Journal of Biological Physics

, Volume 34, Issue 1–2, pp 19–37 | Cite as

Individual-based Modelling: An Essential Tool for Microbiology

Review

Abstract

Micro-organisms play a central role in every ecosystem and in the global biomass cycle. They are strongly involved in many fields of human interest, from medicine to the food industry and waste control. Nevertheless, most micro-organisms remain almost unknown, and nearly 99% of them have not yet been successfully cultured in vitro. Therefore, new approaches and new tools must be developed in order to understand the collective behaviour of microbial communities in any natural or artificial setting. In particular, theoretical and practical methodologies to deal with such systems at a mesoscopic level of description (covering the range from 100 to 108 cells) are required. Individual-based modelling (IBM) has become a widely used tool for describing complex systems made up of autonomous entities, such as ecosystems and social networks. Individual-based models (IBMs) provide some advantages over the traditional whole-population models: (a) they are bottom-up approaches, so they describe the behaviour of a system as a whole by establishing procedural rules for the individuals and for their interactions, and thus allow more realistic assumptions for the model of the individuals than population models do; (b) they permit the introduction of randomness and individual variability, so they can reproduce the diversity found in real systems; and (c) they can account for individual adaptive behaviour to their environmental conditions, so the evolution of the whole system arises from the dynamics that govern individuals in their pursuit of optimal fitness. However, they also present some drawbacks: they lack the clarity of continuous models and may easily become rambling, which makes them difficult to analyse and communicate. All in all, IBMs supply a holistic description of microbial systems and their emerging properties. They are specifically appropriate to deal with microbial communities in non-steady states, and spatially explicit IBMs are particularly appropriate to study laboratory and natural microbiological systems with spatial heterogeneity. In this paper, we review IBM methodology applied to microbiology. We also present some results obtained from the application of Individual Discrete Simulations, an IBM of ours, to the study of bacterial communities, yeast cultures and Plasmodium falciparum-infected erythrocytes in vitro cultures of Plasmodium falciparum-infected erythrocytes.

Keywords

Individual-based Integrative microbiology Spatial heterogeneity Complexity Bacterial lag Microbial community Molecular dynamics Cellular automata Simulation 

References

  1. 1.
    Whitman, W.B., Coleman, D.C., Wiebe, W.J.: Prokaryotes: the unseen majority. Proc. Nat. Acad. Sci. U. S. A. 95, 6578–6583 (1998)CrossRefADSGoogle Scholar
  2. 2.
    Horner-Devine, M.C., Carney, K.M., Bohannan, J.M.: An ecological perspective on bacterial biodiversity. Proc. R. Soc. Lond. B 271, 113–122 (2003)CrossRefGoogle Scholar
  3. 3.
    Maloy, S., Schaechter, M.: The era of microbiology: a golden phoenix. Int. Microbiol. 9(1), 1–7 (2006)Google Scholar
  4. 4.
    Sharma, R., Ranjan, R., Kapardar, R.K., Grover, A.: Unculturable bacterial diversity: an untapped resource. Curr. Sci. Ind. 89(1), 72–76 (2005)Google Scholar
  5. 5.
    Grimm, V., Railsback, S.F.: Individual-based Modeling and Ecology. Princeton University Press, Princeton (2005)MATHGoogle Scholar
  6. 6.
    van Gunsteren, W.F., Bakowies, D., Baron, R., Chandrasekhar, I., Christen, M., Daura, X., Gee, P., Geerke, D.P., Glattli, A., Hunenberger, P.H., Kastenholz, M.A., Ostenbrink, C., Schenk, M., Trzesniak, D., van der Vegt, N.F.A., Yu, H.B.: Molecular dynamics: survey of methods for simulating the activity of proteins. Chem. Rev. 106, 1589–1615 (2006). doi:10.1002/anie.200502655 CrossRefGoogle Scholar
  7. 7.
    Chekmarev, S.F., Palyanov, A.Y., Karplus, M.: Hydrodynamic description of protein folding. Phys. Rev. Let. 100, 018107 (2008). doi:10.1103/PhysRevLett.100.018107 Google Scholar
  8. 8.
    Dodson, G.G., Lane, D.P., Verma, C.S.: Molecular simulations of protein dynamics: new windows on mechanisms in biology. EMBO Rep. 9(2), 144–150 (2008). doi:10.1038/sj.embor.7401160 CrossRefGoogle Scholar
  9. 9.
    Adcock, S.A., McCammon, J.A.: Molecular dynamics: survey of methods for simulating the activity of proteins. Chem. Rev. 106, 1589–1615 (2006)CrossRefGoogle Scholar
  10. 10.
    Jones, D.T., Sternberg, M.J.E., Thornton, J.M.: Introduction. Bioinformatics: from molecules to systems. Philos. Trans. R. Soc. Lond. B 361, 389–391 (2006). doi:10.1098/rstb.2005.1811 CrossRefGoogle Scholar
  11. 11.
    Jou, D.: Introducció a la termodinàmica de processos biològics. Edicions IEC, Barcelona (1985)Google Scholar
  12. 12.
    Schuster, P.: Modeling in biological chemistry. From biochemical kinetics to systems biology. Monatsh. Chem. 139, 427–446 (2008)CrossRefGoogle Scholar
  13. 13.
    Mashego, M.R., Rumbold, K., De Mey, M., Vandamme, E., Soetaert, W., Heijnen, J.J.: Microbial metabolomics: past, present and future methodologies. Biotechnol. Lett. 29, 1–16 (2007). doi:10.1007/s10529-006-9218-0 CrossRefGoogle Scholar
  14. 14.
    Ashburner, M., Ball, C.A., Blake, J.A., Botstein, D., Butler, H., Cherry, J.M., Davis, A.P., Dolinski, K., Dwight, S.S., Eppig, J.T., Harris, M.A., Hill, D.P., Issel-Tarver, L., Kasarskis, A., Lewis, S., Matese, J.C., Richardson, J.E., Ringwald, M., Rubin, G.M., Sherlock, G.: Gene ontology: tool for the unification of biology. Nat. Genet. 25, 25–29 (2000)CrossRefGoogle Scholar
  15. 15.
    Ishii, N., Robert, M., Nakayama, Y., Kanai, A., Tomita, M.: Towards large scale modeling of the microbial cell for computer simulation. J. Biotechnol. 113, 281–294 (2004)CrossRefGoogle Scholar
  16. 16.
    Palsson, B.O., Covert, M.W., Famili, I.: Identifying constraints that govern cell behavior: a key to converting conceptual to computational models in biology? Biotechnol. Bioeng. 84(7), 763–772 (2007)Google Scholar
  17. 17.
    Tomita, M., Hashimoto, K., Takahashi, K., Shimizu, T.S., Matsuzaki, Y., Miyoshi, F., Saito, K., Tanida, S., Yugi, K., Venter, J.C., Hutchison, C.A.: E-CELL: software environment for whole-cell simulation. Bioinformatics 15, 72–84 (1999)CrossRefGoogle Scholar
  18. 18.
    Gillespie, D.T.: Exact stochastic simulation of coupled chemical reactions. J. Phys. Chem. 81, 2340 (1977)CrossRefGoogle Scholar
  19. 19.
    Gibson, M.A., Bruck, J.: Efficient exact stochastic simulation of chemical systems with many species and many channels. J. Phys. Chem. A 104, 1876–1889 (2000)CrossRefGoogle Scholar
  20. 20.
    Swinnen, I.A.M., Bernaerts, K., Dens, E.J.J., Geeraerd, A.H., Van Impe, J.F.: Predictive modelling of the microbial lag phase: a review. Int. J. Food Microbiol. 94, 137–159 (2004)CrossRefGoogle Scholar
  21. 21.
    Raaijmakers, J.M., Vlami, M., Souza, J.T.: Antibiotic production by bacterial biocontrol agents. Antonie van Leeuwenhoek 81, 537–547 (2002)CrossRefGoogle Scholar
  22. 22.
    Seviour, R.J., Mino, T., Onuki, M.: The microbiology of biological phosphorus removal in activated sludge systems. FEMS Microbiol. Rev. 27, 99–127 (2003)CrossRefGoogle Scholar
  23. 23.
    Wood, A.P., Aurikko, J.P., Kelly, D.P.: A challenge for 21st century molecular biology and biochemistry: what are the causes of obligate autotrophy and methanotrophy? FEMS Microbiol. Rev. 28, 335–352 (2004)CrossRefGoogle Scholar
  24. 24.
    Muylaert, K., Van der Gucht, K., Vloemans, N., De Meester, L., Gillis, M., Vyverman, W.: Relationship between bacterial community composition and bottom-up versus top-down variables in four eutrophic shallow lakes. Appl. Environ. Microb. 68(10), 4740–4750 (2002)CrossRefGoogle Scholar
  25. 25.
    Haydon, D.T., Matthews, L., Timms, R., Colegrave, N.: Top-down or bottom-up regulation of intra-host blood-stage malaria: do malaria parasites most resemble the dynamics of prey or predator? Proc. R. Soc. Lond. B 270(1512), 289–298 (2003)CrossRefGoogle Scholar
  26. 26.
    Grimm, V.: Ten years of individual-based modelling in ecology: what have we learned and what could we learn in the future? Ecol. Model. 115(2–3), 129–148 (1999)CrossRefGoogle Scholar
  27. 27.
    Johansson, A., Sumper, D.J.T.: From local interactions to population dynamics in site-based models of ecology. Theor. Popul. Biol. 64, 497–517 (2003)MATHCrossRefGoogle Scholar
  28. 28.
    Barker, G.C., Grimson, M.J.: A cellular automaton model of microbial growth. Binary: Computing in Microbiology 5, 132–137 (1993)Google Scholar
  29. 29.
    Picioreanu, C., Van Loosdrecht, C.M.C., Heijnen, J.J.: A new combined differential discrete cellular automaton approach for biofilm modeling. Biotechnol. Bioeng. 57, 718–731 (1997)CrossRefGoogle Scholar
  30. 30.
    Railsback, S.F.: Concepts from complex adaptive systems as a framework for individual-based modelling. Ecol. Model. 139, 47–62 (2001)CrossRefGoogle Scholar
  31. 31.
    DeAngelis, D.L., Mooij, W.M.: Individual-based modeling of ecological and evolutionary processes. Ann. Rev. Ecolog. Syst. 36, 147–168 (2005)CrossRefGoogle Scholar
  32. 32.
    Grimm, V., Revilla, E., Berger, U., Jeltsch, F., Mooij, W.M., Railsback, S.F., Thulke, H.-H., Weiner, J., Wiegand, T., DeAngelis, D.L.: Pattern-oriented modeling of agent-based complex systems: lessons from ecology. Science 310, 987–991 (2005)CrossRefADSGoogle Scholar
  33. 33.
    van Nes, E.H., Scheffer, M.: A strategy to improve the contribution of complex simulation models to ecological theory. Ecol. Model. 185, 153–164 (2005)CrossRefGoogle Scholar
  34. 34.
    Railsback, S.F., Lytinen, S.L., Jackson, S.K.: Agent-based simulation platforms: review and development recommendations. Simulation 82(9), 609–623 (2006)CrossRefGoogle Scholar
  35. 35.
    Paton, R., Gregory, R., Vlachos, C., Saunders, J., Wu, H.: Evolvable social agents for bacterial systems modeling. IEEE Nanobiosci. 3(3), 208–216 (2004)CrossRefGoogle Scholar
  36. 36.
    Grimm V., et al.: A standard protocol for describing individual-based and agent-based models. J. Ecol. Model. 198(1), 115–126 (2006). doi:10.1016/j.ecolmodel.2006.04.023 CrossRefGoogle Scholar
  37. 37.
    Bermúdez, J., López, D., Valls, J., Wagensberg, J.: On the analysis of microbiological processes by Monte Carlo simulation techniques. CABIOS 5(4), 305–312 (1989)Google Scholar
  38. 38.
    Flierl, G., Grünbaum, D., Levin, S., Olson, D.: From individuals to aggregations: the interplay between behavior and physics. J. Theor. Biol. 196, 397–454 (1999)CrossRefGoogle Scholar
  39. 39.
    Kreft, J.U., Picioreanu, C., Wimpenny, J.W.T., Van Loosdrecht, M.C.M.: Individual-based modelling of biofilms. Microbiology 147, 2897–2912 (2001)Google Scholar
  40. 40.
    Ginovart, M., Lopez, D., Valls, J.: INDISIM, an individual-based discrete simulation model to study bacterial cultures. J. Theor. Biol. 214, 305–314 (2002)CrossRefGoogle Scholar
  41. 41.
    Alpkvist, E., Picioreanu, C., van Loosdrecht, M.C.M., Heyden, A.: Three-dimensional biofilm model with individual cells and continuum EPS Matrix. Biotechnol. Bioeng. 94, 961–979 (2006)CrossRefGoogle Scholar
  42. 42.
    Pfeiffer, T., Bonhoeffer, S.: An evolutionary scenario for the transition to undifferentiated multicellularity. Proc. Natl. Acad. Sci. USA 100(3), 1095–1098 (2003)CrossRefADSGoogle Scholar
  43. 43.
    Drasdo, D., Hoehme, S.: A single-cell based model to tumor growth in-vitro: monolayers and spheroids. Phys. Biol. 2, 133 (2005)CrossRefADSGoogle Scholar
  44. 44.
    Anderson, A.R.A., Weaver, A.M., Cummings, P.T., Quaranta, V.: Tumor morphology and phenotypic evolution driven by selective pressure from the microenvironment. Cell 127(5), 905–915 (2006)CrossRefGoogle Scholar
  45. 45.
    Pierce, S.M., Skalak, T.C., Papin, J.A.: Multiscale biosystems integration: coupling intracellular network analysis with tissue-patterning simulations. IBM J. Res. Develop. 50(6), 601–616 (2006)CrossRefGoogle Scholar
  46. 46.
    Drasdo, D., Hoehme, S., Block, M.: On the role of physics in the growth and pattern formation of multi-cellular systems: what can we learn from individual-cell based models? J. Stat. Phys. 128(1–2), 287–345 (2007)MATHCrossRefADSMathSciNetGoogle Scholar
  47. 47.
    Bagchi, P.: Mesoscale simulation of blood flow in smalll vessels. Biophys. J. 92(6), 1858–1877 (2007). doi:10.1529/biophysj.106.095042 CrossRefADSGoogle Scholar
  48. 48.
    Peck, S.L.: Simulation as experiment: a philosophical reassessment for biological modeling. Trends Ecol. Evol. 19, 530–534 (2004)CrossRefGoogle Scholar
  49. 49.
    Gregory, R., Saunders, J.R., Saunders, V.A.: Rule-based computing system for microbial interactions and communications: evolution in virtual bacterial populations. BioSystems 91, 216–230 (2008)CrossRefGoogle Scholar
  50. 50.
    Long, T., Or, D.: Microbial growth on partially saturated rough surfaces: simulations in idealized roughness networks. Water Resour. Res. 43(2), WO2409 (2007)CrossRefGoogle Scholar
  51. 51.
    Nogueira, E.,Woods, J.D., Harris, C., Field, A.J., Talbot, S.: Phytoplankton co-existence: results from an individual-based simulation model. Ecol. Model. 198, 1–22 (2006)CrossRefGoogle Scholar
  52. 52.
    Gregory, R., Saunders, J.R., Saunders, V.A.: The Paton individual-based model legacy. Biosystems 85, 46–54 (2006)CrossRefGoogle Scholar
  53. 53.
    Gregory, R., Paton, R., Saunders, J., Wu, Q.H.: Parallelising a model of bacterial interaction and evolution. Biosystems 76, 121–131 (2004)CrossRefGoogle Scholar
  54. 54.
    Bankes, S.C.: Agent-based modeling: a revolution? Proc. Natl. Acad. Sci. USA 99(Suppl.), 7199–7200 (2002)CrossRefADSGoogle Scholar
  55. 55.
    Paton, V., Vlachos, L., Wu, Q.H., Saunders, J.R.: Simulated bacterially inspired problem solving the behavioural domain. Nat. Comput. 5, 43–65 (2006)MATHCrossRefMathSciNetGoogle Scholar
  56. 56.
    Giró, A., Padró, J.A., Valls, J., Wagensberg, J.: Monte Carlo simulation of an ecosystem: a matching between two levels of observation. Bull. Math. Biol. 47(1), 111–122 (1985)MathSciNetGoogle Scholar
  57. 57.
    Allen, M.P., Tildesley, D.J.: Computer Simulation of Liquids. Clarendon, Oxford (1987)MATHGoogle Scholar
  58. 58.
    Kreft, J.U., Booth, G., Wimpeney, J.W.T.: BacSim, a simulator for individual-based modelling of bacterial colony growth. Microbiology 144, 3275–3287 (1998)CrossRefGoogle Scholar
  59. 59.
    Solé, R.V., Valls, J.: On structural stability and chaos in biological systems. J. Theor. Biol. 155, 87–102 (1992)CrossRefGoogle Scholar
  60. 60.
    Solé, R.V., Gamarra, J., Ginovart, M., López, D.: Controlling chaos in ecology: from deterministic to individual-based models. Bull. Math. Biol. 61, 1187–1207 (1999)CrossRefGoogle Scholar
  61. 61.
    Keasling, J.D., Kuo, H., Vahanian, G.: A Monte Carlo simulation of the Escherichia coli cell cycle. J. Theor. Biol. 176, 411–430 (1995)CrossRefGoogle Scholar
  62. 62.
    Ratkowsky, D.A., Olley, J., McMeekin, T.A., Ball, A.: Relationship between temperature and growth rate of bacterial cultures. J. Bacteriol. 149, 1–5 (1982)Google Scholar
  63. 63.
    Ratkowsky, D.A., Lowry, R.K., McMeekin, T.A., Stokes, A.N., Chandler, R.E.: Model for bacterial culture growth rate throughout the entire biokinetic temperature range. J. Bacteriol. 154, 1222–1226 (1983)Google Scholar
  64. 64.
    Daves, J.N., Finn, R.K., Wilke, C.R.: Equations of substrate-limited growth: the case for Blackman kinetics. Biotechnol. Bioeng. 15(6), 1159–1177 (1973). doi:10.1002/bit.260150613 CrossRefGoogle Scholar
  65. 65.
    Koch, A.L.: Distribution of cell size in growing cultures of bacteria and applicability of Collins–Richmond principle. J. Gen. Microbiol. 45(3), 409 (1966)Google Scholar
  66. 66.
    Åkerlund, T., Nordström, K., Bernander, R.: Analysis of cell size and DNA content in exponentially growing and stationary-phase batch cultures of Escherichia coli. J. Bacteriol. 177(3), 6791–6797 (1995)Google Scholar
  67. 67.
    Margalef, R.: Perspectives in Ecological Theory. Chicago University Press, Chicago (1968)Google Scholar
  68. 68.
    Wagensberg, J., López, D., Valls, J.: Statistical aspects of biological organization. J. Phys. Chem. Solids 49, 695–700 (1988)CrossRefADSGoogle Scholar
  69. 69.
    Prigogine, I., Waime, J.M.: Biologie et Thermodynamique dès phénomènes irréversibles. Experientia 2, 451–453 (1946)CrossRefGoogle Scholar
  70. 70.
    Ginovart, M., Lopez, D., Valls, J., Silbert, M.: Simulation modelling of bacterial growth in yoghurt. Int. J. Food Microbiol. 73, 415–425 (2002)CrossRefGoogle Scholar
  71. 71.
    Budrene, E.O., Berg, H.C.: Complex patterns formed by motile cells of Escherichia coli. Nature 349, 630–633 (1991)CrossRefADSGoogle Scholar
  72. 72.
    Shapiro, J.A.: Thinking about bacterial populations as multicellular organisms. Annu. Rev. Microbiol. 52, 81–104 (1998)CrossRefGoogle Scholar
  73. 73.
    Ben-Jacob, E., Schochet, O., Tenenbaum, A., Cohen, I., Czirok, A., Vicsek, T.: Generic modelling of cooperative growth patterns in bacterial colonies. Nature 368, 46–49 (1994)CrossRefADSGoogle Scholar
  74. 74.
    Matsushita, M., Fujikawa, H.: Diffusion-limited growth in bacterial colony formation. Physica A 168, 498–506 (1990)CrossRefADSGoogle Scholar
  75. 75.
    Mimura, M., Sakaguchi, H., Matsushita, M.: Reaction-diffusion modelling of bacterial colony patterns. Physica A 282, 283–303 (2000)CrossRefADSGoogle Scholar
  76. 76.
    Tang, W.J., Wu, Q.H., Saunders, J.R.: A novel model for bacterial foraging in varying environments. Lect. Notes Comput. Sci. 3980, 556–565 (2006)CrossRefGoogle Scholar
  77. 77.
    Koch, A.L., Higgins, M.L.: Control of wall band splitting in spectrococcus-faecalis. J. Gen. Microbiol. 130, 735–745 (1984)Google Scholar
  78. 78.
    Sherbaum, O.H.: Synchronous division of microorganisms. J. Gen. Microbiol. 14, 283–310 (1960)Google Scholar
  79. 79.
    López, D., Lorén, J.G., Viñas, M., Bermúdez, J.: Analysis of microcalorimetric curves for bacterial identification. Can. J. Microbiol. 33, 6–11 (1987)CrossRefGoogle Scholar
  80. 80.
    Marincs, F.: On-line monitoring of growth of Escherichia coli in batch cultures by bioluminescence. Appl. Microbiol. Biotechnol. 53, 536–541 (2000)CrossRefGoogle Scholar
  81. 81.
    McMeekin, T.A., Olley, J., Ratkowsky, D.A., Ross, T.: Predictive microbiology: towards the interface and beyond. Int. J. Food Microbiol. 73, 395–407 (2002)CrossRefGoogle Scholar
  82. 82.
    Kolter, R., Siegele, A., Tormo, D.A.: The stationary phase of the bacterial life cycle. Annu. Rev. Microbiol. 47, 855–874 (1993)CrossRefGoogle Scholar
  83. 83.
    Dens, E.J., Bernaerts, K., Standaert, A.R., Van Impe, J.F.: Cell division theory and individual-based modeling of microbial lag part I. The theory of cell division. Int. J. Food Microbiol. 101, 303–318 (2005)CrossRefGoogle Scholar
  84. 84.
    Dens, E.J., Bernaerts, K., Standaert, A.R., Van Impe, J.F.: Cell division theory and individual-based modeling of microbial lag part II. Modeling lag phenomena induced by temperature shifts. Int. J. Food Microbiol. 101, 319–332 (2005)CrossRefGoogle Scholar
  85. 85.
    Ginovart, M., López, D., Valls, J., Silbert, M.: Individual based simulations of bacterial growth on agar plates. Physica A 305, 604–618 (2002)MATHCrossRefADSGoogle Scholar
  86. 86.
    Prats, C., López, D., Giró, A., Ferrer, J., Valls, J.: Individual-based modelling of bacterial cultures to study the microscopic causes of the lag phase. J. Theor. Biol. 241, 939–953 (2006)Google Scholar
  87. 87.
    O’Donnell, A.G., Young, I.M., Rushton, S.P., Shirley, M.D., Crawford, J.W.: Visualization, modelling and prediction in soil microbiology. Nat. Rev. Microbiol. 5(9), 689–699 (2007)CrossRefGoogle Scholar
  88. 88.
    Ginovart, M., López, D., Gras, A.: Individual-based modelling of microbial activity to study mineralization of C and N and nitrification process in soil. Nonlinear Anal.: Real World Appl. 6, 773–795 (2005)MATHCrossRefGoogle Scholar
  89. 89.
    Scheffer, M., Baveco, J.M., DeAngelis, D.L., Rose, K.A., van Nes, E.H.: Super-individuals a simple solution for modelling large populations on an individual basis. Ecol. Model. 80, 161–170 (1995)CrossRefGoogle Scholar
  90. 90.
    Trager, W., Jensen, J.B.: Human malaria parasites in continuous culture. Science 193, 673–675 (1976)CrossRefADSGoogle Scholar
  91. 91.
    Ferrer, J., Vidal, J., Prats, C., Valls, J., Herreros, E., López, D., Giró, A., Gargallo, D.: Individual-based model and simulation of Plasmodium falciparum infected erythrocyte in vitro cultures. J. Theor. Biol. 248, 448–459 (2007)CrossRefGoogle Scholar
  92. 92.
    Martins, A.M.P., Picioreanu, C., Heijen, J.J., van Loosdrecht, M.C.M.: Three-dimensional dual-morphotype species modeling of activated sludge flocs. Environ. Sci. Technol. 38(21), 5632–5641 (2004)CrossRefGoogle Scholar
  93. 93.
    Ginovart, M., López, D., Giró, A., Silbert, M.: Flocculation in brewing yeasts: a computer simulation study. Biosystems 83, 51–55 (2006)CrossRefGoogle Scholar
  94. 94.
    Calleja, G.B.: Cell aggregation. In: Rose, A.H., Harrison, J.S. (eds.) The Yeasts, vol. 2, second ed., pp. 165–238. Academic Press, London (1987)Google Scholar
  95. 95.
    Stratford, M.: Yeast flocculation: a new perspective. In: Rose, A.H. (ed.) Advances in Microbial Physiology, vol. 33, pp. 1–71. Academic Press, London (1992)CrossRefGoogle Scholar
  96. 96.
    Brohan, B., McLoughlin, A.J.: Characterization of the physical properties of yeast flocs. Appl. Microbiol. Biotechnol. 20, 16–22 (1984)Google Scholar

Copyright information

© Springer Science + Business Media B.V. 2008

Authors and Affiliations

  1. 1.Departament de Física i Enginyeria Nuclear, Escola Superior d’Agricultura de BarcelonaUniversitat Politècnica de CatalunyaBarcelonaSpain

Personalised recommendations