Navigating Complex Labyrinths:Optimal Paths from Chemical WavesOliver Steinbock, Agota T6th, Kenneth Showalter*

The properties of excitable media are exploited to find minimum-length paths in complexlabyrinths. Optimal pathways are experimentally determined by the collection of time-lapse position information on chemical waves propagating through mazes prepared withthe Belousov-Zhabotinsky reaction. The corresponding velocity fields provide maps ofoptimal paths from every point in an image grid to a particular target point. Collisions ofwaves that were temporarily separated by obstacles mark boundary lines between sig-nificantly different paths with the same absolute distance. The pathfinding algorithm istested in very complex mazes with a simple reaction-diffusion model.

Propagating waves in spatially distributed,excitable media arise from the coupling of a

positive feedback process with some form oftransport, for example, autocatalytic chem-ical reaction with molecular diffusion (1).Such waves are observed in biological (2),chemical (3), and physicochemical (4) sys-

tems and typically exhibit constant veloci-ties and annihilation in collisions withboundaries or other waves (5, 6). In thisreport we describe how these features giverise to a highly efficient algorithm for thedetermination of optimal paths. Conven-tional pathfinding methods typically rely oniterative searches, in which all possiblepathways between a point of interest and a

target point are successively determinedand the optimal path is then selected (7). Inan excitable medium, a single propagatingwave generates a map for the optimal pathfrom every point of the system to a targetpoint. This feature was first noted by Sep-ulchre et al. (8) in a computational study ofwave propagation in an oscillatory mediumcontaining impenetrable obstacles.

Experimental studies of wave propaga-

tion in complex labyrinths were carried outwith the excitable Belousov-Zhabotinsky(BZ) reaction (9) by means of time-lapsedigital imaging techniques. Planar laby-rinths were made of vinyl-acrylic mem-

branes saturated with BZ reaction mixture(10). We prepared impenetrable barriers bycarefully cutting out rectangular regions ofthe membrane. This simple procedure al-lowed the realization of a variety of geom-

etries and provided an effectively two-di-mensional system without hydrodynamicdisturbances. The composition of the BZsolution was prepared so that no spontane-ous wave initiations occurred; however,

waves could be reproducibly initiated byallowing a silver wire to contact the re-

agent-loaded membrane. The wave propa-gation was monitored by image analysis of

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monochromatic light (wavelength X = 500nm) reflected from the medium. The videosignals were recorded and then digitizedinto image sequences (1 1), which were en-

hanced with standard imaging routines andanalyzed.

Figure 1A shows a 50-frame compositeimage of a chemical wave propagatingthrough a BZ-membrane labyrinth. Thewave splits at each junction during its prop-agation and therefore reaches all points inthe maze that are connected to the startingpoint (at the lower left corner). Wave seg-

ments that propagate into blind channelsultimately vanish on collision with theboundaries. Each snapshot shows waves atan equal distance from their positions in theprevious snapshot, given by the product ofthe constant velocity (2.41 ± 0.18 mm/min) and the time between each two snap-

shots (50 s). To a good approximation,waves propagate according to Huygens'sprinciple, where the front advances a fixeddistance per unit of time in a directionnormal to the front. The effects of curvatureon the wave velocity (12), to be discussedbelow, are slight for mazes with sufficientlylarge length scales as considered here.

Image sequences with higher temporalresolution allow the generation of maps

that give the path length from any locationin the maze to a given target point, withinthe resolution of the composite image. Fig-ure 1B shows a color-coded distance mapbased on higher resolution data, where thecolors are keyed to the elapsed time fromthe wave initiation to the local maximumof excitation at each point. The minimumpath length from any point in the maze tothe target point S is given by the product ofthe elapsed time assigned to that point andthe constant wave velocity. The optimaltransit time and distance for every locationin the labyrinth are determined by a singlepropagating wave, representing a highly ef-ficient accumulation of information in a

parallel manner.

The space-time information compiled inFig. 1B provides the basis for a determina-

SCIENCE * VOL. 267 * 10 FEBRUARY 1995

tion of detailed minimum-length paths.The optimal path from any arbitrary pointto the target point is easily determined ac-

cording to the velocity field, which is gen-

erated by specifying the direction of wave

propagation at each point in the grid. Thevector field derived from Fig. 1 B is shown inFig. 1C, along with several examples ofoptimal paths. One can compute the spe-

cific shortest path connections from thetime-indexed points of the composite imageby searching for the nearest point indexedt - X, where t is the elapsed time from thewave initiation to the current position and

is the time increment between successiveimages.

Figure 1C shows that the path optimiza-tion includes local features such as diagonaltrajectories in the corridors of the labyrinth.The labyrinth is therefore analyzed as a

two-dimensional array of rectangular seg-

ments rather than as a mesh of one-dimen-sional strings. Path boundaries, created bycolliding waves that previously had beenseparated by barriers, are also defined (seeFig. 1B). These boundaries demarcate sig-nificantly different pathways of equal dis-tance to the target point S. The maze isconsequently divided into local domainsthat correspond to families of pathways,where paths in one domain are matched bydifferent paths of equal length in anotherdomain.

Distinct basins of attraction arise fromthe synchronous initiation of several waves

at different locations in a maze. In contrastto experiments with a single initiation, two

different types of annihilation loci occur:

path boundaries from waves originating at a

single initiation point and collision bound-aries from waves originating at differentinitiation points. The latter representboundaries composed of points that are

equidistant to two initiation sites; the in-tersection of three or more boundaries de-fines points equidistant to each of the cor-

responding initiation sites. Both types ofboundaries are shown in Fig. 2, in whichfour waves were initiated almost synchro-nously at sites S1, S2, S3, and S4 near thecorners of the maze.

The basin of attraction associated witheach initiation site is shown by a singlecolor in Fig. 2, where the intensity is in-dexed to the elapsed time from the wave

initiation. Each color boundary represents a

separatrix between two basins. In terms of a

potential surface description, the minimumof each basin lies at the wave initiationpoint and the height at any point on thesurrounding surface is proportional to theelapsed time assigned to that point. Eachseparatrix is defined by the locus of collisiontimes and represents a "watershed" separat-ing trajectories that lead to different targetpoints.

Department of Chemistry, West Virginia University, Mor-gantown, WV 26506, USA.

*To whom correspondence should be addressed.

Examples of shortest distance paths arealso shown in Fig. 2, computed by trackingnearest t 7 points as described above.Four initial points chosen near the intersec-tion of the Si, S2, and S3 basins at (x, ')(14 mm, 16 mm) lead to three differenttarget points with two significaintly differ-ent paths to S2. Three other paths are alsoshown, with initial points chosen to be neara path boundary and a basin separatrix.

A

Br _

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While propagating chemical waves pro-vide vivid examples of path optimization inexcitable media, computational simulationsof such waves allow the approach to bedemonstrated in very complex labyrinths.We use a generic model for excitable media,the txs o-X ariable Barkley model (13), w.xhichhas the form

Uv u f u') ( 1 )

*at =g(f, l) (2)

where the variables u and tr describe thespatiotemporal dynamics of a propagatorspecies and a controller species, respectively(14).

Figure 3 shows the results of simulatedwave propagation through a complex laby-rinth that was developed in 1664 by G. A.Boeckler (15). As in the experiments, thecolor coding corresponds to transit times forthe shortest length path from any point tothe target point, the red door at the topboundary where the wave was initiated.The color map shows very different transittimes for points in each of the four squareareas and the central circular area. The mapindicates times only for the optimal paths(in fact, there are multiple paths from anyparticular starting point to the targetpoint). An example of an optimal pathoriginating from the lower left square (col-ored blue) is shown by the white dashedline. This trajectory was computed from thetime-indexed points according to the pro-cedure described above. Visual inspectionof the labyrinth reveals another, signifi-cantly different pathway, which exceeds theoptimal path in length by only 7'. We alsocarried out an additional calculation toprove that the optimal, shortest length pathis unaffected by switching the point of ori-gin with the target point and vice versa.The optimal path obtained from this "re-

verse" calculation is the same as that shownin Fig. 3.

Throughout this report we have assumedconstant-velocity waves propagating ac-cording to Huygens's principle. This as-sumption is the basis of the vector fieldsderived from the time-indexed grids and thet-T algorithm for determining optimalpaths. In fact, front curvature may signifi-cantly affect the velocity of wave propaga-tion (12). The error introduced by neglect-ing curvature effects is small in the optimal

30

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EE

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x (mm)

Fig. 2. Color map showing a BZ membrane lab-yrinth in which four waves were initiated almostsynchronously (-5-s intervals) at S1, S2, S3, andS4. The area covered by each wave is distin-guished by a different color, with the color inten-sity at any point proportional to the time elapsedsince wave initiation. Obstacles are plotted inblack, and the shortest paths between sevenpoints and various target points are shown in red.The total duration of the experiment was 450 s.

600

500

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l ~~~~~~~~~~~~~~~~op

Fig. 1. (A) Chemical wave propagating through aBZ membrane labyrinth. A sequence of 50 imagesobtained at 50-s intervals was superimposed toform the composite image. The wave was initiatedin the lower left corner of the maze. The total area

of the maze is 3.2 cm by 3.2 cm, with obstaclesappearing as black rectangular segments. (B)Color map representing the time difference be-tween wave initiation and local excitation for allpoints in the labyrinth. A sequence of 250 imagesobtained at 10-s intervals was used to form thetime-indexed composite image (yielding a spacingof 4 pixels between successive front positions).Red, green, yellow, and blue correspond to suc-

cessively longer times over the total elapsed timeof 2500 s. White lines show path boundaries. (C)Velocity field describing the local wave propaga-tion direction. Small black dots represent the ori-

30 gins of vectors. The shortest paths between fivepoints and the target point S are shown in red.

SCIENCE * VOL. 267 * 10 FEBRUARY 1995

300

200

100

0

0 100 200 300 400 500 600X

Fig. 3. Interrogation of a complex labyrinth bywave propagation in the Barkley model (13, 14).Colors indicate propagation times for a wave thatwas initiated at the entrance, shown in red at thetop boundary of the maze. The color coding is thesame as in Fig. 1 over a total elapsed (dimension-less) time of 1174.15. The dashed white lineshows the shortest path from the lower left squareregion (blue) to the maze entrance (red).

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C

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x (mm)

5

A100 paths above; however, errors in the transit

times may become significant in mazes withsmaller length scales.

75 Curvature effects considered in a generalcontext can be exploited to extend thepathfinding capabilities of excitable media.

x50 _EE_The dependence of wave velocity on cur-vature allows the determination of "doorsize" in complex labyrinths with narrow

25 gaps between barriers. The velocity dependson front curvature according to an eikonalrelation (12): c = cO + DK, where c is the

0 _______________________________ normal wave velocity, CO is the planar wave0 25 50 75 100 velocity, D is the diffusion coefficient of the

autocatalyst, and K is the curvature of theB front. For convex waves (K < 0), such as

those exiting narrow channels, the propa-100 gation velocity is reduced by the enhanceddispersion of the autocatalyst. A critical

75t111 i i | pnucleation size with a radius of curvature (r- /1/IK) given by r, = D/co is predictedwhen the normal velocity is reduced to zero.

x 50 Thus, as CO is decreased, rc increases andchannels with radii below rc do not allowwaves to pass. Recent experiments with BZ

25 _-waves propagating through microcapillary2 tubes have demonstrated this behaviorGUS).

0 - Figure 4 shows a labyrinth in which0 25 50 75 100 waves calculated with the Barkley model

were used to determine the relative channelC widths between obstacles. Color maps gen-

erated from low-, medium-, and high-veloc-100 ity waves are shown in Fig. 4, A through C,

respectively. For the slowest wave (Fig.4A), all but two channels have r > rc.

75 III _S _ Because two of the "doors" are too small,the optimal trajectory from the startingpoint in the upper right corner to the target

S 50 point S takes a circuitous route through the"allowed" openings. Above a critical veloc-

25 Asity, the opening at (x, y) = (25, 33) be-comes available for wave propagation (be-cause r > rc) and a more direct path to the

0 W l target point S is determined (Fig. 4B). For25______ 50_____ 75______ 10____ the fastest wave, the opening at (x, y) =

0 25 50 75 100 (75, 66) allows wave propagation and anearly diagonal pathway is determined (Fig.

x 4C). Thus, optimal paths are determinedFig. 4. Interrogation of "door size" in a labyrinth even when some openings are too small,with narrow passages between barriers by wave and the sizes of the openings can be foundpropagation in the Barkley model (13, 14). Dis- of wave propagationtance map and optimal trajectory (white) gener- Propagating wave prop able mediaated by medium with low excitability, a = 0.50, r Propagating waves in excitable media= 1.125 (A); intermediate excitability, a = 0.70, provide an interesting, and potentially use-rc = 0.825 (B); high excitability, a = 0.85, r, - ful, alternative to traditional iterative-0.675 (C). The color coding is the same as in Fig. search methods for determining optimal1 over a total elapsed (dimensionless) time of paths. A relatively simple analysis provides24.15 (A), 15.83 (B), and 14.60 (C). Door sizes [r optimal paths from every point in a grid to= channel radius, (x, y) = maze coordinates]: r = a target point. Thus, a robot in a large1.35, (x, y) = (50,16.5); r = 1.05, (x, y) = (25, 33); warehouse could readily use such a map atr = 1.35, (x, y) -(75, 33); r = 1.35, (x, y) = (50, any point without needing to recompute its

(75, 66); r= 1.35, (x, y)= (50, 83.5). The inte- trajectory at each new location to deliver agration was carried out on a grid of 300 points by parcel to loading dock S. The method can300 points (grid spacing AxY 0.33, time step At easily accommodate changing labyrinths as= 0.0056) with the parameters b = 0.01, e = well, provided that the time scale tor the0.02, and a indicated above. changes is longer than the time required for

recomputing the vector field. In addition,the method automatically discerns whichdoors are wide enough for the parcel, pro-viding an optimal trajectory that ignoresany doors that are too small.

While the computational implementa-tion of the reaction-diffusion algorithm pro-vides intriguing possibilities for practicalapplications, the experimental results pointto possible mechanisms for optimization inbiological systems. Neural pathways, for ex-ample, are known to be extremely complex,yet highly efficient (17). Could path opti-mization in networks of neurons rely on theproperties of excitable media, as suggestedin the optimal path determinations present-ed here? This intriguing possibility provides,in itself, ample reason to develop an under-standing of optimization capabilities of ex-citable media. We believe that the experi-mental approach could also be transferredto microscopic reaction-diffusion systems,such as the oxidation of carbon monoxideon platinum (4). Waves interacting withcomplex boundaries created by photoli-thography have been recently studied withthe use of this system (18), suggesting thatpath optimization could be readily imple-mented on a microscopic scale.

REFERENCES AND NOTES

1. A. T. Winfree, The Geometry of Biological Time(Springer, Berlin, 1980); J. Ross, S. C. Muller, C.Vidal, Science 240, 460 (1988).

2. J. M. Davidenko, A. M. Pertsov, R. Salomonsz, W.Baxter, J. Jalite, Nature 355, 349 (1992); A. Gorelovaand J. Bures, J. Neurobiol. 14, 353 (1983); J. Lech-leiter, S. Girard, E. Peralta, D. Clapham, Science252,123 (1991).

3. R. J. Field and M. Burger, Eds., Oscillations andTraveling Waves in Chemical Systems (Wiley, NewYork, 1985); A. T. Winfree, Science 175, 634 (1972).

4. H. H. Rotermund, S. Jakubith, A. von Oertzen, G.Ertli, Phys. Rev. Lett. 66, 3083 (1991).

5. A. S. Mikhailov, Foundation of Synergetics I: Distrib-uted Active Systems (Springer, Berlin, 1990).

6. Reflection of waves colliding with no-flux boundariesand with other waves has been recently reported in amodel reaction-diffusion system [W. N. Reynolds, J.E. Pearson, S. Ponce-Dawson, Phys. Rev. Lett. 72,2797 (1994); V. Petrov, S. K. Scott, K. Showalter,Philos. Trans. R. Soc. London Ser. A 443, 631(1994)].

7. P. A. Steenbrink, Optimization of Transport Net-works (Wiley, London, 1974); R. Gould, Graph The-ory (Benjamin-Cummings, Menlo Park, CA, 1988); J.Malkevitch and W. Meyer, Graphs, Models, and Fi-nite Mathematics (Prentice-Hall, Englewood Cliffs,NJ, 1974).

8. J. A. Sepulchre, A. Babloyantz, L. Steels, in Pro-ceedings of the International Conference on ArtificialNeural Networks, T. Kohonen, K. Makisara, 0.Simula, J. Kangas, Eds. (Elsevier, Amsterdam,1991), pp. 1265-1268.

9. A. N. Zaikin and A. M. Zhabotinsky, Nature 225, 535(1970).

10. Cut membranes (Gelman Metricel; thickness, 140pm; pore size, 0.45 gm) were soaked with BZ solu-tion and covered with silicone oil in a thermostatedpetri dish maintained at 25.0° + 0.2°C. Initial reagentconcentrations were as follows: [NaBrO3] = 0.2 M,[H2SO4] = 0.4 M, [malonic acid] = 0.17 M, [NaBr]0.1 M, [ferroini = 1.0 mM.

11. 5. C. Muller, Th. Plesser, B. Hess, Physica D024, 71(1987).

12. J. J. Tyson and J. P. Keener, ibid. 32, 327 (1988); V.

SCIENCE * VOL. 267 * 10 FEBRUARY 1995870

S. Zykov, Biophysics 25, 906 (1980).13. D. Barkley, Physica D 49, 61 (1991).14. The functions f and g are defined as f(u, v) = (1 /E)u(1

- u) [u - (v + b)/a] and g(u, v) = u - v. The param-

eter values a = 0.9, b = 0.05, and E = 0.02 werechosen to model an excitable system. The equationswere integrated on a grid of 480 points by 480 points(grid spacing Ax = 1.3, time step At = 0.085) withthe Laplacian operator V2 approximated by a stan-dard nine-point formula. (The same optimal pathwas found in a higher resolution calculation with gridspacing Ax = 0.65 and time step At = 0.011.) Wewere able to realize different complex labyrinths bydigitizing corresponding line drawings and modelingthe borders and obstacles as regions with constant,steady-state values of u and v. These boundary con-ditions correspond to "sink" boundaries and reflect

the experimental system, where all edges of themembrane were in contact with silicone oil.

15. W. H. Matthews, Mazes and Labyrinths (Longmans,London, 1922).

16. A. Toth, V. Gaspar, K. Showalter, J. Phys. Chem. 98,522 (1994).

17. E. R. Kandel, J. H. Schwartz, T. M. Jessell, Princi-ples of Neural Science (Elsevier, New York, ed. 3,1991).

18. M. D. Graham et al., Science 264, 80 (1994).19. This research was supported by the National Sci-

ence Foundation (grant CHE-9222616). We ac-knowledge the donors of the Petroleum ResearchFund, administered by the American Chemical Socidety, for partial support of this research.

26 August 1994; accepted 10 November 1994

Electrochemical Detection of Single MoleculesFu-Ren F. Fan and Allen J. Bard*

The electrochemical behavior of a single molecule can be observed by trapping a smallvolume of a dilute solution of the electroactive species between an ultramicroelectrodetip with a diameter of -15 nanometers and a conductive substrate. A scanning electro-chemical microscope was used to adjust the tip-substrate distance (-10 nanometers),and the oxidation of [(trimethylammonio)methyl] ferrocene (Cp2FeTMA+) to Cp2FeTMA21was carried out. The response was stochastic, and anodic current peaks were observedas the molecule moved into and out of the electrode-substrate gap. Similar experimentswere performed with a solution containing two redox species, ferrocene carboxylate(Cp2FeCOO-) and Os(bpy)321 (bpy is 2,2'-bipyridyl).

A number of techniques have been used inrecent years to detect single molecules or

ions in different environments. These in-clude the detection of ions that are confinedin vacuum in electromagnetic traps (1), sin-gle-molecule spectroscopy of molecules insolid matrices (2), and the detection of mol-ecules on surfaces with high spatial resolu-tion by near-field scanning optical micros-copy (3). These techniques are based largelyon observation of the fluorescent emissionfrom the molecule during repeated photonicexcitation events and provide informationabout energy levels and the environment ofthe molecule. We report single-molecule de-tection (SMD) for an electroactive mole-cule in solution as it repeatedly undergoeselectron-transfer reactions at an electrodeheld at a small distance from a counterelec-trode in a scanning electrochemical micro-scope (SECM) (4, 5).

The principle of this experiment is illus-trated in Fig. 1. To achieve SMD in theSECM, a small tip electrode of nanometerdimensions and with a geometry that pro-vides confinement of the molecule near theactive tip area is held near (-10 nm) a

conductive substrate. The solution concen-

tration is adjusted so that, on average, onlya single molecule will reside in the volume

defined by the tip area and tip-substratespacing d. Consider a tip of radius a = 5 nmheld at d = 10 nm. For a 2 mM solution ofan electroactive species, it is probable thatonly one molecule will be present in the-10`W cm3 volume beneath the tip (5).The time required for the molecule to tran-sit between tip and substrate is of the orderof d2/2D, where D is the diffusion coeffi-cient of the molecule. Thus, a moleculewith D = 5 X 10-6 cm2/s will transit thegap in about 100 ns or undergo 107 roundtrips between tip and substrate per second.If an electron-transfer event occurs at eachcollision with the tip, a current of the orderof 1 pA will flow. In this case, SMD de-pends on the 107 amplification factor pro-vided by the positive feedback of theSECM. The molecule is confined to thespace below the tip because the conductivetip is slightly recessed below the large insu-lating sheath that surrounds it, as shown inFig. 1. Further axial confinement of a

charged species might be afforded by theelectric field in the gap.

For the initial test of SMD by SECM we

chose [(trimethylammonio)methyl] ferro-cene (Cp2FeTMA+) as the electroactivemolecule, because both this species and itsoxidized product are stable in aqueous me-

dia and undergo rapid heterogeneous elec-tron-transfer reactions at electrodes in a

convenient range of potentials. Solutionscontained 2 mM Cp2FeTMA' with

SCIENCE * VOL. 267 * 10 FEBRUARY 1995

NaNO3 as the supporting electrolyte. In allof the experiments, an indium-tin oxide(ITO, Delta Technologies, Stillwater, NewYork) electrode was used as substrate. Thetip was fabricated from an electrochemical-ly sharpened 125-pum Pt-Ir (80%-20%) wirecoated with Apiezon wax or polyethyleneglue. We exposed the end by placing the tipin a scanning tunneling microscope (STM)and allowed it to approach the ITO sub-strate until a set-point current (0.5 nA)began to flow. By constructing the tip inthis manner, we ensured that the tip and itsinsulating sheath would approach the ITOsurface perpendicularly without alignmentdifficulties. Details of the fabrication ofthese tips and their characterization are giv-en elsewhere (5-7). The instrument used inthese studies has been described in (8, 9)and is capable of both STM and SECMmeasurements with a vertical (z direction)resolution of better than 1 A and a currentsensitivity as low as 50 fA when the properfilter is used. The electrochemical cell con-tained a Pt counterelectrode and either asaturated calomel electrode (SCE) or a Ptquasi-reference electrode.

The correct tip geometry is essential forthe successful trapping of a molecule. Infor-mation about the exposed area of the tipand the shapes of the tip and insulatingsheath can be obtained from electrochemi-cal measurements, that is, from determina-tion of the tip current iTas a function of das the tip is moved toward the substrate inan SECM (7). All of the tip currents re-ported were corrected for the backgroundcurrent in the blank solution, that is, asolution containing only supporting elec-trolyte, at large d. A typical approach curveof this type taken with a solution of 2 mMCp2FeTMA' and 2.0 M NaNO3, with thetip held at 0.55 V versus SCE whereCp2FeTMA' oxidation is diffusion-con-trolled, is shown in Fig. 2. The ITO sub-strate was biased at -0.3 V versus SCE, sothat Cp2FeTMA21 generated at the tip wasrapidly reduced back to Cp2FeTMA' at theITO surface. The exposed radius a can befound (5) from the steady-state current iT,when the tip is assumed to be a disk-shaped

Pt-Ir tip

Waxinsulatore

Fp+q FC2 m

ITO substrate 'e

Fig. 1. Idealized schematic illustration of the tipgeometry and the tip-substrate configurationused.

871

Department of Chemistry and Biochemistry, University ofTexas, Austin, TX 78712, USA.

*To whom correspondence should be addressed.

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