Maximum-likelihood position sensing and actively ... - Semantic Scholar

Maximum-likelihood position sensing and actively controlled electrokinetic transport for single-molecule trapping Lloyd Davis, Zbigniew Sikorski, William Robinson, Guoqing Shen, Xiaoxuan Li, Brian Canfield, Isaac Lescano, Bruce Bomar, William Hofmeister, James Germann, Jason King, Yelena White, and Alexander Terekhov Center for Laser Applications, University of Tennessee Space Institute, Tullahoma, TN 37388 ABSTRACT A freely diffusing single fluorescent molecule may be scrutinized for an extended duration within a confocal microscope by actively trapping it within the femtoliter probe region. We present results from computational models and ongoing experiments that research the use of multi-focal pulse-interleaved excitation with time-gated single photon counting and maximum-likelihood estimation of the position for active control of the electrophoretic and/or electro-osmotic motion that re-centers the molecule and compensates for diffusion. The molecule is held within a region with approximately constant irradiance until it photobleaches and/or is replaced by the next molecule. The same photons used for determining the position within the trap are also available for performing spectroscopic measurements, for applications such as the study of conformational changes of single proteins. Generalization of the trap to multi-wavelength excitation and to spectrally-resolved emission is being developed. Also, the effectiveness of the maximum-likelihood position estimates and semi-empirical algorithms for trap control is discussed. Keywords: Single-molecule spectroscopy, single-molecule trapping, maximum-likelihood, photon counting, pulseinterleaved excitation

1. INTRODUCTION Since the first report of the detection of individual single-chromophore molecules in solution [1], applications and advanced techniques for single-molecule detection (SMD) have rapidly developed, including the means for wide-field imaging of single molecules with sub-diffraction precision, for tracking fluorescently-labeled biomolecules in 2 dimensions (2-D) on cellular surfaces [2], and for observing motion between different focal planes [3]. For many applications, confocal microscopy is preferable to wide-field imaging as it provides improved signal-to-noise due to the very small detection volume; it is necessary for two-photon excitation, which offers the advantage of inherent sectioning and no excitation of out-of-plane molecules; and it facilitates monitoring of sub-nanosecond fluorescence lifetimes and sub-millisecond dynamics by use of single-photon avalanche diode (SPAD) detectors for time-resolved single-photon counting and fluorescence correlation spectroscopy. However, a freely diffusing molecule quickly leaves the confocal volume. Tracking and trapping can enable the molecule to be kept within the detection volume until it ultimately photobleaches. A means for 2-D tracking and trapping of a single fluorescent molecule within a confocal microscope by rotating the laser spot about the molecule to sense its position from the phase and amplitude of modulation of the fluorescence signal and then translating the center of rotation or the sample in response to the molecule’s diffusion was suggested some years ago [4] and several groups have since developed the capability for trapping in 2-D by this method [5]. The method has been extended to 3-D tracking of fluorescent particles by use of orbits in different z-sections to track axial motion [69]. In most cases, the response time of the piezoelectric translation stage used to recenter the molecule limits the applicability of the trap to slowly-diffusing species, but the time required for the position sensing, which is determined by the speed of the rotating laser focus and ultimately the photon count rate, can also provide a limitation. The use of acousto-optic beam deflectors to rotate the laser focus at ~50 kHz and electrophoretic and electro-osmotic forces to more quickly recenter the fluorescent particle has been shown to provide advantages for 2-D trapping [10]. For nanoscale objects, electrokinetic forces are stronger than magnetic, ac dielectrophoretic, or optical forces, and potentially enable a feedback system faster than one based on a translation stage.

Single Molecule Spectroscopy and Imaging edited by Jörg Enderlein, Zygmunt K. Gryczynski, Rainer Erdmann Proc. of SPIE Vol. 6862, 68620P, (2008) · 1605-7422/08/$18 · doi: 10.1117/12.763833 Proc. of SPIE Vol. 6862 68620P-1

An alternate means for 3-D trapping, which avoids the time required to mechanically or acousto-optically rotate the laser focus, has been implemented with a piezo-translator for the repositioning [11,12]. Molecule position sensing is accomplished by imaging the collected light from four volumes in a tetrahedral pattern surrounding the excitation volume onto four SPAD detectors and then using the relative signals from these four volumes to determine the most likely molecule position. The optical configuration in this case suffers a loss of signal, firstly as collected fluorescence must be split at a beam splitter before spatially filtering to select light from volumes with different axial (z) depths, and secondly as the two pairs of two optical fibers used to define collection volumes at different transverse (x, y) positions cannot be touching due to the fiber cladding and thus fail to efficiently collect light from the center of the excitation volume, where the signal should be greatest. In this work, we research new configurations for single-molecule position sensing and trapping, which use the relative signals from different volumes in object space for position sensing but avoid the signal loss of Ref. 11 described above, and which use electrokinetic forces similar to Ref. 10 for fast corrective motions, but with extension to trapping in 3-D and in 1-D. We discuss computer simulations, which demonstrate the feasibility and potentially fast response of the new trap configurations, and experimental work towards their implementation. To accomplish discrimination of the signals from different volumes, the beam from a modelocked laser is split and recombined at beam splitters to produce two to four beams that are focused to slightly offset spatial positions and with pulses that are temporally offset, so as to yield sets of pulse-interleaved excitation at the different offset focal volumes. Fluorescence is collected from the entire multi-beam excitation volume onto just one SPAD detector. Time gated photon detection is then used to provide information on the location of the volume from which the photon was emitted and hence the most likely position of the single molecule. As this form of position sensing only requires a single SPAD detector, it is readily scalable to use of more than one detector for multi-color and/or polarization resolved observations on a trapped molecule.

2. 1-D TRAP In the 1-D trap, molecules are confined to a sub-micron channel or ‘nanochannel’ and trapped by corrective electrokinetic motions along the length of this channel. This is of interest because it is conceptually simpler to implement than a 2-D trap, it is potentially faster than the 2-D trap due to the need for a smaller number of photons for position determinations, and it lends insight into the operation of 2-D and 3-D traps. Also, when a molecule in solution is confined to a thin volume between two planar interfaces for trapping in 2-D, it suffers a high rate of collisions with the surfaces [10]. For applications that can tolerate such disturbances, the 1-D trap should be equally useful, because the collision rate should not be not significantly increased by confinement of the solution to a one-dimensional channel. Fig. 1 illustrates the optical set up for the 1-D trap, in which there are two beams focused to Gaussian waists at positions that are offset along the nanochannel. The separation of the two Gaussian beams may be adjusted so that the point of maximum slope of each beam profile is at the center of the trap, so as to provide greatest position determination sensitivity for a given beam size. For 1-photon excitation, this occurs when x = ± σ, where the standard deviation is σ = ω0/2, and ω0 is the beam waist. With this beam spacing, the summed irradiance profile is approximately constant at the center of the trap and the level of excitation of the molecule remains constant even if the molecule deviates slightly from the center of the trap. This is a key advantage for quantitative studies of molecular dynamics and is not the case for the trap of Ref. 11. The voltage applied across the nanochannel V1 − V2 drives the electrophoretic and electro-osmotic motion of the molecule along the nanochannel back to the center of the trap in response to the estimated position of the molecule, which is determined by maximum-likelihood (ML) methods. To accomplish this, time-gated photon counting is used to collect photons from the two time channels that follow each set of excitation pulses. In our experimental set-up, a modelocked picosecond dye laser operating at 76 MHz, with T = 13.2 ns between pulses is used to produce the two offset foci, and hence each time channel has a width of T / 2 = 6.6 ns. If the fluorescence lifetime τ is short compared to the 6.6 ns channel width, then photons generated by pulses at each laser focus will fall in the correct time channel. However, in general a finite fluorescence lifetime will cause cross-talk between the channels wherein the probability that a photon will fall into an incorrect time channel is given by α = (1 + exp(T /(2τ )) )−1 . For τ = 2 ns, the cross-talk is α < 0.04. The SPAD detector timing jitter and electronic timing error will further increase the cross-talk. For the case

Proc. of SPIE Vol. 6862 68620P-2

of the 3-D trap described below, there are 4 time channels each of width T/4 = 3.3 ns and the cross-talk becomes more significant and is α = (1 − exp(3T /(4τ )) ) (1 − exp(T / τ ) ) . The cross-talk is α < 0.04 for τ = 1 ns, but increases sharply for longer lifetimes, as seen in Fig. 2. Hence for our given laser repetition rate, 3-D trapping is effective only for fluorophores with relatively short lifetimes, less than about 2 ns. center

nce profile electrode

1di1 electrode

0.5 0.4

o 0

2

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6

8

10

Fluorescence Lifetime (ns]

Fig. 2. Cross-talk due to photons falling to an incorrect time window. Fig. 1. Configuration of 1-D trap. For the 1-D trap, the numbers of photons collected in each of the time channels N1, N2 are used to find the ML-estimate of the molecule position ~ x as follows: If the molecule is at position x, the probability to detect N1, N2 photons is given by the binomial distribution, Prob( N1 , N 2 | x) =

( N1 + N 2 )! p1 ( x) N1 (1 − p1 ( x)) N 2 , N 1! N 2 !

(1)

where p1(x) is the probability that a photon from a molecule at x will fall into time-channel 1. This is given by p1 ( x) =

(1 − α ) F1 ( x) + α F2 ( x) , F1 ( x) + F2 ( x)

(2)

where F1(x), F2(x) give the dependence of the fluorescence signal from laser focus 1 and 2 on the molecule position. Fig. 3 shows the case where there is no saturation of excitation so that the fluorescence profiles are Gaussian. The MLestimate of the molecule position ~ x is the value of x for which the probability in Eqn. (1) takes its maximum value.

If cross-talk is negligible (α → 0) and the fluorescence profiles are Gaussians centered at x = ± s σ , the following analytic solution for the ML-estimate of x may be found by setting the derivative of Eqn. (1) to zero: ~ x ( N1 , N 2 ) = ln (N1 N 2 ) σ /(2s ) . (3) The numbers of photons N1, N2 are whole numbers, and for a given total number of detected photons N = N1 + N2, there are only N + 1 possible values of N1, N2 and hence the ML-estimate for the position of the molecule can only take on a x = ∞ , or ~ x = −∞ , as set of N + 1 discrete possible values. In the cases where N1 = 0 or N2 = 0, the ML-estimate is ~ may be seen from Eqn. (3), or from the monotonic behavior of the plot of p1(x). However, the use of extreme values for the ML-estimate for such occurrences of N1, N2 causes an overall bias in the ML-estimate, and moreover can lead to the molecule being kicked from the trap. The bias of the ML-estimate is determined from the expected value of the MLestimate when a molecule is actually at x, which is given by ~ x|x =

N

∑

N1 =0

~ x ( N1 , N 2 ) Prob(N1 , N 2 | x ) .


(4)

The solid magenta line in Fig. 4 shows ~ x | x for N = 10 photons, where Gaussian fluorescence profiles are centered at ± σ , (i.e., s = 1), and where we have set ~ x ( N = 0, N = 10) → 5σ , ~ x ( N = 10, N = 0) → −5σ . In this case the 1

2

1

2

Estimated -

sition

magnitude of the displacement from the origin is on average clearly overestimated by the ML method (by a factor of ~2). The green dashed line in the same figure shows the case where we restrict the maximum/minimum estimated positions to equal the beam spacing, i.e., ~ x ( N1 = 0, N 2 = 10) → sσ , ~ x ( N1 = 10, N 2 = 0) → − sσ . In this case, the bias is reduced and the trapping of a molecule can be expected to be more stable.

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Fig. 3. Fluorescence and channel 1 probability profiles.

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Fig. 4. The mean value of the ML-estimate exhibits bias.

The precision of the ML-estimate increases and the bias decreases as the number of photons is increased. Fig. 5 shows, in the four inserts at top and right, the probability density functions for the ML-estimate of the molecule position for the I Die

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case where the cross-talk is α = 4 % , the laser beams are centered at x = ± σ , and the actual molecule position is fixed at x = 0.5 σ . The finite number of discrete values possible for the ML-estimate is evident in these graphs. The standard widths of the distributions are plotted as blue diamond points in the log-log graph of Fig. 5. These fall close to the blue dotted line with a slope of −½ , showing that the uncertainty of the ML-position estimate for a stationary molecule decreases approximately with one over the square root of the number of photons ( 1 / N ). However, while photons are acquired, the molecule position may change due to diffusion and hence the error between the estimated position and the actual position (at the end of the photon acquisition) increases with the number of photons. Also, the feedback electronics introduce a latency or time delay ∆t between the collection of photons and the application of corrective motion. Diffusion causes the position uncertainty σ D to increase according to σ D = 2 Dt = 2 D( N / R + ∆t ) , where D is the diffusion coefficient, N is the number of photons accumulated, and R is the

photon count rate. The slope of the magenta line in Fig. 5 is +½ , corresponding to the N increase in the position uncertainty due to diffusion. The position of the magenta line corresponds to D = 20 µm2/sec, R = 105 /second, and σ = 0.5 µm (i.e., a beam waist of ω0 = 1 µm) and the magenta dotted line corresponds to a latency of ∆t = 10 µs. For these parameters, the minimum net error in the molecule position will be obtained for about 30 photons. In general, the position estimation works best if a small number of photons is used for each ML-estimate of the molecule position, particularly if the counts are momentarily interrupted due to the molecule crossing to a dark state. We have developed independent Monte Carlo simulations in C++ and in Matlab, based on our previous simulation algorithms of fluorescence correlation spectroscopy [13], in order to study different algorithms for trapping, the optimal number of photons or integration time to use for position determinations, and the effects on the trapping of background, latency in the feedback, molecular photophysics, such as triplet crossing and photobleaching, and limitations and empirical scaling of the electrokinetic velocity (about 2 µm/ms maximum). Fig. 6 gives an example in which a single molecule is electrokinetically transported in from the left and then trapped in the region between the two laser foci. The algorithm in this case is as follows: Every 10 µs, the photons in each time channel N1, N2, are counted. A look-up table stores the corrective electrokinetic voltage to be applied for each possible N1, N2 , as determined from the ML-estimate of the molecule position x. The voltage is applied after a delay ∆t that accounts for the circuit latency. When the photon counts remain below a threshold for several time-steps, the full voltage is applied to bring in the next molecule. While trapped, the estimated and actual molecule positions are only weakly correlated, particularly if the number of photons is small. Thus corrective motions have a large random component, as seen in Fig. 6 (right), but the sum of many successive corrections provides an effective drift motion back to the center of the trap.

Typical 1-P Simulation Parameters 1O

Count rate

Power in each beam Net detection efficiency Fluorescence lifetime Beam waist

100

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Fig. 6. Monte Carlo simulation of single-molecule detection with electrokinetic 1-D trapping Progress towards experimental implementation of the 1-D trap includes fabrication of nanochannels in fused silica [14] and detection of single molecules of Streptavidin-Alexa-610 within the device as seen in Fig. 7, and with electrokinetic flow.


fused silica for zero substrate autofluorescence Fig. 7. Experimental set-up for electrokinetic trapping of a single molecule in solution in a nanochannel.

3. 3-D TRAP The principle of the 3-D trap is illustrated in Fig. 8. The beam from a femtosecond Ti-Sapphire laser is split into four beams with different time delays using a double interferometer, shown schematically in Fig. 9, and these beams are focused to four points that are at the vertices of a tetrahedron, which surround the center of the trap. Two-photon fluorescence excited by the focused laser pulses is collected onto a SPAD detector. The pulses from this are sent to a

Double interferorvr 4 laser foci

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6

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laserbeam: N1,N2,

Four paths:

National Instruments PCI-6602 counter timer and custom correlation program

N —+ Prob(N1,N2,N3,N4 x,y,z)

Fig. 8. Configuration for 3-D tracking/trapping.

Al = shortest A2 = Al + 3.3 ns Bl = Al + 6.6 ns B2 = Al + 9.9 ns

laser beam input (40 fs Ti-Sapphire pulses, 76MHz)

Alignment: To adjust foci positions adjust a, fi, y, 8

Fig. 9. Double interferometer to create 4 foci.


time-gated single-photon counting circuit, a photograph of which is shown in the inset. (This same circuit is also used in the 1-D trapping experiments.) The circuit contains a field programmable gate array (FPGA), which is programmed to provide voltages for electrokinetically repositioning the molecule, in response to the photons detected in each time gate, N1, N2, N3, N4. The corrective motion to be applied can be determined by an extension of the ML-estimation methods x, ~ y , ~z is found by maximizing the discussed above for the 1-D trap. In this case, the ML-estimate of the position ~ multinomial probability Prob( N1 , N 2 , N 3 , N 4 | x, y, z ) =

( N1 + N 2 + N 3 + N 4 )! N1 N 2 N3 N 4 p1 p 2 p3 p 4 , N 1! N 2 ! N 3 ! N 4 !

(5)

where p j = p j ( x, y, z ) is the probability that a photon from molecule at x, y, z will fall in channel j. The same issues as were discussed for the 1-D trap, also apply here; namely there is a finite number of discrete possible position estimates, and a possible bias in the ML-position estimates, for a small total number of photons. Two-photon excitation is preferable because it produces a better defined axial localization of the excitation. Without saturation, the fluorescence profile is proportional to the square of the irradiance profile, and for a Gaussian-Lorentzian laser beam of waist ω0 and Rayleigh range z0 the centers of the four laser foci may be adjusted to the positions (0, ±x1, −z1), ( ±x1, 0, +z1) so that the summed fluorescence profile is approximately constant around the origin, as shown in Fig. 10. The procedure shown in Fig. 9 may be used with piezo translation of an immobilized fluorescent bead to determine the 3-D profiles and align the centers of each laser foci. Because the laser profile is elongated (z0 > ω0), the ML determination of the z-position of the molecule is not as precise as that for the x,y-position, but the molecule can tolerate a larger excursion in the z-dimension without being lost from the trap.

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=lOjan,; =33 jan,; =L6pm,; = j1+(zjIzo)2a,I2=Oi4,an Fig. 10. The 4 laser foci may be positioned so that the squared irradiance is approximately constant around the origin. The 3-D view at lower left uses different beam parameters but shows the general elongated shape of the excitation region.


We are researching two configurations for providing electrokinetic motions in 3-D with fast response. Finite volume time domain (FVTD) numerical simulations of the two configurations using ESI CFD-ACE+ software are shown in Fig. 11. The top part of the figure depicts two pairs of two microfabricated electrodes on two glass surfaces with a separation of 100—200 µm. The electric field is approximately uniform over the micron-sized laser focal volume and it may be oriented in any direction by adjustment of the potentials on the four electrodes, so as to provide electrophoretic motion in 3-D. The bottom of the figure shows two microfluidic channels with rectangular cross sections that cross and intersect at their faces. Adjustment of the potentials at the ends of the microchannels alters the electro-osmotic flow of buffer within the channels between the four legs and thereby enables the flow-lines at the center to be oriented in any direction in 3-D.

tetrahedral geometry

form electro- ' phoretic trap .'

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V1=3V,V2=V3=V4=-1 V Electrode separation 200 ttm

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eieeti-oosmotic trap channel 20 x 10 jtm, zeta-potential —40 my, water

Two crossed microchannels for 4-beam 3-D trapping

Fig. 11. Configurations for providing 3-D electrophoretic and electro-osmotic motion by adjustment of V1,V2,V3,V4. Monte Carlo simulations have been used to demonstrate the feasibility of the 3-D trap. Fig. 12 gives an example of simulation results in which a molecule of D = 10 µm2/s, is effectively trapped for 26 ms with the detection of 2973 photons before photobleaching. The trapping algorithm used in this case is that each 10 µs the ML-estimate of the molecule position is determined from the time-channel information of the last 6 photons (which can be conveniently stored in a shift register in the FPGA circuit), and then the voltages are adjusted after another delay of 10 µs to account for the latency of the voltage switching. (Note that with this algorithm it may happen that the position determinations in subsequent time steps include some of the same photons and hence successive computed voltages may be correlated.) To simplify the simulation algorithm, the molecule diffuses on a 3-D Cartesian grid of spacing 20 nm and the applied voltage is that which is necessary to move the molecule along the Cartesian axis in the direction for which the position estimate has the maximum deviation from the center of the trap. The magnitude of the electrokinetic velocity that is applied is either 0 or 2 µm/ms, depending on the magnitude of the deviation. As in the case in the 1-D trapping simulation, the corrective motions have a large random component, but the sum of many successive corrections provides an effective drift motion back to the center of the trap. The parameters used in the simulations for Fig. 12 are: laser beams with ω0 = 0.5 µm, and separation parameters x1 = 0.21 µm, z1 = 0.94 µm, D = 10 µm2/s, fluorescence lifetime τ = 1 ns, photon detection probability 0.02, triplet probability 10−3, triplet lifetime 1 µs, photobleaching probability 10−5; the observed count rate was 1.1×105 /s.


34

x posion of molecule [umi

03

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lime [ms]

lime Ems]

lime Ems]

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0.2

0.8

U =1.62nm ___________________

0.3

x (urn]

0.4

—0.2

0 x (urn]

0.2

0.4

Fig. 12. Actual-position trajectories from a simulation of a molecule trapped in 3-D.

4. CONCLUSIONS We have presented a new approach to trapping a single-molecule in 1-D or 3-D based on multi-focal pulse-interleaved excitation, time-gated single-photon counting, and use of a small numbers of photons for maximum-likelihood based estimation of the position of the molecule. We have implemented a field programmable gate array circuit for processing the detected photon counts and for providing rapid update of voltages for controlling the motion of the molecule. We have analyzed simple geometries for electrophoretic and electro-osmotic 3-D control of the motion of a molecule in solution. High-speed processing of photon counts and a lack of moving parts provide the potential for a significant improvement in single-molecule trapping capabilities. A further advantage is that the net irradiance is approximately spatially constant over a region near the center of the trap, so the molecule sees a uniform excitation rate which is beneficial for quantitative measurements. The trap only requires a single detector and thus is readily expandable to multi-color or multi-channel experiments with multiple detectors; the photons used for position sensing are available for other spectroscopic measurements. The single-molecule trap is expected to open new capabilities in biophysical and biomedical research.

ACKNOWLEDGEMENTS A portion of this research was conducted at the Center for Nanophase Materials Sciences and the SHaRE User Facility, which are sponsored by the Division of Scientific User Facilities, Office of Basic Energy Sciences, U.S. Department of Energy. We thank Claus Daniel at the Center for Nanophase Materials Sciences at Oak Ridge National Laboratory, and Philip Samson, Dmitry Markov, John Wikswo, and Deyu Li at Vanderbilt University. This work is supported by the Center for Laser Applications and DARPA grant W911NF-07-1-0046.


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[3]

[4] [5] [6] [7] [8] [9] [10] [11]

[12] [13]

[14]

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