High flexibility of DNA on short length scales probed by atomic force microscopy
High?exibility of DNA on short length scales probed by atomic force microscopy
PAUL A.WIGGINS1,THIJN VAN DER HEIJDEN2,FERNANDO MORENO-HERRERO2,
ANDREW SPAKOWITZ3,ROB PHILLIPS4,JONATHAN WIDOM5,CEES DEKKER2
AND PHILIP C.NELSON6*
1Whitehead Institute,Cambridge,Massachusetts02142,USA
2Kavli Institute of NanoScience,Delft University of Technology,Lorentzweg1,2628CJ Delft,The Netherlands
3Department of Chemical Engineering,Stanford University,Stanford,California94305,USA
4Division of Engineering and Applied Science,California Institute of Technology,Pasadena,California91125,USA
5Department of Biochemistry,Molecular Biology,and Cell Biology,Northwestern University,Evanston,Illinois60208,USA
6Department of Physics and Astronomy,University of Pennsylvania,Philadelphia,Pennsylvania19104,USA
*e-mail:nelson@https://www.wendangku.net/doc/901738957.html,
Published online:3November2006;doi:10.1038/nnano.2006.63
The mechanics of DNA bending on intermediate length scales(5–100nm)plays a key role in many cellular processes,and is also important in the fabrication of arti?cial DNA structures,but previous experimental studies of DNA mechanics have focused on longer length scales than these.We use high-resolution atomic force microscopy on individual DNA molecules to obtain a direct measurement of the bending energy function appropriate for scales down to5nm.Our measurements imply that the elastic energy of highly bent DNA conformations is lower than predicted by classical elasticity models such as the worm-like chain (WLC)model.For example,we found that on short length scales,spontaneous large-angle bends are many times more prevalent than predicted by the WLC model.We test our data and model with an interlocking set of consistency checks.Our analysis also shows how our model is compatible with previous experiments,which have sometimes been viewed as con?rming the WLC.
The WLC model of DNA conformation is an effective theory that
idealizes the macromolecule as an inextensible elastic rod,and
attributes to its bending deformations a classical(Hooke-law-
type)elastic energy cost.The WLC has come to dominate physical
discussions of double-stranded DNA mechanics,due in part to its
simplicity and its successful description of experiments such as
force spectroscopy on single DNA molecules1–3.Because of these
notable successes,classical elasticity models like the WLC(and its
generalizations to include twist stiffness)have been the framework
for many studies of DNA mechanics.However,previous tests of
the WLC model using DNA stretching,atomic force microscopy
(AFM)and other methods have been largely insensitive to the
details of mechanics in the intermediate-scale regime crucial for
cellular function,from chromosomal DNA packaging,to
transcription,gene regulation and viral packaging4–6.A
quantitative understanding of such interactions requires a model
of DNA bending that is valid on these biologically relevant
length scales.
In the following,we will argue that the successes of the WLC
model in fact arise from the long length scales probed by the classic
experiments,rather than from any true underlying classical elasticity
of DNA,and we will give short-scale measurements that do not
agree with the predictions of the WLC model.It may seem that a
more general model,capable of embracing both our new data and
the classic earlier experiments,would necessarily contain many more
unknown parameters than the WLC model.On the contrary,the
model we propose is as simple as the WLC.
RESULTS
THE WLC
The WLC model can be formulated more precisely by
representing each conformation as a chain of segments of length‘
and assigning to it an elastic energy cost of the form
E WLC?1
2
k B T(j/‘)(u i)2,where u i is the angle between successive
segment orientations t i and t it1(in radians),k B T is the thermal
energy,and j%50nm is an effective elastic constant describing
the chain’s resistance to bending(see Supplementary
Information).We call this bending-energy function classical(or
‘harmonic’),because it is a quadratic function of the strain
variable u i.The probability distribution of bends is then given by
the Boltzmann distribution,g(t it1j t i)?q21exp[2E WLC(u i)/k B T],
where q is a normalization constant.
The choice of a harmonic energy function E WLC makes the
WLC angular distribution gaussian.However,even if E(u)is not
harmonic,the angular distribution g(t itn j t i)will approach a
gaussian form at large separations L?n‘,because the iterated
convolution of any distribution with itself converges to a gaussian
form.That is,even if non-harmonic elastic behaviour is present,
it will be hidden on long length scales by thermal?uctuations,
giving rise to a theory that behaves like the WLC.Thus,to investigate whether E (u )is harmonic,we must measure it directly on the length scale of interest,and not on some much longer scale.
AFM MEASUREMENTS OF DNA CONTOURS
We argued above that to look behind the WLC model,we must examine length scales that are not much longer than the few nanometres scale of the molecule.It is dif?cult to obtain full
three-dimensional views of equilibrium conformations of DNA in solution with the required resolution.However,under appropriate conditions,DNA adheres to a mica surface weakly enough that DNA–mica interactions are believed not to affect the chain statistics 7.We will argue in the following that our molecules indeed adopt two-dimensional equilibrium conformations.We captured these by using AFM,following procedures outlined in ref.8,but with signi?cant improvements in image analysis (see ‘Methods’and Supplementary Information).Figure 1a is a representative image of DNA on mica,taken using tapping-mode AFM in air.The use of ultrasharp silicon tips allows a very high resolution to be obtained;for example,Fig.1b displays a cross-sectional DNA height pro?le with a half width at half maximum of only 2.5nm.
Despite our many precautions,we realized that potential gradients occur upon DNA adsorption that could potentially
a b
c
0.9
150
L R
t′
t
5 nm
Position (nm)
H e i g h t (n m )
Figure 1High-resolution AFM images and tracing.Data shown were taken on V4-grade mica with 12mM Mg 2t(see Supplementary Information for other surfaces and salt concentrations).a ,A 0.5?0.5m m AFM image (256?256pixels)of 2,743-bp dsDNA deposited on mica;z -range 1.5nm.b ,Cross-sectional height pro?le of the DNA molecule,showing a half width at half
maximum of only 2.5nm.c ,Detail of the square section indicated in a ,showing also the chain of points determined by our automated image-analysis routine and illustrating the geometric quantities discussed in the text.Successive points are separated by ‘?2.5nm.The end–end distribution K (R ;L )is the probability distribution of real-space separation R among points separated by contour length L ;in the example in the centre of the ?gure,we take L ?4?2.5nm.The angle distribution G (u ;L )is the probability distribution of angles between tangents (short blue arrows)
separated by L ,which in the example on the right of the ?gure is 3?2.5nm.
100
Separation, L (nm)
200
100
Separation, L (nm)200
1.02
1
?c o s θs ,s +L ?
?(R s ,s +L )2? (n m 2 × 104)
0–I n G (θ; 5 n m )
Bend angle, θ (rad)
a
b
c
Figure 2Checks of equilibrium adsorption and failure of WLC on short length scales.a ,b ,Plots of k (R s ,s tL )2l and k cos u s ,s tL l from experimental data (dots),together with model-independent predictions assuming j ?54nm (curves).c ,Dots:negative logarithm of the observed probability distribution function G (u ;L ?5nm),a measure of the effective bending energy at this length scale in units of k B T .The graph is a histogram,computed using
experimentally observed DNA contours with total length of about 240,000nm,or a total of about 94,000pairs of tangent vectors.The error bars represent
expected p
n error in bin populations due to ?nite sample size.Dashed curve:the same quantity for curves drawn from the distribution appropriate to the WLC with persistence length equal to that used to draw the curves in (a and b ).Although the two distributions agree qualitatively at low de?ection u ,they disagree at large angles:the WLC model predicts far fewer such large
de?ections than were observed.Solid black curve:the same quantity when a sample of WLC con?gurations was generated numerically and converted to
simulated AFM data,then subjected to the same image analysis that yielded the experimental dots.
generate spurious results.For this reason,we?rst made some
detailed,model-independent theoretical predictions and checked
that they were well obeyed by our data.As a?rst check,we
measured the mean-square separation of pairs of points located at
contour length s and stL from the end of the molecule,
averaging over s and over all observed contours.We call this
quantity k(R s,stL)2l,and examine its behaviour as a function of
the contour length separation L between the points(Fig.2a).If
the chain conformations are indeed equilibrated,and the effective
elastic-energy function is local,then this function must take a
particular form(see Supplementary Information).Indeed,as
Fig.2a shows,we found that the data follow this prediction very
well.We can also de?ne the tangent–tangent correlation
k cos u
s,stL
l,where u
s,stL
is the angle between tangent vectors at a
pair of points separated by L.If the chains’conformations are
equilibrated,then this correlation must fall with contour
separation L as e2L/(2j).Figure2b shows that this prediction,
too,is well satis?ed,with j?54nm.T ogether,these two tests
con?rm that,at least over length scales less than200nm,
our contours re?ect equilibrium two-dimensional chain
conformations,extending the observation of Rivetti et al.7.
The statistical measures in Fig.2a,b are model-independent;
they do not distinguish between different forms of the local
elastic energy E(u).Therefore,we next measured the probability
distribution function G(u;5nm)of various bend angles at
points separated by contour length5nm.Figure2c shows the
negative logarithm of this histogram.The resulting curve
measures the effective bending energy E(u)/k B T,coarse-grained
to the length scale‘exp?5nm probed by the experiment.As
explained above,the WLC model predicts that this function will
be quadratic,regardless of the value of‘exp.Instead,Fig.2c
shows that the coarse-grained bending energy is far from being
a quadratic function of the de?ection angle u.At large angles
we see instead that ln G(u)is nearly linear in u.These results
can be stated differently by saying that large angular de?ections
between points separated by5nm were about30times
more frequent in our data than the prediction of the WLC
model(Table1).
ANHARMONIC MODEL
We constructed a local-elasticity model,with an effective
bending-energy function chosen to mimic the behaviour seen in
Fig.2c.This model falls into a large class we have named
‘sub-elastic chain’(SEC),because it describes a polymer chain
with a response to bending that,for large de?ections,is softer
than the usual harmonic model9.One empirical energy function
that summarizes the data is a linear SEC(LSEC):
E LSECeuT?a j u j k B T;e1T
where a is a dimensionless constant that depends on the chosen
segment length‘.We chose‘?2.5nm and adjusted a to?t the
long-distance correlation G(u;30nm),yielding a?6.8.We then
reasoned that,if indeed our contours represent equilibrium
conformations of an elastic body,and in particular if successive
chain elements are independently distributed according to
equation(1),then the angle–angle correlations at all separations
L!‘exp must be computable from that rule,with no further
?tting.Figure3tests our predictions for G(u;L),and also for the
end-to-end distribution K(R;L)de?ned in Fig.1c.The remaining
?ve curves in Fig.3are zero-?t-parameter predictions of our
model and are observed to describe the data very well.The fact
that our simple model passes the interlocking set of hurdles
represented by the six curves in Fig.3is further strong
evidence that we indeed measured spontaneous,equilibrium
conformational?uctuations characterized by equation(1),and
not an instrumental artifact or a surface-adsorption effect.
Moreover,in keeping with the expectations raised earlier,these
?gures show that at large contour-length separations the
distribution of bending angles does converge to that expected
from the WLC model.
Our model correctly reproduces the existing successes of the
WLC model.For example,the force–extension relation in our
model is experimentally indistinguishable from that of the WLC
model,as is the rate of cyclization for linear constructs longer
than a few persistence lengths(see Supplementary Information).
Table1Incidence of bends that are large(!1.1rad)or medium-large(!0.8rad),at contour separations L55nm and10nm.The?rst three rows restate points made in the?gures.Each of the columns labelled‘fraction’show that our data disagree signi?cantly with the predictions of the WLC model(Fig.2c), but agree with our model(Fig.3a).The angles quoted refer to the angle between the vectors t and t0in Fig.1c.Row1:experimental results from our main data.The large absolute numbers emphasize that our conclusions are not merely based on a handful of images.Row2:expected results from a Monte Carlo evaluation of the WLC model with persistence length j554nm.Row3:expected results from Monte Carlo evaluation of our model with the same long-scale behaviour.The remaining rows show results of control experiments and a more detailed calculation(see Supplementary Information).Row4:experimental data obtained using DNA incubated with ligase.Row5:experimental data obtained when DNA was adsorbed to V1-grade mica.Row6:numerical results when the WLC con?gurations generated by the Monte Carlo code(row2)were converted to simulated AFM traces and then sent through our image analysis(solid curve in Fig.2c and Supplementary Information,Fig.S8).
Points separated by L?5nm Points separated by L?10nm No.of
pairs
No.of
large
bends
Fraction
?104
No.of
medium
bends
Fraction
?104
No.of
pairs
No.of
large
bends
Fraction
?104 Experimental data93,895828.77467992,725969100 WLC3,122,109910.296,848223,105,18717,67857 LSEC equation(1)2,922,1112,7569.421,809752,906,27328,77399 Experimental with
ligase
51,303428.24699150,69946792
Experimental with
V1-grade mica
30,59718 5.92638630,263326110 WLC simulated200,152110.5556828185,1641,08959
DISCUSSION
Other authors have already reported that AFM images con?rm that adsorbed DNA follows a two-dimensional WLC distribution 7.Indeed we agree,when DNA is viewed on long length scales.However,we ?nd signi?cant deviations from the WLC model on shorter,biologically relevant,length scales.It is possible in principle that surface adsorption and local defects on mica could conspire to modify the apparent elasticity of DNA.Our many checks and controls make this unlikely,however (for example,those shown in Fig.2a,b).Other experiments not involving AFM,as well as molecular simulations,also point to a modi?cation of the WLC model qualitatively similar to the one we report here (see Supplementary Information).
Our empirical effective bending-energy function,embodied in equation (1),is as simple as the WLC model,and yet unlike the WLC model it accurately describes the observed behaviour of DNA on multiple length scales,at both high and low curvatures.In particular,it is a useful starting point for the description of regulatory loops and other mesoscale DNA complexes.Our generic viewpoint,using models of the SEC type,may also be applicable to other stiff biopolymers.
The form of the effective coarse-grained energy function that we ?nd (equation (1))may come as a surprise,but in fact there is a precedence for functions of this form.For example,
the overstretching transition of DNA reveals a plateau in the force–extension relation,created by a transition between effective links of two different types,with different values of the rise per base pair 10.Another transition leads to a nearly constant axial torque as DNA under tension is twisted 11.Although these dramatic transitions occur in the laboratory at high external stresses,they will also occur spontaneously,albeit infrequently,and may in fact mediate important functions of DNA 12,13.By analogy,we can imagine a transition between two or more conformers with different bend angle (or stiffness),for example,the bent,but still base-paired,state constructed long ago by Crick and Klug 14.Increasing the bending stress on a tract of the molecule could then alter the coexistence between the conformers,leading to a plateau in the stress–strain relation,or in other words an effectively linear energy function for bend.There may be a threshold for the onset of this approximately linear behaviour;the effective elastic energy function may have a harmonic (parabolic)region at small curvature.However,we found that the harmonic-elasticity regime,if any,is small (see Supplementary Information).In any case,the presence of such a regime is not needed to account for previously known facts about DNA mechanics (see Supplementary Information).
An alternative possibility is that ‘bare’DNA elasticity may in fact be harmonic,but it is dressed into a non-harmonic form by electrostatic effects,which have already saturated at the lowest Mg 2tconcentrations we studied.For example,Rouzina and Bloom?eld proposed that the presence of multivalent cations in solution could lead to transient kinking of DNA 15.Because their effect is non-local only over about six base pairs,it would appear local when coarse grained to length scales longer than about 2nm,and so can be described using the methods of this paper.Indeed,one bene?t of our coarse-grained approach is that our empirical effective energy function is useful even before we resolve its mechanistic origin.
One key application of this work is in biological systems,where DNA–protein complexes are important for the molecular-scale function of the cell.Many authors have assumed that for small loops,where thermal ?uctuations may be neglected,classical harmonic rod elasticity is still applicable.On the contrary,we suggest that the WLC model,when it is useful,actually owes its applicability to the existence of thermal ?uctuations;when ?uctuations can be neglected,then we must use a non-harmonic energy function like equation (1).
In summary,we have argued that the short-length-scale bend distribution function must be measured directly,not extrapolated from long-length-scale measurements.We made such measurements,and showed that the probability of spontaneous sharp bending is orders of magnitude higher than predicted by the WLC model.However,we found that a surprisingly simple coarse-grained elasticity theory is quantitatively accurate,both for describing spontaneous conformational ?uctuations of DNA on length scales relevant for looping and nucleosome formation (5nm and longer),and for the force–extension of DNA and other long-length-scale phenomena.
METHODS
SAMPLE PREPARATION AND AFM
DNA samples were prepared for AFM imaging by depositing 5ng 2,743-bp linear double-stranded DNA (pGEM-3Z,Promega)diluted in 6m l of 10mM Tris–HCl (pH 8.0),supplemented with either 6,12,30or 150mM MgCl 2,onto freshly cleaved muscovite-form mica of grade V1and V4(SPI),following the methods of other authors 7,8,16.After approximately 30s,the mica was
washed with MilliQ-?ltered water and blown dry in a gentle stream of nitrogen gas.Samples were imaged in air at room temperature and humidity with a NanoScope IIIa (Digital Instruments),operating in tapping mode with a
a
b 4
2I n K (R ; L )
0–2–I n G (θ; L )
R /L
θ (rad)
Figure 3The non-harmonic elasticity model of equation (1)simultaneously ?ts many statistical properties of the experimental data.a ,Negative logarithm of the probability distribution function G (u ;L )for the angle u between tangents separated by contour length L ,for L ?5nm (orange),10nm (purple)and 30nm (blue).The dots are experimental data;the red dots,and the meaning of the error bars,are the same as in Fig.2c.Solid curves:Monte Carlo evaluation of this correlation function in our model (equation (1)).Dashed curves:Monte Carlo evaluation of the same correlation in the WLC model with persistence length
j ?54nm.The dashed curves are all parabolas.The red dashed line is the same as in Fig.2c.Although the WLC model can reproduce the observed correlation at long separations,at short and medium separation it understates the prevalence of large-angle bends.b ,Logarithm of the probability distribution function K (R ;L ),expressed in nm 21,for the real-space distance R between points separated by contour length L ?7.5nm (orange),15nm (purple)and 50nm (blue).As in a ,the WLC model correctly captures the long-distance behaviour,but understates large bending (leading to large shortening)at separations less than about half a persistence length.
type-E scanner,with a pixel size(grid spacing)of1.95nm.Background
correction consisted of?tting a second-order polynomial to every line in the
AFM image.Tapping-mode SuperSharpSilicon tips,type SSS-NCH-8
(NanoSensors)were used.Some of the experiments were carried out using a
commercial AFM from Nanotec Electronica operating in dynamic mode(see
Supplementary Information).For both the Digital Instruments set-up and the
con?guration from Nanotec Electronica we obtained similar results,thus
excluding artifacts caused by the imaging instrument.
The surface and DNA were free of any salt deposits or protein impurities,
and very clean images of DNA were obtained.Furthermore,we studied the
effect of repairing possible nicks using ligase,examined various qualities of
mica,and systematically varied the concentration of Mg2tin the solution.
None of these variations altered our conclusions(see Supplementary
Information and Table1).We note that Mg2tconcentrations at the lower end
of the range we checked have been found to reduce the persistence length of
DNA only slightly17,18.
IMAGE ANALYSIS
Our image-analysis software was custom developed with the particular goal of
analysing the local bend angles.The DNA molecules were traced automatically
(but with human supervision)using a custom code in Matlab.Chain tracing
was initiated at a user-determined initial point and trial tangent direction.(See
Supplementary Information for a description of the algorithm.)Its output was
a representation of the DNA contours as chains of xy pairs separated by
contour length2.5nm(Fig.1c).
DATA ANALYSIS
T o reduce the effects of noise,we looked at the real-space separations of points
separated by at least three steps(7.5nm;see Fig.3b),and the angular
difference between tangent vectors de?ned by next-nearest neighbours and
separated by at least two steps(5nm;see Figs.1c and3a).We also applied the
same procedures to virtual chains generated by our Monte Carlo code(see
Supplementary Information),and made appropriate comparisons.
We performed extensive tests to support our interpretation of the data.
We checked by eye that our contour-tracking software faithfully followed the
contours,and we sent simulated WLC data,including noise and tip-induced
broadening,through our image-processing and analysis software to con?rm
that the resulting WLC-based contours did not generate distributions
resembling our experimental data(see Fig.2c and Supplementary
Information).Moreover,we checked that the alternative hypothesis of
non-equilibrium adsorption of a WLC-distributed chain to the surface does
not explain our experimental data(see Supplementary Information).
Received23June2006;accepted31August2006;published3November2006.
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Acknowledgements
We thank N.R.Dan,N.H.Dekker,M.Inamdar,R.James,R.D.Kamien,I.Kulic,https://www.wendangku.net/doc/901738957.html,urence,
https://www.wendangku.net/doc/901738957.html,very,J.Maddocks J.Marko,A.Onufriev,P.Purohit,I.Rouzina,J.M.Schurr,R.Seidel,
V.Soghomonian,Z.-G.Wang and Y.Zhang for helpful discussions and correspondence.P.A.W.
acknowledges grant support from an NSF graduate fellowship.F.M.-H.acknowledges support from
La Fundacien Ramon Areces as a postdoctoral fellow.A.J.S.acknowledges funding from the National
Institutes of Health(NIH-GM071552).R.P.and P.A.W.acknowledge the Keck Foundation and NSF
Grant CMS-0301657as well as the NSF-funded Center for Integrative Multiscale Modeling and
Simulation.J.W.acknowledges support from NIH grants R01GM054692and R01GM058617.C.D.
acknowledges the Stichting voor Fundamenteel Onderzoek der Materie(FOM),which is?nancially
supported by the Nederlandse Organisatie voor Wetenschappelijk Onderzoek(NWO).P.N.
acknowledges NSF Grant DMR04-04674and the NSF-funded NSEC on Molecular Function at the
Nano/Bio Interface DMR04-25780.J.W.,R.P.and P.C.N.acknowledge the hospitality of the Kavli
Institute for Theoretical Physics,supported in part by the National Science Foundation under
Grant PHY99-07949.
Correspondence and requests for materials should be addressed to P.C.N.
Supplementary information accompanies this paper on https://www.wendangku.net/doc/901738957.html,/naturenanotechnology.
Author contributions
P.A.W.,T.v.d.H.,F.M.-H.,C.D.and P.N.contributed to the experimental,theoretical and analysis
strategy.T.v.d.H.and F.M.-H.carried out the AFM experiments,and with,C.D.and P.A.W.
contributed to the custom image-processing software.P.A.W.,T.v.d.H.,F.M.-H.,C.D.and P.C.N.
contributed to the analysis.A.S.contributed the analysis and simulations of nonequilibrium
adsorption.P.A.W.,R.P.,J.W.and P.C.N.contributed to the initial formulation of the hypothesis and
its re?nement.P.W.,T.v.d.H.,F.M.-H.,A.S.,C.D.and P.C.N.contributed to the written and graphical
presentation;all authors discussed the results and commented on the manuscript.
Competing?nancial interests
The authors declare that they have no competing?nancial interests.
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