}})=\displaystyle \frac{1}{2}\kappa \displaystyle \sum _{i=1,j\gt i}^{64}{A}_{{ij}}{({d}_{{ij}}-{d}_{{ij}}^{0})}^{2}.\end{eqnarray} \tag{ 1 }$$

Here ${\bf{R}}=\{{{\bf{R}}}_{i}\}$ is the set of all coordinates ${{\bf{R}}}_{i}$ of the particles $i=1,2,\ldots ,64$. κ is the elastic constant which is the same for all the elastic links in the network. The network architecture is specified by the adjacency matrix whose elements A_ij are either 1 or 0 depending on whether a bond exists between a pair of particles i and j. Furthermore, ${d}_{{ij}}=| {{\bf{R}}}_{i}-{{\bf{R}}}_{j}|$ is the distance between the pair of particles i and j, and ${d}_{{ij}}^{0}=| {{\bf{R}}}_{i}^{0}-{{\bf{R}}}_{{j}}^{0}|$ is the equilibrium distance between them; $\{{{\bf{R}}}_{{i}}^{0}\}$ are equilibrium positions of all the nodes. The equilibrium positions ${{\bf{R}}}_{i}^{0}$ of the particles and the matrix of connections A_ij are given in [23] (see also methods).

The machine operates inside a viscous fluid. Its dynamics is overdamped and is described by the equations

$$\begin{eqnarray}&&\displaystyle \frac{{\rm{d}}{{\bf{R}}}_{i}}{{\rm{d}}t}=-\gamma \displaystyle \frac{\partial { \mathcal E }}{\partial {{\bf{R}}}_{i}},\end{eqnarray} \tag{ 2 }$$

where γ is the mobility coefficient, the same for all beads $i=1,\ldots ,64$. Later, thermal fluctuations will be taken into account by introducing noise terms into such relaxation equations. Note that inertial effects can also be neglected for real protein machines, when conformational motions on the time scales larger than a picosecond are considered. Hydrodynamic effects are not treated in the present study, but they can be included too [49].

When the ligand binds, it forms elastic links with three beads ($i=32,40,41)$ located in the hinge area. The ligand itself is treated in our model as an additional particle (i = 65). It is convenient to introduce a binary variable s which takes the value s = 1, if the ligand is attached, and s = 0 otherwise. The expression for the elastic energy, including the ligand at the position ${{\bf{R}}}_{65}$, is

$$\begin{eqnarray}&&{ \mathcal E }({\bf{R}};s)=\displaystyle \frac{1}{2}\kappa \displaystyle \sum _{i=1,j\gt i}^{64}{A}_{{ij}}{({d}_{{ij}}-{d}_{{ij}}^{0})}^{2}+\displaystyle \frac{1}{2}s\kappa \displaystyle \sum _{i=32,40,41}{({d}_{i,65}-{d}_{i,65}^{0})}^{2}.\end{eqnarray} \tag{ 3 }$$

The last ligand interaction term depends on the distances ${d}_{i,65}=| {{\bf{R}}}_{i}-{{\bf{R}}}_{65}|$ between the ligand and the three network beads. Moreover, ${d}_{i,65}^{0}$ are the natural lengths of the three additional links, ${d}_{32,65}^{0}={d}_{40,65}^{0}={d}_{41,65}^{0}={d}_{{lig}}^{0}$. For simplicity, we assume that the ligand links have the same elastic constant κ as the other links in the network. When the ligand is present, the network dynamics is given by equations (2) where the expression (3) for the elastic energy should be used.

Next, the conditions at which binding or release of the ligand take place need to be specified. Moreover, the initial position of the ligand after binding should also be defined. In the present study, we use the ligand binding and detachment conditions which are slightly different from the original formulation [23] and are defined in terms of the hinge angle; such conditions have already been employed in the study of membrane machines [51]. The hinge angle ϕ is introduced through the equation

$$\begin{eqnarray}&&\mathrm{cos}\phi =\displaystyle \frac{({{\bf{R}}}_{7}-{{\bf{R}}}^{\mathrm{cm}})\cdot ({{\bf{R}}}_{64}-{{\bf{R}}}^{\mathrm{cm}})}{| {{\bf{R}}}_{7}-{{\bf{R}}}^{\mathrm{cm}}| | {{\bf{R}}}_{64}-{{\bf{R}}}^{\mathrm{cm}}| }.\end{eqnarray} \tag{ 4 }$$

Where ${{\bf{R}}}^{\mathrm{cm}}$ is the centre of mass of the three beads $i=32,40$ and 41 which form the ligand binding pocket

$$\begin{eqnarray}&&{{\bf{R}}}^{\mathrm{cm}}=\displaystyle \frac{1}{3}({{\bf{R}}}_{32}+{{\bf{R}}}_{40}+{{\bf{R}}}_{41}).\end{eqnarray} \tag{ 5 }$$

Because the two arms of the machine are fairly stiff, the angle ϕ can be interpreted as the angle between the two arms (see figure 2).

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In the equilibrium state of the ligand-free network the hinge angle is relatively large ($\phi ={\phi }_{0}$) and we refer to this state as the open conformation of the machine. We assume that binding of the substrate ligand is possible within a certain interval $[{\phi }_{0}-{{\rm{\Delta }}}_{0},{\phi }_{0}+{{\rm{\Delta }}}_{0}]$ of hinge angles in the vicinity of the open conformation. Within this interval, it can take place with some transition probability w₀ per unit time. We assume that, when the ligand arrives, it becomes located in the centre of mass of the binding pocket formed by the particles $i=32,40,41;$ hence we have ${{\bf{R}}}_{65}^{0}={{\bf{R}}}^{\mathrm{cm}}$ at the moment of binding.

The ligand-network complex will undergo relaxation to its own equilibrium state. This relaxation process will be described by equations (2) where the expression (3) for the elastic energy with s = 1 should be used. Note that, when the ligand is attached to the network, it moves like any other particle; we assume for simplicity that its mobility γ is the same.

When designing the machine, the natural lengths of the ligand links are chosen to be shorter than their typical initial lengths. Therefore, these links tend to shrink. Note that, through the ligand located in the hinge region, the two arms become additionally linked. When the ligand links get shorter, the two arms move closer one to another and the hinge angle decreases. The equilibrium state of the ligand-network complex corresponds therefore to a closed conformation of the machine which is characterized by a relatively small hinge angle $\phi ={\phi }_{1}$.

Figure 3 displays the operation cycle of the designed elastic machine. At the beginning of the cycle (figure 3(a)) the machine is in an open conformation. The substrate ligand (red particle) arrives into the ligand binding pocket in the hinge region and establishes elastic links (figure 3(b)) to three neighbouring beads (blue colour). This induces a transition to the the closed conformation (figure 3(c)) which corresponds to the equilibrium state of the ligand-machine complex. Now the ligand is converted to the product particle which is instantaneously released (figure 3(d)). After that the free machine returns to its equilibrium open conformation and the cycle can be repeated. The detailed operation of the machine can be observed in video 1.

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The reaction is introduced by assuming that in the vicinity of the closed conformation the nature of the bound ligand can suddenly change, so that it gets transformed to a product particle which is immediately released. This transition occurs with the probability w₁ per unit time within the interval $[{\phi }_{1}-{{\rm{\Delta }}}_{1},{\phi }_{1}+{{\rm{\Delta }}}_{1}]$ of the hinge angle. The release of the product is characterized by sudden disappearance of the three elastic links between the ligand and the network. After the transition the elastic energy of the network is described by equation (3) with s = 0. As we assume, the product is immediately evacuated and therefore the reverse binding of the product to the machine cannot occur. We also neglect the possibility that the substrate dissociates before it gets converted to the product.

Our aim is to imitate some aspects of the functioning of a real molecular motor, the muscle myosin. In each of its cycles, the myosin performs a power stroke, induced by binding of ATP and the hydrolysis reaction. Under the power stroke, the myosin head is attached to the actin filament and generates a force to drag it. After the power stroke the head gets detached and the protein returns to its initial conformation. Because the myosin holds the filament during the forward part of the cycle and is detached from it during the backward part, a ratchet effect is naturally implemented. As we will show, similar operation can be achieved by using the elastic machine described in the previous section. Note, however, that an important aspect of myosin motor operation would still be absent in our model. The myosin actively grasps the actin filament, because of the conformational changes in the actin binding cleft that are ligand-induced. Our model is too primitive to incorporate such a mechanism. Instead, a different interaction between the motor and the filament will be employed.

Before we proceed to the detailed formulation, we want to show how the motor would work. Video 2 gives an example of the motor operation in absence of thermal fluctuations. When the ligand binds, a power stroke is executed. As the machine moves forward approaching its closed conformation, the swinging arm comes very near the filament and establishes an attractive bond with it, thereby dragging the filament forward. While holding the filament, it moves forward, until the ligand conversion into the product and its release occur. In the second part of the cycle, when the machine returns to its equilibrium open conformation, the swinging arm moves along a path farther away from the filament and no attractive bond is formed, so the machine and the filament stay detached from each other.

Figure 4(a) illustrates the set-up of our model. As shown in figure 4(a), the filament is positioned along the x-axis of our chosen coordinate system. It can only move back or forth along this axis. The swinging free arm of the elastic machine is assumed to have a special node (i = 64) that can interact with the filament. The other arm of the machine is immobilized by fixing the nodes, $i=1,2,\;{\rm{and}}\;10$ at ${{\bf{R}}}_{1}=\{{x}_{1},{y}_{1},{z}_{1}\}$, ${{\bf{R}}}_{2}=\{{x}_{2},{y}_{2},{z}_{2}\}$ and ${{\bf{R}}}_{10}=\{{x}_{10},{y}_{10},{z}_{10}\}$.

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The trajectory of the end particle (i = 64) of the machine as it undergoes cyclic conformational changes is displayed in figure 4(b). The trajectory of the end particle is an almost planar loop, lying approximately on the x-z plane. Figure 5(a) shows the time dependence of the distance h₆₄ of the interacting node from the filament. For the chosen position of the machine, the end particle remains close to the filament during the forward part of the cycle (s = 1) while it is relatively far away during the reverse part (s = 0). Additionally, figure 5(b) shows the time dependence of the position x₆₄ of the end particle along the filament. During the power stroke (s = 1), the arm moves in the forward direction, whereas the reverse motion occurs during the rest of the operation cycle for the motor.

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Note that the time of the power stroke is much shorter than the time needed for the arm to return to its initial position. This is because, during the power stroke, additional attractive interactions between the arm and the head of the machine are present through the ligand located in the hinge.

Real actin filaments are formed by a periodic linear arrangement of monomers representing single actin proteins. In our model system, we do not want to reproduce such complex structure. Instead, it is assumed that the filament represents a rigid rod whose motions are constrained to a line chosen as the x-axis in our coordinate frame. Along the filament, periodically spaced force centres are located. Their positions are

$$\begin{eqnarray}&&{x}_{n}(t)=X(t)+{na},\quad n=0,\pm 1,\pm 2,\cdots ,\pm m,\end{eqnarray} \tag{ 6 }$$

where a is the spatial period and X(t) is the coordinate of the entire rigid filament at time t. Each of the force centres is capable of interacting with the machine tip.

To model the interaction between the filament and the machine tip, Lennard–Jones potentials are used. The total interaction potential ${{ \mathcal V }}^{\mathrm{int}}$ between the end particle ($i=64)$ of the machine and the filament is given by a sum of pair interation potentials

$$\begin{eqnarray}&&{{ \mathcal V }}^{\mathrm{int}}({{\bf{R}}}_{64},X)=\displaystyle \sum _{n=-m}^{m}{{ \mathcal V }}^{\mathrm{int}}({\rho }_{n}),\end{eqnarray} \tag{ 7 }$$

where

$$\begin{eqnarray}{{ \mathcal V }}^{\mathrm{int}}({\rho }_{n})=\left\{\begin{array}{ll}{V}_{\mathrm{LJ}}({\rho }_{n})-{V}_{\mathrm{LJ}}({l}_{c}) & \quad {\rm{}}\;{\rm{if}}\;{\rm{}}{\rho }_{n}\lt {l}_{c};\\ 0 & \quad {\rm{otherwise}},\end{array}\right.\end{eqnarray} \tag{ 8 }$$

and

$$\begin{eqnarray}&&{V}_{\mathrm{LJ}}({\rho }_{n})=\sigma \left[{\left(\displaystyle \frac{C}{{\rho }_{n}}\right)}^{12}-2{\left(\displaystyle \frac{C}{{\rho }_{n}}\right)}^{6}\right],\end{eqnarray} \tag{ 9 }$$

Here ${\rho }_{n}$ is the distance between the end particle (i = 64) and the nth force centre of the filament, ${\rho }_{n}^{2}={h}_{64}^{2}+{(X+{na}-{x}_{64})}^{2}$. The coefficient σ specifies the interaction strength, while the parameters C and l_c determine the range of the interaction.

Note that the interaction potential is a periodic function of the filament position X and it has the period a. Indeed, the total interaction with the (infinite) filament is not changed if the filament is shifted by distance a corresponding to the separation between individual force centres. Figure 6 shows the interaction potential (equation (7)) as a function of the filament position X for three different distances h₆₄. The periodicity of the effective interaction potential is determined by the lattice distance a, of the force centres on the filament.

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Thus, the equations of motion for the particles ($i=1,\cdots ,64$) of the motor and the filament have the form

$$\begin{eqnarray}&&\displaystyle \frac{{\rm{d}}{{\bf{R}}}_{i}}{{\rm{d}}t}=-\gamma \displaystyle \frac{\partial { \mathcal E }({\bf{R}};s)}{\partial {{\bf{R}}}_{i}}+{\vec{\zeta }}_{i}(t);(i\ne 1,2,10,64),\end{eqnarray} \tag{ 10 }$$

$$\begin{eqnarray}&&\displaystyle \frac{{\rm{d}}{{\bf{R}}}_{64}}{{\rm{d}}t}=-\gamma \displaystyle \frac{\partial { \mathcal E }({\bf{R}};s)}{\partial {{\bf{R}}}_{64}}-\gamma \displaystyle \frac{\partial {{ \mathcal V }}_{\mathrm{int}}({{\bf{R}}}_{64},X)}{\partial {{\bf{R}}}_{64}}+{\vec{\zeta }}_{64}(t),\end{eqnarray} \tag{ 11 }$$

$$\begin{eqnarray}&&\displaystyle \frac{{\rm{d}}{{\bf{R}}}_{1}}{{\rm{d}}t}=\displaystyle \frac{{\rm{d}}{{\bf{R}}}_{2}}{{\rm{d}}t}=\displaystyle \frac{{\rm{d}}{{\bf{R}}}_{10}}{{\rm{d}}t}=0,\end{eqnarray} \tag{ 12 }$$

$$\begin{eqnarray}&&\displaystyle \frac{{\rm{d}}X}{{\rm{d}}t}=-{\rm{\Gamma }}\displaystyle \frac{\partial {{ \mathcal V }}_{\mathrm{int}}({{\bf{R}}}_{64},X)}{\partial X}-{\rm{\Gamma }}{F}_{\mathrm{ext}}+\chi (t).\end{eqnarray} \tag{ 13 }$$

Here ${ \mathcal E }$ is the elastic energy (3) of the machine and ${{ \mathcal V }}_{\mathrm{int}}$ in the interaction given by equation (7). The mobilities of the machine particles and of the filament are γ and Γ, respectively. We have added to the last equation the external force F_ext which may be applied to the filament.

The last terms in the equations (10), (11) and (13) take into account thermal fluctuations. In equations (10) and (11) , ${\vec{\zeta }}_{i}(t)=\{{\zeta }_{i}^{x}(t),{\zeta }_{i}^{y}(t),{\zeta }_{i}^{z}(t)\}$ are independent white vector noises having the correlations

$$\begin{eqnarray}&&\langle {\zeta }_{i}^{\alpha }(t){\zeta }_{j}^{\beta }(t^{\prime} )\rangle =2\gamma {k}_{{\rm{B}}}T{\delta }_{{ij}}{\delta }_{\alpha \beta }\delta (t-t^{\prime} ){\rm{}}\;{\rm{where}}\;{\rm{}}\alpha ,\beta =x,y,z,\end{eqnarray} \tag{ 14 }$$

k_B is the Boltzmann constant and T is the temperature. In equation (13) $\chi (t)$ is a white noise with the correlation function

$$\begin{eqnarray}&&\langle \chi (t)\chi (t^{\prime} )\rangle =2{\rm{\Gamma }}{k}_{{\rm{B}}}T\delta (t-t^{\prime} ).\end{eqnarray} \tag{ 15 }$$

The discrete variable s takes two values : s = 0 corresponds to the free machine, whereas s = 1 corresponds to the ligand-bound machine. Binding of the ligand, i.e., transition from s = 0 to s = 1, is a stochastic event which takes place with the probability rate w₀ within the interval $[{\phi }_{0}-{{\rm{\Delta }}}_{0},{\phi }_{0}+{{\rm{\Delta }}}_{0}]$ of the hinge angle given by equation (4). The release of the ligand, i.e., transition from s = 1 to s = 0, can take place with the probability rate w₁ inside the interval $[{\phi }_{1}-{{\rm{\Delta }}}_{1},{\phi }_{1}+{{\rm{\Delta }}}_{1}]$ of the hinge angle. Note that generally the rates w₀ and w₁ depend on temperature T. In our model system, such possible dependence is not however taken into account. Our investigations will be always performed near to the saturation regime, with binding or release of a ligand always taking place once the respective window has been entered. The focus will be on the fluctuation effects controlled by the temperature parameter in the model.

When s = 1, the machine includes a ligand particle (i = 65) and the equation of motion for this particle has the form

$$\begin{eqnarray}&&\displaystyle \frac{{\rm{d}}{{\bf{R}}}_{65}}{{\rm{d}}t}=-\gamma \displaystyle \frac{\partial { \mathcal E }({\bf{R}};s)}{\partial {{\bf{R}}}_{65}}+{\vec{\zeta }}_{65}(t),\end{eqnarray} \tag{ 16 }$$

where ${ \mathcal E }({\bf{R}};s)$ is given by equation (3). The ligand particle has the same mobility γ and is subject to the same kind of noise (equation (14)) as the machine particles. When the ligand arrives, i.e., at the moment of the transition from s = 0 to s = 1, its initial position is ${{\bf{R}}}_{65}={{\bf{R}}}^{\mathrm{cm}}$ where ${{\bf{R}}}^{\mathrm{cm}}$ is given by equation (5). When the reverse transition from s = 1 to s = 0 takes place, the ligand is immediately removed and we do not track its position any further. A new ligand particle is bound to the machine as the substrate and removed from it as the product in each next operation cycle.

In our numerical simulations, the dimensionless form of the model is employed. To obtain it, time is measured in units of $\tau =1/\kappa \gamma$ whereas the coordinates $\{{{\bf{R}}}_{i}\}$ and X as well as the distances h₆₄ and x₆₄ are measured in the units of $L=a/4$ where a is the spacing between consecutive force centres on the filament. Below the same notations are used for the rescaled variables. In the dimensionless form, the explicit evolution equations of the model are

$$\begin{eqnarray}&&\displaystyle \frac{{\rm{d}}{{\bf{R}}}_{i}}{{\rm{d}}t}=-\displaystyle \sum _{j=1}^{64}{A}_{{ij}}({{\bf{R}}}_{i}-{{\bf{R}}}_{j})\left(\displaystyle \frac{{d}_{{ij}}-{d}_{{ij}}^{0}}{{d}_{{ij}}}\right)+{\vec{\xi }}_{i}(t);\quad {\rm{for}}\ i\ne 1,2,10,64,\end{eqnarray} \tag{ 17 }$$

$$\begin{eqnarray}\displaystyle \frac{{\rm{d}}{{\bf{R}}}_{64}}{{\rm{d}}t} & = & -\displaystyle \sum _{j=1}^{64}{A}_{64,j}({{\bf{R}}}_{64}-{{\bf{R}}}_{j})\left(\displaystyle \frac{{d}_{64,j}-{d}_{64,j}^{0}}{{d}_{64,j}}\right)+\displaystyle \sum _{n=-m}^{m}\;{f}_{{int}}\quad ({\rho }_{n})\;\displaystyle \frac{\partial {\rho }_{n}}{\partial {{\bf{R}}}_{64}}+{\vec{\xi }}_{64}(t)\quad ;{\rm{}}\;{\rm{for}}\;{\rm{}}i=64,\end{eqnarray} \tag{ 18 }$$

$$\begin{eqnarray}&&\displaystyle \frac{{\rm{d}}{{\bf{R}}}_{i}}{{\rm{d}}t}=0;{\rm{for}}\ i=1,2,10,\end{eqnarray} \tag{ 19 }$$

$$\begin{eqnarray}&&\displaystyle \frac{{\rm{d}}{{\bf{R}}}_{65}}{{\rm{d}}t}=-\displaystyle \sum _{j=32,\;40,\;41}({{\bf{R}}}_{65}-{{\bf{R}}}_{j})\left(\displaystyle \frac{{d}_{65,j}-{d}_{65,j}^{0}}{{d}_{65,j}}\right)+{\vec{\xi }}_{65}(t);{\rm{for}}\;i=65({\rm{ligand}}),\end{eqnarray} \tag{ 20 }$$

$$\begin{eqnarray}&&\displaystyle \frac{{\rm{d}}X}{{\rm{d}}t}=\mu \displaystyle \sum _{n=-m}^{m}\;{f}_{{int}}\quad ({\rho }_{n})\quad \displaystyle \frac{\partial {\rho }_{n}}{\partial X}-\mu {f}_{\mathrm{ext}}+\eta (t),\end{eqnarray} \tag{ 21 }$$

$$\begin{eqnarray}&&{f}_{\mathrm{int}}=\epsilon \left[{\left(\displaystyle \frac{c}{{\rho }_{n}}\right)}^{13}-{\left(\displaystyle \frac{c}{{\rho }_{n}}\right)}^{7}\right]\;H({\rho }_{n}-{l}_{c})\end{eqnarray} \tag{ 22 }$$

where ${d}_{{ij}}=| {{\bf{R}}}_{i}-{{\bf{R}}}_{j}|$ and ${\rho }_{n}=\sqrt{{h}_{64}^{2}+{(X+4n-{x}_{64})}^{2}}$; we use the step function H(z) = 1 for z > 0 and H(z) = 0 for z ≤ 0. The dimensionless interaction strength coefficient is $\epsilon =12\sigma /\kappa \quad a\quad C$ and the dimensionless characteristic interaction distance is $c=4C/a$. The coefficient $\mu ={\rm{\Gamma }}/\gamma$ is the ratio of the mobilities of the machine particles and the filament. The rescaled external force is ${f}_{\mathrm{ext}}=(4/\kappa a){F}_{\mathrm{ext}}$

The rescaled equations include new noises ${\vec{\xi }}_{i}(t)=(\tau /L){\vec{\zeta }}_{i}(t)$ and $\eta (t)=(\tau /L)\chi (t)$ with the correlation functions

$$\begin{eqnarray}&&\langle {\xi }_{i}^{\alpha }({t}_{1}){\xi }_{j}^{\beta }({t}_{2})\rangle =2{\rm{\Theta }}{\delta }_{{ij}}{\delta }_{\alpha \beta }\delta ({t}_{1}-{t}_{2}){\rm{}}\;{\rm{where}}\;{\rm{}}\ \alpha ,\beta =x,y,z,\end{eqnarray} \tag{ 23 }$$

$$\begin{eqnarray}&&\langle \eta ({t}_{1})\eta ({t}_{2})\rangle =2\mu {\rm{\Theta }}\delta ({t}_{1}-{t}_{2}).\end{eqnarray} \tag{ 24 }$$

where Θ is the rescaled temperature given by ${\rm{\Theta }}=16{k}_{{\rm{B}}}T/\kappa {a}^{2}$. Note that the transition rate constants for the binding and release of the ligand also become rescaled and the dimensionless transition rate constants are ${\nu }_{0}={w}_{0}/\kappa \gamma$ and ${\nu }_{1}={w}_{1}/\kappa \gamma$.

When conversion of the substrate into the product is excluded, the ligand binds to the machine and stays indefinitely long within it. Therefore, the motor can only exhibit thermal fluctuations characteristic for its ligand-bound equilibrium state (s = 1). In video 3, we show a simulation of the motor in this case under relatively weak thermal noise (${\rm{\Theta }}=0.005$). The swinging arm of the motor gets attached to the filament and performs equilibrium thermal fluctuations together with it. As could be indeed expected under equilibrium conditions, no uni-directional movement of the filament is observed.

If the motor is still at equilibrium (no substrate conversion), but the temperature is increased (${\rm{\Theta }}=0.030$), the behaviour becomes different (see video 4). Now, the arm of the motor cannot firmly hold onto the filament and, as a result, the filament easily slides against it. In contrast to video 3, motions of the arm and the filament seem to be independent in this case.

The behaviour of the motor gets changed dramatically when the reaction, i.e., the conversion of the substrate into product, is included. Video 5 shows the motor operating under these conditions at a low level of thermal fluctuations (${\rm{\Theta }}=0.005$).

If the temperature is increased to ${\rm{\Theta }}=0.030$, it can be observed (Video 6) that the arm of the motor begins to slide against the filament, similar to what is seen in video 4. This means that in this case the motor can only weakly affect the intrinsic independent Brownian motion of the filament.

Figures 7(a) and (b) display trajectories of the bead i = 64, located at the end of the arm, in the plane of the variables h₆₄ (distance to the filament) and ϕ (hinge angle) at two temperatures ${\rm{\Theta }}=0.005$ and ${\rm{\Theta }}=0.03$, respectively. Here, the red colour indicates that the ligand is present inside the machine (s = 1) and the green colour corresponds to the free machine (s = 0). For comparison, we also show in figure 7(c) the trajectories in absence of thermal noise, i.e., ${\rm{\Theta }}=0$. For each temperature, the data for 10 consecutive cycles is taken. Examining figures 7(a) and (b), one can notice that the motor arm is persistently cycling in a counter-clockwise direction. We can also notice that even at the relatively high temperature (${\rm{\Theta }}=0.030$) in figure 7(b) the motions of the arm remain well-defined, so that the two branches with s = 0 and s = 1 do not overlap.

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Suppose first that the motor arm is fixed, so that the distance of the bead i = 64 from the filament, h₆₄, and the projection of its position on the filament, x₆₄, are both constant. Then the filament is effectively performing thermal Brownian motion in a periodic potential ${V}_{\mathrm{int}}(X)={V}_{\mathrm{int}}(X-{x}_{64},{h}_{64})$. This potential is given by equation (22) and displayed for three different values of h₆₄ in figure 6. As seen in figure 6, the height of the barrier strongly depends on the distance (h₆₄) between the arm and the filament. At the typical distance, ${h}_{64}\approx 0.4$, which is characteristic for the ligand-bound state (s = 1), the barrier height is about ${V}_{\mathrm{max}}\approx 0.02$. When the ligand is absent (s = 0), the separation gets increased above ${h}_{64}\gtrsim 2.0$ and, at such distances the interaction potential V_int vanishes. This means that in the ligand-free state the motor cannot exhibit any action on the filament.

Suppose now the motor arm moves in such a way that its separation from the filament h₆₄ remains constant, but its position x₆₄ with respect to the filament changes, ${x}_{64}={x}_{64}(t)$. Then the filament will experience a travelling periodic potential ${V}_{\mathrm{int}}(X,t)={V}_{\mathrm{int}}(X-{x}_{64}(t),{h}_{64})$.

At low temperatures ${\rm{\Theta }}\ll {V}_{\mathrm{max}}$ the filament gets locked into a travelling trough and moves together with it. This means that the moving motor arm is able to hold the filament and transport it by a power stroke when the ligand is bound (s = 1). In the other part of the cycle s = 0, the interaction is absent and therefore the arm can move back without the filament. This operation mode of the motor can be described as the strong coupling regime and can be observed in video 5. In such a regime, the motor works almost like a deterministic ratchet. The cycling machine generates a flashing travelling potential which is present in one part of the cycle, dragging the filament, and absent in the other part.

The above explanation of the motor operation in the strong coupling regime is only approximate and qualitative. For instance, one has to further take into account that the interaction potential changes gradually with the distance h₆₄ to the filament which varies within the operation cycle (see figure 6) and is also affected by thermal fluctuations. Moreover the motion of the arm and therefore the operation of the machine are also affected by the interactions with the filament. When the motor arm drags the filament in the power stroke, the reaction force acts on the arm and modifies its motion. Hence, the flashing potential is not external and independent of the filament dynamics.

Figure 8(a) displays the dependence of the filament coordinate X on time in the strong coupling regime. Additionally, highlighted narrow stripes in this figure indicate the intervals of time during which the motor is in the ligand-bound state (s = 1). The upward steps in the figure correspond to the directed translational motion of the filament , i.e. to the power strokes. It can be clearly seen that each such step takes place when the motor goes through the state s = 1. Due to thermal fluctuations, there is however a significant variation in the heights of the steps and the times between them. It can be also noticed that the durations of the power stroke, where the motor arm moves the filament, are much shorter than the intervals between them. This is because, during the power stroke, a ligand is present inside the hinge region and additional contracting links are established through it, making the dynamics faster. Note that our simulations are performed in the substrate saturation regime where the next substrate binds almost instantaneously when this is allowed and there is no waiting time for this.

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When the temperature is increased (or interactions with the filament made weaker), the weak coupling regime is encountered (see video 6). Now the temperature is always higher than the height of the barriers separating the troughs and therefore the moving arm of the motor cannot hold the filament. Essentially in this case the interactions with the motor arm provide only a small fluctuating perturbation to the intrinsic Brownian motion of the filament. This perturbation, however, still tends to drive the filament in a definite direction. Indeed, when the arm moves forward in the ligand-bound state (s = 1), it is more often located closer to the filament, as compared to the other half (s = 0) of the cycle when the arm moves back. Therefore, on the average some net driving force is generated. Figure 8(c) displays the dependence of the filament coordinate X on time in the weak coupling regime.

The transition from the strong coupling to the weak coupling regimes under an increase of temperature proceeds through the intermittent operation mode (see figure 8(b)). In this mode, the firm grip of the arm and its sliding motion against the filament are randomly alternating. Roughly, the intermittent regime is defined by the condition ${\rm{\Theta }}\approx {V}_{\mathrm{max}}$. Figure 9 shows the displacement of the filament with time at three different temperatures over roughly 50 cycles.

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We have performed systematic long simulations, covering 300 motor cycles each at different temperatures and determined the corresponding mean filament velocities v_mean. The results for the mean velocity v_mean and its statistical dispersion at different temperatures are shown in figure 10.

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Finally, figure 11 displays histograms of filament displacement per single cycle at the temperatures corresponding to different operation regimes. At relatively low temperature (figure 11(a)), almost all cycles result in the filament translation along the forward direction. There is however a large dispersion in step sizes. Occasional back steps in this regime are explained by Brownian motion of the filament during those parts of the cycle when the machine is detached from the filament. In the intermediate regime (figure 11(b)), a significant fraction of cycles results in the back steps, although the prevalence of forward steps is still clear. In the weak coupling regime (figure 11(c)), the distribution of steps is almost symmetric. Nonetheless, a slight bias towards the forward steps can still be observed, explaining the persistence of the directed filament motion.

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In the above analysis, the rescaled temperature ${\rm{\Theta }}=16{k}_{{\rm{B}}}T/\kappa {a}^{2}$ has been varied in a broad interval to control the level of fluctuations in the system. While this variation may correspond to changes in the real temperature T, it can be also achieved by modifying the stiffness constant κ and thus making the EN more or less soft.

Real protein motors are employed by biological cells not only to transport the load along the filaments, but also to generate force, as is indeed the case for muscle myosin. In the latter operation mode, the force generated by the motor counterbalances the external force, so that the net translational motion is absent. In this section, we investigate the operation of the proposed synthetic motor under the action of an external force.

Suppose that an external force f_ext is applied to the filament in the direction opposite to the direction of its motor-induced translational motion (see equation (21)). There are basically two effects of such external force. When the motor arm is attached to the filament, such a force will slow down the forward motion of the motor arm and the filament. In the other half of the cycle, when the arm is detached, this force induces a drift of the filament in the opposite direction.

Video 7 illustrates the motor operation in the presence of a strong external force, close to the stall condition, in the absence of thermal fluctuations. As can be observed, the motor pulls the attached filament in the forward direction for part of the cycle in the ligand-bound state. However, when the ligand is released and the motor arm moves away from the filament the attractive force between motor and filament decreases allowing the external force to dominate for a while and pushing the filament back.

When the external force is increased, the backward motion of the filament in the ligand-free state of the cycle becomes stronger. Under the stall condition, the active motor-induced forward motion of the filament should be on the average compensated by the force-induced backward motion. As a result, the net filament displacement will be vanishing.

By running simulations covering about 300 cycles each at different intensities of applied external force, we could determine the dependence of the mean filament velocity on the external force (see figure 12). As can be seen in figure 12, the mean propagation velocity of the filament indeed decreases and tends to zero, as the external force is increased. The stall condition could not be however reached in our simulations. The fluctuations in the propagation velocity were so rapidly increasing that extremely long simulations had to be performed to reliably identify the stall. Such long simulations were however beyond our capacity.

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