Skip to main content
SpringerLink
Log in

Mechanisms and Kinetics of Amyloid Aggregation Investigated by a Phenomenological Coarse-Grained Model

  • Chapter
  • First Online:
Computational Modeling of Biological Systems

Part of the book series: Biological and Medical Physics, Biomedical Engineering ((BIOMEDICAL))

Abstract

Amyloid fibrils are ordered polypeptide aggregates that have been implicated in several neurodegenerative pathologies, such as Alzheimer’s, Parkinson’s, Huntington’s, and prion diseases, [1, 2] and, more recently, also in biological functionalities. [3, 4, 5] These findings have paved the way for a wide range of experimental and computational studies aimed at understanding the details of the fibril-formation mechanism. Computer simulations using low-resolution models, which employ a simplified representation of protein geometry and energetics, have provided insights into the basic physical principles underlying protein aggregation in general [6, 7, 8] and ordered amyloid aggregation. [9, 10, 11, 12, 13, 14, 15] For example, Dokholyan and coworkers have used the Discrete Molecular Dynamics method [16, 17] to shed light on the mechanisms of protein oligomerization [18] and the conformational changes that take place in proteins before the aggregation onset. [19, 20] One challenging observation, which is difficult to observe by computer simulations, is the wide range of aggregation scenarios emerging from a variety of biophysical measurements. [21, 22] Atomistic models have been employed to study the conformational space of amyloidogenic polypeptides in the monomeric state, [23, 24, 25] the very initial steps of amyloid formation, [26, 27, 28, 29, 30, 31, 32] and the structural stability of fibril models. [33, 34, 35) However, all-atom simulations of the kinetics of fibril formation are beyond what can be done with modern computers.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Dobson, C.M.: Protein folding and misfolding. Nature 426, 884–890 (2003).

    Google Scholar 

  2. Lansbury, P.T., Lashuel, H.A.: A century-old debate on protein aggregation and neurodegeneration enters the clinic. Nature 443, 774–779 (2006).

    Article  ADS  Google Scholar 

  3. Fowler, D.M., Koulov, A.V., Balch, W.E., Kelly, J.W.: Functional amyloid–from bacteria to humans. Trends Biochem. Sci. 32, 217–224 (2007).

    Article  Google Scholar 

  4. Maji, S.K., Perrin, M.H., Sawaya, M.R., Jessberger, S., Vadodaria, K., Rissman, R.A., Singru, P.S., Nilsson, K.P.R., Simon, R., Schubert, D., et al.: Functional amyloids as natural storage of peptide hormones in pituitary secretory granules. Science 325, 328–332 (2009).

    Article  ADS  Google Scholar 

  5. Greenwald, J., Riek, R.: Biology of amyloid: structure, function, and regulation. Structure 18, 1244–1260 (2010).

    Article  Google Scholar 

  6. Broglia, R.A., Tiana, G., Pasquali, S., Roman, H.E., Vigezzi, E.: Folding and aggregation of designed proteins. Proc. Natl. Acad. Sci. USA 95, 12930–12933 (1998).

    Article  ADS  Google Scholar 

  7. Gupta, P., Hall, C.K., Voegler, A.C.: Effect of denaturant and protein concentrations upon protein refolding and aggregation: a simple lattice model. Protein Sci. 7, 2642–2652 (1998).

    Article  Google Scholar 

  8. Harrison, P.M., Chan, H.S., Prusiner, S.B., Cohen, F.E.: Thermodynamics of model prions and its implications for the problem of prion protein folding. J. Mol. Biol. 286, 593–606 (1999).

    Article  Google Scholar 

  9. Urbanc, B., Cruz, L., Yun, S., Buldyrev, S.V., Bitan, G., Teplow, D.B., Stanley, H.E.: In silico study of amyloid beta-protein folding and oligomerization. Proc. Natl. Acad. Sci. USA 101, 17345–17350 (2004).

    Article  ADS  Google Scholar 

  10. Sørensen, J., Periole, X., Skeby, K.K., Marrink, S.J., Schiøtt, B.: Protofibrillar assembly toward the formation of amyloid fibrils. J. Chem. Phys. Lett. 2, 2385–2390 (2011).

    Article  Google Scholar 

  11. Jang, H., Hall, C.K., Zhou, Y.: Assembly and kinetic folding pathways of a tetrameric betasheet complex: molecular dynamics simulations on simplified off-lattice protein models. Biophys. J. 86(1 Pt 1), 31–49 (2004).

    Article  Google Scholar 

  12. Dima, R.I., Thirumalai, D.: Exploring protein aggregation and self-propagation using lattice models: phase diagram and kinetics. Protein Sci. 11(5), 1036–1049 (2002).

    Article  Google Scholar 

  13. Malolepsza, E., Boniecki, M., Kolinski, A., Piela, L.: Theoretical model of prion propagation: a misfolded protein induces misfolding. Proc. Natl. Acad. Sci. USA 102, 7835–7840 (2005).

    Article  ADS  Google Scholar 

  14. Khare, S.D., Ding, F., Gwanmesia, K.N., Dokholyan, N.V.: Molecular origin of polyglutamine aggregation in neurodegenerative diseases. PLoS Comput. Biol. 1, 230–235 (2005).

    Article  Google Scholar 

  15. Chen, Y., Dokholyan, N.V.: A single disulfide bond differentiates aggregation pathways of beta2-microglobulin. J. Mol. Biol. 354, 473–482 (2005).

    Article  Google Scholar 

  16. Dokholyan, N.V., Buldyrev, S.V., Stanley, H.E., Shakhnovich, E.I.: Discrete molecular dynamics studies of the folding of a protein-like model. Fold. Des. 3, 577–587 (1998).

    Article  Google Scholar 

  17. Ding, F., Buldyrev, S.V., Dokholyan, N.V.: Folding Trp-cage to NMR resolution native structure using a coarse-grained protein model. Biophys. J. 88, 147–155 (2005).

    Article  ADS  Google Scholar 

  18. Ding, F., Dokholyan, N.V., Buldyrev, S.V., Stanley, H.E., Shakhnovic, E.I.: Molecular dynamics simulation of the SH3 domain aggregation suggests a generic amyloidogenesis mechanism. J. Mol. Biol. 324, 851–857 (2002).

    Article  Google Scholar 

  19. Ding, F., Borreguero, J.M., Buldyrey, S.V., Stanley, H.E., Dokholyan, N.V.: Mechanism for the alpha-helix to beta-hairpin transition. Proteins 53, 220–228 (2003).

    Article  Google Scholar 

  20. Ding, F., LaRocque, J.J., Dokholyan, N.V.: Direct observation of protein folding, aggregation, and a prion-like conformational conversion. J. Biol. Chem. 48, 40235–40240 (2005).

    Article  Google Scholar 

  21. Gosal, W.S., Morten, I.J., Hewitt, E.W., Smith, D.A., Thomson, N.H., Radford, S.E.: Competing pathways determine fibril morphology in the self-assembly of beta2-microglobulin into amyloid. J. Mol. Biol. 351, 850–864 (2005).

    Article  Google Scholar 

  22. Plakoutsi, G., Bemporad, F., Calamai, M., Taddei, N., Dobson, C.M., Chiti, F.: Evidence for a mechanism of amyloid formation involving molecular reorganisation within native-like precursor aggregates. J. Mol. Biol. 351, 910–922 (2005).

    Article  Google Scholar 

  23. Vitalis, A., Wang, X., Pappu, R.V.: Quantitative characterization of intrinsic disorder in polyglutamine: insights from analysis based on polymer theories. Biophys. J. 93, 1923–1937 (2007).

    Article  ADS  Google Scholar 

  24. Vitalis, A., Lyle, N., Pappu, R.V.: Thermodynamics of beta-sheet formation in polyglutamine. Biophys. J. 97, 303–311 (2009).

    Article  ADS  Google Scholar 

  25. Vitalis, A., Caflisch, A.: Micelle-like architecture of the monomer ensemble of Alzheimer’s amyloid- peptide in aqueous solution and its implications for A_ aggregation. J. Mol. Biol. 403, 148–165 (2010).

    Article  Google Scholar 

  26. Gsponer, J., Haberthr, U., Caflisch, A.: The role of side-chain interactions in the early steps of aggregation: Molecular dynamics simulations of an amyloid-forming peptide from the yeast prion Sup35. Proc. Natl. Acad. Sci. USA 100, 5154–5159 (2003).

    Article  ADS  Google Scholar 

  27. Hwang, W., Zhang, S., Kamm, R.D., Karplus, M.: Kinetic control of dimer structure formation in amyloid fibrillogenesis. Proc. Natl. Acad. Sci. USA 101, 12916–12921 (2004).

    Article  ADS  Google Scholar 

  28. de la Paz, M.L., de Mori, G.M.S., Serrano, L., Colombo, G.: Sequence dependence of amyloid fibril formation: insights from molecular dynamics simulations. J. Mol. Biol. 349, 583–596 (2005).

    Article  Google Scholar 

  29. Cecchini, M., Curcio, R., Pappalardo, M., Melki, R., Caflisch, A.: A molecular dynamics approach to the structural characterization of amyloid aggregation. J. Mol. Biol. 357, 1306–1321 (2006).

    Article  Google Scholar 

  30. Strodel, B., Whittleston, C.S., Wales, D.J.: Thermodynamics and kinetics of aggregation for the GNNQQNY peptide. J. Am. Chem. Soc. 129, 16005–16014 (2007).

    Article  Google Scholar 

  31. De Simone, A., Esposito, L., Pedone, C., Vitagliano, L.: Insights into stability and toxicity of amyloid-like oligomers by replica exchange molecular dynamics analyses. Biophys. J. 95, 1965–1973 (2008).

    Article  Google Scholar 

  32. Bellesia, G., Shea, J.E.: What determines the structure and stability of KFFE monomers, dimers, and protofibrils? Biophys. J. 96, 875–886 (2009).

    Article  ADS  Google Scholar 

  33. Ma, B., Nussinov, R.: Stabilities and conformations of Alzheimer’s beta-amyloid peptide oligomers (Abeta 16–22, Abeta 16–35, and Abeta 10–35): sequence effects. Proc. Natl. Acad. Sci. USA 99, 14126–14131 (2002).

    Article  ADS  Google Scholar 

  34. Buchete, N.V., Tycko, R., Hummer, G.: Molecular dynamics simulations of Alzheimer’s beta-amyloid protofilaments. J. Mol. Biol. 353, 804–821 (2005).

    Article  Google Scholar 

  35. Wu, C., Bowers, M.T., Shea, J.E.: Molecular structures of quiescently grown and brain-derived polymorphic fibrils of the Alzheimer amyloid abeta9–40 peptide: a comparison to agitated fibrils. PLoS Comput. Biol. 6, e1000693 (2010).

    Article  Google Scholar 

  36. Wu, C., Shea, J.E.: Coarse-grained models for protein aggregation. Curr. Opin. Struct. Biol. 21, 209–220 (2011).

    Article  Google Scholar 

  37. Pellarin, R., Caflisch, A.: Interpreting the aggregation kinetics of amyloid peptides. J. Mol. Biol. 360, 882–892 (2006).

    Article  Google Scholar 

  38. Müller, M., Katsov, K., Schick, M.: Biological and synthetic membranes: what can be learned from a coarse-grained description? Phys. Rep. 434, 113–176 (2006).

    Article  ADS  Google Scholar 

  39. Zhang, J., Muthukumar, M.: Simulations of nucleation and elongation of amyloid fibrils. J. Chem. Phys. 130, 035102 (2009).

    Article  ADS  Google Scholar 

  40. Auer, S., Dobson, C.M., Vendruscolo, M., Maritan, A.: Self-templated nucleation in peptide and protein aggregation. PLos Comput. Biol. 4, e1000222 (2008).

    Article  Google Scholar 

  41. Li, M.S., Klimov, D.K., Straub, J.E., Thirumalai, D.: Probing the mechanisms of fibril formation using lattice models. J. Chem. Phys. 129, 175101 (2008).

    Article  ADS  Google Scholar 

  42. Nguyen, H.D., Hall, C.K.: Molecular dynamics simulations of spontaneous fibril formation by random-coil peptides. Proc. Natl. Acad. Sci. USA 101, 16180–16185 (2004).

    Article  ADS  Google Scholar 

  43. Bellesia, G., Shea, J.E.: Self-assembly of beta-sheet forming peptides into chiral fibrillar aggregates. J. Chem. Phys. 126, 245104 (2007).

    Article  ADS  Google Scholar 

  44. MacKerell, A.D.J., Feig, M., Brooks, C.L.: Improved treatment of the protein backbone in empirical force fields. J. Am. Chem. Soc. 126, 698–699 (2004).

    Article  Google Scholar 

  45. Zhou, Y., Karplus, M.: Interpreting the folding kinetics of helical proteins. Nature 401, 400–403 (1999).

    ADS  Google Scholar 

  46. Brooks, B.R., Brooks, C.L., Mackerell, A.D., Nilsson, L., Petrella, R.J., Roux, B., Won, Y., Archontis, G., Bartels, C., Boresch, S. et al.: CHARMM: the biomolecular simulation program J. Comput. Chem. 30, 1545–1614 (2009).

    Article  Google Scholar 

  47. Fändrich, M.: Absolute correlation between lag time and growth rate in the spontaneous formation of several amyloid-like aggregates and fibrils. J. Mol. Biol. 365, 1266–1270 (2007).

    Article  Google Scholar 

  48. Hortschansky, P., Schroeckh, V., Christopeit, T., Zandomeneghi, G., Fändrich, M.: The aggregation kinetics of Alzheimer’s beta-amyloid peptide is controlled by stochastic nucleation. Protein Sci. 14, 1753–1759 (2005).

    Article  Google Scholar 

  49. Christopeit, T., Hortschansky, P., Schroeckh, V., Guhrs, K., Zandomeneghi, G., Fändrich, M.: Mutagenic analysis of the nucleation propensity of oxidized Alzheimer’s beta-amyloid peptide. Protein Sci. 14, 2125–2131 (2005).

    Article  Google Scholar 

  50. Pellarin, R., Guarnera, E., Caflisch, A.: Pathways and intermediates of amyloid fibril formation. J. Mol. Biol. 374, 917–924 (2007).

    Article  Google Scholar 

  51. Nilsberth, C., Westlind-Danielsson, A., Eckman, C.B., Condron, M.M., Axelman, K., Forsell, C. et al.: The ‘Arctic’ APP mutation (E693G) causes Alzheimer’s disease by enhanced Abeta protofibril formation. Nat. Neurosci. 4, 887–893 (2001).

    Article  Google Scholar 

  52. Conway, K.A., Lee, S.J., Rochet, J.C., Ding, T.T., Williamson, R.E., Lansbury, P.T.: Acceleration of oligomerization, not fibrillization, is a shared property of both alpha-synuclein mutations linked to early-onset Parkinson’s disease: implications for pathogenesis and therapy. Proc. Natl. Acad. Sci. USA 97, 571–576 (2000).

    Article  ADS  Google Scholar 

  53. Sabate, R., Estelrich, J.: Evidence of the existence of micelles in the fibrillogenesis of betaamyloid peptide. J. Phys. Chem. B 109, 11027–11032 (2005).

    Article  Google Scholar 

  54. Lomakin, A., Chung, D.S., Benedek, G.B., Kirschner, D.A., Teplow, D.B.: On the nucleation and growth of amyloid beta-protein fibrils: detection of nuclei and quantitation of rate constants. Proc. Natl. Acad. Sci. USA 93, 1125–1129 (1996).

    Article  ADS  Google Scholar 

  55. Serio, T.R., Cashikar, A.G., Kowal, A.S., Sawicki, G.J., Moslehi, J.J., Serpell, L. et al.: Nucleated conformational conversion and the replication of conformational information by a prion determinant. Science 289, 1317–1321 (2000).

    Article  ADS  Google Scholar 

  56. Fowler, D.M., Koulov, A.V., Alory-Jost, C., Marks, M.S., Balch, W.E., Kelly, J.W.: Functional amyloid formation within mammalian tissue. PLoS Biol. 4, e6 (2006).

    Article  Google Scholar 

  57. Lomakin, A., Teplow, D.B., Kirschner, D.A., Benedek, G., Kinetic theory of fibrillogenesis of amyloid beta-protein. Proc. Natl. Acad. Sci. USA 94, 7942–7947 (1997).

    Article  ADS  Google Scholar 

  58. Nielsen, L., Khurana, R., Coats, A., Frokjaer, S., Brange, J., Vyas, S., Uversky, V.N., Fink, A.L.: Effect of environmental factors on the kinetics of insulin fibril formation: elucidation of the molecular mechanism. Biochemistry 40, 6036–6046 (2001).

    Article  Google Scholar 

  59. Wasmer, C., Soragni, A., Sabate, R., Lange, A., Riek, R., Meier, B.H.: Infectious and noninfectious amyloids of the HET-s(218–289) prion have different NMR spectra. Angew. Chem. Int. Ed. 47, 5839–5841 (2008).

    Article  Google Scholar 

  60. Dzwolak, W., Grudzielanek, S., Smirnovas, V., Ravindra, R., Nicolini, C., Jansen, R., Loksztejn, A., Porowski, S., Winter, R.: Ethanol-perturbed amyloidogenic self-assembly of insulin: looking for origins of amyloid strains. Biochemistry 44, 8948–8958 (2005).

    Article  Google Scholar 

  61. Petkova, A.T., Leapman, R.D., Guo, Z., Yau, W.M., Mattson, M.P., Tycko, R.: Self-propagating, molecular-level polymorphism in Alzheimer’s beta-amyloid fibrils. Science 307, 262–265 (2005).

    Article  ADS  Google Scholar 

  62. Paravastu, A.K., Petkova, A.T., Tycko, R.: Polymorphic fibril formation by residues 10–40 of the Alzheimer’s beta-amyloid peptide. Biophys. J. 90, 4618–4629 (2006).

    Article  ADS  Google Scholar 

  63. Meinhardt, J., Sachse, C., Hortschansky, P., Grigorieff, N., Fändrich, M.: Abeta(1–40) fibril polymorphism implies diverse interaction patterns in amyloid fibrils. J. Mol. Biol. 386, 869–877 (2009).

    Article  Google Scholar 

  64. Pellarin, R., Schuetz, P., Guarnera, E., Caflisch, A.: Amyloid fibril polymorphism is under kinetic control. J. Am. Chem. Soc. 132, 14960–14970 (2010).

    Article  Google Scholar 

  65. Goldsbury, C., Frey, P., Olivieri, V., Aebi, U., Müller, S.A.: Multiple assembly pathways underlie amyloid-beta fibril polymorphisms. J. Mol. Biol. 352, 282–298 (2005).

    Article  Google Scholar 

  66. Ellis, J.R.: Macromolecular crowding: obvious but underappreciated. Trends Biochem. Sci. 26, 597–604 (2001).

    Article  Google Scholar 

  67. Lopes, D., Meister, A., Gohlke, A., Hauser, A., Blume, A., Winter, R.: Mechanism of islet amyloid polypeptide fibrillation at lipid interfaces studied by infrared reflection absorption spectroscopy. Biophys. J. 93, 3132–3141 (2007).

    Article  ADS  Google Scholar 

  68. Chi, E., Ege, C., Winans, A., Majewski, J., Wu, G., Kjaer, K., Lee, K.: Lipid membrane templates the ordering and induces the fibrillogenesis of Alzheimer’s disease amyloid-beta peptide. Proteins 72, 1–24 (2008).

    Article  Google Scholar 

  69. Engel, M.F.M., Khemtémourian, L., Kleijer, C., Meeldijk, H., Jacobs, J., Verkleij, A. et al.: Membrane damage by human islet amyloid polypeptide through fibril growth at the membrane. Proc. Natl. Acad. Sci. USA 105, 6033–6038 (2008).

    Article  ADS  Google Scholar 

  70. Ellis, R.J., Minton, A.P.: Protein aggregation in crowded environments. Biol. Chem. 387, 485–497 (2006).

    Article  Google Scholar 

  71. Munishkina, L.A., Cooper, E.M., Uversky, V.N., Fink, A.L.: The effect of macromolecular crowding on protein aggregation and amyloid fibril formation. J. Mol. Recognit. 17, 456–464 (2004).

    Article  Google Scholar 

  72. Munishkina, L.A., Ahmad, A., Fink, A.L., Uversky, V.N.: Guiding protein aggregation with macromolecular crowding. Biochemistry 47, 8993–9006 (2008).

    Article  Google Scholar 

  73. Friedman, R., Pellarin, R., Caflisch, A.: Amyloid aggregation on lipid bilayers and its impact on membrane permeability. J. Mol. Biol. 387, 407–415 (2009).

    Article  Google Scholar 

  74. Friedman, R., Pellarin, R., Caflisch, A.: Soluble protofibrils as metastable intermediates in simulations of amyloid fibril degradation induced by lipid vesicles. J. Phys. Chem. Lett. 1, 471–474 (2010).

    Article  Google Scholar 

  75. Volles, M.J., Lee, S.J., Rochet, J.C., Shtilerman, M.D., Ding, T.T., Kessler, J.C., Lansbury, P.T.: Vesicle permeabilization by protofibrillar alpha-synuclein: implications for the pathogenesis and treatment of Parkinson’s disease. Biochemistry 40, 7812–7819 (2001).

    Article  Google Scholar 

  76. Sharp, J., Forrest, J., Jones, R.: Surface denaturation and amyloid fibril formation of insulin at model lipid-water interfaces. Biochemistry 41, 15810–15819 (2002).

    Article  Google Scholar 

  77. Khemtémourian, L., Engel, M.F.M., Liskamp, R.M.J., Höppener, J.W.M., Killian, J.A.: The N-terminal fragment of human islet amyloid polypeptide is non-fibrillogenic in the presence of membranes and does not cause leakage of bilayers of physiologically relevant lipid composition. Biochim. Biophys. Acta. 1798, 1805–1811 (2010).

    Article  Google Scholar 

  78. Lashuel, H., Lansbury, P.: Are amyloid diseases caused by protein aggregates that mimic bacterial pore-forming toxins? Q. Rev. Biophys. 39, 167–201 (2006).

    Article  Google Scholar 

  79. Martins, I.C., Kuperstein, I., Wilkinson, H., Maes, E., Vanbrabant, M., Jonckheere, W., Gelder, P.V., Hartmann, D., D’Hooge, R., Strooper, B.D. et al.: Lipids revert inert Abeta amyloid fibrils to neurotoxic protofibrils that affect learning in mice. EMBO J 27, 224–233 (2008).

    Article  Google Scholar 

  80. Friedman, R., Caflisch, A.: Surfactant effects on amyloid aggregation kinetics. J. Mol. Biol. 414, 303–312 (2011).

    Article  Google Scholar 

  81. Magno, A., Caflisch, A., Pellarin, R.: Crowding effects on amyloid aggregation kinetics. J. Phys. Chem. Lett. 1, 3027–3032 (2010).

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Biochemistry, University of Zürich, Winterthurerstrasse 190, CH-8057, Zürich, Switzerland

    Andrea Magno, Riccardo Pellarin & Amedeo Caflisch

Authors
  1. Andrea Magno
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Riccardo Pellarin
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Amedeo Caflisch
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Amedeo Caflisch .

Editor information

Editors and Affiliations

  1. , Biochemistry and Biophysics, University of North Carolina at Chapel H, Mason Farm Rd 120, Chapel Hill, 27599-7260, North Carolina, USA

    Nikolay V Dokholyan

Rights and permissions

Reprints and permissions

Copyright information

© 2012 Springer Science+Business Media, LLC

About this chapter

Cite this chapter

Magno, A., Pellarin, R., Caflisch, A. (2012). Mechanisms and Kinetics of Amyloid Aggregation Investigated by a Phenomenological Coarse-Grained Model. In: Dokholyan, N. (eds) Computational Modeling of Biological Systems. Biological and Medical Physics, Biomedical Engineering. Springer, Boston, MA. https://doi.org/10.1007/978-1-4614-2146-7_8

Download citation

  • DOI: https://doi.org/10.1007/978-1-4614-2146-7_8

  • Published:

  • Publisher Name: Springer, Boston, MA

  • Print ISBN: 978-1-4614-2145-0

  • Online ISBN: 978-1-4614-2146-7

  • eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)

Publish with us

Policies and ethics

Search

Navigation