Electrostatic stiffening and induced persistence length for coassembled molecular bottlebrushes

A self-consistent field analysis for tunable contributions to the persistence length of isolated semiflexible polymer chains including electrostatically driven coassembled deoxyribonucleic acid (DNA) bottlebrushes is presented. When a chain is charged, i.e., for polyelectrolytes, there is, in addition to an intrinsic rigidity, an electrostatic stiffening effect, because the electric double layer resists bending. For molecular bottlebrushes, there is an induced contribution due to the grafts. We explore cases beyond the classical phantom main-chain approximation and elaborate molecularly more realistic models where the backbone has a finite volume, which is necessary for treating coassembled bottlebrushes. We find that the way in which the linear charge density or the grafting density is regulated is important. Typically, the stiffening effect is reduced when there is freedom for these quantities to adapt to the curvature stresses. Electrostatically driven coassembled bottlebrushes, however, are relatively stiff because the chains have a low tendency to escape from the compressed regions and the electrostatic binding force is largest in the convex part. For coassembled bottlebrushes, the induced persistence length is a nonmonotonic function of the polymer concentration: For low polymer concentrations, the stiffening grows quadratically with coverage; for semidilute polymer concentrations, the brush chains retract and regain their Gaussian size. When doing so, they lose their induced persistence length contribution. Our results correlate well with observed physical characteristics of electrostatically driven coassembled DNA-bioengineered protein-polymer bottlebrushes.

Figure 1

Illustration of the two-gradient cylindrical coordinate system used in the SCF calculations. The $z$ and $r$ coordinates are indicated as well as the radius of curvature $R$ of the chain, which is drawn as a thick black line. Below we will use a lattice model (here not indicated) where the $z$ and the $r$ coordinates have discrete values. The cross-sectional grit $(z,r)$ is shown in Fig. 2.

Figure 2

Illustration of models for the cross section of the main-chain (backbone) (cf. Fig. 1). The cylindrical two-gradient coordinate system $(z,r)$ as well as the radius of curvature $R$ of the main chain are indicated only in panel A (cf. Fig. 1) [(a), (b)] Phantom chains. [(c)–(h)] Backbone chain with finite size (here $a=5$ sites), and segment type S. (a) No constraints; grafts (not shown) emanate from the gray site. (b) Spatial constraints: half of the chains grafted at the gray site have to stay in the red region and the other half of the chains are fixed to the green half-space (regions overlap at $r=R$). (c) Chain with fixed size (black): annealed grafting in gray sites. (d) Chain with fixed size: quenched grafting; equal number of chains grafted on green and red sites. (e) Chain with fixed size; homogeneous adsorption energy ${\chi }_{\mathrm{S}}<0$ for $K$ stickers. (f) Main chain with fixed size and fixed charge density (blue sites have fixed valency). (g) Chain with fixed size: annealed charges in gray zones (total charge is fixed). (h) Chain with fixed size: quenched charge distribution. Equal amount of charge in red and green regions. More details are in the text. See also Table 1 for extra info and how the models are being used.

Figure 3

(a) The electrostatic stiffening, $l_{\mathrm{p}}$ in units of lattice sites, as a function of the salt concentration ${\phi }_s$, for a polymer chain with linear charge density $v=-1.5$ charges per lattice site in log-log coordinates. (b) The electrostatic stiffening $l_{\mathrm{p}}$ in units of lattice sites as a function of the linear charge density for a given volume fraction of salt ${\phi }_s=0.001$ in log-log coordinates. Q is quenched charge density, A is annealed charge density, Ph is the main chain that has minimum volume of one lattice site (phantom main chain), and S is the fixed surface charge density around the DNA chain. 1:1 electrolyte. All interaction parameters are taken to be athermal. The indicated slopes give the prediction of Fixman and Odijk [3, 4]. See Fig. 2 and Table 1 for details of the models used.

Figure 4

(a) The induced persistence length, $l_{\mathrm{p}}$, in units of lattice sites, as a function of the degree of polymerization of the side chains for a given distance between the grafts $h=2$ lattice sites in log-log coordinates. (b) The induced persistence length $l_{\mathrm{p}}$, in units of lattice sites as a function of the distance between the side chains for a given length of the side chains $N=400$ in double logarithmic coordinates. Q is quenched mobility of side chains, A is annealed mobility. Ph, the volume of the main chain, is ignored: First segment of the side chains is fixed to ($M_z/2,R$) (phantom chain). H is the model in which chains cannot translocate with any of the segments from the convex to the concave side. See Fig. 2 and Table 1 for details of the models used. The theoretically expected slopes 2 in panel (a) and $-2$ in panel (b) are indicated.

Figure 5

The adsorbed amount of ${\mathrm{C}}_4{\mathrm{K}}_{12}$ copolymers in number of segments per unit length of the DNA chain as a function of the volume fraction of copolymer in solution in lin-log coordinates. The solid lines are for the electrostatically driven adsorption [Fig. 2, $v=-3]$. The ionic strengths ${\phi }_s=1\times {10}^{-4};1\times {10}^{-3};5\times {10}^{-3}$ are indicated. The dashed lines are for the classical case with fixed adsorption energy [Fig. 2]. The adsorption energy ${\chi }_{\mathrm{S}}=-18,-15,-12$ are indicated. Crossing point of the isotherms are numbered 1–6 and these points represent conditions for which the bending rigidity of the self-assembled bottlebrush are computed, which are presented in Fig. 6.

Figure 6

The bending rigidity of the self-assembled bottlebrush in lattice units as a function of the amount of ${\mathrm{C}}_4{\mathrm{K}}_{12}$ copolymers that are physisorbed per unit length (lattice units) onto the DNA chain in log-log coordinates. Solid lines are for the charged case. Three curves are given for ionic strengths ${\phi }_s={10}^{-4}$, ${10}^{-3}$, and $5\times {10}^{-3}$, as indicated. The dashed line is for the fixed adsorption energy case. The symbols with numbers correspond to the crossings of the isotherms in Fig. 5. The slope of the dashed line is 1.8. The dotted line (with same slope) is to guide the eye.

Figure 7

The tunable contribution to the persistence length $l_{\mathrm{p}}$ in lattice units for coassembled DNA bottlebrushes as a function of the bulk volume fraction of ${\mathrm{C}}_4{\mathrm{K}}_{12}$ for ${\phi }_s=0.001$ in lin-log coordinates. The charge $v=-3$ around the DNA chain is homogeneously distributed [cf. Fig. 2]. The focus is on the behavior near the overlap concentration.