Folding of Electrostatically Charged Beads-on-a-String: An Experimental Realization of a Theoretical Model

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Entry by Emily Redston, AP 225, Fall 2011

Work in progress


Folding of Electrostatically Charged Beads-on-a-String: An Experimental Realization of a Theoretical Model by Reches, M., Snyder, P.W., and Whitesides, G.M., Proc. Natl. Acad. Sci. USA, 2009, 106, 17644-17649.


"beads-on-a-string" model, electrostatic interactions, polymers, polymer folding, persistence length, triboelectricity, self-assembly


The folding of linear polymers in solution is a subject of enormous importance in areas ranging from materials science to molecular biology. In exploring folding, theorists have developed models at every level of complexity. One of the simplest and most useful of these conceptual models is the “beads-on-a-string” model (a cornerstone of theoretical polymer science). This model represents each monomer of the polymer as a bead, and the backbone of the chain as a flexible string. It has been the basis for many computational models for folding. All theoretical models are, however, necessarily incomplete, and their failure to capture the full complexity of reality stimulates the development of more complex theory. Here the authors defied the conventional strategy of using complex theory to to try to rationalize an even more complex reality; they developed a very simple experimental system to match the simplest theory. They designed a physical model of beads-on-a-string, based on the folding of flexible strings of electrostatically charged beads in two dimensions.

Using a physical system composed of beads of two materials threaded in a defined sequence on a flexible string, they are able to examine the predictions of theoretical beads-on-a-string models. It is a very nice design for several reasons: (1) it is 2-D, (2) the interactions among the beads are electrostatic, (3) the shapes of the beads and properties of the string can be controlled, and (3) the agitation of the beads is well defined. Examination and comparison of two models—one physical and one theoretical— offers a new approach to understanding folding, collapse, and molecular recognition at an abstract level, with particular opportunity to explore the influence of the flexibility of the string and the shape of the beads on the pattern and rate of folding. This system, although much simpler than molecular polymers in 3-D solution, still includes the inevitable nonlinearities of a real physical system. It is, thus, an analog computer designed to extend and to simulate 2-D calculations of beads-on-a-string models of polymer folding and collapse.

The Model

Figure 1. The beads-on-a-string experimental model.

They made the beads-on-a-string system by threading sequences of spherical (diameter = 6.35 mm) or cylindrical (diameter = 6.35 mm, length = 14.2 mm) Nylon and Teflon beads on a thin flexible string. Beads of this size are easy to handle and machine. To increase the visual contrast between the Teflon and Nylon beads, Nylon beads were stained with a neutral organic dye. These beads are separated by smaller (≈3 mm) PMMA spherical beads. The smaller, uncharged beads define the distances between the larger beads, and control the flexibility of the string.

Upon agitation on a surface (made of paper) located in the middle of the triboelectric series, Teflon and Nylon beads develop electrostatic charges of similar magnitudes and at similar rates, but with opposite electrical polarities: Teflon charges negatively and Nylon positively. The smaller PMMA beads essentially uncharged. The resulting electrostatic interactions cause the string to fold.(Fig. 1A). The surface on which the beads charged was planar and axially symmetrical, with a slight curvature (radius ≈3 cm) at the perimeter (Fig. 1B). This geometry avoided interactions of the beads with the corners of a sharply defined frame

A Short-Chain RNA Model

We exploited our ability to control the strength of the interactions among the cylindrical beads by changing their surface area to create a physical analog of an RNA hairpin. Short, palindromic sequences of RNA are convenient molecules to model because they represent a subset of biologically relevant polymers for which existing theory can predict stabilities and structures of some folded states accurately (23).

We threaded sequences composed of long (length ≈14 mm) and short (length ≈7 mm) cylindrical beads; we assumed that the net charge on the shorter cylindrical beads would be smaller than that on the longer cylinders, and the interactions between these beads would correspondingly be weaker. Our intention was that the stronger interactions between long cylindrical beads would be analogous to those between base pairs joined by three hydrogen bonds (GC), and that the weaker interactions between shorter cylinders would model those involving two hydrogen bonds (AU). The total charge on the shorter cylinders was smaller than that on the longer ones, but the areal charge on the two lengths of beads was the same: short Nylon cylinders charged to +2,500 ± 200 pC (7 pC/mm2) and short Teflon cylinders charged to −2,800 ± 300 pC (8 pC/mm2).

Using these cylindrical beads, we constructed a palindromic sequence analogous to GGCAUAAUAGCC (24). We used one spacer bead to introduce stiffness comparable to that of RNA molecules (the persistence length of RNA molecules is ≈700 Å) (24, 25). We studied the folding of this sequence of beads by agitating it on a paper surface; it folded repeatedly, within 5 min, into the same hairpin conformation. This 2-D conformation persisted for at least 1 h under agitation; we infer that it represents the global minimum for this sequence (Fig. 3), and it corresponds to the structure predicted by theory and shown to exist in RNA by experiment (24, 26). Theoretical calculations for the inverted sequence AAUGCGGCGAUU predict that it would not spontaneously fold into a hairpin structure (i.e., the calculated free energy for formation of a hairpin from a linear sequence in solution is positive) (25). In agreement with this theoretical calculation, the analogous sequence did not fold into a single stable conformation. Instead, it sampled numerous different structures in several experiments (Fig. 3).

Figure 1. The beads-on-a-string experimental model.

“Polymer Chain” Models