Abstract

Estimating the location of single molecules from microscopy images is a key step in many quantitative single molecule data analysis techniques. Different algorithms have been advocated for the fitting of single molecule data, particularly the nonlinear least squares and maximum likelihood estimators. Comparisons were carried out to assess the performance of these two algorithms in different scenarios. Our results show that both estimators, on average, are able to recover the true location of the single molecule in all scenarios we examined. However, in the absence of modeling inaccuracies and low noise levels, the maximum likelihood estimator is more accurate than the nonlinear least squares estimator, as measured by the standard deviations of its estimates, and attains the best possible accuracy achievable for the sets of imaging and experimental conditions that were tested. Although neither algorithm is consistently superior to the other in the presence of modeling inaccuracies or misspecifications, the maximum likelihood algorithm emerges as a robust estimator producing results with consistent accuracy across various model mismatches and misspecifications. At high noise levels, relative to the signal from the point source, neither algorithm has a clear accuracy advantage over the other. Comparisons were also carried out for two localization accuracy measures derived previously. Software packages with user-friendly graphical interfaces developed for single molecule location estimation (EstimationTool) and limit of the localization accuracy calculations (FandPLimitTool) are also discussed.

© 2009 Optical Society of America

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2008 (3)

X. S. Xie, P. J. Choi, G. W. Li, N. K. Lee, and G. Lia, "Single-molecule approach to molecular biology in living bacterial cells," Annu. Rev. Biophys. 37, 417-444 (2008).
[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]

J. S. Biteen, M. A. Thompson, N. K. Tselentis, G. R. Bowman, L. Shapiro, and W. E. Moerner, "Super-resolution imaging in live Caulobacter crescentus cells using photoswitchable EYFP," Nat. Methods. 5, 947-949 (2008).
[CrossRef] [PubMed]

2007 (3)

B. Zhang, J. Zerubia, and J. C. Olivo-Marin, "Gaussian approximations of fluorescence microscope point-spread function models," Appl. Opt. 46, 1819-1829 (2007).
[CrossRef] [PubMed]

J. H. Kim and R. G. Larson, "Single-molecule analysis of 1D diffusion and transcription elongation of T7 RNA polymerase along individual stretched DNA molecules," Nucleic Acids Res. 35, 3848-3858 (2007).
[CrossRef] [PubMed]

W. E. Moerner, "New directions in single-molecule imaging and analysis," Proc. Natl. Acad. Sci. U.S.A. 104, 12596-12602 (2007).
[CrossRef] [PubMed]

2006 (4)

S. T. Hess, T. P. K. Girirajan, and M. D. Mason, "Ultra-high resolution imaging by fluorescence photoactivation localization microscopy," Biophys. J. 91, 4258-4272 (2006).
[CrossRef] [PubMed]

E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, and H. F. Hess, "Imaging intracellular fluorescent proteins at nanometer resolution," Science 313, 1642-1645 (2006).
[CrossRef] [PubMed]

M. J. Rust, M. Batest, X. Zhuang, "Sub-diffraction-limit imaging by stochastic optical reconstruction microscopy (STORM)," Nat. Meth. 3, 793-796 (2006).
[CrossRef]

S. Ram, E. S. Ward, and R. J. Ober, "A stochastic analysis of performance limits for optical microscopes," Multidim. Syst. Sign. Process. 17, 27-57 (2006).
[CrossRef]

2005 (2)

P. H. M. Lommerse, B. E. Snaar-Jagalska, H. P. Spaink, and T. Schmidt, "Single-molecule diffusion measurements of H-Ras at the plasma membrane of live cells reveal microdomain localization upon activation," J. Cell. Sci. 118, 1799-1809 (2005).
[CrossRef] [PubMed]

S. Ram, E. S. Ward, and R. J. Ober, "How accurately can a single molecule be localized in three dimensions using a fluorescence microscope?" Proc. SPIE. 5699, 426-435 (2005).
[CrossRef] [PubMed]

2004 (5)

K. Murase, T. Fujiwara, Y. Umemura, K. Suzuki, R. Iino, H. Yamashita, M. Saito, H. Murakoshi, K. Ritchie, and A. Kusumi, "Ultrafine membrane compartments for molecular diffusion as revealed by single molecule techniques," Biophys. J. 86, 4075-4093 (2004).
[CrossRef] [PubMed]

P. H. M. Lommerse, G. A. Blab, L. Cognet, G. S. Harms, B. E. Snaar-Jagalska, H. P. Spaink, and T. Schmidt, "Single-molecule imaging of the H-Ras membrane-anchor reveals domains in the cytoplasmic leaflet of the cell membrane," Biophys. J. 86, 609-616 (2004).
[CrossRef]

R. J. Ober, S. Ram, and E. S. Ward, "Localization accuracy in single-molecule microscopy," Biophys. J. 86, 1185-1200 (2004).
[CrossRef] [PubMed]

X. Qu, D. Wu, L. Mets, and N. F. Scherer, "Nanometer-localized multiple single-molecule fluorescence microscopy," Proc. Natl. Acad. Sci. U.S.A. 101, 11298-11303 (2004).
[CrossRef] [PubMed]

H. Park, G. T. Hanson, S. R. Duff, and P. R. Selvin, "Nanometer localization of single ReAsH molecules," J. Microsc. 216, 199-205 (2004).
[CrossRef] [PubMed]

2003 (2)

M. Dahan, S. Levi, C. Luccardini, P. Rostaing, B. Riveau, and A. Triller, "Diffusion dynamics of glycine receptors revealed by single-quantum dot tracking," Science 302, 442-445 (2003).
[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]

2002 (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]

2001 (2)

J. Markham and J. A. Conchello, "Fast maximum-likelihood image-restoration algorithms for three-dimensional fluorescence microscopy," J. Opt. Soc. Am. 18, 1062-1071 (2001).
[CrossRef]

M. K. Cheezum, W. F. Walker, and W. H. Guilford, "Quantitative comparison of algorithms for tracking single fluorescent particles," Biophys. J. 81, 2378-2388 (2001).
[CrossRef] [PubMed]

1997 (1)

P. J. Verveer and T. M. Jovin, "Efficient superresolution restoration algorithms using maximum a posteriori estimations with application to fluorescence microscopy," J. Opt. Soc. Am. 14, 1696-1706 (1997).
[CrossRef]

Bates, M.

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]

Batest, M.

M. J. Rust, M. Batest, X. Zhuang, "Sub-diffraction-limit imaging by stochastic optical reconstruction microscopy (STORM)," Nat. Meth. 3, 793-796 (2006).
[CrossRef]

Betzig, E.

E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, and H. F. Hess, "Imaging intracellular fluorescent proteins at nanometer resolution," Science 313, 1642-1645 (2006).
[CrossRef] [PubMed]

Biteen, J. S.

J. S. Biteen, M. A. Thompson, N. K. Tselentis, G. R. Bowman, L. Shapiro, and W. E. Moerner, "Super-resolution imaging in live Caulobacter crescentus cells using photoswitchable EYFP," Nat. Methods. 5, 947-949 (2008).
[CrossRef] [PubMed]

Blab, G. A.

P. H. M. Lommerse, G. A. Blab, L. Cognet, G. S. Harms, B. E. Snaar-Jagalska, H. P. Spaink, and T. Schmidt, "Single-molecule imaging of the H-Ras membrane-anchor reveals domains in the cytoplasmic leaflet of the cell membrane," Biophys. J. 86, 609-616 (2004).
[CrossRef]

Bonifacino, J. S.

E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, and H. F. Hess, "Imaging intracellular fluorescent proteins at nanometer resolution," Science 313, 1642-1645 (2006).
[CrossRef] [PubMed]

Bowman, G. R.

J. S. Biteen, M. A. Thompson, N. K. Tselentis, G. R. Bowman, L. Shapiro, and W. E. Moerner, "Super-resolution imaging in live Caulobacter crescentus cells using photoswitchable EYFP," Nat. Methods. 5, 947-949 (2008).
[CrossRef] [PubMed]

Cheezum, M. K.

M. K. Cheezum, W. F. Walker, and W. H. Guilford, "Quantitative comparison of algorithms for tracking single fluorescent particles," Biophys. J. 81, 2378-2388 (2001).
[CrossRef] [PubMed]

Choi, P. J.

X. S. Xie, P. J. Choi, G. W. Li, N. K. Lee, and G. Lia, "Single-molecule approach to molecular biology in living bacterial cells," Annu. Rev. Biophys. 37, 417-444 (2008).
[CrossRef] [PubMed]

Cognet, L.

P. H. M. Lommerse, G. A. Blab, L. Cognet, G. S. Harms, B. E. Snaar-Jagalska, H. P. Spaink, and T. Schmidt, "Single-molecule imaging of the H-Ras membrane-anchor reveals domains in the cytoplasmic leaflet of the cell membrane," Biophys. J. 86, 609-616 (2004).
[CrossRef]

Conchello, J. A.

J. Markham and J. A. Conchello, "Fast maximum-likelihood image-restoration algorithms for three-dimensional fluorescence microscopy," J. Opt. Soc. Am. 18, 1062-1071 (2001).
[CrossRef]

Dahan, M.

M. Dahan, S. Levi, C. Luccardini, P. Rostaing, B. Riveau, and A. Triller, "Diffusion dynamics of glycine receptors revealed by single-quantum dot tracking," Science 302, 442-445 (2003).
[CrossRef] [PubMed]

Davidson, M. W.

E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, and H. F. Hess, "Imaging intracellular fluorescent proteins at nanometer resolution," Science 313, 1642-1645 (2006).
[CrossRef] [PubMed]

Duff, S. R.

H. Park, G. T. Hanson, S. R. Duff, and P. R. Selvin, "Nanometer localization of single ReAsH molecules," J. Microsc. 216, 199-205 (2004).
[CrossRef] [PubMed]

Forkey, J. N.

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]

Fujiwara, T.

K. Murase, T. Fujiwara, Y. Umemura, K. Suzuki, R. Iino, H. Yamashita, M. Saito, H. Murakoshi, K. Ritchie, and A. Kusumi, "Ultrafine membrane compartments for molecular diffusion as revealed by single molecule techniques," Biophys. J. 86, 4075-4093 (2004).
[CrossRef] [PubMed]

Girirajan, T. P. K.

S. T. Hess, T. P. K. Girirajan, and M. D. Mason, "Ultra-high resolution imaging by fluorescence photoactivation localization microscopy," Biophys. J. 91, 4258-4272 (2006).
[CrossRef] [PubMed]

Goldman, Y. E.

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]

Guilford, W. H.

M. K. Cheezum, W. F. Walker, and W. H. Guilford, "Quantitative comparison of algorithms for tracking single fluorescent particles," Biophys. J. 81, 2378-2388 (2001).
[CrossRef] [PubMed]

Ha, T.

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]

Hanson, G. T.

H. Park, G. T. Hanson, S. R. Duff, and P. R. Selvin, "Nanometer localization of single ReAsH molecules," J. Microsc. 216, 199-205 (2004).
[CrossRef] [PubMed]

Harms, G. S.

P. H. M. Lommerse, G. A. Blab, L. Cognet, G. S. Harms, B. E. Snaar-Jagalska, H. P. Spaink, and T. Schmidt, "Single-molecule imaging of the H-Ras membrane-anchor reveals domains in the cytoplasmic leaflet of the cell membrane," Biophys. J. 86, 609-616 (2004).
[CrossRef]

Hess, H. F.

E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, and H. F. Hess, "Imaging intracellular fluorescent proteins at nanometer resolution," Science 313, 1642-1645 (2006).
[CrossRef] [PubMed]

Hess, S. T.

S. T. Hess, T. P. K. Girirajan, and M. D. Mason, "Ultra-high resolution imaging by fluorescence photoactivation localization microscopy," Biophys. J. 91, 4258-4272 (2006).
[CrossRef] [PubMed]

Huang, B.

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]

Iino, R.

K. Murase, T. Fujiwara, Y. Umemura, K. Suzuki, R. Iino, H. Yamashita, M. Saito, H. Murakoshi, K. Ritchie, and A. Kusumi, "Ultrafine membrane compartments for molecular diffusion as revealed by single molecule techniques," Biophys. J. 86, 4075-4093 (2004).
[CrossRef] [PubMed]

Jovin, T. M.

P. J. Verveer and T. M. Jovin, "Efficient superresolution restoration algorithms using maximum a posteriori estimations with application to fluorescence microscopy," J. Opt. Soc. Am. 14, 1696-1706 (1997).
[CrossRef]

Kim, J. H.

J. H. Kim and R. G. Larson, "Single-molecule analysis of 1D diffusion and transcription elongation of T7 RNA polymerase along individual stretched DNA molecules," Nucleic Acids Res. 35, 3848-3858 (2007).
[CrossRef] [PubMed]

Kusumi, A.

K. Murase, T. Fujiwara, Y. Umemura, K. Suzuki, R. Iino, H. Yamashita, M. Saito, H. Murakoshi, K. Ritchie, and A. Kusumi, "Ultrafine membrane compartments for molecular diffusion as revealed by single molecule techniques," Biophys. J. 86, 4075-4093 (2004).
[CrossRef] [PubMed]

Larson, D. R.

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]

Larson, R. G.

J. H. Kim and R. G. Larson, "Single-molecule analysis of 1D diffusion and transcription elongation of T7 RNA polymerase along individual stretched DNA molecules," Nucleic Acids Res. 35, 3848-3858 (2007).
[CrossRef] [PubMed]

Lee, N. K.

X. S. Xie, P. J. Choi, G. W. Li, N. K. Lee, and G. Lia, "Single-molecule approach to molecular biology in living bacterial cells," Annu. Rev. Biophys. 37, 417-444 (2008).
[CrossRef] [PubMed]

Levi, S.

M. Dahan, S. Levi, C. Luccardini, P. Rostaing, B. Riveau, and A. Triller, "Diffusion dynamics of glycine receptors revealed by single-quantum dot tracking," Science 302, 442-445 (2003).
[CrossRef] [PubMed]

Li, G. W.

X. S. Xie, P. J. Choi, G. W. Li, N. K. Lee, and G. Lia, "Single-molecule approach to molecular biology in living bacterial cells," Annu. Rev. Biophys. 37, 417-444 (2008).
[CrossRef] [PubMed]

Lia, G.

X. S. Xie, P. J. Choi, G. W. Li, N. K. Lee, and G. Lia, "Single-molecule approach to molecular biology in living bacterial cells," Annu. Rev. Biophys. 37, 417-444 (2008).
[CrossRef] [PubMed]

Lindwasser, O. W.

E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, and H. F. Hess, "Imaging intracellular fluorescent proteins at nanometer resolution," Science 313, 1642-1645 (2006).
[CrossRef] [PubMed]

Lippincott-Schwartz, J.

E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, and H. F. Hess, "Imaging intracellular fluorescent proteins at nanometer resolution," Science 313, 1642-1645 (2006).
[CrossRef] [PubMed]

Lommerse, P. H. M.

P. H. M. Lommerse, B. E. Snaar-Jagalska, H. P. Spaink, and T. Schmidt, "Single-molecule diffusion measurements of H-Ras at the plasma membrane of live cells reveal microdomain localization upon activation," J. Cell. Sci. 118, 1799-1809 (2005).
[CrossRef] [PubMed]

P. H. M. Lommerse, G. A. Blab, L. Cognet, G. S. Harms, B. E. Snaar-Jagalska, H. P. Spaink, and T. Schmidt, "Single-molecule imaging of the H-Ras membrane-anchor reveals domains in the cytoplasmic leaflet of the cell membrane," Biophys. J. 86, 609-616 (2004).
[CrossRef]

Luccardini, C.

M. Dahan, S. Levi, C. Luccardini, P. Rostaing, B. Riveau, and A. Triller, "Diffusion dynamics of glycine receptors revealed by single-quantum dot tracking," Science 302, 442-445 (2003).
[CrossRef] [PubMed]

Markham, J.

J. Markham and J. A. Conchello, "Fast maximum-likelihood image-restoration algorithms for three-dimensional fluorescence microscopy," J. Opt. Soc. Am. 18, 1062-1071 (2001).
[CrossRef]

Mason, M. D.

S. T. Hess, T. P. K. Girirajan, and M. D. Mason, "Ultra-high resolution imaging by fluorescence photoactivation localization microscopy," Biophys. J. 91, 4258-4272 (2006).
[CrossRef] [PubMed]

McKinney, S. A.

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]

Mets, L.

X. Qu, D. Wu, L. Mets, and N. F. Scherer, "Nanometer-localized multiple single-molecule fluorescence microscopy," Proc. Natl. Acad. Sci. U.S.A. 101, 11298-11303 (2004).
[CrossRef] [PubMed]

Moerner, W. E.

J. S. Biteen, M. A. Thompson, N. K. Tselentis, G. R. Bowman, L. Shapiro, and W. E. Moerner, "Super-resolution imaging in live Caulobacter crescentus cells using photoswitchable EYFP," Nat. Methods. 5, 947-949 (2008).
[CrossRef] [PubMed]

W. E. Moerner, "New directions in single-molecule imaging and analysis," Proc. Natl. Acad. Sci. U.S.A. 104, 12596-12602 (2007).
[CrossRef] [PubMed]

Murakoshi, H.

K. Murase, T. Fujiwara, Y. Umemura, K. Suzuki, R. Iino, H. Yamashita, M. Saito, H. Murakoshi, K. Ritchie, and A. Kusumi, "Ultrafine membrane compartments for molecular diffusion as revealed by single molecule techniques," Biophys. J. 86, 4075-4093 (2004).
[CrossRef] [PubMed]

Murase, K.

K. Murase, T. Fujiwara, Y. Umemura, K. Suzuki, R. Iino, H. Yamashita, M. Saito, H. Murakoshi, K. Ritchie, and A. Kusumi, "Ultrafine membrane compartments for molecular diffusion as revealed by single molecule techniques," Biophys. J. 86, 4075-4093 (2004).
[CrossRef] [PubMed]

Ober, R. J.

S. Ram, E. S. Ward, and R. J. Ober, "A stochastic analysis of performance limits for optical microscopes," Multidim. Syst. Sign. Process. 17, 27-57 (2006).
[CrossRef]

S. Ram, E. S. Ward, and R. J. Ober, "How accurately can a single molecule be localized in three dimensions using a fluorescence microscope?" Proc. SPIE. 5699, 426-435 (2005).
[CrossRef] [PubMed]

R. J. Ober, S. Ram, and E. S. Ward, "Localization accuracy in single-molecule microscopy," Biophys. J. 86, 1185-1200 (2004).
[CrossRef] [PubMed]

Olenych, S.

E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, and H. F. Hess, "Imaging intracellular fluorescent proteins at nanometer resolution," Science 313, 1642-1645 (2006).
[CrossRef] [PubMed]

Olivo-Marin, J. C.

Park, H.

H. Park, G. T. Hanson, S. R. Duff, and P. R. Selvin, "Nanometer localization of single ReAsH molecules," J. Microsc. 216, 199-205 (2004).
[CrossRef] [PubMed]

Patterson, G. H.

E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, and H. F. Hess, "Imaging intracellular fluorescent proteins at nanometer resolution," Science 313, 1642-1645 (2006).
[CrossRef] [PubMed]

Qu, X.

X. Qu, D. Wu, L. Mets, and N. F. Scherer, "Nanometer-localized multiple single-molecule fluorescence microscopy," Proc. Natl. Acad. Sci. U.S.A. 101, 11298-11303 (2004).
[CrossRef] [PubMed]

Ram, S.

S. Ram, E. S. Ward, and R. J. Ober, "A stochastic analysis of performance limits for optical microscopes," Multidim. Syst. Sign. Process. 17, 27-57 (2006).
[CrossRef]

S. Ram, E. S. Ward, and R. J. Ober, "How accurately can a single molecule be localized in three dimensions using a fluorescence microscope?" Proc. SPIE. 5699, 426-435 (2005).
[CrossRef] [PubMed]

R. J. Ober, S. Ram, and E. S. Ward, "Localization accuracy in single-molecule microscopy," Biophys. J. 86, 1185-1200 (2004).
[CrossRef] [PubMed]

Ritchie, K.

K. Murase, T. Fujiwara, Y. Umemura, K. Suzuki, R. Iino, H. Yamashita, M. Saito, H. Murakoshi, K. Ritchie, and A. Kusumi, "Ultrafine membrane compartments for molecular diffusion as revealed by single molecule techniques," Biophys. J. 86, 4075-4093 (2004).
[CrossRef] [PubMed]

Riveau, B.

M. Dahan, S. Levi, C. Luccardini, P. Rostaing, B. Riveau, and A. Triller, "Diffusion dynamics of glycine receptors revealed by single-quantum dot tracking," Science 302, 442-445 (2003).
[CrossRef] [PubMed]

Rostaing, P.

M. Dahan, S. Levi, C. Luccardini, P. Rostaing, B. Riveau, and A. Triller, "Diffusion dynamics of glycine receptors revealed by single-quantum dot tracking," Science 302, 442-445 (2003).
[CrossRef] [PubMed]

Rust, M. J.

M. J. Rust, M. Batest, X. Zhuang, "Sub-diffraction-limit imaging by stochastic optical reconstruction microscopy (STORM)," Nat. Meth. 3, 793-796 (2006).
[CrossRef]

Saito, M.

K. Murase, T. Fujiwara, Y. Umemura, K. Suzuki, R. Iino, H. Yamashita, M. Saito, H. Murakoshi, K. Ritchie, and A. Kusumi, "Ultrafine membrane compartments for molecular diffusion as revealed by single molecule techniques," Biophys. J. 86, 4075-4093 (2004).
[CrossRef] [PubMed]

Scherer, N. F.

X. Qu, D. Wu, L. Mets, and N. F. Scherer, "Nanometer-localized multiple single-molecule fluorescence microscopy," Proc. Natl. Acad. Sci. U.S.A. 101, 11298-11303 (2004).
[CrossRef] [PubMed]

Schmidt, T.

P. H. M. Lommerse, B. E. Snaar-Jagalska, H. P. Spaink, and T. Schmidt, "Single-molecule diffusion measurements of H-Ras at the plasma membrane of live cells reveal microdomain localization upon activation," J. Cell. Sci. 118, 1799-1809 (2005).
[CrossRef] [PubMed]

P. H. M. Lommerse, G. A. Blab, L. Cognet, G. S. Harms, B. E. Snaar-Jagalska, H. P. Spaink, and T. Schmidt, "Single-molecule imaging of the H-Ras membrane-anchor reveals domains in the cytoplasmic leaflet of the cell membrane," Biophys. J. 86, 609-616 (2004).
[CrossRef]

Selvin, P. R.

H. Park, G. T. Hanson, S. R. Duff, and P. R. Selvin, "Nanometer localization of single ReAsH molecules," J. Microsc. 216, 199-205 (2004).
[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]

Shapiro, L.

J. S. Biteen, M. A. Thompson, N. K. Tselentis, G. R. Bowman, L. Shapiro, and W. E. Moerner, "Super-resolution imaging in live Caulobacter crescentus cells using photoswitchable EYFP," Nat. Methods. 5, 947-949 (2008).
[CrossRef] [PubMed]

Snaar-Jagalska, B. E.

P. H. M. Lommerse, B. E. Snaar-Jagalska, H. P. Spaink, and T. Schmidt, "Single-molecule diffusion measurements of H-Ras at the plasma membrane of live cells reveal microdomain localization upon activation," J. Cell. Sci. 118, 1799-1809 (2005).
[CrossRef] [PubMed]

P. H. M. Lommerse, G. A. Blab, L. Cognet, G. S. Harms, B. E. Snaar-Jagalska, H. P. Spaink, and T. Schmidt, "Single-molecule imaging of the H-Ras membrane-anchor reveals domains in the cytoplasmic leaflet of the cell membrane," Biophys. J. 86, 609-616 (2004).
[CrossRef]

Sougrat, R.

E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, and H. F. Hess, "Imaging intracellular fluorescent proteins at nanometer resolution," Science 313, 1642-1645 (2006).
[CrossRef] [PubMed]

Spaink, H. P.

P. H. M. Lommerse, B. E. Snaar-Jagalska, H. P. Spaink, and T. Schmidt, "Single-molecule diffusion measurements of H-Ras at the plasma membrane of live cells reveal microdomain localization upon activation," J. Cell. Sci. 118, 1799-1809 (2005).
[CrossRef] [PubMed]

P. H. M. Lommerse, G. A. Blab, L. Cognet, G. S. Harms, B. E. Snaar-Jagalska, H. P. Spaink, and T. Schmidt, "Single-molecule imaging of the H-Ras membrane-anchor reveals domains in the cytoplasmic leaflet of the cell membrane," Biophys. J. 86, 609-616 (2004).
[CrossRef]

Suzuki, K.

K. Murase, T. Fujiwara, Y. Umemura, K. Suzuki, R. Iino, H. Yamashita, M. Saito, H. Murakoshi, K. Ritchie, and A. Kusumi, "Ultrafine membrane compartments for molecular diffusion as revealed by single molecule techniques," Biophys. J. 86, 4075-4093 (2004).
[CrossRef] [PubMed]

Thompson, M. A.

J. S. Biteen, M. A. Thompson, N. K. Tselentis, G. R. Bowman, L. Shapiro, and W. E. Moerner, "Super-resolution imaging in live Caulobacter crescentus cells using photoswitchable EYFP," Nat. Methods. 5, 947-949 (2008).
[CrossRef] [PubMed]

Thompson, R. E.

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]

Triller, A.

M. Dahan, S. Levi, C. Luccardini, P. Rostaing, B. Riveau, and A. Triller, "Diffusion dynamics of glycine receptors revealed by single-quantum dot tracking," Science 302, 442-445 (2003).
[CrossRef] [PubMed]

Tselentis, N. K.

J. S. Biteen, M. A. Thompson, N. K. Tselentis, G. R. Bowman, L. Shapiro, and W. E. Moerner, "Super-resolution imaging in live Caulobacter crescentus cells using photoswitchable EYFP," Nat. Methods. 5, 947-949 (2008).
[CrossRef] [PubMed]

Umemura, Y.

K. Murase, T. Fujiwara, Y. Umemura, K. Suzuki, R. Iino, H. Yamashita, M. Saito, H. Murakoshi, K. Ritchie, and A. Kusumi, "Ultrafine membrane compartments for molecular diffusion as revealed by single molecule techniques," Biophys. J. 86, 4075-4093 (2004).
[CrossRef] [PubMed]

Verveer, P. J.

P. J. Verveer and T. M. Jovin, "Efficient superresolution restoration algorithms using maximum a posteriori estimations with application to fluorescence microscopy," J. Opt. Soc. Am. 14, 1696-1706 (1997).
[CrossRef]

Walker, W. F.

M. K. Cheezum, W. F. Walker, and W. H. Guilford, "Quantitative comparison of algorithms for tracking single fluorescent particles," Biophys. J. 81, 2378-2388 (2001).
[CrossRef] [PubMed]

Wang, W.

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]

Ward, E. S.

S. Ram, E. S. Ward, and R. J. Ober, "A stochastic analysis of performance limits for optical microscopes," Multidim. Syst. Sign. Process. 17, 27-57 (2006).
[CrossRef]

S. Ram, E. S. Ward, and R. J. Ober, "How accurately can a single molecule be localized in three dimensions using a fluorescence microscope?" Proc. SPIE. 5699, 426-435 (2005).
[CrossRef] [PubMed]

R. J. Ober, S. Ram, and E. S. Ward, "Localization accuracy in single-molecule microscopy," Biophys. J. 86, 1185-1200 (2004).
[CrossRef] [PubMed]

Webb, W. W.

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]

Wu, D.

X. Qu, D. Wu, L. Mets, and N. F. Scherer, "Nanometer-localized multiple single-molecule fluorescence microscopy," Proc. Natl. Acad. Sci. U.S.A. 101, 11298-11303 (2004).
[CrossRef] [PubMed]

Xie, X. S.

X. S. Xie, P. J. Choi, G. W. Li, N. K. Lee, and G. Lia, "Single-molecule approach to molecular biology in living bacterial cells," Annu. Rev. Biophys. 37, 417-444 (2008).
[CrossRef] [PubMed]

Yamashita, H.

K. Murase, T. Fujiwara, Y. Umemura, K. Suzuki, R. Iino, H. Yamashita, M. Saito, H. Murakoshi, K. Ritchie, and A. Kusumi, "Ultrafine membrane compartments for molecular diffusion as revealed by single molecule techniques," Biophys. J. 86, 4075-4093 (2004).
[CrossRef] [PubMed]

Yildiz, A.

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]

Zerubia, J.

Zhang, B.

Zhuang, X.

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]

M. J. Rust, M. Batest, X. Zhuang, "Sub-diffraction-limit imaging by stochastic optical reconstruction microscopy (STORM)," Nat. Meth. 3, 793-796 (2006).
[CrossRef]

Annu. Rev. Biophys. (1)

X. S. Xie, P. J. Choi, G. W. Li, N. K. Lee, and G. Lia, "Single-molecule approach to molecular biology in living bacterial cells," Annu. Rev. Biophys. 37, 417-444 (2008).
[CrossRef] [PubMed]

Appl. Opt. (1)

Biophys. J. (6)

P. H. M. Lommerse, G. A. Blab, L. Cognet, G. S. Harms, B. E. Snaar-Jagalska, H. P. Spaink, and T. Schmidt, "Single-molecule imaging of the H-Ras membrane-anchor reveals domains in the cytoplasmic leaflet of the cell membrane," Biophys. J. 86, 609-616 (2004).
[CrossRef]

K. Murase, T. Fujiwara, Y. Umemura, K. Suzuki, R. Iino, H. Yamashita, M. Saito, H. Murakoshi, K. Ritchie, and A. Kusumi, "Ultrafine membrane compartments for molecular diffusion as revealed by single molecule techniques," Biophys. J. 86, 4075-4093 (2004).
[CrossRef] [PubMed]

S. T. Hess, T. P. K. Girirajan, and M. D. Mason, "Ultra-high resolution imaging by fluorescence photoactivation localization microscopy," Biophys. J. 91, 4258-4272 (2006).
[CrossRef] [PubMed]

M. K. Cheezum, W. F. Walker, and W. H. Guilford, "Quantitative comparison of algorithms for tracking single fluorescent particles," Biophys. J. 81, 2378-2388 (2001).
[CrossRef] [PubMed]

R. J. Ober, S. Ram, and E. S. Ward, "Localization accuracy in single-molecule microscopy," Biophys. J. 86, 1185-1200 (2004).
[CrossRef] [PubMed]

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]

J. Cell. Sci. (1)

P. H. M. Lommerse, B. E. Snaar-Jagalska, H. P. Spaink, and T. Schmidt, "Single-molecule diffusion measurements of H-Ras at the plasma membrane of live cells reveal microdomain localization upon activation," J. Cell. Sci. 118, 1799-1809 (2005).
[CrossRef] [PubMed]

J. Microsc. (1)

H. Park, G. T. Hanson, S. R. Duff, and P. R. Selvin, "Nanometer localization of single ReAsH molecules," J. Microsc. 216, 199-205 (2004).
[CrossRef] [PubMed]

J. Opt. Soc. Am. (2)

J. Markham and J. A. Conchello, "Fast maximum-likelihood image-restoration algorithms for three-dimensional fluorescence microscopy," J. Opt. Soc. Am. 18, 1062-1071 (2001).
[CrossRef]

P. J. Verveer and T. M. Jovin, "Efficient superresolution restoration algorithms using maximum a posteriori estimations with application to fluorescence microscopy," J. Opt. Soc. Am. 14, 1696-1706 (1997).
[CrossRef]

Multidim. Syst. Sign. Process. (1)

S. Ram, E. S. Ward, and R. J. Ober, "A stochastic analysis of performance limits for optical microscopes," Multidim. Syst. Sign. Process. 17, 27-57 (2006).
[CrossRef]

Nat. Meth. (1)

M. J. Rust, M. Batest, X. Zhuang, "Sub-diffraction-limit imaging by stochastic optical reconstruction microscopy (STORM)," Nat. Meth. 3, 793-796 (2006).
[CrossRef]

Nat. Methods. (1)

J. S. Biteen, M. A. Thompson, N. K. Tselentis, G. R. Bowman, L. Shapiro, and W. E. Moerner, "Super-resolution imaging in live Caulobacter crescentus cells using photoswitchable EYFP," Nat. Methods. 5, 947-949 (2008).
[CrossRef] [PubMed]

Nucleic Acids Res. (1)

J. H. Kim and R. G. Larson, "Single-molecule analysis of 1D diffusion and transcription elongation of T7 RNA polymerase along individual stretched DNA molecules," Nucleic Acids Res. 35, 3848-3858 (2007).
[CrossRef] [PubMed]

Proc. Natl. Acad. Sci. U.S.A. (2)

X. Qu, D. Wu, L. Mets, and N. F. Scherer, "Nanometer-localized multiple single-molecule fluorescence microscopy," Proc. Natl. Acad. Sci. U.S.A. 101, 11298-11303 (2004).
[CrossRef] [PubMed]

W. E. Moerner, "New directions in single-molecule imaging and analysis," Proc. Natl. Acad. Sci. U.S.A. 104, 12596-12602 (2007).
[CrossRef] [PubMed]

Proc. SPIE. (1)

S. Ram, E. S. Ward, and R. J. Ober, "How accurately can a single molecule be localized in three dimensions using a fluorescence microscope?" Proc. SPIE. 5699, 426-435 (2005).
[CrossRef] [PubMed]

Science (3)

M. Dahan, S. Levi, C. Luccardini, P. Rostaing, B. Riveau, and A. Triller, "Diffusion dynamics of glycine receptors revealed by single-quantum dot tracking," Science 302, 442-445 (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]

E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, and H. F. Hess, "Imaging intracellular fluorescent proteins at nanometer resolution," Science 313, 1642-1645 (2006).
[CrossRef] [PubMed]

Science. (1)

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]

Other (6)

S. M. Kay, Fundamentals of statistical signal processing (Prentice Hall, 1993).

"EstimationTool," http://www4.utsouthwestern.edu/wardlab/EstimationTool.

"FandPLimitTool," http://www4.utsouthwestern.edu/wardlab/FandPLimitTool.

M. Born and E. Wolf, Principles of Optics (Cambridge University Press, Cambridge, UK, 1999).

P. Torok and F.-J. Kao, Optical Imaging and Microscopy (Springer, 2003).

S. Zacks, Theory of Statistical Inference (John Wiley and Sons, 1971).

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Figures (7)

Fig. 1.
Fig. 1.

Comparison of standard deviation of estimates from the nonlinear least squares and maximum likelihood estimators in the ideal case. Panels A(B) and C(D) show the results when the Airy (Gaussian) pixelated profile was used both to generate and fit the single molecule images. Panels A(C) and B(D) show the mean (standard deviations) of the x 0 coordinate estimates from the nonlinear least squares (⋄) and maximum likelihood (∗) estimators. (∙) indicates the true x 0 coordinate value and (∘) the PLAM or limit of the localization accuracy of x 0. For each value of the expected number of photons at the detector plane, 1000 images of a stationary single molecule were generated. Location coordinates (x 0, y 0) were estimated from each image using both estimators. Values for the width, photon detection rate, and background parameters, were fixed to the values used to generate the images. The following numerical values were used when simulating the single molecule images. Pixel size: 13 µm×13 µm, pixel array size: 13×13, magnification M=100. The single molecule image was centered within the pixel array. The mean of the background noise component was set to zero. When the Airy pixelated profile was used to generate the single molecule images or to fit them, wavelength λ=520 nm, and numerical aperture na =1.3. When the Gaussian pixelated profile was used to generate the single molecule images or to fit them, the width parameter of the Gaussian pixelated profile was calculated as σ=1.323λ/2πna =84.225 nm.

Fig. 2.
Fig. 2.

Comparison of standard deviation of estimates from the nonlinear least squares and maximum likelihood estimators in the presence of additive Poisson noise and absence of model mismatches or misspecifications. Panels A(B) and C(D) show the results when the Airy(Gaussian) pixelated profile was used both to generate and fit the single molecule images. Panels A(C) and B(D) show the mean (standard deviations) of the x 0 coordinate estimates from the nonlinear least squares (⋄) and maximum likelihood (∗) estimators. (∙) indicates the true x 0 coordinate value and (∘) the PLAM or limit of the localization accuracy of x0. For each value of the total background photon count, 1000 images of a stationary single molecule were generated. Location coordinates (x 0, y 0) were estimated from each image using both estimators. Values for the width, photon detection rate, and background parameters, were fixed to the values used to generate the images. The following numerical values were used when simulating the single molecule images. Pixel size: 13 µm×13 µm, pixel array size: 13×13, magnification M=100. The single molecule image was centered within the pixel array. When the Airy pixelated profile was used to generate the single molecule images or to fit them, wavelength λ=520 nm, and numerical aperture na =1.3. When the Gaussian pixelated profile was used to generate the single molecule images or to fit them, the width parameter of the Gaussian pixelated profile was calculated as σ=1.323λ/2πna =84.225 nm.

Fig. 3.
Fig. 3.

Comparison of standard deviation of estimates from the nonlinear least squares and maximum likelihood estimators in the presence of extraneous Poisson and Gaussian noise sources and the absence of model mismatches. Panels A(C) and B(D) shows the mean(standard deviations) of the x 0 coordinate estimates from the nonlinear least squares (⋄) and maximum likelihood (∗) estimators. (∙) indicates the true x 0 coordinate value, and (∘) the PLAM or limit of the localization accuracy of x 0. In panels A and C, 1000 images of a stationary single molecule were generated using the Airy pixelated profile and readout noise with standard deviation of 4e - for each value of the total background photon count. In panels B and D, 1000 images of a stationary single molecule were generated using the Airy pixelated profile and fixed background of 2 photons/pixel/s for each value of the standard deviation of Gaussian noise. The mean of the Gaussian noise component was set to zero in all cases. Location coordinates (x 0, y 0) were estimated from each image using both estimators. Values for the width, photon detection rate, and background parameters, were fixed to the values used to generate the images. The following numerical values were used when simulating the single molecule images. Pixel size: 13 µm×13 µm, pixel array size: 13×13, expected number of photons from the single molecule at the detector plane: 1000 photons, magnification M=100, wavelength λ=520 nm, numerical aperture na =1.3. The single molecule image was centered within the pixel array.

Fig. 4.
Fig. 4.

Comparison of standard deviation of estimates from the nonlinear least squares and maximum likelihood estimators when the point source is away from the center of the image array. Panels A and B show the difference between the mean of the x 0 coordinate estimates from both algorithms and the true x 0 coordinate value. Panels C and D show the standard deviations of the x 0 coordinate estimates from both estimators. (⋄) show the results from the nonlinear least squares estimator, (∗) the results from the maximum likelihood estimator, and (∘) the PLAM or limit of the localization accuracy of x 0. The dashed line indicates the center of the image array in absolute coordinates. For each position of the point source, 1000 images of a stationary single molecule were generated using either the Airy (panels A and C) or Gaussian (panels B and D) pixelated profile. Location coordinates (x 0, y 0) were estimated from each image using both estimators. Values for the width, photon detection rate, and background parameters, were fixed to the values used to generate the images. The following numerical values were used when simulating the single molecule images. Pixel size: 13 µm×13 µm, pixel array size: 13×13, expected number of photons from the single molecule at the detector plane: 1000 photons, magnification M=100. When the Airy pixelated profile was used to generate the single molecule images or to fit them, wavelength λ=520 nm, and numerical aperture na =1.3. When the Gaussian pixelated profile was used to generate the single molecule images or to fit them, the width parameter of the Gaussian pixelated profile was calculated as σ=1.323λ/2πna =84.225 nm. The x 0 coordinate of the center of the pixel array in the object space is 845nm.

Fig. 5.
Fig. 5.

Comparison of the accuracies of the nonlinear least squares and maximum likelihood estimators when the data is generated and fit with the same type of profile but the width parameter is misspecified. Panels A(B) and C(D) show the results when the Airy (Gaussian) pixelated profile was used both to generate the images of the single molecule and to fit them. Panels A(C) and B(D) show the mean (standard deviations) of the x 0 coordinate estimates from both the nonlinear least squares (⋄) and the maximum likelihood (∗) estimators. (∙) indicates the true x 0 coordinate value. (∘) indicates the PLAM or limit of the localization accuracy of x 0. The dashed line indicates the correct value for the width parameter. Values for the photon detection rate and background parameters were fixed to their true values, i.e., the values used to generate the images. In both cases, sets of 1000 images of a stationary single molecule were generated. The location coordinates (x 0, y 0) were estimated from each image using both algorithms. The width parameter of the profile being fit was misspecified by various amounts between sets of images. The following numerical values were used when simulating the single molecule images. Pixel size: 13 µm×13 µm, pixel array size: 13×13, magnification M=100. The single molecule image was centered within the pixel array. The mean of the background noise component was set to zero. When the Airy pixelated profile was used to generate the single molecule images, wavelength λ=520 nm, and numerical aperture na =1.3, resulting in α=0.0157 nm -1. When the Gaussian pixelated profile was used to generate the single molecule images, the width parameter of the Gaussian pixelated profile was calculated asσ=1.323λ/2πna =84.225 nm.

Fig. 6.
Fig. 6.

Comparison of the accuracies of the nonlinear least squares and maximum likelihood estimators when Gaussian pixelated profiles are used to fit data generated using Airy pixelated profiles. Panels A(C) and B(D) show the mean (standard deviation) of the x 0 coordinate estimates from both the nonlinear least squares (⋄) and the maximum likelihood (∗) estimators. (∗) indicates the true x 0 coordinate value. (∘) indicates the PLAM or limit of the localization accuracy of x 0. In panels A and C, sets of 1000 images were generated for each value of the expected photon count at the detector plane. In panels B and D, sets of 1000 images were generated with the expected photon count at the detector plane set to 500 photons. In both cases, the location coordinates (x 0, y 0) were estimated from each image using both the nonlinear least squares and maximum likelihood estimators. Values for the photon detection rate and background parameters were fixed to their true values, i.e., the values used to generate the images. In panels A and C the value for the width parameter was also fixed to the true value. All images were generated using Airy pixelated profiles and fit using Gaussian pixelated profiles. In panels B and D, the width parameter of the Gaussian profile was misspecified by varying amounts between sets of images. The dashed line indicates the best approximate for the Gaussian width parameter. The following numerical values were used when simulating the single molecule images. Pixel size: 13 µm×13 µm, pixel array size: 13×13, magnification M=100. The single molecule image was centered within the pixel array. The mean of the background noise component is set to zero. For the Airy pixelated profile used to generate the single molecule images, wavelength λ=520 nm, and numerical aperture na =1.3, resulting in α=0.0157 nm -1. For the Gaussian pixelated profile used to fit the images (in the absence of misspecifications), the width parameter was calculated as σ=1.323λ/2πna =84.225 nm.

Fig. 7.
Fig. 7.

Comparison of the FLAM and PLAM with estimates of the localization error according to Thompson et. al. [17]. The PLAM for the Airy pixelated profile (∘) and estimates of the localization error given by Eq. 1 [17] (⊗) are plotted as a function of pixel size. For comparison, the FLAM (⊙) and the corresponding localization error estimate by Thompson et. al. for the ideal case (×), calculated as s/√N, are also plotted. For each pixel size, a set of 1000 single molecule images were generated using the Airy pixelated profile. The standard deviation, calculated independently for each pixel size, of the single molecule location coordinate estimates from each image obtained using both the nonlinear least squares (⋄) and maximum likelihood (∗) estimators are also shown. The numerical values used in the calculations are as follows. Pixel array size: closest match to 169µm×169µm total detector area, magnification M=100, expected number of photons from the single molecule: 500 photons. The single molecule image was centered within the detector area. The mean of the background noise component is set to zero. For the Airy profile, wavelength λ=520 nm, and numerical aperture na =1.3. For the calculations of the localization error based on Eq. 1, s is calculated as 1.323λ/2πna =84.225 nm, N=500 (expected number of photons from the single molecule), and a is obtained by dividing the pixel size by the magnification.

Equations (8)

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( Δ x ) 2 = s 2 N + a 2 / 12 N + 8 π s 4 b 2 a 2 N 2 ,
𝓘 θ , k : = S θ , k + B k + W k ,
μ θ , k : = ( A · Δ t ) C k f θ ( r ) dr ,
f θ ( r ) : = [ J 1 ( α M r r 0 ) ] 2 π r r 0 2 ,
f θ ( r ) : = 1 2 π M 2 σ 2 · e ( ( x M x 0 ) 2 2 ( M σ ) 2 ( y M y 0 ) 2 2 ( M σ ) 2 ) ,
S = k = 1 K [ z k ( μ θ , k + b Δ t ) ] 2 ,
( θ z 1 , , z K ) : = k = 1 K ln ( 1 2 π σ w , k · l = 0 ( [ μ θ , k + b Δ t ] l e ( μ θ , k + b Δ t ) l ! e 1 2 ( z k l η w , k σ w , k ) 2 ) )
( θ z 1 , , z K ) : = k = 1 K ( ( z k ) · ln ( μ θ , k + b Δ t ) ( μ θ , k + b Δ t ) ) ,

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