We derive a thermodynamic identity that allows one to infer the change in the number of screening ions that are associated with a charged macromolecule as the macromolecule is continuously stretched. Applying this identity to force-extension data on both single-stranded and double-stranded DNA, we find that the number of polymer-associated ions depends nontrivially on both the bulk salt concentration and the bare rigidity of the polymer, with single-stranded DNA exhibiting a relatively large decrease in ion excess upon stretching. We rationalize these observations using simple models for polyelectrolyte extension.

Ligand associations play a significant role in biochemical processes, typically through stabilizing a particular conformation of a folded biomolecule. Here, we demonstrate the ability to measure the changes in the number of ligands associated with a single, stretched biomolecule as it undergoes a conformational change. We do this by combining thermodynamic theory with single-molecule measurements that directly track biomolecular conformation. We utilize this technique to determine the changes in the ionic atmosphere of a DNA hairpin undergoing a force-destabilized folding transition. We find that the number of counterions liberated upon DNA unfolding is a nonmonotonic function of the monovalent salt concentration of the solution, contrary to predictions from common nucleic acid models. This demonstrates that previously unobserved phenomena can be measured with our ligand counting approach.

We study the problem of counterion condensation for ellipsoidal macroions, a geometry well-suited for modeling liquid crystals, anisotropic vesicles, and polymers. We find that the ions within an ellipsoid's condensation layer are relatively unrestricted in their motions, and consequently work to establish a quasiequipotential at its surface. This simplifies the application of Alexander et al.'s procedure, enabling us to obtain accurate analytic estimates for the critical valence of a general ellipsoid in the weak screening limit. Interestingly, we find that the critical valence of an eccentric ellipsoid is always larger than that of the sphere of equal volume, implying that counterion condensation provides a force resisting the deformation of spherical macroions. This contrasts with a recent study of flexible spherical macroions, which observed a preference for deformation into flattened shapes when considering only linear effects. Our work suggests that the balance of these competing forces might alter the nature of the transition.T he behavior of weakly charged macroions in biological and soft materials is well described by the DLVO theory, which assumes very weak variations of the electrostatic potential, less than k B T, over scales comparable to the screening length. 1 In this limit, the counterions (or salt ions) within one screening length of a macroion are not bound to its surface, but are free to move. This approximation breaks down near highly charged macroions, where the counterions are bound to the surface and form a condensation layer. 2 The distribution and behavior of the counterions within this layer are not wellcharacterized by mean field analysis. Instead, they are highly localized, and can be considered part of a macroion-condensed counterion composite, which moves about as a single entity. 2−7 Considering the averaged field of this composite, one can construct a generalized DLVO theory based on the effective, renormalized charge. 8−10The degree of charge renormalization depends upon the shape of a macroion. As explained by Zimm and Le Bret, in the zero salt concentration limit, no condensation occurs for an isolated spherical macroion, because the attractive energy gained through condensation onto such a macroion is always less than the entropy associated with ion liberation. 11 In contrast, a finite fraction and complete counterion condensation occurs for cylindrical and planar macroions, respectively, in the same limit. 12,13 Counterion condensation is expected for all geometries under finite salt concentration conditions. 4,14−19 The most drastic, qualitative change occurs for the spherical geometry, for which a finite fraction of counterions now condense. This was first explained by Alexander et al., who obtained the condensation fraction by simply requiring the surface potential to equate to the free ion entropy. 2 These behaviors are of interest in that various experimental measures relating to macromolecule solutions, including the osmotic pressure, structure factor, and compressibilit...

We measure the ensemble averaged deformation of an enzyme for an oscillating applied force. From the low frequency divergence of the mechanical susceptibility for the hinge motion of Guanylate Kinase we obtain a non-equilibrium phase diagram in the frequency -force plane. A phase line separates linear elasticity dynamics from softer (viscoelastic) dynamics. The hinge motion corresponds to crossing this phase line (not to a soft linear elastic mode). The phase line is dramatically shifted in the closed state compared to the open state of the enzyme.Introduction. Virtually all enzymes couple catalysis to conformational motion. Evident in motor proteins, mechano-chemical coupling is also the basis for substrate specificity [1,2] and activity regulation [3][4][5] in these molecular machines. Conformational changes coupled to catalysis are often large, with amplitudes of ∼ 1 nm compared to an overall size of the enzyme of ∼ 5 nm. The folded protein being a solid, these huge strains pose an interesting question about the nature of conformational dynamics and the associated materials properties of the system. Surely this motion is not within the linear elasticity regime, and a specific nonlinear mechanism for these transitions, called "cracking", was proposed some years ago [6,7]. Cracking connects initial and final states through a local melting and refolding event. Translated into stress-strain relations, this should give rise to an interesting dynamics. Here we measure directly the stress-strain relations for the hinge motion of the enzyme Guanylate Kinase (GK), a molecule quite similar to the enzyme considered in [6,7]. Specifically, we report comparative nano-rheology [8] measurements on the open (no substrate) and closed (with the substrate GMP) forms of the enzyme and find that while the linear elasticity regime is almost the same, the nonlinear behavior, or what we have called the viscoelastic regime [9,10] is dramatically different in the two cases. A phase diagram in the frequency -force plane maps out regions of linear elastic vs softer (viscoelastic) dynamics. The existence of this phase line may be a universal feature of the dynamics of enzymes.

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