Every angler knows that a fishing line breaks easily at the place of a tight knot. In their report (1) "How strong is a covalent bond?" Michel Grandbois et al. present a "fly fishing method" for measuring the force needed to break long polysaccharide chains. In these experiments, a single molecule was attached with covalent bonds (Si-C, Si-O) to the tip of an atomic force microscope (AFM) at one end and to a substrate at the other. The polymers they used were very long (thousands of sugar

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... and were relaxed in a solution before attachment to the surfaces. Persistence length of such polymers roughly corresponds to the dimension of the composing monomers, as demonstrated in a report by S. B. Smith et al. (2), in which single-stranded DNA was used. Chains of this size (a thousandfold longer that their persistence length) are usually knotted and frequently will have more than one knot (3). This knotty property of long polymers was not discussed by Grandbois et al. (1), and it casts doubt on the interpretations of the data and the conclusions in the report. Grandbois et al. observed a breaking force that was much smaller than one would expect if the C-C or C-O bonds within the polysaccaride chains breaking. They then infer that the polymer itself did not break, but that the multiple bonds at each end (attaching the polymer to the AFM and to the substrate) were breaking, one at a time (1). This interpretation is rather unlikely, because such oneby-one breaking of identical covalent bonds should produce a succession of peaks of roughly the same height (4). Grandbois et al., however, observed a continuous steep increase between the consecutive peaks. Also, Grandbois et al. state that the observed breaking force was smaller than the theoretical strength of Si-C bonds. Therefore, it is likely that the actual attaching bonds did not break. What, then, were the small peaks of increasing height that they observed and analyzed? We hypothesize a third possibility: that the observed peaks preceding the final breakage could be the "signatures" of progressive tightening of complex knots in the polysaccharide chain, whereby this tightening could be opposed by entanglement with other chains attached in the vicinity. Why, then, might the change of the attaching chemistry change the strength of the polymer, which would remain chemically unchanged? Small differences in a strength of a solvent (caused, for example, by the presence of mercaptoethylamine) could affect the tightness of the formed knots: A good solvent would loosen the knot, and thus change the breaking force. If Grandbois et al. were in fact measuring the breakpoints of knotted polysaccharide chains, why would the breaking force of a knotted polymer then be measured at about half the expected breaking force of an unknotted polymer? In an attempt to answer this question, we performed an experiment with a fishing line that had a nominal resistance of 10 kg. After tying a trefoil knot in the line, we tested the subsequent resistance. The line broke, exactly at the knot, under a weight of about 6 kg. Response: Science is based on the falsification of hypotheses, and we are grateful for this comment by Stasiak et al., which challenges the hypothesis we proposed in our report by launching a competitor. In contrast to the assumption made by Stasiak et al., it was not the case that the rupture peaks that we observed were always in increasing order. In fact, we frequently found peaks in descending order (Fig. 1) . The insert in figure 2 in our report (1) may have mislead readers; however, it is only one example among many. Also, because the insert shows a close-up view of the trace at the position marked by the arrow, the microruptures shown at about 2 nN and not, as Stasiak et al. assume, at half the force. We agree with the assumption made by Stasiak et al. that knotted polymers will break at lower forces than unentangled ones. This topic was addressed recently by Saita et al. (2). Polymers, however, break only once, and we do not see how one could use this property of knotted lines to explain the multiple breaking of molecular bonds that we observed in our investigations. The tightening of a molecular knot will manifest itself in force curves in several ways. Let us, for ease of discussion, distinguish between equilibrium and nonequilibrium processes. In cases where the knotted states are separated by energy barriers on the order of the thermal energy, kT, one would expect a fully reversible behavior: the knots would tighten or disentangle during the experiment. With changing external force, the average time the polymer spends in either of the two states would change. The apparent stiffness of a segment would thus be decreased slightly. But, as a result of the ratio of knotted length to contour length of the polymer, this effect would be marginal and, most likely, not detectable. In cases where the energy barriers between the knotted states were so high that a disentanglement would not occur spontaneously during the experiment (for example, as a result of steric restrictions caused by sidegroups, as suggested by Stasiak et al.), the tightening of the knots could indeed result in discontinuous force scans. The discontinuities under discussion correspond to energy barriers on the order of covalent bonds, so the restrictions would have to be severe. Following the reasoning of Stasiak et al., the restriction imposed by bulky side groups would, on pulling of the polymer, greatly increase with decreasing size of the loop of the knot. This would be a percolation-type problem. Thus, only when the diameter of the mesh became comparable to the size of the side group would an energy much higher than kT be built up. These conditions would result in a very narrow window of forces and polymer elongation (after the discontinuity) within which this effect could contribute to the force curves. We can estimate the range of this effect as follows: Let us assume that a loop is tightened and is stuck at a side group. This situation would be possible only if the loop had Fig. 1. Multiple bond rupture events of a covalently attached polysaccharide showing rupture forces in random order. Experimental conditions were the same as those in figure 2 of our report (1).