Optimal flexibility for conformational transitions in macromolecules
📝 Abstract
Conformational transitions in macromolecular complexes often involve the reorientation of lever-like structures. Using a simple theoretical model, we show that the rate of such transitions is drastically enhanced if the lever is bendable, e.g. at a localized “hinge’’. Surprisingly, the transition is fastest with an intermediate flexibility of the hinge. In this intermediate regime, the transition rate is also least sensitive to the amount of “cargo’’ attached to the lever arm, which could be exploited by molecular motors. To explain this effect, we generalize the Kramers-Langer theory for multi-dimensional barrier crossing to configuration dependent mobility matrices.
💡 Analysis
Conformational transitions in macromolecular complexes often involve the reorientation of lever-like structures. Using a simple theoretical model, we show that the rate of such transitions is drastically enhanced if the lever is bendable, e.g. at a localized “hinge’’. Surprisingly, the transition is fastest with an intermediate flexibility of the hinge. In this intermediate regime, the transition rate is also least sensitive to the amount of “cargo’’ attached to the lever arm, which could be exploited by molecular motors. To explain this effect, we generalize the Kramers-Langer theory for multi-dimensional barrier crossing to configuration dependent mobility matrices.
📄 Content
Many biological functions depend on transitions in the global conformation of macromolecules, and the associated kinetic rates can be under strong evolutionary pressure. For instance, the directed motion of molecular motors is based on power strokes [1], protein binding to DNA can require DNA bending [2] or spontaneous partial unwrapping of DNA from histones [3,4], and the functioning of some ribozymes depends on global transitions in the tertiary structure [5]. These and other examples display two generic features: (i) A long segment within the molecule or complex is turned during the transition, e.g. an RNA stem in a ribozyme, the DNA as it unwraps from histones or bends upon protein binding, or the lever arm of a molecular motor relative to the attached head. (ii) The segment has a certain bending flexibility. Here, we use a minimal physical model to study the coupled dynamics of the transition and the bending fluctuations.
Our model, illustrated in Fig. 1, demonstrates explicitly how even a small bending flexibility can drastically accelerate the transition. Furthermore, if the flexibility arises through a localized “hinge”, e.g. in the protein structure of some molecular motors [6,7] or an interior loop in an RNA stem, we find that the transition rate is maximal at an intermediate hinge stiffness. Thus, in situations where rapid transition rates are crucial, molecular evolution could tune a hinge stiffness to the optimal value. We find that an intermediate stiffness is optimal also from the perspective of robustness, since it renders the transition rate least sensitive to changes in the drag on the lever arm, incurred e.g. by different cargos transported by a molecular motor.
Our finding of an optimal rate is reminiscent of a phenomenon known as resonant activation [8,9], where a transition rate displays a peak as a function of the characteristic timescale of fluctuations in the potential barrier. However, we will see that the peak in our system has a different origin: a trade-off between the accelerating effect of the bending fluctuations and a decreasing average mobility of the reaction coordinate. The standard Kramers-Langer theory [10] for multi-dimensional transition processes is not sufficient to capture this trade-off. A generalization of the theory to the case of configurationdependent mobility matrices turns out to be essential to understand the peak at intermediate stiffness.
Model.-We model the conformational transition as a thermally activated change in the attachment angle ϕ of a macromolecular lever, see Fig. 1. The lever has two segments connected by a hinge with stiffness ǫ, which renders the lever preferentially straight, but allows thermal fluctuations in the bending angle θ. The energy function V (ϕ, θ) of this ‘Two-Segment Lever’ (TSL) is
where k B T is the thermal energy unit. The hinge, described by the first term, serves not only as a simple model for a protein or RNA hinge, but also as a zeroth-order approximation to a more continuously distributed flexibility; see below. The second term is the potential on the attachment angle ϕ, which produces a metastable minimum at (ϕ, θ) = (0, 0). The thermallyassisted escape from this minimum passes through the transition state at (ϕ, θ) = (b/a, 0) with a barrier height ∆V = b 3 k B T /6 [20]. In the present context, inertial forces are negligible, i.e. it is sufficient to consider the stochastic dynamics of the TSL in the overdamped limit. We localize the friction forces to the ends of the two segments, as indicated by the beads in Fig. 1(a). The length of the first segment defines our length unit and ρ denotes the relative length of the second segment. Similarly, we choose our time unit such that the friction coefficient of the first bead is unity, and denote the coefficient of the second bead by ξ. To describe the Brownian dynamics of the TSL, we derive the Fokker-Planck equation for the time-evolution of the configurational probability density p(ϕ, θ, t). In general, the derivation of the correct dynamic equations can be a nontrivial task for stochastic systems with constraints [11,12]. For instance, implementing fixed segment lengths through the limit of stiff springs, leads to Fokker-Planck equations with equilibrium distributions that depend on the way in which the limit is taken [12]. However, for our overdamped system, we can avoid this problem by imposing the desired equilibrium distribution, i.e. the Boltzmann distribution p = exp(-V /k B T ), which together with the well-defined deterministic equations of motion uniquely determines the Fokker-Planck equation for the TSL.
The deterministic equations of motion take the form qk = -M kl ∂V /∂q l with the coordinates (q 1 , q 2 ) = (ϕ, θ) and a mobility matrix M. We obtain M with a standard Lagrange procedure: Given linear friction, M is the inverse of the friction matrix, which in turn is the Hessian matrix of the dissipation function [13]. This yields
The Fokker-Planck equat
This content is AI-processed based on ArXiv data.