The relationship between curvature, flexibility and persistence length in the tropomyosin coiled-coil

Abstract

The inherent flexibility of rod-like tropomyosin coiled-coils is a significant factor that constrains tropomyosin's complex positional dynamics on actin filaments. Flexibility of elongated straight molecules typically is assessed by persistence length, a measure of lengthwise thermal bending fluctuations. However, if a molecule's equilibrium conformation is curved, this formulation yields an "apparent" persistence length ( approximately 100nm for tropomyosin), measuring deviations from idealized straight conformations which then overestimate actual dynamic flexibility. To obtain the "dynamic" persistence length, a true measurement of flexural stiffness, the average curvature of the molecule must be taken into account. Different methods used in our studies for measuring the dynamic persistence length directly from Molecular Dynamics (MD) simulations of tropomyosin are described here in detail. The dynamic persistence length found, 460+/-40nm, is approximately 12-times longer than tropomyosin and 5-times the apparent persistence length, showing that tropomyosin is considerably stiffer than previously thought. The longitudinal twisting behavior of tropomyosin during MD shows that the amplitude of end-to-end twisting fluctuation is approximately 30 degrees when tropomyosin adopts its near-average conformation. The measured bending and twisting flexibilities are used to evaluate different models of tropomyosin motion on F-actin.

Figure 1

Tropomyosin bending. (A) Bending of an homogeneous curved rod. θ is the overall curvature, used to measure the flexibility of straight rods. δ is the angular deflection from the averaged curved shape, needed to measure the flexibility of curved rods. R is the curvature radius and s is the arc length, where R = s/θ. (B) The average structure of tropomyosin over 1 ns intervals taken at different times along the MD simulation. For comparison, the Lorenz-Holmes model is shown in grey. (C) The structure of tropomyosin averaged over the entire MD (chains A and B of the coiled coil in blue and cyan), shown docked onto F-actin. Amino acid residues facing actin are in red.

Figure 2

Fluctuations of tropomyosin twisting during MD. (A, B) The [x,y] coordinates correspond to the position of the tropomyosin C-terminus, after superposing the 15 N-terminal residues to a common reference oriented along the z-axis (i.e., the N-terminus projects to the [0,0] position). The position of the C-terminus in the average tropomyosin structure is [−82Å, 31Å] (indicated by a white diamond). A) The end-to-end twisting angle ψ observed in each [x, y] position (averaged over the MD, see Methods section) is shown as color map (dark blue: ψ=40°; dark red: ψ=150°). B) Same as panel A, but showing the standard deviation of ψ during the MD (dark blue: stdev(ψ)=10°; dark red: stdev(ψ)=35°).

Figure 3

Tangent correlation plot of the angle θ(s) (see Fig. 1A) along EM images of negatively stained tropomyosin molecules, used to get the apparent persistence length ξ_a from the inverse slope of the linear regression (see Equation 6).

Figure 4

Dynamic persistence length of tropomyosin. (A) Tangent correlation plot of the angle δ(s,t) (see Fig. 1A) along conformers of the MD simulation, used to get the overall dynamic persistence length ξ_d from the inverse slope of the linear regression (see Equation 7). (B) The local dynamic persistence length, based on the fluctuations of δ(t) over a 9 residues window centered on each residue.