Abstract

There is a huge abundance of viruses and membrane vesicles in seawater. We describe a new full-field, incoherently illuminated, shot-noise limited, common-path interferometric detection method that we couple with the analysis of Brownian motion to detect, quantify, and differentiate biotic nanoparticles. We validated the method with calibrated nanoparticles and homogeneous DNA or RNA viruses. The smallest virus size that we characterized with a suitable signal-to-noise ratio was around 30 nm in diameter. Analysis of Brownian motions revealed anisotropic trajectories for myoviruses.We further applied the method for vesicles detection and for analysis of coastal and oligotrophic samples from Tara Oceans circumnavigation.

© 2016 Optical Society of America

Full Article  |  PDF Article

Corrections

20 September 2016: A correction was made to the acknowledgments.

11 October 2016: A correction was made to the funding section.


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References

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2015 (1)

P. Bork, C. Bowler, C. de Vargas, G. Gorsky, E. Karsenti, and P. Wincker, “Tara Oceans. Tara Oceans studies plankton at planetary scale. Introduction,” Science 348(6237), 873 (2015).
[Crossref] [PubMed]

2014 (1)

S. J. Biller, F. Schubotz, S. E. Roggensack, A. W. Thompson, R. E. Summons, and S. W. Chisholm, “Bacterial vesicles in marine ecosystems,” Science 343(6167), 183–186 (2014).
[Crossref] [PubMed]

2013 (5)

J. P. Payet and C. A. Suttle, “To kill or not to kill: the balance between lytic and lysogenic viral infection is driven by trophic status,” Limnol. Oceanogr. 58(2), 465–474 (2013).
[Crossref]

G. F. Steward, A. I. Culley, J. A. Mueller, E. M. Wood-Charlson, M. Belcaid, and G. Poisson, “Are we missing half of the viruses in the ocean?” ISME J. 7(3), 672–679 (2013).
[Crossref] [PubMed]

P. Forterre, N. Soler, M. Krupovic, E. Marguet, and H. W. Ackermann, “Fake virus particles generated by fluorescence microscopy,” Trends Microbiol. 21(1), 1–5 (2013).
[Crossref] [PubMed]

M. Heidernätsch, M. Bauer, and G. Radons, “Characterizing N-dimensional anisotropic Brownian motion by the distribution of diffusivities,” J. Chem. Phys. 139(18), 184105 (2013).
[Crossref] [PubMed]

J. R. Brum, R. O. Schenck, and M. B. Sullivan, “Global morphological analysis of marine viruses shows minimal regional variation and dominance of non-tailed viruses,” ISME J. 7(9), 1738–1751 (2013).
[Crossref] [PubMed]

2012 (4)

A. Yurt, G. G. Daaboul, J. H. Connor, B. B. Goldberg, and M. S. Ünlü, “Single nanoparticle detectors for biological applications,” Nanoscale 4(3), 715–726 (2012).
[Crossref] [PubMed]

A. Mitra, F. Ignatovich, and L. Novotny, “Real-time optical detection of single human and bacterial viruses based on dark-field interferometry,” Biosens. Bioelectron. 31(1), 499–504 (2012).
[Crossref] [PubMed]

K. Holmfeldt, D. Odić, M. B. Sullivan, M. Middelboe, and L. Riemann, “Cultivated single-stranded DNA phages that infect marine Bacteroidetes prove difficult to detect with DNA-binding stains,” Appl. Environ. Microbiol. 78(3), 892–894 (2012).
[Crossref] [PubMed]

M. Breitbart, “Marine viruses: truth or dare,” Annu. Rev. Mar. Sci. 4(1), 425–448 (2012).
[Crossref] [PubMed]

2011 (4)

R. Danovaro, C. Corinaldesi, A. Dell’anno, J. A. Fuhrman, J. J. Middelburg, R. T. Noble, and C. A. Suttle, “Marine viruses and global climate change,” FEMS Microbiol. Rev. 35(6), 993–1034 (2011).
[Crossref] [PubMed]

C. A. Lopez, G. G. Daaboul, R. S. Vedula, E. Ozkumur, D. A. Bergstein, T. W. Geisbert, H. E. Fawcett, B. B. Goldberg, J. H. Connor, and M. S. Unlü, “Label-free multiplexed virus detection using spectral reflectance imaging,” Biosens. Bioelectron. 26(8), 3432–3437 (2011).
[Crossref] [PubMed]

E. Karsenti, S. G. Acinas, P. Bork, C. Bowler, C. De Vargas, J. Raes, M. Sullivan, D. Arendt, F. Benzoni, J. M. Claverie, M. Follows, G. Gorsky, P. Hingamp, D. Iudicone, O. Jaillon, S. Kandels-Lewis, U. Krzic, F. Not, H. Ogata, S. Pesant, E. G. Reynaud, C. Sardet, M. E. Sieracki, S. Speich, D. Velayoudon, J. Weissenbach, P. Wincker, and Tara Oceans Consortium, “A holistic approach to marine eco-systems biology,” PLoS Biol. 9(10), e1001177 (2011).
[Crossref] [PubMed]

L. He, S. K. Ozdemir, J. Zhu, W. Kim, and L. Yang, “Detecting single viruses and nanoparticles using whispering gallery microlasers,” Nat. Nanotechnol. 6(7), 428–432 (2011).
[Crossref] [PubMed]

2010 (3)

S. Wang, X. Shan, U. Patel, X. Huang, J. Lu, J. Li, and N. Tao, “Label-free imaging, detection, and mass measurement of single viruses by surface plasmon resonance,” Proc. Natl. Acad. Sci. U.S.A. 107(37), 16028–16032 (2010).
[Crossref] [PubMed]

V. Filipe, A. Hawe, and W. Jiskoot, “Critical evaluation of Nanoparticle Tracking Analysis (NTA) by NanoSight for the measurement of nanoparticles and protein aggregates,” Pharm. Res. 27(5), 796–810 (2010).
[Crossref] [PubMed]

A. Kulp and M. J. Kuehn, “Biological functions and biogenesis of secreted bacterial outer membrane vesicles,” Annu. Rev. Microbiol. 64(1), 163–184 (2010).
[Crossref] [PubMed]

2009 (1)

F. Rohwer and R. V. Thurber, “Viruses manipulate the marine environment,” Nature 459(7244), 207–212 (2009).
[Crossref] [PubMed]

2008 (2)

C. P. D. Brussaard, S. W. Wilhelm, F. Thingstad, M. G. Weinbauer, G. Bratbak, M. Heldal, S. A. Kimmance, M. Middelboe, K. Nagasaki, J. H. Paul, D. C. Schroeder, C. A. Suttle, D. Vaqué, and K. E. Wommack, “Global-scale processes with a nanoscale drive: the role of marine viruses,” ISME J. 2(6), 575–578 (2008).
[Crossref] [PubMed]

F. Vollmer, S. Arnold, and D. Keng, “Single virus detection from the reactive shift of a whispering-gallery mode,” Proc. Natl. Acad. Sci. U.S.A. 105(52), 20701–20704 (2008).
[Crossref] [PubMed]

2007 (3)

A. Patel, R. T. Noble, J. A. Steele, M. S. Schwalbach, I. Hewson, and J. A. Fuhrman, “Virus and prokaryote enumeration from planktonic aquatic environments by epifluorescence microscopy with SYBR Green I,” Nat. Protoc. 2(2), 269–276 (2007).
[Crossref] [PubMed]

C. A. Suttle, “Marine viruses--major players in the global ecosystem,” Nat. Rev. Microbiol. 5(10), 801–812 (2007).
[Crossref] [PubMed]

J. S. Hwang and W. E. Moerner, “Interferometry of a single nanoparticle using the Gouy phase of a focused laser beam,” Opt. Commun. 280(2), 487–491 (2007).
[Crossref]

2006 (1)

Y. Han, A. M. Alsayed, M. Nobili, J. Zhang, T. C. Lubensky, and A. G. Yodh, “Brownian motion of an ellipsoid,” Science 314(5799), 626–630 (2006).
[Crossref] [PubMed]

2004 (1)

K. Wen, A. C. Ortmann, and C. A. Suttle, “Accurate Estimation of Viral Abundance by Epifluorescence Microscopy,” Appl. Environ. Microbiol. 70(7), 3862–3867 (2004).
[Crossref] [PubMed]

2002 (2)

2000 (2)

W. M. Balch, J. Vaughn, J. Novotny, D. T. Drapeau, R. Vaillancourt, J. Lapierre, and A. Ashe, “Light Scattering by Viral Suspensions,” Limnol. Oceanogr. 45(2), 492–498 (2000).
[Crossref]

Y. Bettarel, T. Sime-Ngando, C. Amblard, and H. Laveran, “A comparison of methods for counting viruses in aquatic systems,” Appl. Environ. Microbiol. 66(6), 2283–2289 (2000).
[Crossref] [PubMed]

1996 (1)

J. C. Crocker and D. G. Grier, “Methods of digital video microscopy for colloidal studies,” J. Colloid Interface Sci. 179(1), 298–310 (1996).
[Crossref]

1971 (1)

L. A. Simon, “Phage shows its claws,” New Sci. 49, 670–673 (1971).

Acinas, S. G.

E. Karsenti, S. G. Acinas, P. Bork, C. Bowler, C. De Vargas, J. Raes, M. Sullivan, D. Arendt, F. Benzoni, J. M. Claverie, M. Follows, G. Gorsky, P. Hingamp, D. Iudicone, O. Jaillon, S. Kandels-Lewis, U. Krzic, F. Not, H. Ogata, S. Pesant, E. G. Reynaud, C. Sardet, M. E. Sieracki, S. Speich, D. Velayoudon, J. Weissenbach, P. Wincker, and Tara Oceans Consortium, “A holistic approach to marine eco-systems biology,” PLoS Biol. 9(10), e1001177 (2011).
[Crossref] [PubMed]

Ackermann, H. W.

P. Forterre, N. Soler, M. Krupovic, E. Marguet, and H. W. Ackermann, “Fake virus particles generated by fluorescence microscopy,” Trends Microbiol. 21(1), 1–5 (2013).
[Crossref] [PubMed]

Alsayed, A. M.

Y. Han, A. M. Alsayed, M. Nobili, J. Zhang, T. C. Lubensky, and A. G. Yodh, “Brownian motion of an ellipsoid,” Science 314(5799), 626–630 (2006).
[Crossref] [PubMed]

Amblard, C.

Y. Bettarel, T. Sime-Ngando, C. Amblard, and H. Laveran, “A comparison of methods for counting viruses in aquatic systems,” Appl. Environ. Microbiol. 66(6), 2283–2289 (2000).
[Crossref] [PubMed]

Arendt, D.

E. Karsenti, S. G. Acinas, P. Bork, C. Bowler, C. De Vargas, J. Raes, M. Sullivan, D. Arendt, F. Benzoni, J. M. Claverie, M. Follows, G. Gorsky, P. Hingamp, D. Iudicone, O. Jaillon, S. Kandels-Lewis, U. Krzic, F. Not, H. Ogata, S. Pesant, E. G. Reynaud, C. Sardet, M. E. Sieracki, S. Speich, D. Velayoudon, J. Weissenbach, P. Wincker, and Tara Oceans Consortium, “A holistic approach to marine eco-systems biology,” PLoS Biol. 9(10), e1001177 (2011).
[Crossref] [PubMed]

Arnold, S.

F. Vollmer, S. Arnold, and D. Keng, “Single virus detection from the reactive shift of a whispering-gallery mode,” Proc. Natl. Acad. Sci. U.S.A. 105(52), 20701–20704 (2008).
[Crossref] [PubMed]

Ashe, A.

W. M. Balch, J. Vaughn, J. Novotny, D. T. Drapeau, R. Vaillancourt, J. Lapierre, and A. Ashe, “Light Scattering by Viral Suspensions,” Limnol. Oceanogr. 45(2), 492–498 (2000).
[Crossref]

Balch, W. M.

W. M. Balch, J. Vaughn, J. Novotny, D. T. Drapeau, R. Vaillancourt, J. Lapierre, and A. Ashe, “Light Scattering by Viral Suspensions,” Limnol. Oceanogr. 45(2), 492–498 (2000).
[Crossref]

Bauer, M.

M. Heidernätsch, M. Bauer, and G. Radons, “Characterizing N-dimensional anisotropic Brownian motion by the distribution of diffusivities,” J. Chem. Phys. 139(18), 184105 (2013).
[Crossref] [PubMed]

Beaurepaire, E.

Belcaid, M.

G. F. Steward, A. I. Culley, J. A. Mueller, E. M. Wood-Charlson, M. Belcaid, and G. Poisson, “Are we missing half of the viruses in the ocean?” ISME J. 7(3), 672–679 (2013).
[Crossref] [PubMed]

Benzoni, F.

E. Karsenti, S. G. Acinas, P. Bork, C. Bowler, C. De Vargas, J. Raes, M. Sullivan, D. Arendt, F. Benzoni, J. M. Claverie, M. Follows, G. Gorsky, P. Hingamp, D. Iudicone, O. Jaillon, S. Kandels-Lewis, U. Krzic, F. Not, H. Ogata, S. Pesant, E. G. Reynaud, C. Sardet, M. E. Sieracki, S. Speich, D. Velayoudon, J. Weissenbach, P. Wincker, and Tara Oceans Consortium, “A holistic approach to marine eco-systems biology,” PLoS Biol. 9(10), e1001177 (2011).
[Crossref] [PubMed]

Bergstein, D. A.

C. A. Lopez, G. G. Daaboul, R. S. Vedula, E. Ozkumur, D. A. Bergstein, T. W. Geisbert, H. E. Fawcett, B. B. Goldberg, J. H. Connor, and M. S. Unlü, “Label-free multiplexed virus detection using spectral reflectance imaging,” Biosens. Bioelectron. 26(8), 3432–3437 (2011).
[Crossref] [PubMed]

Bettarel, Y.

Y. Bettarel, T. Sime-Ngando, C. Amblard, and H. Laveran, “A comparison of methods for counting viruses in aquatic systems,” Appl. Environ. Microbiol. 66(6), 2283–2289 (2000).
[Crossref] [PubMed]

Biller, S. J.

S. J. Biller, F. Schubotz, S. E. Roggensack, A. W. Thompson, R. E. Summons, and S. W. Chisholm, “Bacterial vesicles in marine ecosystems,” Science 343(6167), 183–186 (2014).
[Crossref] [PubMed]

Boccara, A. C.

Bork, P.

P. Bork, C. Bowler, C. de Vargas, G. Gorsky, E. Karsenti, and P. Wincker, “Tara Oceans. Tara Oceans studies plankton at planetary scale. Introduction,” Science 348(6237), 873 (2015).
[Crossref] [PubMed]

E. Karsenti, S. G. Acinas, P. Bork, C. Bowler, C. De Vargas, J. Raes, M. Sullivan, D. Arendt, F. Benzoni, J. M. Claverie, M. Follows, G. Gorsky, P. Hingamp, D. Iudicone, O. Jaillon, S. Kandels-Lewis, U. Krzic, F. Not, H. Ogata, S. Pesant, E. G. Reynaud, C. Sardet, M. E. Sieracki, S. Speich, D. Velayoudon, J. Weissenbach, P. Wincker, and Tara Oceans Consortium, “A holistic approach to marine eco-systems biology,” PLoS Biol. 9(10), e1001177 (2011).
[Crossref] [PubMed]

Bowler, C.

P. Bork, C. Bowler, C. de Vargas, G. Gorsky, E. Karsenti, and P. Wincker, “Tara Oceans. Tara Oceans studies plankton at planetary scale. Introduction,” Science 348(6237), 873 (2015).
[Crossref] [PubMed]

E. Karsenti, S. G. Acinas, P. Bork, C. Bowler, C. De Vargas, J. Raes, M. Sullivan, D. Arendt, F. Benzoni, J. M. Claverie, M. Follows, G. Gorsky, P. Hingamp, D. Iudicone, O. Jaillon, S. Kandels-Lewis, U. Krzic, F. Not, H. Ogata, S. Pesant, E. G. Reynaud, C. Sardet, M. E. Sieracki, S. Speich, D. Velayoudon, J. Weissenbach, P. Wincker, and Tara Oceans Consortium, “A holistic approach to marine eco-systems biology,” PLoS Biol. 9(10), e1001177 (2011).
[Crossref] [PubMed]

Bratbak, G.

C. P. D. Brussaard, S. W. Wilhelm, F. Thingstad, M. G. Weinbauer, G. Bratbak, M. Heldal, S. A. Kimmance, M. Middelboe, K. Nagasaki, J. H. Paul, D. C. Schroeder, C. A. Suttle, D. Vaqué, and K. E. Wommack, “Global-scale processes with a nanoscale drive: the role of marine viruses,” ISME J. 2(6), 575–578 (2008).
[Crossref] [PubMed]

Breitbart, M.

M. Breitbart, “Marine viruses: truth or dare,” Annu. Rev. Mar. Sci. 4(1), 425–448 (2012).
[Crossref] [PubMed]

Brum, J. R.

J. R. Brum, R. O. Schenck, and M. B. Sullivan, “Global morphological analysis of marine viruses shows minimal regional variation and dominance of non-tailed viruses,” ISME J. 7(9), 1738–1751 (2013).
[Crossref] [PubMed]

Brussaard, C. P. D.

C. P. D. Brussaard, S. W. Wilhelm, F. Thingstad, M. G. Weinbauer, G. Bratbak, M. Heldal, S. A. Kimmance, M. Middelboe, K. Nagasaki, J. H. Paul, D. C. Schroeder, C. A. Suttle, D. Vaqué, and K. E. Wommack, “Global-scale processes with a nanoscale drive: the role of marine viruses,” ISME J. 2(6), 575–578 (2008).
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Chisholm, S. W.

S. J. Biller, F. Schubotz, S. E. Roggensack, A. W. Thompson, R. E. Summons, and S. W. Chisholm, “Bacterial vesicles in marine ecosystems,” Science 343(6167), 183–186 (2014).
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Claverie, J. M.

E. Karsenti, S. G. Acinas, P. Bork, C. Bowler, C. De Vargas, J. Raes, M. Sullivan, D. Arendt, F. Benzoni, J. M. Claverie, M. Follows, G. Gorsky, P. Hingamp, D. Iudicone, O. Jaillon, S. Kandels-Lewis, U. Krzic, F. Not, H. Ogata, S. Pesant, E. G. Reynaud, C. Sardet, M. E. Sieracki, S. Speich, D. Velayoudon, J. Weissenbach, P. Wincker, and Tara Oceans Consortium, “A holistic approach to marine eco-systems biology,” PLoS Biol. 9(10), e1001177 (2011).
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Connor, J. H.

A. Yurt, G. G. Daaboul, J. H. Connor, B. B. Goldberg, and M. S. Ünlü, “Single nanoparticle detectors for biological applications,” Nanoscale 4(3), 715–726 (2012).
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C. A. Lopez, G. G. Daaboul, R. S. Vedula, E. Ozkumur, D. A. Bergstein, T. W. Geisbert, H. E. Fawcett, B. B. Goldberg, J. H. Connor, and M. S. Unlü, “Label-free multiplexed virus detection using spectral reflectance imaging,” Biosens. Bioelectron. 26(8), 3432–3437 (2011).
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Corinaldesi, C.

R. Danovaro, C. Corinaldesi, A. Dell’anno, J. A. Fuhrman, J. J. Middelburg, R. T. Noble, and C. A. Suttle, “Marine viruses and global climate change,” FEMS Microbiol. Rev. 35(6), 993–1034 (2011).
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Crocker, J. C.

J. C. Crocker and D. G. Grier, “Methods of digital video microscopy for colloidal studies,” J. Colloid Interface Sci. 179(1), 298–310 (1996).
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Culley, A. I.

G. F. Steward, A. I. Culley, J. A. Mueller, E. M. Wood-Charlson, M. Belcaid, and G. Poisson, “Are we missing half of the viruses in the ocean?” ISME J. 7(3), 672–679 (2013).
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Daaboul, G. G.

A. Yurt, G. G. Daaboul, J. H. Connor, B. B. Goldberg, and M. S. Ünlü, “Single nanoparticle detectors for biological applications,” Nanoscale 4(3), 715–726 (2012).
[Crossref] [PubMed]

C. A. Lopez, G. G. Daaboul, R. S. Vedula, E. Ozkumur, D. A. Bergstein, T. W. Geisbert, H. E. Fawcett, B. B. Goldberg, J. H. Connor, and M. S. Unlü, “Label-free multiplexed virus detection using spectral reflectance imaging,” Biosens. Bioelectron. 26(8), 3432–3437 (2011).
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Danovaro, R.

R. Danovaro, C. Corinaldesi, A. Dell’anno, J. A. Fuhrman, J. J. Middelburg, R. T. Noble, and C. A. Suttle, “Marine viruses and global climate change,” FEMS Microbiol. Rev. 35(6), 993–1034 (2011).
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de Vargas, C.

P. Bork, C. Bowler, C. de Vargas, G. Gorsky, E. Karsenti, and P. Wincker, “Tara Oceans. Tara Oceans studies plankton at planetary scale. Introduction,” Science 348(6237), 873 (2015).
[Crossref] [PubMed]

E. Karsenti, S. G. Acinas, P. Bork, C. Bowler, C. De Vargas, J. Raes, M. Sullivan, D. Arendt, F. Benzoni, J. M. Claverie, M. Follows, G. Gorsky, P. Hingamp, D. Iudicone, O. Jaillon, S. Kandels-Lewis, U. Krzic, F. Not, H. Ogata, S. Pesant, E. G. Reynaud, C. Sardet, M. E. Sieracki, S. Speich, D. Velayoudon, J. Weissenbach, P. Wincker, and Tara Oceans Consortium, “A holistic approach to marine eco-systems biology,” PLoS Biol. 9(10), e1001177 (2011).
[Crossref] [PubMed]

Dell’anno, A.

R. Danovaro, C. Corinaldesi, A. Dell’anno, J. A. Fuhrman, J. J. Middelburg, R. T. Noble, and C. A. Suttle, “Marine viruses and global climate change,” FEMS Microbiol. Rev. 35(6), 993–1034 (2011).
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Drapeau, D. T.

W. M. Balch, J. Vaughn, J. Novotny, D. T. Drapeau, R. Vaillancourt, J. Lapierre, and A. Ashe, “Light Scattering by Viral Suspensions,” Limnol. Oceanogr. 45(2), 492–498 (2000).
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Dubois, A.

Fawcett, H. E.

C. A. Lopez, G. G. Daaboul, R. S. Vedula, E. Ozkumur, D. A. Bergstein, T. W. Geisbert, H. E. Fawcett, B. B. Goldberg, J. H. Connor, and M. S. Unlü, “Label-free multiplexed virus detection using spectral reflectance imaging,” Biosens. Bioelectron. 26(8), 3432–3437 (2011).
[Crossref] [PubMed]

Filipe, V.

V. Filipe, A. Hawe, and W. Jiskoot, “Critical evaluation of Nanoparticle Tracking Analysis (NTA) by NanoSight for the measurement of nanoparticles and protein aggregates,” Pharm. Res. 27(5), 796–810 (2010).
[Crossref] [PubMed]

Follows, M.

E. Karsenti, S. G. Acinas, P. Bork, C. Bowler, C. De Vargas, J. Raes, M. Sullivan, D. Arendt, F. Benzoni, J. M. Claverie, M. Follows, G. Gorsky, P. Hingamp, D. Iudicone, O. Jaillon, S. Kandels-Lewis, U. Krzic, F. Not, H. Ogata, S. Pesant, E. G. Reynaud, C. Sardet, M. E. Sieracki, S. Speich, D. Velayoudon, J. Weissenbach, P. Wincker, and Tara Oceans Consortium, “A holistic approach to marine eco-systems biology,” PLoS Biol. 9(10), e1001177 (2011).
[Crossref] [PubMed]

Forterre, P.

P. Forterre, N. Soler, M. Krupovic, E. Marguet, and H. W. Ackermann, “Fake virus particles generated by fluorescence microscopy,” Trends Microbiol. 21(1), 1–5 (2013).
[Crossref] [PubMed]

Fuhrman, J. A.

R. Danovaro, C. Corinaldesi, A. Dell’anno, J. A. Fuhrman, J. J. Middelburg, R. T. Noble, and C. A. Suttle, “Marine viruses and global climate change,” FEMS Microbiol. Rev. 35(6), 993–1034 (2011).
[Crossref] [PubMed]

A. Patel, R. T. Noble, J. A. Steele, M. S. Schwalbach, I. Hewson, and J. A. Fuhrman, “Virus and prokaryote enumeration from planktonic aquatic environments by epifluorescence microscopy with SYBR Green I,” Nat. Protoc. 2(2), 269–276 (2007).
[Crossref] [PubMed]

Geisbert, T. W.

C. A. Lopez, G. G. Daaboul, R. S. Vedula, E. Ozkumur, D. A. Bergstein, T. W. Geisbert, H. E. Fawcett, B. B. Goldberg, J. H. Connor, and M. S. Unlü, “Label-free multiplexed virus detection using spectral reflectance imaging,” Biosens. Bioelectron. 26(8), 3432–3437 (2011).
[Crossref] [PubMed]

Goldberg, B. B.

A. Yurt, G. G. Daaboul, J. H. Connor, B. B. Goldberg, and M. S. Ünlü, “Single nanoparticle detectors for biological applications,” Nanoscale 4(3), 715–726 (2012).
[Crossref] [PubMed]

C. A. Lopez, G. G. Daaboul, R. S. Vedula, E. Ozkumur, D. A. Bergstein, T. W. Geisbert, H. E. Fawcett, B. B. Goldberg, J. H. Connor, and M. S. Unlü, “Label-free multiplexed virus detection using spectral reflectance imaging,” Biosens. Bioelectron. 26(8), 3432–3437 (2011).
[Crossref] [PubMed]

Gorsky, G.

P. Bork, C. Bowler, C. de Vargas, G. Gorsky, E. Karsenti, and P. Wincker, “Tara Oceans. Tara Oceans studies plankton at planetary scale. Introduction,” Science 348(6237), 873 (2015).
[Crossref] [PubMed]

E. Karsenti, S. G. Acinas, P. Bork, C. Bowler, C. De Vargas, J. Raes, M. Sullivan, D. Arendt, F. Benzoni, J. M. Claverie, M. Follows, G. Gorsky, P. Hingamp, D. Iudicone, O. Jaillon, S. Kandels-Lewis, U. Krzic, F. Not, H. Ogata, S. Pesant, E. G. Reynaud, C. Sardet, M. E. Sieracki, S. Speich, D. Velayoudon, J. Weissenbach, P. Wincker, and Tara Oceans Consortium, “A holistic approach to marine eco-systems biology,” PLoS Biol. 9(10), e1001177 (2011).
[Crossref] [PubMed]

Grier, D. G.

J. C. Crocker and D. G. Grier, “Methods of digital video microscopy for colloidal studies,” J. Colloid Interface Sci. 179(1), 298–310 (1996).
[Crossref]

Han, Y.

Y. Han, A. M. Alsayed, M. Nobili, J. Zhang, T. C. Lubensky, and A. G. Yodh, “Brownian motion of an ellipsoid,” Science 314(5799), 626–630 (2006).
[Crossref] [PubMed]

Hawe, A.

V. Filipe, A. Hawe, and W. Jiskoot, “Critical evaluation of Nanoparticle Tracking Analysis (NTA) by NanoSight for the measurement of nanoparticles and protein aggregates,” Pharm. Res. 27(5), 796–810 (2010).
[Crossref] [PubMed]

He, L.

L. He, S. K. Ozdemir, J. Zhu, W. Kim, and L. Yang, “Detecting single viruses and nanoparticles using whispering gallery microlasers,” Nat. Nanotechnol. 6(7), 428–432 (2011).
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Heidernätsch, M.

M. Heidernätsch, M. Bauer, and G. Radons, “Characterizing N-dimensional anisotropic Brownian motion by the distribution of diffusivities,” J. Chem. Phys. 139(18), 184105 (2013).
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Heldal, M.

C. P. D. Brussaard, S. W. Wilhelm, F. Thingstad, M. G. Weinbauer, G. Bratbak, M. Heldal, S. A. Kimmance, M. Middelboe, K. Nagasaki, J. H. Paul, D. C. Schroeder, C. A. Suttle, D. Vaqué, and K. E. Wommack, “Global-scale processes with a nanoscale drive: the role of marine viruses,” ISME J. 2(6), 575–578 (2008).
[Crossref] [PubMed]

Hewson, I.

A. Patel, R. T. Noble, J. A. Steele, M. S. Schwalbach, I. Hewson, and J. A. Fuhrman, “Virus and prokaryote enumeration from planktonic aquatic environments by epifluorescence microscopy with SYBR Green I,” Nat. Protoc. 2(2), 269–276 (2007).
[Crossref] [PubMed]

Hingamp, P.

E. Karsenti, S. G. Acinas, P. Bork, C. Bowler, C. De Vargas, J. Raes, M. Sullivan, D. Arendt, F. Benzoni, J. M. Claverie, M. Follows, G. Gorsky, P. Hingamp, D. Iudicone, O. Jaillon, S. Kandels-Lewis, U. Krzic, F. Not, H. Ogata, S. Pesant, E. G. Reynaud, C. Sardet, M. E. Sieracki, S. Speich, D. Velayoudon, J. Weissenbach, P. Wincker, and Tara Oceans Consortium, “A holistic approach to marine eco-systems biology,” PLoS Biol. 9(10), e1001177 (2011).
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Holmfeldt, K.

K. Holmfeldt, D. Odić, M. B. Sullivan, M. Middelboe, and L. Riemann, “Cultivated single-stranded DNA phages that infect marine Bacteroidetes prove difficult to detect with DNA-binding stains,” Appl. Environ. Microbiol. 78(3), 892–894 (2012).
[Crossref] [PubMed]

Huang, X.

S. Wang, X. Shan, U. Patel, X. Huang, J. Lu, J. Li, and N. Tao, “Label-free imaging, detection, and mass measurement of single viruses by surface plasmon resonance,” Proc. Natl. Acad. Sci. U.S.A. 107(37), 16028–16032 (2010).
[Crossref] [PubMed]

Hwang, J. S.

J. S. Hwang and W. E. Moerner, “Interferometry of a single nanoparticle using the Gouy phase of a focused laser beam,” Opt. Commun. 280(2), 487–491 (2007).
[Crossref]

Ignatovich, F.

A. Mitra, F. Ignatovich, and L. Novotny, “Real-time optical detection of single human and bacterial viruses based on dark-field interferometry,” Biosens. Bioelectron. 31(1), 499–504 (2012).
[Crossref] [PubMed]

Iudicone, D.

E. Karsenti, S. G. Acinas, P. Bork, C. Bowler, C. De Vargas, J. Raes, M. Sullivan, D. Arendt, F. Benzoni, J. M. Claverie, M. Follows, G. Gorsky, P. Hingamp, D. Iudicone, O. Jaillon, S. Kandels-Lewis, U. Krzic, F. Not, H. Ogata, S. Pesant, E. G. Reynaud, C. Sardet, M. E. Sieracki, S. Speich, D. Velayoudon, J. Weissenbach, P. Wincker, and Tara Oceans Consortium, “A holistic approach to marine eco-systems biology,” PLoS Biol. 9(10), e1001177 (2011).
[Crossref] [PubMed]

Jaillon, O.

E. Karsenti, S. G. Acinas, P. Bork, C. Bowler, C. De Vargas, J. Raes, M. Sullivan, D. Arendt, F. Benzoni, J. M. Claverie, M. Follows, G. Gorsky, P. Hingamp, D. Iudicone, O. Jaillon, S. Kandels-Lewis, U. Krzic, F. Not, H. Ogata, S. Pesant, E. G. Reynaud, C. Sardet, M. E. Sieracki, S. Speich, D. Velayoudon, J. Weissenbach, P. Wincker, and Tara Oceans Consortium, “A holistic approach to marine eco-systems biology,” PLoS Biol. 9(10), e1001177 (2011).
[Crossref] [PubMed]

Jiskoot, W.

V. Filipe, A. Hawe, and W. Jiskoot, “Critical evaluation of Nanoparticle Tracking Analysis (NTA) by NanoSight for the measurement of nanoparticles and protein aggregates,” Pharm. Res. 27(5), 796–810 (2010).
[Crossref] [PubMed]

Kandels-Lewis, S.

E. Karsenti, S. G. Acinas, P. Bork, C. Bowler, C. De Vargas, J. Raes, M. Sullivan, D. Arendt, F. Benzoni, J. M. Claverie, M. Follows, G. Gorsky, P. Hingamp, D. Iudicone, O. Jaillon, S. Kandels-Lewis, U. Krzic, F. Not, H. Ogata, S. Pesant, E. G. Reynaud, C. Sardet, M. E. Sieracki, S. Speich, D. Velayoudon, J. Weissenbach, P. Wincker, and Tara Oceans Consortium, “A holistic approach to marine eco-systems biology,” PLoS Biol. 9(10), e1001177 (2011).
[Crossref] [PubMed]

Karsenti, E.

P. Bork, C. Bowler, C. de Vargas, G. Gorsky, E. Karsenti, and P. Wincker, “Tara Oceans. Tara Oceans studies plankton at planetary scale. Introduction,” Science 348(6237), 873 (2015).
[Crossref] [PubMed]

E. Karsenti, S. G. Acinas, P. Bork, C. Bowler, C. De Vargas, J. Raes, M. Sullivan, D. Arendt, F. Benzoni, J. M. Claverie, M. Follows, G. Gorsky, P. Hingamp, D. Iudicone, O. Jaillon, S. Kandels-Lewis, U. Krzic, F. Not, H. Ogata, S. Pesant, E. G. Reynaud, C. Sardet, M. E. Sieracki, S. Speich, D. Velayoudon, J. Weissenbach, P. Wincker, and Tara Oceans Consortium, “A holistic approach to marine eco-systems biology,” PLoS Biol. 9(10), e1001177 (2011).
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Keng, D.

F. Vollmer, S. Arnold, and D. Keng, “Single virus detection from the reactive shift of a whispering-gallery mode,” Proc. Natl. Acad. Sci. U.S.A. 105(52), 20701–20704 (2008).
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Kim, W.

L. He, S. K. Ozdemir, J. Zhu, W. Kim, and L. Yang, “Detecting single viruses and nanoparticles using whispering gallery microlasers,” Nat. Nanotechnol. 6(7), 428–432 (2011).
[Crossref] [PubMed]

Kimmance, S. A.

C. P. D. Brussaard, S. W. Wilhelm, F. Thingstad, M. G. Weinbauer, G. Bratbak, M. Heldal, S. A. Kimmance, M. Middelboe, K. Nagasaki, J. H. Paul, D. C. Schroeder, C. A. Suttle, D. Vaqué, and K. E. Wommack, “Global-scale processes with a nanoscale drive: the role of marine viruses,” ISME J. 2(6), 575–578 (2008).
[Crossref] [PubMed]

Krupovic, M.

P. Forterre, N. Soler, M. Krupovic, E. Marguet, and H. W. Ackermann, “Fake virus particles generated by fluorescence microscopy,” Trends Microbiol. 21(1), 1–5 (2013).
[Crossref] [PubMed]

Krzic, U.

E. Karsenti, S. G. Acinas, P. Bork, C. Bowler, C. De Vargas, J. Raes, M. Sullivan, D. Arendt, F. Benzoni, J. M. Claverie, M. Follows, G. Gorsky, P. Hingamp, D. Iudicone, O. Jaillon, S. Kandels-Lewis, U. Krzic, F. Not, H. Ogata, S. Pesant, E. G. Reynaud, C. Sardet, M. E. Sieracki, S. Speich, D. Velayoudon, J. Weissenbach, P. Wincker, and Tara Oceans Consortium, “A holistic approach to marine eco-systems biology,” PLoS Biol. 9(10), e1001177 (2011).
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Kuehn, M. J.

A. Kulp and M. J. Kuehn, “Biological functions and biogenesis of secreted bacterial outer membrane vesicles,” Annu. Rev. Microbiol. 64(1), 163–184 (2010).
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Kulp, A.

A. Kulp and M. J. Kuehn, “Biological functions and biogenesis of secreted bacterial outer membrane vesicles,” Annu. Rev. Microbiol. 64(1), 163–184 (2010).
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Lapierre, J.

W. M. Balch, J. Vaughn, J. Novotny, D. T. Drapeau, R. Vaillancourt, J. Lapierre, and A. Ashe, “Light Scattering by Viral Suspensions,” Limnol. Oceanogr. 45(2), 492–498 (2000).
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Laveran, H.

Y. Bettarel, T. Sime-Ngando, C. Amblard, and H. Laveran, “A comparison of methods for counting viruses in aquatic systems,” Appl. Environ. Microbiol. 66(6), 2283–2289 (2000).
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Li, J.

S. Wang, X. Shan, U. Patel, X. Huang, J. Lu, J. Li, and N. Tao, “Label-free imaging, detection, and mass measurement of single viruses by surface plasmon resonance,” Proc. Natl. Acad. Sci. U.S.A. 107(37), 16028–16032 (2010).
[Crossref] [PubMed]

Lopez, C. A.

C. A. Lopez, G. G. Daaboul, R. S. Vedula, E. Ozkumur, D. A. Bergstein, T. W. Geisbert, H. E. Fawcett, B. B. Goldberg, J. H. Connor, and M. S. Unlü, “Label-free multiplexed virus detection using spectral reflectance imaging,” Biosens. Bioelectron. 26(8), 3432–3437 (2011).
[Crossref] [PubMed]

Lu, J.

S. Wang, X. Shan, U. Patel, X. Huang, J. Lu, J. Li, and N. Tao, “Label-free imaging, detection, and mass measurement of single viruses by surface plasmon resonance,” Proc. Natl. Acad. Sci. U.S.A. 107(37), 16028–16032 (2010).
[Crossref] [PubMed]

Lubensky, T. C.

Y. Han, A. M. Alsayed, M. Nobili, J. Zhang, T. C. Lubensky, and A. G. Yodh, “Brownian motion of an ellipsoid,” Science 314(5799), 626–630 (2006).
[Crossref] [PubMed]

Marguet, E.

P. Forterre, N. Soler, M. Krupovic, E. Marguet, and H. W. Ackermann, “Fake virus particles generated by fluorescence microscopy,” Trends Microbiol. 21(1), 1–5 (2013).
[Crossref] [PubMed]

Middelboe, M.

K. Holmfeldt, D. Odić, M. B. Sullivan, M. Middelboe, and L. Riemann, “Cultivated single-stranded DNA phages that infect marine Bacteroidetes prove difficult to detect with DNA-binding stains,” Appl. Environ. Microbiol. 78(3), 892–894 (2012).
[Crossref] [PubMed]

C. P. D. Brussaard, S. W. Wilhelm, F. Thingstad, M. G. Weinbauer, G. Bratbak, M. Heldal, S. A. Kimmance, M. Middelboe, K. Nagasaki, J. H. Paul, D. C. Schroeder, C. A. Suttle, D. Vaqué, and K. E. Wommack, “Global-scale processes with a nanoscale drive: the role of marine viruses,” ISME J. 2(6), 575–578 (2008).
[Crossref] [PubMed]

Middelburg, J. J.

R. Danovaro, C. Corinaldesi, A. Dell’anno, J. A. Fuhrman, J. J. Middelburg, R. T. Noble, and C. A. Suttle, “Marine viruses and global climate change,” FEMS Microbiol. Rev. 35(6), 993–1034 (2011).
[Crossref] [PubMed]

Mitra, A.

A. Mitra, F. Ignatovich, and L. Novotny, “Real-time optical detection of single human and bacterial viruses based on dark-field interferometry,” Biosens. Bioelectron. 31(1), 499–504 (2012).
[Crossref] [PubMed]

Moerner, W. E.

J. S. Hwang and W. E. Moerner, “Interferometry of a single nanoparticle using the Gouy phase of a focused laser beam,” Opt. Commun. 280(2), 487–491 (2007).
[Crossref]

Mueller, J. A.

G. F. Steward, A. I. Culley, J. A. Mueller, E. M. Wood-Charlson, M. Belcaid, and G. Poisson, “Are we missing half of the viruses in the ocean?” ISME J. 7(3), 672–679 (2013).
[Crossref] [PubMed]

Nagasaki, K.

C. P. D. Brussaard, S. W. Wilhelm, F. Thingstad, M. G. Weinbauer, G. Bratbak, M. Heldal, S. A. Kimmance, M. Middelboe, K. Nagasaki, J. H. Paul, D. C. Schroeder, C. A. Suttle, D. Vaqué, and K. E. Wommack, “Global-scale processes with a nanoscale drive: the role of marine viruses,” ISME J. 2(6), 575–578 (2008).
[Crossref] [PubMed]

Nobili, M.

Y. Han, A. M. Alsayed, M. Nobili, J. Zhang, T. C. Lubensky, and A. G. Yodh, “Brownian motion of an ellipsoid,” Science 314(5799), 626–630 (2006).
[Crossref] [PubMed]

Noble, R. T.

R. Danovaro, C. Corinaldesi, A. Dell’anno, J. A. Fuhrman, J. J. Middelburg, R. T. Noble, and C. A. Suttle, “Marine viruses and global climate change,” FEMS Microbiol. Rev. 35(6), 993–1034 (2011).
[Crossref] [PubMed]

A. Patel, R. T. Noble, J. A. Steele, M. S. Schwalbach, I. Hewson, and J. A. Fuhrman, “Virus and prokaryote enumeration from planktonic aquatic environments by epifluorescence microscopy with SYBR Green I,” Nat. Protoc. 2(2), 269–276 (2007).
[Crossref] [PubMed]

Not, F.

E. Karsenti, S. G. Acinas, P. Bork, C. Bowler, C. De Vargas, J. Raes, M. Sullivan, D. Arendt, F. Benzoni, J. M. Claverie, M. Follows, G. Gorsky, P. Hingamp, D. Iudicone, O. Jaillon, S. Kandels-Lewis, U. Krzic, F. Not, H. Ogata, S. Pesant, E. G. Reynaud, C. Sardet, M. E. Sieracki, S. Speich, D. Velayoudon, J. Weissenbach, P. Wincker, and Tara Oceans Consortium, “A holistic approach to marine eco-systems biology,” PLoS Biol. 9(10), e1001177 (2011).
[Crossref] [PubMed]

Novotny, J.

W. M. Balch, J. Vaughn, J. Novotny, D. T. Drapeau, R. Vaillancourt, J. Lapierre, and A. Ashe, “Light Scattering by Viral Suspensions,” Limnol. Oceanogr. 45(2), 492–498 (2000).
[Crossref]

Novotny, L.

A. Mitra, F. Ignatovich, and L. Novotny, “Real-time optical detection of single human and bacterial viruses based on dark-field interferometry,” Biosens. Bioelectron. 31(1), 499–504 (2012).
[Crossref] [PubMed]

Odic, D.

K. Holmfeldt, D. Odić, M. B. Sullivan, M. Middelboe, and L. Riemann, “Cultivated single-stranded DNA phages that infect marine Bacteroidetes prove difficult to detect with DNA-binding stains,” Appl. Environ. Microbiol. 78(3), 892–894 (2012).
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Ogata, H.

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K. Wen, A. C. Ortmann, and C. A. Suttle, “Accurate Estimation of Viral Abundance by Epifluorescence Microscopy,” Appl. Environ. Microbiol. 70(7), 3862–3867 (2004).
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L. He, S. K. Ozdemir, J. Zhu, W. Kim, and L. Yang, “Detecting single viruses and nanoparticles using whispering gallery microlasers,” Nat. Nanotechnol. 6(7), 428–432 (2011).
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C. A. Lopez, G. G. Daaboul, R. S. Vedula, E. Ozkumur, D. A. Bergstein, T. W. Geisbert, H. E. Fawcett, B. B. Goldberg, J. H. Connor, and M. S. Unlü, “Label-free multiplexed virus detection using spectral reflectance imaging,” Biosens. Bioelectron. 26(8), 3432–3437 (2011).
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A. Patel, R. T. Noble, J. A. Steele, M. S. Schwalbach, I. Hewson, and J. A. Fuhrman, “Virus and prokaryote enumeration from planktonic aquatic environments by epifluorescence microscopy with SYBR Green I,” Nat. Protoc. 2(2), 269–276 (2007).
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S. Wang, X. Shan, U. Patel, X. Huang, J. Lu, J. Li, and N. Tao, “Label-free imaging, detection, and mass measurement of single viruses by surface plasmon resonance,” Proc. Natl. Acad. Sci. U.S.A. 107(37), 16028–16032 (2010).
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J. P. Payet and C. A. Suttle, “To kill or not to kill: the balance between lytic and lysogenic viral infection is driven by trophic status,” Limnol. Oceanogr. 58(2), 465–474 (2013).
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E. Karsenti, S. G. Acinas, P. Bork, C. Bowler, C. De Vargas, J. Raes, M. Sullivan, D. Arendt, F. Benzoni, J. M. Claverie, M. Follows, G. Gorsky, P. Hingamp, D. Iudicone, O. Jaillon, S. Kandels-Lewis, U. Krzic, F. Not, H. Ogata, S. Pesant, E. G. Reynaud, C. Sardet, M. E. Sieracki, S. Speich, D. Velayoudon, J. Weissenbach, P. Wincker, and Tara Oceans Consortium, “A holistic approach to marine eco-systems biology,” PLoS Biol. 9(10), e1001177 (2011).
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E. Karsenti, S. G. Acinas, P. Bork, C. Bowler, C. De Vargas, J. Raes, M. Sullivan, D. Arendt, F. Benzoni, J. M. Claverie, M. Follows, G. Gorsky, P. Hingamp, D. Iudicone, O. Jaillon, S. Kandels-Lewis, U. Krzic, F. Not, H. Ogata, S. Pesant, E. G. Reynaud, C. Sardet, M. E. Sieracki, S. Speich, D. Velayoudon, J. Weissenbach, P. Wincker, and Tara Oceans Consortium, “A holistic approach to marine eco-systems biology,” PLoS Biol. 9(10), e1001177 (2011).
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S. J. Biller, F. Schubotz, S. E. Roggensack, A. W. Thompson, R. E. Summons, and S. W. Chisholm, “Bacterial vesicles in marine ecosystems,” Science 343(6167), 183–186 (2014).
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Appl. Opt. (1)

Biosens. Bioelectron. (2)

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FEMS Microbiol. Rev. (1)

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Figures (4)

Fig. 1
Fig. 1

Principle of the interferometry method.A) Experimental set up: The sample to be analyzed fills the micro-cuvette (in white) bounded by coverslips and located at the focus of the oil immersion objective. A bare LED illuminates the cuvette. The common path interferometer uses the incoherent flux of a LED for a light source and a single microscope objective whose focal plane is imaged on the camera CMOS chip. B) Principle of the common path detection. The sample is illuminated by an incoherent source, so the spatial coherence expands over a diffraction spot whose size is linked to the numerical aperture of the microscope objective. The illuminating wave, restricted to this diffraction spot, is able to interfere with the light scattered by the particles that are located close (within the depth of field) to the objective focus F. Due to the Gouy phase shift the interference is constructive or destructive for a particle located slightly after (blue dot and dotted line) or before (red dot and dotted line) the focus. C) Calculated signal amplitude (in arbitrary units) as a function of the distance of the scattering particle to the focus F of the microscope objective (red crosses). The amplitude of the incoming field is the blue dotted line. At the focus the scattered particle field is in quadrature with the incoming wave so the interference signal is zero (black dot and dotted line).

Fig. 2
Fig. 2

Signal Processing.For a given spot the intensity of the scattered signal was measured after subtracting from the raw image the average of the stack of 200 images (before filtering, red curve). The normalized convolution then reduces the noise without affecting the signal (after filtering, blue curve). On the left images before and after filtering

Fig. 3
Fig. 3

Calibration with 50 nm silica beads. (A) Histogram of scattered signals observed with beads (concentration: 1010 particles/mL). (B) Histogram of particle sizes. Note that the uncertainly corresponds to about 7%, while it is about 30% in A) because the scatterd amplitude is proportional to the third power of the diameter. (C) Correlation between the number of particles measured and dilutions. (D) Beads diameters measured through the scattering level as function of their diameters measured through the Brownian motion (N = 606). As expected (see text) the dispersion of the measurements is much larger in this later case. We considered particles that experienced at least 10 measured jumps. The colored bar corresponds to the number of particles per unit area.

Fig. 4
Fig. 4

Analysis of membrane vesicles and Tara-oceans samples.A) Plot of diameters (N = 4,788) computed with scattered signals function of diameters computed by Brownian motion of membrane vesicles secreted by P. tricornutum. The line of slope 1 corresponds to a refractive index of 1.5. The colored bar corresponds to the number of particles per unit area. B) Analysis of Tara Oceans samples from the Dalmatian coast (Station 23) Plot of the particle diameters (N = 4,247) computed with their scattered signals and Brownian motion. Lines indicate the refractive indexes of the different particles (viruses and vesicles). The colored bar corresponds to the number of particles per unit area (for A and B). We intentionally threshold the DiameterBM to 20nm, corresponding to our detection noise. C) Analysis of Tara Oceans sample from an oligotrophic open ocean region (Station 98). Plot of the particle diameters (N = 8,136) computed with their scattered signals and Brownian motion. Lines indicate the refractive indexes of the two types of particles, small viruses and vesicles. D): Electron micrograph of viral fraction from Tara Oceans station 98 (concentrated 200 times) showing abundance of small particles. Scale bar: 100 nm.

Tables (2)

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Table 1 Morphological properties of known viruses and sizes determination by interferometry

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Table 2 Coefficient of anisotropy for Brownian trajectories (computed with trajectories of at least 10 jumps)

Equations (1)

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< cos ( θ 1 + θ 2 ) > = < cos ( θ 1 ) > < cos ( θ 2 ) > < sin ( θ 1 ) > < sin ( θ 2 ) > = < cos ( θ ) > 2 < sin ( θ ) > 2 = g 2

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