Flexible, charged biopolymers in monovalent and mixed-valence salt: Regimes of anomalous electrostatic stiffening and of salt insensitivity

The conformations of biological polyelectrolytes (PEs), such as polysaccharides, proteins, and nucleic acids, affect how they behave and interact with other biomolecules. Relative to neutral polymers, PEs in solution are more locally rigid due to intrachain electrostatic repulsion, the magnitude of which depends on the concentration of added salt. This is typically quantified using the Odijk-Skolnick-Fixman (OSF) electrostatic-stiffening model, in which salt-dependent Debye-Hückel (DH) screening modulates intrachain repulsion. However, the applicability of this approach to flexible PEs has long been questioned. To investigate this, we use high-precision single-molecule elasticity measurements to infer the scaling with salt of the local stiffness of three flexible biopolymers (hyaluronic acid, single-stranded RNA, and single-stranded DNA) in both monovalent and mixed-valence salt solutions. In monovalent salt, we collapse the data across all three polymers by accounting for charge spacing, and find a common power-law scaling of the electrostatic persistence length with ionic strength with an exponent of $0.66\pm 0.02$. This result rules out simple OSF pictures of electrostatic stiffening. It is roughly compatible with a modified OSF picture developed by Netz and Orland; alternatively, we posit the exponent can be explained if the relevant electrostatic screening length is the interion spacing rather than the DH length. In mixed salt solutions, we find a regime where adding monovalent salt, in the presence of multivalent salt, does not affect PE stiffness. Using coarse-grained simulations, and a three-state model of condensed, chain-proximate, and bulk ions, we attribute this regime to a “jacket” of ions surrounding the PE that regulates the chain's effective charge density as ionic strength varies. The size of this jacket in simulations is again consistent with a screening length controlled by interion spacing rather than the DH length. Taken together, our results describe a unified picture of the electrostatic stiffness of polyelectrolytes in the mixed-valence salt conditions of direct relevance to cellular and intercellular biological systems.

Figure 1

(a) Extension $L$ versus applied force $f$ for a single HA tether at 10 mM NaCl (blue triangles), 50 mM NaCl (green squares), and 1000 mM NaCl (yellow circles), all with 10 mM MOPS buffer, pH 7. Lines indicate best fits to Eq. (A6), from which we extract estimates of the crossover force $f_c$, giving $0.33\pm 0.009\mathrm{pN}, 0.40\pm 0.009\mathrm{pN}$, and $0.48\pm 0.006\mathrm{pN}$ for the 10, 50, and 1000 mM curves, respectively. Inset shows a schematic of a tethered polymer stretched using magnetic tweezers. (b) Force-extension curves from (a), cofit (multiplicatively shifted to achieve best overlap) onto a reference curve for HA in 50 mM salt. Fit unitless shift factors in force are $\overline{f}=$ 0.58, 0.79, and 1.18.

Figure 2

(a) Scaling with $I$ of inverse crossover force $1/\left(\overline{f}{f}_c^{\mathrm{ref}}\right)$ for ssRNA (gray), ssDNA (yellow), and HA (blue and red). ssDNA and ssRNA data are fit with one-parameter power laws with exponents $\alpha =0.62\pm 0.09$ and $0.70\pm 0.03$, respectively (solid yellow and gray lines). Raw HA data (blue) are fit by Eq. (A4) with $\alpha$ = 0.71, $I_c$ = 16 mM (solid blue line). Plateau-subtracted HA data (red) are fit by a single parameter power law with exponent $\alpha =0.71\pm 0.02$ (solid red line). (b) Multiplying the inverse crossover force by $d^2$ collapses the data from the three PEs onto a common curve (with $d^2$ = 1, 0.49, and $0.36{\mathrm{nm}}^2$ for HA, ssDNA, and ssRNA). The combined data set is fit by a power law with exponent $\alpha =0.66\pm 0.02$ (solid black line).

Figure 3

Inverse crossover force $1/\left(\overline{f}{f}_c^{\mathrm{ref}}\right)$ as 1:1 salt is added to increase $I$, with and without a small constant amount of multivalent salt. (a) HA, with and without 0.1 mM 3:1 salt. Blue: 1:1 salt only. Red: combined trivalent data set. Blue points and associated fit are reproduced from Fig. 2. The plateau value has been subtracted from HA data. (b) ssDNA with and without 1 mM 2:1 salt. Blue: 1:1 salt only. Orange: ${\mathrm{CaCl}}_2$. Blue points and associated fit are reproduced from Fig. 2.

Figure 4

Simulated chain average end-to-end extension $R$ under no applied force, as a function of ionic strength $I$. (a) HA-like chain with (red squares) and without (blue circles) 0.1 mM 3:1 salt. (b) ssNA-like chain with (orange squares) and without (blue circles) 1 mM 2:1 salt. Error bars show standard error of the mean, approximately $\pm 3\%$ and $\pm 6\%$ for the HA and ssNA systems, respectively.

Figure 5

Charge enclosed in a cylinder centered at the chain, from simulation, as a function of distance $r$ from the chain. (a) HA-like chain (225 negative charges) with 0.1 mM 3:1 salt and 4–16 mM 1:1 salt. (b) ssNA-like chain (200 negative charges) with 1 mM 2:1 salt and 4–50 mM 1:1 salt. Insets are zoomed in on region near the chain and linearly scaled to show the precipitous decrease in total enclosed charge due to a layer of closely associated multivalent counterions.

Figure 6

Illustration of three-state model for ions interacting with the PE. An ion may be (1) condensed on the charged chain, (2) free to move within a charge-neutral local jacket whose composition is dictated by Donnan equilibrium, or (3) free in the bulk solution.

Figure 7

Number of ions within a cylinder of radius $R_{\mathrm{cyl}}=u{x}_i$. Symbols show ion numbers from simulation, solid curves show Donnan model predictions. Blue: monovalent cations, green: monovalent anions, red and orange: tri- and divalent cations. Dot-dashed lines show the number of expected cations if all ions were uniformly distributed. (a) HA-like chain $+$ 0.1 mM 3:1 salt. (b) ssNA-like chain $+$ 1 mM 2:1 salt.

Figure 8

Renormalized chain charge spacing $d_{\mathrm{eff}}$ predicted from the Donnan-condensation three-state model. (a) HA $+$ 0.1 mM 3:1 salt. (b) ssDNA $+$ 1 mM 2:1 salt. Thin dotted red line shows $d_{\mathrm{eff}}$ neglecting monovalent condensation. Dashed black line shows the structural charge spacing $d$.

Figure 9

$I^{-0.66}$ scaling of inverse crossover force in monovalent salt (black line), reproduced from Fig. 2, alongside data on HA $+$ 0.1 mM 3:1 salt and ssDNA $+$ 1 mM 2:1 salt, which show plateaus when multiplied by the structural charge spacing $d^2$ (HA: purple points, ssDNA: orange points), but collapse nearly onto the expected curve when $d_{\mathrm{eff}}^2$ is used instead (HA: dark red circles, ssDNA: yellow circles).

Figure 10

Comparison of condensation models used to calculate $I$-dependent $d_{\mathrm{eff}}$ to replace structural charge spacing $d$ in the normalization. Red points are multiplied by $d^2$. Yellow circles use $d_{\mathrm{eff}}$ calculated from our three-state model. Green triangles use $d_{\mathrm{eff}}$ calculated from a two-state model with no locally enhanced screening. Orange squares show result of the BAA model [22]. Black line shows $\alpha$ = 0.66 power-law fit to all monovalent-only data. (a) HA $+$ 0.1 mM 3:1 salt. (b) ssDNA $+$ 1 mM 2:1 salt.

Figure 11

Cofitting onto reference curves. (a–c) $n$ force-extension curves at various concentrations of 1:1 salt are cofit onto reference curves on the same PE at 50 mM NaCl. Salt ranges from 1 mM to 1000 mM, with low-salt data in blue and high-salt data in yellow. (a) HA, $n$ = 274. (b) ssDNA, $n$ = 59. (c) ssRNA, $n$ = 33. (d–f) Reference curves fit by piecewise functions [Eqs. (A6) and (A7)]. Crossover at $f_c^{\mathrm{ref}}$ is indicated by an orange cross. Low force power law of exponent $\gamma$, green, is extended to $f>f_c^{\mathrm{ref}}$ with a dashed line to show deviation above the crossover. Yellow curves show high-force fits. (d) HA, $\gamma =0.62, f_c^{\mathrm{ref}}=0.50\pm 0.01\mathrm{pN}$. (e) ssDNA, $\gamma =0.57, f_c^{\mathrm{ref}}=0.64\pm 0.03\mathrm{pN}$. (f) ssRNA, $\gamma =0.60, f_c^{\mathrm{ref}}=0.50\pm 0.01\mathrm{pN}$. $\overline{f}$ and $\overline{L}$ denote shift factors in force and length; $L_c$ denotes chain contour length, estimated from WLC fit.

Figure 12

Cofitting $n$ force-extension curves with multivalent salt (black) onto reference curves taken in 54 mM monovalent salt (yellow). (a) HA $+$ 0.1 mM 3:1 salt, $n=170$. (b) ssDNA $+$ 1 mM 2:1 salt, $n=63$. $\overline{f}$ and $\overline{L}$ denote shift factors in force and length.

Figure 13

Comparison of inverse crossover force values obtained from different reference curves. For HA (left), we have already subtracted the plateau value from the quantity plotted on the $y$-axis. Light blue: $I_{\mathrm{ref}}=24\mathrm{mM}$ (27 mM for ssRNA), purple: $I_{\mathrm{ref}}=54\mathrm{mM}$ (57 mM for ssRNA), black: $I_{\mathrm{ref}}=504\mathrm{mM}$ (507 mM for ssRNA). If cofitting procedures are equivalent, division by $f_c^{\mathrm{ref}}$ will collapse data referenced to different curves.