Chain entropy: spoiler or benefactor in pattern recognition?

A lthough how life began on our planet will always be of intrigue, polymers must have certainly existed before the beginning of life, because life as we now know it requires replicating polymers. There is little doubt (1) that replication of polymers, e.g., DNA, involves complexation among molecules consisting of complementary specific chemical sequences. Such processes of recognition of chemical patterns occur even within one macromolecule leading to tertiary structures (2–6) originating from patterned primary structures, e.g., protein folding. Thus macromolecular recognition of specified chemical patterns, mediated through a combination of coulombic, hydrogen bonding, hydrophobic and hydrophilic, and van der Waals forces, has manipulated evolution of humans and other organisms in this planet over three and half billion years. Yet, little progress has been made in this problem of tremendous significance. The most straightforward approach to this problem is to directly solve a specific example with appropriate details of potentials between all pairs of atoms in the system. But this computational methodology, in its present status, cannot address the large-scale aspects of macromolecular recognition, as seen overwhelmingly in biological contexts (1). The alternate method is to coarse-grain nonessential microscopic details and consider only the bare essentials of potential interactions and accompanying entropy of selfassembled complexes. Only recently, general principles guiding the macromolecular pattern-recognition process were addressed (7–9). The most important principle of pattern recognition by polymers is the entropic frustration leading to topological dereliction (7–9). According to this principle, the entropy associated with different ways of cooperatively arranging various moieties of macromolecular complexes leads to rugged and hilly free-energy landscapes such that the unavoidable presence of lowest free-energy states at intermediate stages of pattern recognition can actually make the approach to the final fully recognized state much delayed, and that the distance between different paths diverges with time. It is therefore necessary to temporally modify the free-energy landscape to guide the pattern recognition to be successful and efficient. By using dynamic Monte Carlo simulations of a two-component heteropolymer bearing statistical patterns of sequences in the proximity of a heterogeneous surface bearing statistical patches of chemistries with short-range interaction toward the polymer, a vivid demonstration of the above-mentioned theme of pattern recognition is reported in this issue of PNAS by Golumbfskie et al. (10). Consider the following simple but exact argument, which illustrates entropic frustration and the devastating role it plays in pattern recognition. Let us imagine a scenario of recognition of a pattern imprinted on a surface by a polymer containing chemical groups complementary to the surface pattern. Somewhere intermediate in the evolution of this process, many pairings would have occurred and many other pairings remain to be made. At this intermediate juncture, let us follow the details of sequentially making three new pairs in the immediate future (Fig. 1). Consider only three units (labeled i 5 1, 2, 3, and green) making up a linear pattern with spacing between i 5 1 and i 5 2 being b (in units of polymer segment length), and that between i 5 2 and i 5 3 being 2b in threedimensional space. (The shape of the pattern, chosen here to illustrate the argument for a specific case, can be generalized to any situation). Let us proceed to monitor the recognition of this pattern by a portion of a polymer with three special groups (labeled ip 5 1, 2, 3, and red) complementary to green groups. For specificity, we assume that there are m segments between ip 5 1 and ip 5 2 and 2 m segments between ip 5 2 and ip 5 3. Also ip 5 1 segment complexes with i 5 1, as shown in Fig. 1a. Assuming that the polymer obeys Gaussian statistics, and the gain in energy per pairing is 2«, the free-energy F corresponding to the configuration of Fig. 1a is 2«. The second contact between the polymer and the pattern can take place in four possible ways. Each of ip 5 2 and ip 5 3 can be paired with either i 5 2 or i 5 3. These are shown in Fig. 1 b–e. When ip 5 2 is paired with i 5 2, the end-to-end distance of the polymer spacer with m segments is b, and the entropy of the loop in Fig. 1b is 23kBby2m, apart from uninteresting constants, where kB is the Boltzmann constant. Therefore, the free energy of the topological state of Fig. 1b is 22« 1 fs, where fs 5 3kBTby2m is the free energy associated with the loop arising from entropy. Similarly, the complexes with contacts (ip 5 2; i 5 3), (ip 5 3; i 5 3), and (ip 5 3; i 5 2) have free energies 22« 1 9fs, 22« 1 3fs, and 22« 1 fsy3, respectively. It must be noted that the topological state of Fig. 1e (ip 5 3; i 5 2) is the most stable as far as two-contact complexes are considered (and for b2ym . 2 ln 3, if we keep track of all numerical prefactors). Proceeding now to consider the threecontact complexes, there are only two such complexes, as shown in Fig. 1 f and g, with respective free energies 23« 1 3fs and 23« 1 11 fs. Assuming sequential pairing, the complex with full registry given in Fig. 1f can arise only from Fig. 1 b and d. The complex of Fig. 1g can arise from Fig. 1 c and e. This free-energy landscape is sketched in Fig. 2, where free energies F of various topological states in the present specific example are plotted against time, which is proportional to the number of sequential pairings. Although the most stable complex with only two contacts is that of Fig. 1e, its further evolution to the three-contact state is to a state with higher free energy and not to the lowest free-energy state. It is obvious from the configurations of the complexes shown in Fig. 1 f and g that these configurations cannot be directly converted to each other, marked as a forbidden transition in Fig. 2. The only way the complex of Fig. 1g can relax to that of Fig. 1f is to trace its trajectory backward to its original state and try again to avoid the free-energy minimum state at an intermediate time.