Abstract
In this paper we suggest that, under suitable conditions, supervised learning can provide the basis to formulate at the microscopic level quantitative questions on the phenotype structure of multicellular organisms. The problem of explaining the robustness of the phenotype structure is rephrased as a real geometrical problem on a fixed domain. We further suggest a generalization of path integrals that reduces the problem of deciding whether a given molecular network can generate specific phenotypes to a numerical property of a robustness function with complex output, for which we give heuristic justification. Finally, we use our formalism to interpret a pointedly quantitative developmental biology problem on the allowed number of pairs of legs in centipedes.
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Notes
- 1.
Refer to Remarks 5 an 6 for outstanding issues related to the definition of geometric path integrals.
References
Arnold, V.I.: Mathematical Methods of Classical Mechanics, 2nd edn. Springer, New York (2010)
Altland, A., Simons, B.D.: Condensed Matter Field Theory. Cambridge University Press, Cambridge (2010)
Basu, S., Pollack, R., Roy, M.F.: Algorithms in Real Algebraic Geometry. Springer, New York (2003)
Berenstein, C.A.: Review of “The universality of the Radon transform” by L. Ehrenpreis, MR2019604 (2007d:58047)
Bochnak, J., Coste, M., Roy, M.F.: Real Algebraic Geometry. Springer, New York (1998)
Calvert, V., Tang, Y., Boveia, V., Wulfkuhle, J., Schutz-Geschwender, A., Olive, M., Liotta, L., Petricoin, E.: Development of multiplexed protein profiling and detection using near infrared detection of reverse-phase protein microarrays. Clin. Proteomics 1, 81–90 (2004)
Ehrenpreis, L.: Fourier Analysis in Several Complex Variables. Wiley-Interscience, New York (1970)
Ehrenpreis, L.: The Universality of the Radon Transform. Oxford University Press, Oxford (2003)
Feynman, R.P., Hibbs, A.R.: Quantum Mechanics and Path Integrals. McGraw-Hill, New York (1965)
Gunawardena, J.: Models in systems biology: The parameter problem and the meanings of robustness. In: Lodhi, H., Muggleton, S. (eds.) Elements of Computational Systems Biology. Wiley, New York (2009)
Hastie, T., Tibshirani, R., Friedman, J.: The Elements of Statistical Learning, 2nd edn. Springer, New York (2008)
Houle, D., Govindaraju, D.R., Omholt, S.: Phenomics: The next challenge. Nat. Rev. Genet. 11, 855–866 (2010)
Irish, J.M., Kotecha, N., Nolan, G.P.: Mapping normal and cancer cell signalling networks: towards single-cell proteomics. Nat. Rev., Cancer 6, 146–155 (2006)
Izenman, A.J.: Modern Multivariate Statistical Techniques: Regression, Classification, and Manifold Learning. Springer, New York (2008)
Kumano-go, N., Fujiwara, D.: Feynman path integrals and semiclassical approximation. RIMS Kokyuroku Bessatsu B5, 241–263 (2008)
Kashiwara, M., Kawai, T., Kimura, T.: Foundations of Algebraic Analysis. Princeton Mathematical Series, vol. 37. Princeton University Press, Princeton (1986)
Kitano, H.: Towards a theory of biological robustness. Mol. Syst. Biol. 3, 137 (2007)
Kleinert, H.: Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets, 4th edn. World Scientific, Singapore (2004)
Minelli, A.: The Development of Animal Form. Cambridge University Press, Cambridge (2003)
Schulman, L.S.: Techniques and Applications of Path Integration. Dover, New York (2005)
Voit, E.O.: Computational Analysis of Biochemical Systems. Cambridge University Press, Cambridge (2000)
Wray, G.A.: Evolutionary dissociations between homologous genes and homologous structures. In: Bock, G.R., Cardew, G. (eds.) Homology, pp. 189–206. Wiley, Chichester (1999)
Acknowledgement
We would like to thank Mirco Mannucci, Roman Buniy, and the referee for very useful comments.
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This paper is dedicated to the memory of Leon Ehrenpreis 1930–2010.
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Napoletani, D., Petricoin, E., Struppa, D.C. (2012). Geometric Path Integrals. A Language for Multiscale Biology and Systems Robustness. In: Sabadini, I., Struppa, D. (eds) The Mathematical Legacy of Leon Ehrenpreis. Springer Proceedings in Mathematics, vol 16. Springer, Milano. https://doi.org/10.1007/978-88-470-1947-8_16
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DOI: https://doi.org/10.1007/978-88-470-1947-8_16
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