Entropic barrier of topologically immobilized DNA in hydrogels

The single most intrinsic property of nonrigid polymer chains is their ability to adopt enormous numbers of chain conformations, resulting in huge conformational entropy. When such macromolecules move in media with restrictive spatial constraints, their trajectories are subjected to reductions in their conformational entropy. The corresponding free energy landscapes are interrupted by entropic barriers separating consecutive spatial domains which function as entropic traps where macromolecules can adopt their conformations more favorably. Movement of macromolecules by negotiating a sequence of entropic barriers is a common paradigm for polymer dynamics in restrictive media. However, if a single chain is simultaneously trapped by many entropic traps, it has recently been suggested that the macromolecule does not undergo diffusion and is localized into a topologically frustrated dynamical state, in apparent violation of Einstein’s theorem. Using fluorescently labeled λ-DNA as the guest macromolecule embedded inside a similarly charged hydrogel with more than 95% water content, we present direct evidence for this new state of polymer dynamics at intermediate confinements. Furthermore, using a combination of theory and experiments, we measure the entropic barrier for a single macromolecule as several tens of thermal energy, which is responsible for the extraordinarily long extreme metastability. The combined theory–experiment protocol presented here is a determination of single-molecule entropic barriers in polymer dynamics. Furthermore, this method offers a convenient general procedure to quantify the underlying free energy landscapes behind the ubiquitous phenomenon of movement of single charged macromolecules in crowded environments.

Movement of charged macromolecules in aqueous media with crowded spatial constraints is ubiquitous in biology, biotechnology, and separation science (1–7). A familiar scenario is the movement of messenger RNA inside the milieu of a nucleus of a mammalian cell, which is a thick Coulomb soup, toward the nuclear pore complex in the process of initiating protein synthesis. Another well-known example is the gel electrophoresis used to identify and separate polynucleotides of different lengths. Furthermore, hydrogel-based technology is extensively implemented to perform targeted delivery of drugs and genes (2–6). Parallel to these aqueous systems, movement of macromolecules in congested nonaqueous polymer melts containing nanoparticles, and infiltration of polymer melts through restricted media, are of tremendous interest in terms of fundamental understanding and applications (8–15). Although entropic barriers associated with conformational changes of the moving macromolecules are often invoked in the above examples of both aqueous and nonaqueous polymer systems, the quantitative nature of such entropic barriers remains to be established (16–42). Despite the above-mentioned now-standard experimental protocols and substantial applications, an understanding of how macromolecules navigate themselves in crowded environments continues to be a persistent challenge.Adding fuel to the flames, a recent discovery (41, 42) shows that large charged macromolecules such as λ-DNA and sodium poly(styrene sulfonate) can be immobilized in hydrogels which are essentially water at ambient temperatures, as a strong deviation from Einstein’s theorem on diffusion. In this paper, we present a combination of theory and fluorescence-based single-molecule tracking to directly observe the movement of guest macromolecules and quantify the entropic barriers associated with their movement. We provide direct evidence for the emergence of the recently proposed nondiffusive topologically frustrated dynamical state at intermediate confinements and determine the associated entropic barrier responsible for extraordinarily long-lived metastable dynamical states.The difficulty in attaining an adequate understanding of entropically driven movement of large macromolecules in soft background media stems from the interdependence between the conformational fluctuations of the macromolecule subjected to transport and the spatial correlation of spatial constraints in the crowded background medium. As an example, consider a large macromolecule (guest) embedded inside a host hydrogel (Fig. 1A). Let the radius of gyration of the guest molecule of N monomers be Rg and the cross-link density of the hydrogel be such that the average mesh size (correlation length for local monomer concentration of the host) is ξ. If Rg≲ξ, the guest molecule only brushes against the host matrix and diffuses. On the other hand, if Rg≫ξ, a single guest molecule is partitioned among many meshes, resulting in its localization. In each occupied mesh, the partitioned portion of the guest under confinement can adopt numerous conformations with a confinement free energy. As a result, each occupied mesh functions as a free energy trap relative to its neighboring matrix domains that are inaccessible to the guest. Accounting for excluded volume interactions and ignoring correlations, the confinement free energy is proportional to M2/ξ3, where M≪N is the number of monomers inside a trap of volume ξ3 (42). The localization effect occurs within two boundary regions. One region corresponds to weaker confinement, Rg≲ξ, as mentioned above. At the other boundary region of transition into reptation, where the free energy traps cease to exist (M2/ξ3→0), we expect ξ to be so small that its corresponding three-dimensional mesh does not have enough space for the guest, namely, M→0. Let us define a length ℓc, comparable to that of only a small number of monomers, corresponding to this condition (vanishing confinement free energy). Furthermore, in the limit of very strong confinements with no free energy traps, the guest is confined inside an essentially one-dimensional tube of a diameter comparable to the entanglement length ℓe. The precise value of ℓc and its relation to ℓe are presently unknown. For brevity, we use ℓ in Fig. 1 to denote both ℓc and ℓe, and leave the detailed values of ℓc and ℓe and their relation to persistence length for future investigation (see Conclusions and Perspective). The movement of the guest molecule is dictated by the relative value of Rg to ξ, and the relative value of ξ to the length ℓ defined above. Different possible scenarios based on Rg/ξ and ξ/ℓ are portrayed in Fig. 1B.Open in a separate windowFig. 1.(A) Cartoon of a large charged macromolecule of radius of gyration Rg embedded in a hydrogel of mesh size ξ. (B) Sketch of the dependence of the diffusion coefficient D of the guest molecule on degree of confinement (which increases with a decrease in ξ). The four conventional regimes of polymer dynamics, namely, Zimm, Rouse, entropic barrier, and reptation, and their conditions of experimental relevance are denoted by 1, 2, 3, and 5, respectively. The corresponding scaling laws in these regimes connecting D and the degree of polymerization N of the polymer chain are given in Eq. 1. Regime 4 is the newly hypothesized nondiffusive (D≃0, as denoted by the red horizontal line) topologically frustrated state at intermediate confinements. Direct observation of this new dynamical state and measurement of the entropic barrier responsible for polymer localization are presented in Results. (C) Sketch of partitioning of a single chain into multiple deep entropic traps.Based on experiments during the past seven decades and buttressed by theories (43, 44), four regimes (1, 2, 3, and 5) have been widely recognized. These are described, respectively, by the Stokes–Einstein–Zimm, Rouse, entropic barrier, and reptation models. In all these regimes, the guest molecule undergoes diffusion. The dependence of the diffusion coefficient D on the degree of polymerization N of the polymer molecule is given by the scaling laws as (Fig. 1B)[1]where ν is the size exponent defined through Rg≈Nν, and A is a nonuniversal numerical factor denoting the local structure of the host medium (16). The above four regimes of polymer diffusion have been validated by a preponderance of experimental data in the past (17–20, 22–33).Furthermore, in a recent development, room temperature experiments on charged macromolecules (such as λ-DNA and sodium poly(styrene sulfonate)) embedded inside similarly charged hydrogels at intermediate confinements (Rg≫ξ≫ℓ, regime 4 in Fig. 1B) have revealed an apparent breakdown of even the phenomenon of diffusion, and the emergence of a new nondiffusive topologically frustrated dynamical state. When Rg≫ξ≫ℓ (regime 4 in Fig. 1B), each macromolecule is so large in comparison with the mesh size that it is partitioned among many meshes. Since ξ≫ℓ, each mesh is an entropic trap holding a significant portion of the macromolecule and allowing local Rouse–Zimm dynamics for that portion. Furthermore, that portion can move to a neighboring empty mesh only through an entropic barrier as in regime 3 (where a single entropic barrier is operative for the whole molecule). As a consequence, many sets of entropic barriers must be simultaneously negotiated for the center of mass of one molecule to diffuse. The free energy landscape for the situation in regime 4 is cartooned in Fig. 1C. Theoretical analysis (41) of the simultaneous crossing of multiple entropic barriers by a single chain shows that the net effective free energy barrier U arising only from conformational entropy isU≃kBT ln196ξ8ℓ8,[2]where kB is the Boltzmann constant, and T is the absolute temperature. Thus the barrier U can be enormous depending on ξ/ℓ. For example, for ξ/ℓ=50, U≃27kBT. Such large barriers result in extremely long times for a single chain to diffuse a distance comparable to its own size. As a result, the macromolecule is practically locked into an immobile state in regime 4. Therefore, in regime 4, the time evolution of the mean square displacement of the center of mass Rcm(t) of the guest molecule is hypothesized to be nondiffusive according to⟨Rcm(t)−Rcm(0)2⟩≈t0, (Rg≫ξ≫ℓ).[3]This additional dynamical state is included in Fig. 1B, where the red horizontal line denotes the nondiffusive regime 4.Although the above hypothesis of the emergence of a nondiffusive topologically frustrated long-lived metastable state at intermediate confinements is supported by dynamic light scattering experiments (41, 42), direct evidence for this phenomenon and quantification of the collective entropic barrier responsible for this effect are yet to be established. These are the primary goals of the present paper. Using a combination of theory and statistical analysis of thousands of trajectories of fluorescently labeled λ-DNA inside poly(acrylamide-coacrylate) hydrogels with more than 95% water content, we have measured the entropic barrier that results in the new topologically frustrated state, in addition to finding direct evidence for the immobility of λ-DNA at intermediate confinements.