Principal-components analysis of shape fluctuations of single DNA molecules

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   Fig. 1.

   Shape fluctuations of λ-DNA. (a) Series of images showing equilibrium conformational fluctuations of fluorescently labeled λ-DNA. Here every eighth image from the full movie provided in SI Movie 1 is shown. (Scale bar, 2 μm.) (b) Experimentally determined first 16 PCs of conformational fluctuations, ordered by their associated eigenvalues. Symmetry forbids the existence of (1, 1) modes (analogous to the 2p hydrogen wavefunctions), because amplitude in these modes leads to a displacement of the center of mass. The color scale maps the most positive and most negative excursions to red and blue colors, respectively, and the color corresponding to zero at the edges of the PCs varies from panel to panel. (Scale bar, 2 μm.) (c) PCs predicted by a random-walk model. Because of the rotational symmetry, experimental and theoretical eigenfunctions are often aligned along different axes. (d) Stiffness of 45 of the low-energy modes. All modes with l ≠ 0 are 2-fold degenerate. A random-walk model, taking into account finite imaging resolution, yields a similar spectrum of eigenvalues. AU, arbitrary units. (e) Fraction of the total variance of the data accounted for by the first p modes. The first 34 modes account for 90% of the variance.

   
   
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   Fig. 2.

   Linear and nonlinear dynamics of the PCs. (a) Covariance matrix of amplitudes in the first 15 PCs at τ = 18 ms. The off-diagonal elements indicate that the PCs are not eigenstates of the time-evolution operator. The only significant transitions conserve l (i.e., are vertical on Fig. 1 d) and change n by ± 1. (b) Power-law scaling of the relaxation times in the first 45 PCs. The relaxation time was extracted from the short-time autocorrelation of the mode amplitudes. (c) Nonlinear couplings in the first 5 PCs. Each correlation function ρ̃_pq ⁽³⁾(τ) in the 5 × 5 array probes the effect of amplitude in mode a _q on the magnitude of the thermal fluctuations ξ_p ². Strong nonlinear interactions are shaded pink, and weak ones are in blue. Each black line is the calculation for a single molecule of λ-DNA, and the red lines are the ensemble average. Each box has a time axis of τ = (−450, 450 ms) and a vertical axis ρ̃_pq ⁽³⁾ = (−0.1, 0.3). A table of values of ρ̃_pq ⁽³⁾(0) is provided in SI Appendix.