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A Laplacian characterization of phytoplankton shape

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Phytoplankton exhibit pronounced morphological diversity, impacting a range of processes. Because these impacts are challenging to quantify, however, phytoplankton are often approximated as spheres, and when effects of non-sphericity are studied it is usually experimentally or via geometrical approximations. New methods for quantifying phytoplankton size and shape generally, so all phytoplankton are analyzable by the same procedure, can complement advances in microscopic imagery and automated classification to study the influence of shape in phytoplankton. Here we apply to phytoplankton a technique for defining the size of arbitrary shapes based on the Laplacian—the operator that governs processes, such as nutrient uptake and fluid flow, where phytoplankton shape is expected to have the greatest effect. Deviations from values given by spherical approximation are a measure of phytoplankton shape and indicate the fitness increases for phytoplankton conferred by their non-spherical shapes. Comparison with surface-to-volume quotients suggests the Laplacian-based metric is insensitive to small-scale features which can increase surface area without affecting key processes, but is otherwise closely related to surface-area-to-volume, demonstrating this metric is a meaningful measure. While our analysis herein is limited to axisymmetric phytoplankton due to relative sparsity of 3D information about other phytoplankton shapes, the definition and method are directly generalizable to 3D shape data, which will in the near future be more readily available.

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  • Culverhouse PF, Herry V, Reguera B, Gonzalez-Gil S, Williams R, Fonda S, Cabrini M, Parisini T, Ellis R (2001) Dinoflagellate categorisation by artificial neural network (DiCANN). In: Hallegraeff GM, Blackburn SI, Bolck CJ, Lewis RJ (eds) Harmful algal blooms. Intergovernmental Oceanographic Commission of UNESCO, Vigo, pp 195–198

    Google Scholar 

  • Culverhouse PF, Williams R, Benfield M, Flood PR, Sell AF, Mazzocchi MG, Buttino I, Sieracki M (2006) Automatic image analysis of plankton: future perspectives. Mar Ecol Prog Ser 312:297–309

    Article  Google Scholar 

  • Cussler EL (2009) Diffusion: mass transfer in fluid systems. Cambridge University Press, Cambridge

    Book  Google Scholar 

  • Estep K, MacIntyre F (1989) Counting, sizing, and identification of algae using image analysis. Sarsia 74(4):261–268

    Article  Google Scholar 

  • Evans LC (2010) Partial differential equations, 2nd edn. American Mathematical Society, Providence

    MATH  Google Scholar 

  • Field CB, Behrenfeld MJ, Randerson JT, Falkowski P (1998) Primary production of the biosphere: integrating terrestrial and oceanic components. Science 281(5374):237–240

    Article  Google Scholar 

  • Gran HH (1912) Pelagic plant life. In: Murray J, Hjort J (eds) Depths of the ocean. Macmillan, London, pp 307–386

    Google Scholar 

  • Hense BA, Gais P, Jütting U, Scherb H, Rodenacker K (2008) Use of fluorsecence information for automated phytoplankton investigation by image analysis. J Plankton Res 30(5):587–606

    Article  Google Scholar 

  • Hillebrand H, Dürselen CD, Kirschtel D, Pollingher U, Zohary T (1999) Biovolume calculation for pelagic and benthic microalgae. J Phycol 35(2):403–424

    Article  Google Scholar 

  • Horiuchi T, Akiba T, Kakui Y (2004) Development of a continuous imaging system equipped with fluorescent imaging for classification of phytoplankton. In: MTTS/IEEE TECHNO-OCEAN ’04, vol 3, pp 1410–1413

  • Jennings BR, Parslow K (1988) Particle size measurement: the equivalent spherical diameter. Proc R Soc Lond A Math Phys Eng Sci 419(1856):137–149

    Article  Google Scholar 

  • Jones SE, Buchbinder BR, Aharon I (2000) Three-dimensional mapping of cortical thickness using Laplaces equation. Hum Brain Mapp 11:12–32

    Article  Google Scholar 

  • Kang L, Yang C, Gao Y (2009) Improved shape description using radon transform and application in phytoplankton identification. In: 2nd IEEE international conference on broadband network & multimedia technology, pp 477–481

  • Karp-Boss L, Boss E (2016) The elongated, the squat and the spherical: selective pressures for phytoplankton shape. In: Gilbert PM, Kana TM (eds) Aquatic microbial ecology and biogeochemistry: a dual perspective. Springer International Publishing, Cham, Switzerland, pp 25–34

  • Karp-Boss L, Boss E, Jumars PA (1996) Nutrient fluxes to planktonic osmotrophs in the presence of fluid motion. Oceanogr Mar Biol 34:71–108

    Google Scholar 

  • Lavoie M, Levasseur M, Babin M (2015) Testing the potential ballast role for dimethylsulfoniopropionate in marine phytoplankton: a modeling study. J Plankton Res 37(4):699–711

    Article  Google Scholar 

  • Lewis WM (1976) Surface/volume ratio: implications for phytoplankton morphology. Science 192:885–887

    Article  Google Scholar 

  • McKown JS, Malaika J (1950) Effect of particle shape on settling velocity at low Reynols numbers. Trans Am Geophys Union 31:74–82

    Article  Google Scholar 

  • Moberg E, Sosik H (2012) Distance maps to estimate cell volume from two-dimensional plankton images. Limnol Oceanogr Methods 10:278–288

    Article  Google Scholar 

  • Naselli-Flores L, Padisák J, Albay M (2007) Shape and size in phytoplankton ecology: do they matter? Hydrobiologia 578(1):157–161

    Article  Google Scholar 

  • Nguyen HV, Karp-Boss L, Jumars PA, Fauci L (2011) Hydrodynamic effects of spines: a different spin. Limnol Oceanogr Fluids Environ 1:110–119

    Article  Google Scholar 

  • Olson RJ, Sosik HM (2007) A submersible imaging-in-flow instrument to analyze nano-and microplankton: Imaging FlowCytobot. Limnol Oceanogr Methods 5(6):195–203

    Article  Google Scholar 

  • Padisák J, Soróczki-Pintér É, Rezner Z (2003) Sinking properties of some phytoplankton shapes and the relation of form resistance to morphological diversity of plankton—an experimental study. Aquat Biodivers 500:243–257

    Article  Google Scholar 

  • Reynolds CS (2006) The ecology of phytoplankton. Cambridge University Press, Cambridge

    Book  Google Scholar 

  • Roland C, Grace JR, Weber ME (2005) Bubbles, drops, and particles. Courier 375 Corporation

  • Rodenacker K, Hense B, Gais P (2006) Automatic analysis of aqueous specimens for phytoplankton structure recognition and population estimation. Microsc Res Tech 69(9):708–720

    Article  Google Scholar 

  • Roselli L, Paparella F, Stanca E, Basset A (2015) New datadriven method from 3D confocal microscopy for calculating phytoplankton cell biovolume. J Microsc 258(3):200–211

    Article  Google Scholar 

  • Sardet C (2015) Plankton: wonders of the drifting world. University of Chicago Press, Chicago

    Book  Google Scholar 

  • Sommer U (1998) Silicate and the functional geometry of marine phytoplankton. J Plankton Res 20(9):1853–1859

    Article  Google Scholar 

  • Sosik HM, Olson RJ (2007) Automated taxonomic classification of phytoplankton sampled with imaging-in-flow cytometry. Limnol Oceanogr Methods 5(6):204–216

    Article  Google Scholar 

  • Sosik HM, Peacock EE, Brownlee EF (2015) WHOI-Plankton, annotated plankton images - data set for developing and evaluating classification methods. doi:10.1575/1912/7341

  • Strong C (2012) Atmospheric influence on Arctic marginal ice zone position and width in the Atlantic sector, February–April 1979–2010. Clim Dyn 39(12):3091–3102

    Article  Google Scholar 

  • Tett P, Barton ED (1995) Why are there about 5000 species of phytoplankton in the sea? J Plankton Res 17(8):1693–1704

    Article  Google Scholar 

  • Visser AW, Jonsson PR (2000) On the reorientation of non-spherical prey particles in a feeding current. J Plankton Res 22(4):761–777

    Article  Google Scholar 

  • Vogel S (1996) Life in moving fluids: the physical biology of flow. Princeton University Press, Princeton

    Google Scholar 

  • Walsby AE, Holland DP (2006) Sinking velocities of phytoplankton measured on a stable density gradient by laser scanning. J R Soc Interface 3:429–439

    Article  Google Scholar 

  • Young KD (2006) The selective value of bacterial shape. Microbiol Mol Biol Rev 70(3):660–703

    Article  Google Scholar 

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It is a pleasure to thank Heidi Sosik, Lee Karp-Boss, and Emmanuel Boss for invaluable feedback. This research was primarily funded through National Science Foundation Awards EPS-1208732 and OCE-1315201. This study was also supported by National Science Foundation Graduate Research Fellowship Program (Grant No. 2388357).

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Correspondence to B. B. Cael.

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Cael, B.B., Strong, C. A Laplacian characterization of phytoplankton shape. J. Math. Biol. 76, 1327–1338 (2018).

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