DNA condensation by field-induced non-equilibrium noise.

Gel electrophoresis of DNA molecules is an important tool for the separation of DNA molecules of different length. For the electrical field strengths typically used in electrophoresis experiments, the DNA molecules are stretched-out along the field direction1. This is consistent with theoretical studies of the polarization energy of polyelectrolyte molecules, which find that it should increase as the cube of the length of a molecule if it is aligned with the field direction2. It was a surprise therefore that for AC electrical fields in the 104-105 V/m range and frequencies in the 300-700 Hz range, λ-phage DNA molecules confined in channels seemed to collaps3. Condensation phenomena of DNA solutions in AC electrical fields had actually been reported earlier4,5. Interestingly, the condensation is observed only over a limited frequency range and a limited range of field strengths. In particular, for fields above 105 V/m, the molecules were stretched6. The condensation was attributed by the authors of Ref.3 to polarization attraction between sections of the DNA molecule with opposite curvature. Under conditions of thermodynamic equilibrium, condensation of DNA can be induced by adding low concentrations of polyvalent counterions7 to the solution. It is believed that correlated, out-of-phase charge fluctuations of the counterions condensed on neighboring chains generate an effective attractive force, a mechanism - first proposed by Asakura and Oosawa (AO) in 19588 - that is akin to van der Waals attraction. For monovalent counterions, this attraction is not strong enough to overcome the electrostatic repulsion, at least under ambient conditions. We will demonstrate below that AC electrical fields in a limited frequency window and a limited field strength window do generate enhanced coupled charge-curvature fluctuations. These field-enhanced fluctuations significantly amplify the AO fluctuation attraction mechanism and is, we argue, responsible for the observed collapse. Before discussing the field-enhanced fluctuations, we first examine the time-averaged polarization energy of a semi-flexible polyelectrolyte molecule under AC conditions and discuss why the electrophoretic stretching effect9 weakens as the drive frequency increases. Next, we show that curvature and charge fluctuations are strongly coupled under AC conditions, followed by a discussion of the AO attraction under these conditions, after which we conclude. The paper certainly does not present anything like an exact theory of this problem. DNA molecules in AC electrical fields are subject to viscous drag, dielectrophoretic effects10 and field-induced hydrodynamic flows11. Specifically, we will make a number of simplifying assumptions to illustrate the mechanism that is being proposed: (i) the collection of condensed counterions along the DNA backbone is treated as a conducting, one-dimensional ionic “wire” with no charge exchange with the surrounding medium, (ii), we neglect electrokinetic effects apart from those associated with the condensed counterions, (iii) we assume that the frequency is sufficiently high so the center of mass of the DNA molecule does not “follow” the AC electrical field12. We also will follow the worm-like chain model of DNA and treat it as a uniform semi-flexible polyelectrolyte with a persistence length lp of about 500A and a “bare” negative charge per unit length ρ0 = −e/b, with b ~ 1.7A. Because of Manning condensation7, positively charged, mobile counterions are condensed along the molecule with a charge per unit length e/b - e/lb. The effective charge per unit length - e/lb of the molecule is significantly smaller than the bare charge per unit length. The Bjerrum length lb is defined here by the condition e2 /ewlB = kBT. The remaining neutralizing counterions are not bound. Under these restrictions, an applied AC electrical field will drive an oscillating electrical current of counter-ions along the curving DNA molecule. Let s be the arc distance along the molecule, t(s,t) the local tangent unit vector and ρ(s,t) the counterion charge-density along the molecule induced by a time-dependent AC electrical field E→(t). If there is no ion exchange between the condensed counterions and the aqueous medium then ρ(s,t) must obey a one-dimensional Nernst-Planck (NP) equation. For ρ(s,t) small compared to ρ0 the NP equation can be linearized, giving: ∂ρ∂t=∂∂s(D∂ρ∂s−μρ0eE→(t).t^(s,t)) (1) Here, μ is the counterion mobility related to the ion diffusion coefficient D by D = μkBT. The term in brackets is (minus) the counter-ion current For now assume that the molecule has a fixed, time-independent shape. For an AC electrical field with frequency Ω, the general solution of Eq. (1) for a time-independent shape of the molecule is easy to find: ρΩ(s)=−e−iπ/4ρ0eE→Ω2kTDΩ∫ds′exp(−eiπ/4ΩD|s−s′|)(dt^(s′)ds′) (2) with ρ(s,t) = Re{ρΩb (s)exp(iΩt)}, as can be verified by substitution. In order to satisfy the zero current boundary conditions at any endpoints of the molecule, solutions of the homogeneous equation iΩρΩ=Dd2ρΩds2have to be added but we will assume a circular DNA molecule without endpoints. Note that the curvature of the chain acts as a “source” term for the induced charge density. It follows from Eq.(2) that the length scale ξ=D/Ω plays the role of a correlation length: induced polarization charges are correlated with the chain curvature dt^(s)ds within arc distances less than ξ. For a 300 Hz AC field, ξ is in the range of microns, assuming Na+ counterions. The central role of this correlation length, which was emphasized earlier by Cohen13, can be better appreciated by considering a special conformation: a kink in the DNA molecule, which would be generated by stretching the DNA loop along the field direction (it could also be produced by certain DNA-binding proteins or by specific DNA sequences). First assume that the DNA molecule has the shape of a single kink, located at s=0, connected to two infinite, perfectly straight sections. The appropriate solution of Eq.2: ρΩ(s)/ρ0~−e−iπ/4eE→Ω.Δt^ξ2kTexp(−e−iπ/4|s|/ξ) (3) is proportional to the change Δt of the tangent vector on opposite sides of the kink. For field strengths of 104 V/m and frequencies in the 100 Hz range, the RHS is of the order of one at s =0, i.e., sufficient for charge reversal of the DNA chain. For higher field strengths, the linearization approximation is apparently invalid. The charge density decays, in an oscillatory fashion, over a distance of the order of ξ. The amplitude of the charge oscillation localized around the kink is QΩ~ρ0eE→Ω.Δt^2kTξ2 (4) For ξ one μm and a field strength of 104 V/m, this corresponds to about 104 elementary charges distributed over a distance of the order of ξ. Next, assume a DNA loop has a length L large compared to the correlation length. The loop is stretched along the field direction, with two kinks on opposite ends separated by two strands of length L/2. Polarization charges with opposite sign accumulate at the two kinks at the two ends of the loop, producing an effective tension TΩ~EΩQΩ~ρ0eξ22kBTEΩ2 that is proportional to the square of the field strength and inversely proportional to the drive frequency. As discussed in the Appendix A, the introduction of such a tension is equivalent to the statement that under AC conditions the DNA chain is effectively a string of polarizable dielectric “units”, each with a length ξ, whose dipole moment tries to align with the field (see also Ref.13). If one increased the length of the loop from zero to L then the work done by the field would be of the order of TΩL / 2. In Appendix A we compute the polarization energy by standard methods and confirm that it indeed is of order TΩL. For field strengths of the order 104 V/m and 300 Hz frequencies, the tension is of the order of picoNewtons (pN). Next, fold the loop to form a smaller double loop, and stretch the double loop along the field, so there are two positively charged kinks and two negatively charged kinks linked by four parallel strands with a length L/4. The total work 2 ×(TΩL / 4) done by the field separating the polarization charges is the same as before. In terms of polarization energy, there is thus an approximate degeneracy between the single and double loop (see also Ref.13). The argument can be extended to multiple doublings until the separation between the kinks is of the order of the correlation length. In Appendix A we show that if the correlation length is larger than L, then the single loop has a much lower polarization energy. One might expect the total free energy of the single loop to be lower than that of the folded loops, since that would seem to interfere least with conformational chain fluctuations. That, as we shall see below, need not be the case.