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

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 predict stabilities and structures of some folded states accurately.

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), and it 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).

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.

Figure 2


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 our 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. It allows us, for example, to examine a many-body electrostatic system, which has no analytical solution, without the use of a mean-field theory (30).

This experimental system provides examples of folding at two extremes of possible energy landscapes: (i) for short sequences (10 beads), a flat landscape with a single global minimum, in which folding is dominated by a search through conformations and (ii) for longer sequences (40 beads), a more rugged landscape in which the polymer can be trapped kinetically in local minima. This system makes it possible to explore how sequences sample conformational space; we are interested in, for example, the lifetime of correct and incorrect pairs on the pathway to a minimum, and whether longer sequences can be designed that fold to a single global minimum—in analogy to more complex problems in folding.

In considering more complex theoretical modeling, we note that 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). The sequences of beads can, therefore, be described as chains of quenched charges (i.e., the specific sequence of the chain is determined during the polymerization stage, and therefore the charge distribution is frozen) (31, 32). In this case, theories of spin glasses that deal with the nature of quenched disordered systems could be applicable (2, 32, 33).

The principles demonstrated here are a starting point for the study of more specific problems; for example, the effect of point mutation on the folding of hairpin-forming sequences in RNA and the folding of primitive “proto-proteins” using limited sets of amino acids in prebiotic chemistry (34, 35).

The physical system we describe is the simplest that we can design to simulate behaviors characteristic of polymers. Comparison of results from it with results of computations and simulations based on electrostatically interacting 2-D beads-on-a-string will allow scientists interested in collapse and folding of molecular polymers to explore the physical bases of these problems using a different perspective.