Global cross-over dynamics of semiflexible polymers

We present a mean-field dynamical theory for semiflexible polymers which can precisely capture, without fitting parameters, recent fluorescence correlation spectroscopy results on single monomer kinetics of DNA strands in solution. Our approach works globally, covering three decades of strand length and five decades of time: it includes the complex cross-overs occurring between stiffness-dominated and flexible bending modes, along with larger-scale rotational and center-of-mass motion. The accuracy of the theory stems in part from long-range hydrodynamic coupling between the monomers, which makes a mean-field description more realistic. Its validity extends even to short, stiff fragments, where we also test the theory through Brownian hydrodynamics simulations. For a polymer in solution, hydrodynamics introduces long-range coupling between different points on the chain contour, its strength falling off with inverse distance like 1/r. Though this has long been recognized as a crucial factor in understanding the dynamics of flexible polymers [1], its importance in the case of semiflexible and stiff chains is not fully appreciated. The cause is often expediency: even without taking long-range coupling into account, the nature of the local interactions—governed by bending stiffness and inextensibility—presents a formidable problem in constructing a theory of semiflexible polymer dynamics. The most widely used model—the Kratky-Porod or worm-like chain (WLC) [2]—yields nonlinear equations of motion, and thus all dynamical theories of the WLC have been based on approximations. One approach is the weakly-bending assumption: deriving equations of motion from a perturbation analysis around the rigid rod limit [3–7]. This is particularly relevant for certain biopolymers like actin, where the large persistence length, lp ∼ O(1μm), means that a broad dynamical regime, consisting of motion on length scales much smaller than lp, will be dominated by the bending stiffness. Yet for less rigid cases like double-stranded DNA, where lp ≈ 50 nm, many empirical situations will involve complex cross-overs between stiffness-dominated and flexible regimes at different time scales. Weakly-bending approaches cannot provide an accurate description of these cross-overs, which hydrodynamics makes even more challenging to model: the fluctuation modes at all length scales are coupled due to the long-range interactions. The need for a comprehensive, quantitatively accurate theory of semiflexible polymer dynamics is made more urgent by advances in single-molecule experimental techniques. Fluorescence correlation spectroscopy (FCS) can already probe double-stranded DNA kinetics at the level of a single monomer [8–10]. The most recent results, by Petrov et. al. [10], reveal a rich sequence of dynamical behaviors for the motion of a tagged end in DNA fragments varying in length from L ≈ 30−7000 nm. In our work, we show that a single theory, without any fitting parameters, can give an excellent description of these experimental results over the entire range, covering the whole cross-over between stiff and flexible chain dynamics. Our approach is based on a Gaussian mean-field theory (MFT) [11–15], which is in itself surprising: Gaussian models are usually considered tools for flexible polymers, with limited applicability as one nears the rigid rod limit. What we demonstrate is that hydrodynamics—the complicating element in the theory—is what underlies its success: the MFT becomes more accurate because of the long-range interactions. We specifically concentrate on highly stiff chains, the most difficult terrain for a Gaussian theory, and investigate the strengths and limitations of the MFT approximation. Since the experimental results for short-time dynamics of stiff fragments are not well resolved, we additionally validate the theory through Brownian hydrodynamics (BD) simulations. Though the MFT is restricted to quantities which are spatially-averaged over all coordi-