Spontaneous Oscillations of Collective Molecular Motors

We analyze a simple stochastic model to describe motor molecules which cooperate in large groups and present a physical mechanism which can lead to oscillatory motion if the motors are elastically coupled to their environment. Beyond a critical fuel concentration, the non-moving state of the system becomes unstable with respect to a mode with angular frequency ω. We present a perturbative description of the system near the instability and demonstrate that oscillation frequencies are determined by the typical timescales of the motors. PACS Numbers: 87.10.+e, 05.40.+j Typeset using REVTEX 1 Motor proteins are highly specialized macromolecules which can consume mechanical energy to induce motion and to generate forces. These molecules are involved in active transport processes, cell locomotion and muscle contraction [1,2]. A typical motor molecule specifically attaches to a certain protein filament which serves as a track for its motion. In the presence of fuel, which in the cell ist adenosinetriphosphate (ATP), the motor starts moving in a direction defined by the polarity of the track filament. Experimental methods to study the physical properties of motor proteins have been developed in recent years [3–7]. These methods allow to measure forces and velocities of individual motors or small groups of motors. Normally, at given fuel concentration and temperature, motor molecules will generate a constant average force leading to a constant average velocity [7]. In some cases, however, biological motors are used to drive oscillatory motion. Insect flight muscles e.g. generate oscillating forces in order to move the wings with high frequency. While for some insects (e.g. butterflies) the timing of these oscillations is controlled by a periodic external nerve signal, others (e.g. bees and wasps) generate oscillations within the muscle. The mechanisms which lead to these oscillations are not understood but they seem to be related to the process of force-generation [8]. The purpose of this article is to describe theoretically a physical mechanism by which motor collections acting on a spring can give rise to oscillatory motion. Oscillations occur as the result of cooperation of many motors while individual motors only generate timeindependent velocities. The microscopic mechanisms which are responsible for the force generation of molecular motors are not understood. In order to indentify possible physical mechanisms which allow such a motor molecule to convert the chemical energy of a fuel to motion and mechanical work, simple physical models have been suggested [9–16]. One idea [10] is to simplify the internal degrees of freedom of the motor to two different states of a particles which moves along a one-dimensional coordinate x. The interaction of the particle with the track is described by periodic potentials which reflect the periodic structure along the surface of the 2 track. Within such a two-state model motion is induced if the potentials are asymmetric with respect to x → −x (which reflects the polarity of the track) and if detailed balance of the transitions between states is broken [10]. New phenomena occur if the collective motion of many particles is considered [17–19]. In a simple model for collective motion, particles are rigidly coupled to a common structure which they set in motion collectively [18]. As a result of cooperation, motion can in this case even occur in a symmetric system via spontaneous symmetry breaking. Instabilities and dynamical phase transitions also occur in asymmetric systems revealing the rich behaviour of many-motor systems. At these transitions, a steady state becomes unstable and the system chooses a new steady state. These transitions correspond therefore to dynamical instabilities at angular frequency ω = 0. In the following, we demonstrate that many coupled motors also can induce instabilities with ω 6= 0 towards a state with periodically varying velocity. Such instabilities occur if the system is elastically coupled to its environment. This implies, that the complete system is connected to its environment via a spring attached to the common backbone, see Fig. 1 [20]. A single particle reaches in this case a steady state with zero velocity and a position at which the average force generated by the motor is balanced by the elastic force of the spring [5]. We are interested in the behavior of many motors coupled together. Let us first review the most simple two-state model which we are going to discuss [10,18]. Motors are described by particles which move in two periodic potentials Wσ(x) with period l. Transitions between states σ = 1, 2 occur with rates ω1(x) and ω2(x) which also are l-periodic. The excitation amplitude can in general be written as [10] ω1(x) = ω2(x)[exp[(W1−W2)/T ]+Ωθ(x)], where T denotes temperature measured in units of the Boltzmann constant. We assume that the periodic function θ(x) is given. In principle it can be calculated from a specific model for the reaction kinetics of the fuel coupled to the particle [21]. The amplitude Ω is related to the difference of chemical potentials ∆μ ≡ μATP −μADP −μP of the fuel and its products which is the chemical driving force of the motors. Here, we assume that the fuel is ATP which 3 is hydrolyzed to adenosinediphosphate (ADP) and phosphate (P): ATP ⇀↽ ADP + P . For ∆μ = 0, Ω = 0 and the rates ω1 and ω2 obey detailed balance ω1/ω2 = exp[(W1−W2)/T ]. As soon as ∆μ 6= 0, Ω becomes nonzero. For small ∆μ/T , Ω ∼ ∆μ; for large ∆μ/T , Ω ∼ CATP is proportional to the fuel concentration. We now study the behavior of many particles which are rigidly connected to a common backbone, see Fig. 1. Particles have a fixed spacing q which for simplicity is assumed to be incommensurate with the period l, i.e. l/q is irrational. In the limit of an infinite system one can then introduce densities Pσ(ξ) with σ = 1, 2 which give the probability to find a particle at position ξ = x mod l relative to the beginning of the potential period in state σ. These densities are not independent but obey P1(ξ) + P2(ξ) = 1/l and ∫ l 0 dξ(P1 + P2) = 1. The equations of motion for this system read [18] ∂tP1 + v∂ξP1 = −ω1P1 + ω2P2 ∂tP2 + v∂ξP2 = ω1P1 − ω2P2 (1) fext = λv +KX + ∫ l 0 dξP1∂ξ(W1 −W2) . (2) Eq. (1) describes the dynamics of the density P1(ξ) resultig from motion of the backbone with velocity v = ∂tX and the transitions between the states. Eq. (2) is a force balance: The externally applied force per particle fext(t) is balanced by viscous drag with damping coefficient λ, the average force excerted by the potentials and an additional elastic force KX of a spring of length X with elastic modulus KN where N is the number of particles. Note, that Eqns. (1) and (2) only depend on the difference W1 −W2 of the two potentials and we can choose W2 to be constant without loss of generality. As a consequence of the spring action, a nonmoving solution to Eqns. (1) and (2) exists for fext = 0, v = 0, P1 = R ≡ ω2/(αl), where α(ξ) ≡ ω1(ξ) + ω2(ξ) and X = X0 ≡ − ∫ l 0 dξR∂ξ(W1 − W2)/K. We first study the linear stability of this solution and determine the instability threshold with respect to oscillations. With the ansatz P1(ξ, t) = R(ξ) + p(ξ) exp(st), v(t) = u exp(st) and X(t) = X0 + u exp(st)/s, one finds using Eq. (1) and (2) to linear order in p and the velocity amplitude u: 4 p(ξ) = −u ∂ξR s+ α(ξ) . (3) The possible values of the complex eigenvalue s = −τ + iω are determined by λ+ K s = ∫ l 0 dξ ∂ξR ∂ξ(W1 −W2) s+ α(ξ) . (4) The non-moving state is unstable if τ < 0. The instability occurs for τ = 0, where the real part of s vanishes. This happens at a threshold value Ω = Ωc(K) for which α(ξ) = αc(ξ). This critical value together with the frequency ωc at the instability is determined by λ = ∫ l 0 dξ αc α2 c + ω 2 c ∂ξR ∂ξ(W1 −W2) (5) K = ∫ l 0 dξ ω c α2 c + ω 2 c ∂ξR ∂ξ(W1 −W2) . (6) In the limit K = 0 where no elastic element is present, it follows from Eq. (6) that ωc = 0. Eq. (5) now determines Ωc(0) which is the condition for the instability of the nonmoving state as previously described in Ref. [18]. In the case of a symmetric system, this instability leads to spontaneous motion via a symmetry-breaking transition. For nonzero K the instability occurs for Ωc(K) = Ωc(0) + δΩc with δΩc ∼ K for small K and the system starts to oscillate with finite angular frequency ωc ∼ K . Note, that beyond a maximal value K > Kmax, the resting state is stable for any value of Ω. Fig. 2 (a) and (b) show examples for these oscillations which have been calculated numerically for constant deexcitation rate ω2, λω2l /U = 0.1 and piecewise linear and symmetric potentials Wσ(ξ), see Fig. 1 (a) with a = l/2. In this case Kmaxl /U ≃ 2.8. The function θ(ξ) is chosen as shown in Fig. 1 (a) with d/l = 0.1. Fig. 2 (a) displays the position X versus time t for an excitation level Ω = 0.1 slightly above threshold and elastic modulus Kl/U = 0.2. After an initial relaxation period, motion is almost sinusoidal. An example for small elastic molulus Kl/U = 0.002 and same Ω is shown in Fig. 2 (b). The periodic motion in this case shows cusp-like maxima which correspond to sudden changes of the velocity. These cusps are related to discontinuities of the velocity as a function of external force for K = 0 [18]. 5 We now give an analytical description of oscillations in the vicinity of the instability. Anticipating, that the motion is periodic with period tP ≡ 2π/ω, we can write P1(ξ, t) = ∑ ∞ k=−∞ P1(ξ, k)e , v(t) = ∑ ∞ k=−∞ vke , and fext(t) = ∑ ∞ k=−∞ fke , which defines the Fourier coefficients P1(ξ, k), vk and fk. Using this representation, one can derive the nonlinear relation between velocity and external force fk = F (1) kl vl + F (2) klmvlvm + F (3) klmnvlvmvn +O(v ) . (7) The coefficients F (n) k,k1,..,kn can be calculated by first rewriting Eq. (1) as P1(ξ, k) = δk,0R(ξ)− ∑ lm δk,l+m α + iωk vl∂ξP1(ξ,m) . (8) Inserting the ansatz P1(ξ, k) = Rδk,0 + P (1) kl (ξ)vl + P (2) klm(ξ)vlvm +O(v ) , (9) into Eq. (8), one obtains a recursion relation for the functions P (n) k,k1,..,kn : P (ξ)k,k1,..,kn = − ∑ l δk,kn+l α+ iωk ∂ξP (n−1) l,k1,..,kn−1 (10) Using Eq. (2), one finds