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 rings) 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
one-by-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.
Andrzej Stasiak
Akos Dobay
Jacques
Dubochet
Laboratoire
d'Analyse Ultrastructurale
Université de Lausanne,
CH-1015
Lausanne, Switzerland,
E-mail: andrzej.stasiak{at}lau.unil.ch
Giovanni Dietler
Institut de Physique de la
Matiere Condensée
Université de Lausanne
REFERENCES
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M. Grandbois,
M. Beyer,
M. Rief,
H. Clausen-Schaumann,
H. E. Gaub,
Science
283,
1727
(1999)
.
-
S. B. Smith,
Y. Cui,
C. Bustamante,
271,
795
(1996)
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T. Deguchi and
K. Tsurusaki,
Phys. Rev. E
55,
6245
(1997)
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A. D. Mehta,
M. Rief,
J. A. Spudich,
D. A. Smith,
R. M. Simmons,
Science
283,
1689
(1999)
;
U. Bockelmann,
B. Essevaz-Roulet,
F. Heslot,
Phys. Rev. Lett.
79,
4489
(1997)
.
23 April 1999; accepted 21 June
1999
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 micro-ruptures
shown at about 2 nN and not, as Stasiak et al. assume, at
half the force.
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).
[View Larger Version of this Image (12K GIF file)]
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
side-groups, 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 reached a diameter that was comparable
to the size of the side group; otherwise, as a result of thermal
excitations, the side group would easily pass through the loop. On
pulling, the barrier would be overcome and the loop would be tightened
further. A second side group could be pulled through the loop
(resulting in a subsequent discontinuity in the force scan) only if the
loop did not close smaller than the size of the side group. Because the
diameter of a loop would decrease linearly with the length of the
polymer, this second event could not occur "later" than a few
angstroms. Triple or higher discontinuities, which we predominantly
observed in the experiment, cannot be explained by the tightening of
knots.
We do find, in our histograms of polymer elongation after a
discontinuity [figure 3 in (1)], a fraction of about 5%
of events for which this criterion would hold, and so we cannot exclude
the possibility that these events might reflect the tightening of
knots. The remaining 95% of events, however, which show polymer elongation of more than 10 Å, cannot be explained in this way.
A knot can have a complex topology, like that which results from trying
to pull a tight knot in a rope through an entangled mess made by the
rest of the rope (a situation that all of us fishermen are familiar
with). However, the analogy to molecular events is not valid because
molecular knots have no friction, only activation barriers of the kind
we discuss above. The conformation of the disentangled mess of a hemp
rope does not change spontaneously as one tries to pull the knot
through. In the case of a molecular mess, however, thermal
excitations would create a multitude of conformations during pulling
that would allow a molecular knot to move through the mess.
In analogy to DeGennes's description of polymer diffusion
(3), this might be called "forced reptation."
We are still convinced that our explanation of the data in our report
(1) is correct. Nevertheless, this discussion has yielded an
interesting catch, and it may inspire further thought and
experimentation.
Hermann E. Gaub
Lehrstuhl für
Angewandte Physik,
Ludwig-Maximilians-Universität,
D-80799
München, Germany
E-mail:
gaub{at}physik.uni-muenchen.de
Hauke Clausen-Schaumann
Lehrstuhl für Angewandte
Physik,
Ludwig-Maximilians-Universität,
D-80799
München, Germany
Martin Beyer
Institut für Physikalische
und Theoretische Chemie,
Technische Universität München,
85748 Garching, Germany
Matthias Rief
Department of Biochemistry,
School of Medicine,
Stanford University,
Stanford, CA 94305-5307, USA
Michel Grandbois
Department of Physics and Astronomy,
University of Missouri,
Columbia, MO 65211, USA
REFERENCES
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M. Grandbois,
M. Beyer,
M. Rief,
H. Clausen-Schaumann,
H. E. Gaub,
Science
283,
1727
(1999)
.
-
A. M. Saitta,
P. D. Soper,
E. Wasserman,
M. L. Klein,
Nature
399,
46
(1999)
; Y. Arai, R. Yasuda, K. Akashi, Y. Harada, H. Miyata, K. Kinosita, H. Itoh, ibid.,
p. 446.
-
P. DeGennes, Scaling Concepts in Polymer Physics
(Cornell Univ. Press, Ithaca, NY, 1979).
19 May 1999: accepted 21 June 1999
