## Precision analysis for standard deviation measurements of immobile single fluorescent molecule images

Optics Express, Vol. 18, Issue 7, pp. 6563-6576 (2010)

http://dx.doi.org/10.1364/OE.18.006563

Acrobat PDF (1149 KB)

### Abstract

Standard deviation measurements of intensity profiles of stationary single fluorescent molecules are useful for studying axial localization, molecular orientation, and a fluorescence imaging system’s spatial resolution. Here we report on the analysis of the precision of standard deviation measurements of intensity profiles of single fluorescent molecules imaged using an EMCCD camera. We have developed an analytical expression for the standard deviation measurement error of a single image which is a function of the total number of detected photons, the background photon noise, and the camera pixel size. The theoretical results agree well with the experimental, simulation, and numerical integration results. Using this expression, we show that single-molecule standard deviation measurements offer nanometer precision for a large range of experimental parameters.

© 2010 Optical Society of America

## 1. Introduction

1. R. E. Thompson, D. R. Larson, and W. W. Webb,“Precise nanometer localization analysis for individual fluorescent probes,” Biophys. J. **82**, 2775–2783 (2002). [CrossRef] [PubMed]

2. A. Yildiz, J. N. Forkey, S. A. McKinney, T. Ha, Y. E. Goldman, and P. R. Selvin, “Myosin V walks hand-overhand: Single fluorophore imaging with 1.5-nm localization,” Science **300**, 2061–2065 (2003). [CrossRef] [PubMed]

3. Y. M. Wang, R. H. Austin, and E. C. Cox, “Single molecule measurements of repressor protein 1D diffusion on DNA,” Phys. Rev. Lett. **97**, 048302 (2006). [CrossRef] [PubMed]

4. C. Joo, H. Balci, Y. Ishitsuka, C. Buranachai, and T. Ha, “Advances in single-molecule fluorescence methods for molecular biology,” Annu. Rev. Biochem. **77**, 51–76 (2008). [CrossRef] [PubMed]

5. A. M. van Oijen, J. Khler, J. Schmidt, M. Mller, and G. J. Brakenhoff, “3-Dimensional super-resolution by spectrally selective imaging,” Chem. Phys. Lett. **292**, 183–187 (1998). [CrossRef]

6. M. Speidel, A. Jonas, and E.-L. Florin, “Three-dimensional tracking of fluorescent nanoparticles with sub-nanometer precision by use of off-focus imaging,” Opt. Lett. **28** (2003). [CrossRef] [PubMed]

7. B. Huang, W. Wang, M. Bates, and X. Zhuang, “Three-dimensional super-resolution imaging by stochastic optical reconstruction microscopy,” Science **319**, 810–813 (2008). [CrossRef] [PubMed]

8. K. Adachi, R. Yasuda, H. Noji, Y. Harada, M. Yoshida, and K. Kinosita, “Stepping rotation of F1-ATPase visualized through angle-resolved single-fluorophore imaging,” Proc. Natl. Acad. Sci. USA **97**, 7243–7247 (2000). [CrossRef] [PubMed]

9. J. Enderlein, E. Toprak, and P. R. Selvin, “Polarization effect on position accuracy of fluorophore localization,” Opt. Express **14**, 8111–8120 (2006). [CrossRef] [PubMed]

10. F. Aguet, S. Geissbühler, I. Märki, T. Lasser, and M. Unser, “Super-resolution orientation estimation and localization of fluorescent dipoles using 3-D steerable filters,” Opt. Express **17**, 6829–6848 (2009). [CrossRef] [PubMed]

11. T. J. Holmes, D. Briggs, and A. A. Tarif, “Blind deconvolution,” in *Handbook of Biological Confocal Microscopy*, J. B. Pawley, ed. (Springer, New York, 2006), pp. 468–487. [CrossRef]

1. R. E. Thompson, D. R. Larson, and W. W. Webb,“Precise nanometer localization analysis for individual fluorescent probes,” Biophys. J. **82**, 2775–2783 (2002). [CrossRef] [PubMed]

2. A. Yildiz, J. N. Forkey, S. A. McKinney, T. Ha, Y. E. Goldman, and P. R. Selvin, “Myosin V walks hand-overhand: Single fluorophore imaging with 1.5-nm localization,” Science **300**, 2061–2065 (2003). [CrossRef] [PubMed]

5. A. M. van Oijen, J. Khler, J. Schmidt, M. Mller, and G. J. Brakenhoff, “3-Dimensional super-resolution by spectrally selective imaging,” Chem. Phys. Lett. **292**, 183–187 (1998). [CrossRef]

6. M. Speidel, A. Jonas, and E.-L. Florin, “Three-dimensional tracking of fluorescent nanoparticles with sub-nanometer precision by use of off-focus imaging,” Opt. Lett. **28** (2003). [CrossRef] [PubMed]

7. B. Huang, W. Wang, M. Bates, and X. Zhuang, “Three-dimensional super-resolution imaging by stochastic optical reconstruction microscopy,” Science **319**, 810–813 (2008). [CrossRef] [PubMed]

8. K. Adachi, R. Yasuda, H. Noji, Y. Harada, M. Yoshida, and K. Kinosita, “Stepping rotation of F1-ATPase visualized through angle-resolved single-fluorophore imaging,” Proc. Natl. Acad. Sci. USA **97**, 7243–7247 (2000). [CrossRef] [PubMed]

9. J. Enderlein, E. Toprak, and P. R. Selvin, “Polarization effect on position accuracy of fluorophore localization,” Opt. Express **14**, 8111–8120 (2006). [CrossRef] [PubMed]

10. F. Aguet, S. Geissbühler, I. Märki, T. Lasser, and M. Unser, “Super-resolution orientation estimation and localization of fluorescent dipoles using 3-D steerable filters,” Opt. Express **17**, 6829–6848 (2009). [CrossRef] [PubMed]

11. T. J. Holmes, D. Briggs, and A. A. Tarif, “Blind deconvolution,” in *Handbook of Biological Confocal Microscopy*, J. B. Pawley, ed. (Springer, New York, 2006), pp. 468–487. [CrossRef]

1. R. E. Thompson, D. R. Larson, and W. W. Webb,“Precise nanometer localization analysis for individual fluorescent probes,” Biophys. J. **82**, 2775–2783 (2002). [CrossRef] [PubMed]

2. A. Yildiz, J. N. Forkey, S. A. McKinney, T. Ha, Y. E. Goldman, and P. R. Selvin, “Myosin V walks hand-overhand: Single fluorophore imaging with 1.5-nm localization,” Science **300**, 2061–2065 (2003). [CrossRef] [PubMed]

12. A. Yildiz, M. Tomishige, R. D. Vale, and P. R. Selvin, “Kinesin walks hand-over-hand,” Science **303**, 676–678 (2004). [CrossRef]

*N*, the standard deviation of the background noise

*σ*, and the camera’s finite pixel size

_{b}*a*. We have obtained an analytical expression for the PSF SD measurement error as a function of these parameters. Our SD measurements have achieved nanometer resolution for a wide range of experimental conditions. This expression for the SD measurement error will provide confidence in determining a particle’s axial position and molecular orientation from measurements using a single-molecule imaging system of known resolution.

## 2. Theory

### 2.1. Formulating SD measurement error, ∆*s*, by *χ*^{2} minimization

*N*photons, from a common distribution emitted by a point light source. We include the additional experimental effects of photon count fluctuation per PSF, background noise, and camera pixelation in our study.

13. N. Bobroff, “Position measurement with a resolution and noise-limited instrument,” Rev. Sci. Instrum. **57**, 1152–1157 (1986). [CrossRef]

**82**, 2775–2783 (2002). [CrossRef] [PubMed]

**82**, 2775–2783 (2002). [CrossRef] [PubMed]

*N, a*, and

*σ*(

_{b}*b*in prior studies) beginning with one dimension and extending to two dimensions.

*χ*

^{2}(

*s*) is proportional to the sum of squared errors between the observed photon count at pixel

*i, y*, and the expected photon count

_{i}*N*(

_{i}*x,s*), of a PSF. Here

*x*and

*s*are the measured position and SD of the PSF, respectively, while

*x*

_{0}and

*s*

_{0}are the true location and the theoretical SD of the molecule:

*σ*is the expected photon count uncertainty at pixel

_{i,photon}*i*without accounting for photon-to-camera count conversion (described in the following section). In this article, we emphasize the SD error and assume that the location measurement errors are negligible, i.e.

*x*=

*x*

_{0}. For simplicity,

*N*(

_{i}*x*

_{0},

*s*) is denoted as

*N*in this article unless otherwise specified.

_{i}*σ*at pixel

_{i,photon}*i*: one is the Poisson-distributed photon shot noise of the PSF where the variance is the mean expected photon count of the pixel,

*N*, and the other is the SD of the background noise,

_{i}*σ*, expressed in photons. The variances of the two sources add to yield

_{b}*s*from

*s*

_{0}, ∆

*s*=

*s*-

*s*

_{0}, is obtained by setting

*dχ*

^{2}(

*s*)/

*ds*to 0, expanding

*N*about

_{i}*s*

_{0}, and keeping the first order term in ∆

*s*. Appendix A shows the detailed derivation of ∆

*s*from

*dχ*

^{2}(

*s*)/

*ds*= 0. The mean squared value of ∆

*s*is

*s*, ∆

*s*, is the PSF SD error that we calculate in this article.

_{rms}### 2.2. Modifying *σ*_{i,photon} to include camera count conversion effects

_{i,photon}

*σ*appears. Below we derive the uncertainty in photon counts,

_{i,photon}*σ*, to use in place of

_{i}*σ*in Eq. (3) for experiments where EMCCD camera count conversions are involved.

_{i,photon}*f*(

*n*

^{*}) [14

14. M. H. Ulbrich and E. Y. Isacoff, “Subunit counting in membrane-bound proteins,” Nature Methods **4**, 319–321 (2007). [PubMed]

*n*

^{*}is the camera counts in the distribution and

*M*is the photon multiplication factor of the camera. Here we use

^{*}to denote camera counts in order to differentiate from photon counts. The

*n*

^{*}distribution has a mean of

*M*and a variance of

*M*

^{2}.

*i*, with a variance of

*σ*

_{b}^{2}and a mean of 〈

*b*〉. The total background variance in camera counts is the sum of the background count variance

*σ*

_{b}^{2}

*M*

^{2}, and the variance introduced by the average number of background photons, 〈

*b*〉, each with a variance of

*M*

^{2}: (

*σ*

_{b}^{2}+ 〈

*b*〉)

*M*

^{2}.

*i*is

*σ*we have

_{i}### 2.3. Expressing ∆s in photon counts

*x*

_{0}= 0 for simplicity and without loss in generality. We approximate the pixel summation in Eq. (8) by an integral going from negative to positive infinity, and we estimate 〈(∆

*s*)

^{2}〉 at the two extrema of

*σ*

_{i}^{2}: the high photon count regime where

*σ*

_{b}^{2}+ 〈

*b*〉 can be neglected, and the high background noise regime where 2

*N*, can be neglected. In the high photon count regime,

_{i}*s*)

^{2}〉 =

*s*

_{0}

^{2}/2

*N*. The total 1D 〈(∆

*s*)

^{2}〉 is the sum of Eqs. (10) and (11) (without the pixelation effect discussed below)

*s*)

^{2}〉 by summing these results for both extema of

*σ*

_{i}^{2}is validated by numerical calculation results shown in Fig. 2, and is in accordance with Ref. [1

**82**, 2775–2783 (2002). [CrossRef] [PubMed]

*s*)

^{2}〉. Each photon in a PSF is associated with two variances with respect to the centroid. One is the mean variance of the PSF,

*s*

_{0}

^{2}, and the other is due to the fact that each photon is further binned into a pixel that has an intensity profile described by a uniform distribution with a width corresponding to the pixel size

*a*. The variance of this distribution is

*a*

^{2}/12. Thus, the total variance of a photon due to pixelation is the sum of the two,

*s*should be (

*s*

_{0}

^{2}+

*a*

^{2}/12)

^{1/2}and for theoretical formulations, the expected SD of a PSF should include the pixelation effect. We have verified that

*s*

_{0}

^{†2}increases with

*a*according to Eq. (13) by simulation. Plugging Eq. (13) into Eq. (12) we have for 1D

*s*)

^{2}〉 calculation to 2D where

*s*, which for the remainder of this article, represents the SD in either the

_{x,y}*x*or

*y*direction of the imaging plane, and

*s*

_{0x}and

*s*

_{0y}are the theoretical SD values in the

*x*and

*y*directions, respectively,

*s*)

_{x,y}^{2}〉 can be obtained by numerically integrating Eq. (8), incorporating the transition region between the high photon count and the high background noise regimes. The numerical integration results are shown in Fig. 2 to be consistently higher than the analytical calculation results by ≈ 15%.

## 3. Methods

### 3.1. Experimental setup

*Eclipse*TE2000-S inverted microscope (Nikon, Melville, NY) attached to an iXon back-illuminated EMCCD camera (DV897ECS-BV, Andor Technology, Belfast, Northern Ireland). Prism-type Total Internal Reflection Fluorescence (TIRF) microscopy was used to excite the fluorophores with a linearly polarized 532 nm laser line (I70C-SPECTRUM Argon/Krypton laser, Coherent Inc., Santa Clara, CA) focused to a 40

*μ*m × 20

*μ*m region on fused-silica surfaces (Hoya Corporation USA, San Jose, CA). The incident angle at the fused-silica water interface was 68° – 71° with respect to the normal. The laser was pulsed with illumination intervals between 1 ms and 500 ms and excitation intensity between 0.3 kW/cm

^{2}and 2.6 kW/cm

^{2}. By combining laser power and pulsing interval variations we obtained 50 to 3000 photons per PSF. A Nikon 100X TIRF objective (Nikon, 1.49 NA, oil immersion) was used in combination with a 2X expansion lens, giving a pixel size of 79 nm.

*λ*/2NA = 580 nm/2.9 ≈ 200 nm and theoretical

*s*

_{0}= FWHM/2.35 ≈ 85 nm. Including the pixelation effect [Eq. (13)], the measured PSF SD

*s*

_{0x,0y}

^{†}, for our imaging system should be 88 nm. Due to random fluctuations in the emission polarization direction of streptavidin-Cy3 molecules attached to surfaces [8

8. K. Adachi, R. Yasuda, H. Noji, Y. Harada, M. Yoshida, and K. Kinosita, “Stepping rotation of F1-ATPase visualized through angle-resolved single-fluorophore imaging,” Proc. Natl. Acad. Sci. USA **97**, 7243–7247 (2000). [CrossRef] [PubMed]

*s*

_{0x,0y}

^{†}values from 90 nm to 140 nm.

*μ*l of 0.04 nM streptavidin-Cy3 powder dissolved in 0.5X TBE buffer (45 mM Tris, 45 mM Boric Acid, 1 mM EDTA, pH 7.0). A coverslip flattened the droplet and its edges were sealed with nail polish. The fused-silica chips were cleaned using oxygen plasma before use. We inspected for possible surface fluorescence contaminations by imaging the TBE buffer alone; no impurities were found on either the fused-silica surface or in the buffer. The immobilization of the adsorbed molecules was verified by centroid vs time measurements.

### 3.2. Data acquisition and selection

17. Y. M. Wang, J. Tegenfeldt, W. Reisner, R. Riehn, X.-J. Guan, L. Guo, I. Golding, E. C. Cox, J. Sturm, and R. H. Austin, “Single-molecule studies of repressor-DNA interactions show long-range interactions,” Proc. Natl. Acad. Sci. USA **102**, 9796–9801 (2005). [CrossRef] [PubMed]

*f*

_{0}was the amplitude and 〈

*b*〉 was the mean background value. A background pixel’s total count is the sum of the floor, electronic readout noise, and background fluorescence counts. For the 〈

*b*〉 in this article, the floor value, determined by the lowest background pixel value, has already been subtracted. With this fitting, the PSF’s SD values in both the

*x*and

*y*directions, its measured location (

*x*

_{0},

*y*

_{0}), and the image’s mean background value were obtained.

*N*that fluctuated less than 20% from the experimental mean 〈

*N*〉, of the monomer. The PSF

*N*count restriction is necessary for precise SD error analysis at

*N*by using a statistically sufficient number of PSFs with consistent

*N*. (3) PSFs with signal-to-noise ratios (

*I*

_{0}/√

*I*

_{0}+

*σ*

_{b}^{2}) larger than 2.5, where

*I*

_{0}is the peak PSF photon count (total photon count minus 〈

*b*〉) and

*σ*

_{b}^{2}is the background variance in photons. (4) Mean 〈

*s*〉 and 〈

_{x}*s*〉 obtained by Gaussian fitting of the

_{y}*s*and

_{x}*s*distributions of all valid images did not differ by more than 10 nm, or ±5% of the mean SD value to minimize polarization effects of Cy3. (5) The mean SD values 〈

_{y}*s*〉 were between 95 nm and 135 nm to minimize defocusing effects. These constraints on

_{x,y}*s*and

_{x}*s*are necessary for obtaining the expression for ∆

_{y}*s*, as a function of

_{rms}*N*, with minimal variations in the other parameters.

*M*. In order to obtain

*M*for each experimental setting, the center nine pixel values of the PSF were evaluated if the molecule’s average signal-to-noise ratio was greater than 3. According to Eq. (7),

*N*

_{i}^{*}〉 and

*σ*

_{i}^{*}are the Gaussian fitted mean and standard deviation of the measured camera count distribution of pixel

*i*, respectively. Here 〈

*N*

_{i}^{*}〉 is the mean camera count that includes background fluorescence and electronic noise counts.

### 3.3. PSF and background simulations

*s*

_{0x,0y}of each simulated PSF was determined by the experimental means 〈

*s*〉. The observed fluctuation in the number of photons

_{x,y}*N*, was incorporated. The generated photons of each PSF were binned into 15 × 15 pixels with a pixel size of 79 nm. Then each photon count in a pixel was converted into camera count using Eq. 4 with a

*M*value of one. Random background photons at each pixel were generated using the corresponding experimental background distribution function. Although the exact experimental background distributions were used for the simulations, the numerical integrations and analytical calculations were computed using the theoretical variance and the mean of all background counts,

*σ*

_{b}^{2}and 〈

*b*〉, respectively, rather than their fitted values. The background counts are primarily drawn from two types of distributions: a full Gaussian with a high mean or a truncated Gaussian with a low mean, depending on the background fluorescence level of each specific experiment. The final simulated PSFs with background noise were fitted to a 2D Gaussian [Eq. (16)] to obtain the centroid and SD values of the PSF.

*s*data point, 1000 iterations (2000 iterations for Fig. 3) were performed and the Gaussian fitted SDs of the

_{x,y,rms}*s*distributions were the simulated ∆

_{x,y}*s*results.

_{x,y,rms}## 4. Results

*s*using four different methods: (1) experimental measurements, (2) simulations, (3) numerical integrations of Eq. (8), and (4) analytical calculations using Eq. (15).

_{x,rms}*N*. These molecules have similar mean SD 〈

*s*〉 values of 110 nm, 111 nm, and 107 nm, respectively. In order to demonstrate the decreasing SD error with increasing

_{x}*N*, each representative image was chosen such that the 2D SD value was the sum of the mean SD 〈

*s*〉, and one standard deviation of the molecule’s

_{x}*s*distribution ∆

_{x}*s*(SD

_{x,rms}_{image}= 〈

*s*〉 + ∆

_{x}*s*). To clearly illustrate the change in the SD error, which is measured as the PSF SD minus 〈

_{x,rms}*s*〉, the 1D intensity profiles of the PSFs are plotted in Fig. 1B as opposed to their 2D intensity profiles for clarity. The 1D intensity values were obtained by averaging transverse pixel intensity values of the PSF at each longitudinal pixel

_{x}*i*. The measured 2D SD

_{image}values deviate from their respective means, 〈

*s*〉 values, by 10.3 nm, 7.2 nm, and 2.7 nm. As expected, the 2D SD error decreases with increasing

_{x}*N*.

*s*obtained by using experimental measurements, simulations, numerical integrations, and analytical calculations. Each experimental ∆

_{x,rms}*s*data point is the standard deviation of the

_{x,rms}*s*distribution for a single streptavidin-Cy3 monomer. A simulation was performed for each experimental data point. The parameters were based upon experimental results including fluctuations in a PSF’s total detected photons, background distribution, and the

_{x}*s*

_{0x,0y}values determined by the mean experimental 〈

*s*〉 after subtracting for the pixelation effect [Eq. (13)]. The finite bandwidth of the emission filter was also taken into consideration by simulating each photon as being drawn from a PSF whose width is varied according to a Gaussian distribution centered about

_{x,y}*s*

_{0x,0y}(with SD of 2 nm). Numerical integrations and analytical calculations used the same 〈

*N*〉,

*s*

_{0x,0y},

*σ*, and 〈

_{b}*b*〉 as those in the corresponding experimental data point. For all

*N*, the numerically integrated ∆

*s*results are ≈ 15% higher than the theoretical results while the experimental results are ≈ 57% higher. The causes of these discrepancies are discussed in the following section. The simulations agree well with the experimental results.

_{x,rms}*s*. Figure 3 shows ∆

_{x,rms}*s*vs

_{x,rms}*a*/

*s*

_{0}studied by simulations using

*s*

_{0x}=

*s*

_{0y}=

*s*

_{0}= 120 nm,

*N*= 500 photons,

*σ*= 1 photon, and 〈

_{b}*b*〉 = 4 photons. The generated photons of each PSF were binned into 19 × 19 pixels and subsequently converted into camera counts following the same procedure described above for Fig. 2. As

*a*/

*s*

_{0}increases, there is an initial decline in ∆

*s*until rising at

_{x,rms}*a*/

*s*

_{0}≈ 0.73. Beyond

*a*/

*s*

_{0}≈ 0.73, ∆

*s*increases slightly and then continues the decline again at

_{x,rms}*a*/

*s*

_{0}≈ 1.18. This decline after

*a*/

*s*

_{0}≈ 1.18 disagrees with theory, which suggests an increase in ∆

*s*beyond the theoretical minimum of

_{x,rms}*a/s*

_{0}= 1.18 (vertical dashed line). The overall decreasing ∆

*s*trend after the theoretical minimum occurs because when the pixel size increases, the measured PSF SD is increasingly affected by the width of the pixel and approaches the SD of the pixel; thus, variations among measured SD values decrease. Eventually, at sufficiently large pixel sizes where the whole PSF is contained within one pixel, the measured SD will be the SD of the pixel, inferred by the top-hat distribution function, and the measured SD error will be zero. The analytical calculation does not take this large pixelation effect into consideration; consequently, these results and those of the simulations begin to rapidly diverge.

_{x,rms}*s*minimum occurs at

_{x,rms}*a*/

*s*

_{0}= 0.73, rather than at the theoretical minimum of

*a*/

*s*

_{0}= 1.18 due to the pixel size effect described above. Our experimental settings of

*a*= 79 nm and

*s*

_{0}= 120 nm yield

*a*/

*s*

_{0}= 0.66 and is close to the simulated ∆

*s*minimum. We have also performed additional simulations using different parameter sets where the theoretical minimum always preceded the continued decline in ∆

_{x,rms}*s*. According to Fig. 3 and our other simulations, a good

_{x,rms}*a*/

*s*

_{0}range for future studies should be between ≈ 0.5 and 1, as is usually the case. Future ∆

*s*studies using different pixel sizes should take this discrepancy into account.

_{x,rms}*s*minimum at

_{x,rms}*a*/

*s*

_{0}= 0.73 is different from the theoretical ∆

*x*minimum at

_{rms}*a*/

*s*

_{0}= 0.88 described in Ref. [1

**82**, 2775–2783 (2002). [CrossRef] [PubMed]

*s*minimum at

_{x,rms}*a*/

*s*

_{0}= 1.10 calculated from Eq. 18, using our set of parameter values. Future studies should take this difference into consideration by selecting an optimal pixel size.

## 5. Discussion and Extensions

**82**, 2775–2783 (2002). [CrossRef] [PubMed]

*s*, for dimeric fluorophores and mobile molecules in future studies.

_{x,y,rms}### 5.1. Causes for discrepancies

*N*. There are four reasons for these discrepancies:

- In the ∆
*s*calculation (Appendix A), the_{rms}*N*distribution function is assumed to be a Gaussian for all pixels of the PSF [Eq. (21)]. This assumption will only be statistically accurate for center pixels of PSFs with high_{i}*N*. For peripheral pixels, especially for PSFs with low*N*, the*N*distribution function approaches a Poisson with a low mean, rather than a Gaussian. These different_{i}*N*distributions, which have been verified by simulation, were not considered in the analytical calculations._{i} - In simulations, we attempted to model the background count distribution exactly, whereas in numerical integrations and analytical calculations, the shape of the background count distribution was not considered, and therefore did not influence the results.

*a*/

*s*

_{0}values.

### 5.2. Modifications to centroid error analysis

**82**, 2775–2783 (2002). [CrossRef] [PubMed]

*s*

_{0}, should be modified to include the pixelation effect (

*s*

_{0}

^{2}+

*a*

^{2}/12)

^{1/2}, with respect to both directions. We have modified the PSF centroid measurement error to be

### 5.3. Interpreting ∆*s*_{x,y,rms} in SD measurement applications

_{x,y,rms}

11. T. J. Holmes, D. Briggs, and A. A. Tarif, “Blind deconvolution,” in *Handbook of Biological Confocal Microscopy*, J. B. Pawley, ed. (Springer, New York, 2006), pp. 468–487. [CrossRef]

*z*away from the focal plane can be expressed as [5

5. A. M. van Oijen, J. Khler, J. Schmidt, M. Mller, and G. J. Brakenhoff, “3-Dimensional super-resolution by spectrally selective imaging,” Chem. Phys. Lett. **292**, 183–187 (1998). [CrossRef]

6. M. Speidel, A. Jonas, and E.-L. Florin, “Three-dimensional tracking of fluorescent nanoparticles with sub-nanometer precision by use of off-focus imaging,” Opt. Lett. **28** (2003). [CrossRef] [PubMed]

7. B. Huang, W. Wang, M. Bates, and X. Zhuang, “Three-dimensional super-resolution imaging by stochastic optical reconstruction microscopy,” Science **319**, 810–813 (2008). [CrossRef] [PubMed]

*D*≈ 400 nm is the imaging depth of a typical single-molecule imaging system. Consequently, by error propagation, the precision in the SD measurement of a single image, ∆

*s*, can be used to determine the localization error associated with the molecule’s axial position, ∆

_{rms}*z*:

*x*and

*y*directions [8

**97**, 7243–7247 (2000). [CrossRef] [PubMed]

9. J. Enderlein, E. Toprak, and P. R. Selvin, “Polarization effect on position accuracy of fluorophore localization,” Opt. Express **14**, 8111–8120 (2006). [CrossRef] [PubMed]

10. F. Aguet, S. Geissbühler, I. Märki, T. Lasser, and M. Unser, “Super-resolution orientation estimation and localization of fluorescent dipoles using 3-D steerable filters,” Opt. Express **17**, 6829–6848 (2009). [CrossRef] [PubMed]

*s*and

_{x}*s*to the orientation is developed, the error in measuring

_{y}*s*and

_{x}*s*, once again by error propagation, can be used to calculate an error associated with the reported orientation of the molecule.

_{y}### 5.4. ∆*s*_{x,y,rms} calculation for future SD measurement applications

_{x,y,rms}

*N*distribution function at each pixel may be different from the Gaussian assumption for stationary molecules in Eq. (9). A new

_{i}*N*distribution function for each specific case can be obtained and a new

_{i}*σ*

_{i}^{2}formula [Eq. (7)] can be derived. Using the new

*N*distribution function and

_{i}*σ*

_{i}^{2}, the SD error for these cases can be obtained following the same procedure outlined in the theory (Sec. 2).

## 6. Conclusion

## Appendix

## A.

*y*. At large

_{i}*N*of a few hundred photons, the

*y*probability distribution function at each of the center nine pixels of the PSF is a Gaussian, while at the peripheral pixels, the

_{i}*y*probability distribution function is better approximated by a Poisson with a low mean. Here we assume that our

_{i}*N*is significantly larger than 100 photons and the

*y*probability distribution functions for all PSF pixels are Gaussian functions

_{i}*y*=

_{i}*N*(

_{i}*x*

_{0},

*s*

_{0})-

*y*and

_{i}*σ*

_{i}^{2}is

*σ*

_{i}^{2}, photon as in Eq. (1). For Gaussian distributed

*y*, we have

_{i}*s*,

*s*to the left-hand side,

*y*/

_{i}*σ*

_{i}^{2}term, we get

*y*only, so we have

_{i}## B.

*s*

_{x}_{2})〉. In 2D, the expected counts at pixel

*i,j*is given by

*N*with respect to

_{i}*s*and evaluating at

_{x}*s*

_{0x},

*i*and

*j*are continuous from negative to positive infinity. There are two limits to the approximation, one being the high photon count limit and the other being the high background noise limit. At the high photon count limit,

*s*

_{0x,0y}by (

*s*

_{0x,0y}

^{2}+

*a*

^{2}/12)

^{1/2}to incorporate the pixelation effect, we arrive at Eq. (15).

## Acknowledgements

## References and links

1. | R. E. Thompson, D. R. Larson, and W. W. Webb,“Precise nanometer localization analysis for individual fluorescent probes,” Biophys. J. |

2. | A. Yildiz, J. N. Forkey, S. A. McKinney, T. Ha, Y. E. Goldman, and P. R. Selvin, “Myosin V walks hand-overhand: Single fluorophore imaging with 1.5-nm localization,” Science |

3. | Y. M. Wang, R. H. Austin, and E. C. Cox, “Single molecule measurements of repressor protein 1D diffusion on DNA,” Phys. Rev. Lett. |

4. | C. Joo, H. Balci, Y. Ishitsuka, C. Buranachai, and T. Ha, “Advances in single-molecule fluorescence methods for molecular biology,” Annu. Rev. Biochem. |

5. | A. M. van Oijen, J. Khler, J. Schmidt, M. Mller, and G. J. Brakenhoff, “3-Dimensional super-resolution by spectrally selective imaging,” Chem. Phys. Lett. |

6. | M. Speidel, A. Jonas, and E.-L. Florin, “Three-dimensional tracking of fluorescent nanoparticles with sub-nanometer precision by use of off-focus imaging,” Opt. Lett. |

7. | B. Huang, W. Wang, M. Bates, and X. Zhuang, “Three-dimensional super-resolution imaging by stochastic optical reconstruction microscopy,” Science |

8. | K. Adachi, R. Yasuda, H. Noji, Y. Harada, M. Yoshida, and K. Kinosita, “Stepping rotation of F1-ATPase visualized through angle-resolved single-fluorophore imaging,” Proc. Natl. Acad. Sci. USA |

9. | J. Enderlein, E. Toprak, and P. R. Selvin, “Polarization effect on position accuracy of fluorophore localization,” Opt. Express |

10. | F. Aguet, S. Geissbühler, I. Märki, T. Lasser, and M. Unser, “Super-resolution orientation estimation and localization of fluorescent dipoles using 3-D steerable filters,” Opt. Express |

11. | T. J. Holmes, D. Briggs, and A. A. Tarif, “Blind deconvolution,” in |

12. | A. Yildiz, M. Tomishige, R. D. Vale, and P. R. Selvin, “Kinesin walks hand-over-hand,” Science |

13. | N. Bobroff, “Position measurement with a resolution and noise-limited instrument,” Rev. Sci. Instrum. |

14. | M. H. Ulbrich and E. Y. Isacoff, “Subunit counting in membrane-bound proteins,” Nature Methods |

15. | J. Hynecek and T. Nishiwaki, “Excess noise and other important characteristics of low light level imaging using charge multiplying CCDs,” IEEE Trans. on Electron Devices |

16. | J. R. Taylor, |

17. | Y. M. Wang, J. Tegenfeldt, W. Reisner, R. Riehn, X.-J. Guan, L. Guo, I. Golding, E. C. Cox, J. Sturm, and R. H. Austin, “Single-molecule studies of repressor-DNA interactions show long-range interactions,” Proc. Natl. Acad. Sci. USA |

18. | M. Born and E. Wolf, |

19. | S. H. DeCenzo, M. C. DeSantis, and Y. M. Wang, “Single-image measurements of unresolved dimolecular separations,” In preparation (2010). |

20. | S. K. G. Zareh, M. C. DeSantis, and Y. M. Wang, “Direct observation and analysis of 3D-diffusing fluorescent proteins in solution using single-image measurements,” In preparation (2010). |

**OCIS Codes**

(100.6640) Image processing : Superresolution

(100.6890) Image processing : Three-dimensional image processing

(110.2960) Imaging systems : Image analysis

(180.2520) Microscopy : Fluorescence microscopy

(180.6900) Microscopy : Three-dimensional microscopy

**ToC Category:**

Image Processing

**History**

Original Manuscript: January 25, 2010

Revised Manuscript: March 1, 2010

Manuscript Accepted: March 2, 2010

Published: March 15, 2010

**Virtual Issues**

Vol. 5, Iss. 7 *Virtual Journal for Biomedical Optics*

**Citation**

Michael C. DeSantis, Shawn H. DeCenzo, Je-Luen Li, and Y. M. Wang, "Precision analysis for standard deviation measurements of immobile single
fluorescent molecule images," Opt. Express **18**, 6563-6576 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-7-6563

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### References

- R. E. Thompson, D. R. Larson, and W. W. Webb, "Precise nanometer localization analysis for individual fluorescent probes," Biophys. J. 82, 2775-2783 (2002). [CrossRef] [PubMed]
- A. Yildiz, J. N. Forkey, S. A. McKinney, T. Ha, Y. E. Goldman, and P. R. Selvin, "Myosin V walks hand-overhand: Single fluorophore imaging with 1.5-nm localization," Science 300, 2061-2065 (2003). [CrossRef] [PubMed]
- Y. M. Wang, R. H. Austin, and E. C. Cox, "Single molecule measurements of repressor protein 1D diffusion on DNA," Phys. Rev. Lett. 97, 048302 (2006). [CrossRef] [PubMed]
- C. Joo, H. Balci, Y. Ishitsuka, C. Buranachai, and T. Ha, "Advances in single-molecule fluorescence methods for molecular biology," Annu. Rev. Biochem. 77, 51-76 (2008). [CrossRef] [PubMed]
- A. M. van Oijen, J. Köhler, J. Schmidt, M. Müller, and G. J. Brakenhoff, "3-Dimensional super-resolution by spectrally selective imaging," Chem. Phys. Lett. 292, 183-187 (1998). [CrossRef]
- M. Speidel, A. Jonas, and E.-L. Florin, "Three-dimensional tracking of fluorescent nanoparticles with subnanometer precision by use of off-focus imaging," Opt. Lett. 28 (2003). [CrossRef] [PubMed]
- B. Huang, W. Wang, M. Bates, and X. Zhuang, "Three-dimensional super-resolution imaging by stochastic optical reconstruction microscopy," Science 319, 810-813 (2008). [CrossRef] [PubMed]
- K. Adachi, R. Yasuda, H. Noji, Y. Harada, M. Yoshida, and K. Kinosita, "Stepping rotation of F1-ATPase visualized through angle-resolved single-fluorophore imaging," Proc. Natl. Acad. Sci. USA 97, 7243-7247 (2000). [CrossRef] [PubMed]
- J. Enderlein, E. Toprak, and P. R. Selvin, "Polarization effect on position accuracy of fluorophore localization," Opt. Express 14, 8111-8120 (2006). [CrossRef] [PubMed]
- F. Aguet, S. Geissbühler, I. Märki, T. Lasser, and M. Unser, "Super-resolution orientation estimation and localization of fluorescent dipoles using 3-D steerable filters," Opt. Express 17, 6829-6848 (2009). [CrossRef] [PubMed]
- T. J. Holmes, D. Briggs, and A. A. Tarif, "Blind deconvolution," in Handbook of Biological Confocal Microscopy, J. B. Pawley, ed. (Springer, New York, 2006), pp. 468-487. [CrossRef]
- A. Yildiz, M. Tomishige, R. D. Vale, and P. R. Selvin, "Kinesin walks hand-over-hand," Science 303, 676-678 (2004). [CrossRef]
- N. Bobroff, "Position measurement with a resolution and noise-limited instrument," Rev. Sci. Instrum. 57, 1152-1157 (1986). [CrossRef]
- M. H. Ulbrich and E. Y. Isacoff, "Subunit counting in membrane-bound proteins," Nature Methods 4, 319-321 (2007). [PubMed]
- J. Hynecek and T. Nishiwaki, "Excess noise and other important characteristics of low light level imaging using charge multiplying CCDs," IEEE Trans. on Electron Devices 50, 239-245 (2003). [CrossRef]
- J. R. Taylor, An Introduction to Error Analysis (University Science Books, California, 1997).
- Y. M. Wang, J. Tegenfeldt, W. Reisner, R. Riehn, X.-J. Guan, L. Guo, I. Golding, E. C. Cox, J. Sturm, and R. H. Austin, "Single-molecule studies of repressor-DNA interactions show long-range interactions," Proc. Natl. Acad. Sci. USA 102, 9796-9801 (2005). [CrossRef] [PubMed]
- M. Born and E. Wolf, Principles of Optics (Cambridge University Press, Cambridge, UK, 1999).
- S. H. DeCenzo, M. C. DeSantis, and Y. M. Wang, "Single-image measurements of unresolved dimolecular separations," In preparation (2010).
- S. K. G. Zareh, M. C. DeSantis, and Y. M. Wang, "Direct observation and analysis of 3D-diffusing fluorescent proteins in solution using single-image measurements," In preparation (2010).

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