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, RNA, 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.

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

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. In particular, with their model, they are able 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 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 remain 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 helps avoid interactions of the beads with the corners of a sharply defined frame.

A Short-Chain RNA Model

Figure 2 Folding of short chain RNA. (A) The letters represent sequences of nucleosides. The two sequences are related by inversion of purines and pyrimidines. (B) Theoretical calculations predicted hairpin loop structures (with different free energies of folding) for each sequence. (C) Time lapse photographs captured the folding process for the analogous sequences composed of long (≈14 mm) and short (≈7 mm) cylindrical beads. The images represent the folding of the two sequences, GGCAUAAUAGCC (Left), which folded to one stable conformation (highlighted in red) in eight experiments, and AAUGCGGCGAUU (Right), the structure of which evolved continuously throughout eight independent experiments.

The authors exploited their 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 accurately predict stabilities and structures of some folded states.

They threaded sequences composed of long (length ≈14 mm) and short (length ≈7 mm) cylindrical beads. They assumed that the net charge on the shorter cylindrical beads would be smaller than that on the longer cylinders, and thus the interactions between these beads would correspondingly be weaker. Their 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).

Using these cylindrical beads, they constructed a palindromic sequence analogous to GGCAUAAUAGCC. They used one spacer bead to introduce stiffness comparable to that of RNA molecules. They 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. The authors believe that this represents the global minimum for this sequence (Fig. 2). This hairpin conformation corresponds to the structure predicted by theory and shown to exist in RNA by experiment. 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). 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. 2).


The authors presented a very simple yet elegant physical model following the beads-on-a-string theoretical model in this paper. Comparison with results of computations and simulations based on electrostatically interacting 2-D beads-on-a-string will allow others interested in collapse and folding of molecular polymers to explore the physical bases of these problems using a different perspective. The system emerging from this work—one comprising strings of macroscopic monomers—bridges simple theoretical and physical models of polymers. Theory and experiment provide alternative and complementary approaches to the study of polymer conformation. Many of the interactions that make the physical system complex are well-defined and (to a degree) under control. This control provides the foundations for a system that is experimentally practical, but also retains nonlinearities of the types—although not of the specific detail—that are important in molecular systems. There is still room from improvement since, as the authors note, the charge distribution in this system does not change as the sequence folds (i.e., the sign of charge on the beads remains the same during agitation). Nonetheless, I believe these authors have approached this research from a unique perspective, going for a more simple approach where others have gotten lost in the complexities of other models. They developed a very intuitive model that has a lot of potential to yield some interesting results.