“Quantitative genetics state-space modeling of phenotypic plasticity and evolution”

Authors: Rolf Ergon,
Affiliation: University of South-Eastern Norway
Reference: 2019, Vol 40, No 1, pp. 51-69.

Keywords: Reaction norms, reference environment, multivariate breeder's equation, evolving equilibrium, genetic assimilation

Abstract: Living organisms adapt to changes in environment by phenotypic plasticity and evolution by natural selection (or they migrate). At detailed genetic levels these phenomena are complicated, and quantitative genetics attempts to capture essential processes at a higher abstraction level. Phenotypic plasticity is then commonly modeled by reaction norms, which describe how individual traits in a population are expressed in response to changes in environmental variables. The mean reaction norms are evolvable, and here I present a general quantitative genetics state-space model for evolutionary reaction norm dynamics. Reaction norms make use of a reference environment, which is traditionally set to zero. This is problematic when the reference environment is the environment a population is adapted to, for the reason that this environment is a population property, which in itself may be evolvable. With reference to Ergon (2018), I describe models that take such evolvability into account. The resulting models are fundamentally different from most engineering system models, where given reference values are constant, and therefore without consequences can be set to zero. For simplicity I assume only temporal variations in environment, although there obviously are a lot of spatial variations in nature, and I assume that no mutations are involved. Fundamentals from quantitative evolutionary theory are given in appendices.

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DOI forward links to this article:
[1] Rolf Ergon (2022), doi:10.4173/mic.2022.3.1
[2] Rolf Ergon (2022), doi:10.4173/mic.2022.4.2
[3] Rolf Ergon (2023), doi:10.1002/ece3.10194
[4] Rolf Ergon (2023), doi:10.4173/mic.2023.3.1
[1] Arnold, S.J., Burger, R., Hohenlohe, P.A., Ajie, B.C., and Jones, A.G. (2008). Arnold, S, J., Burger, R., Hohenlohe, P.A., Ajie, B.C., and Jones, A.G. Understanding the Evolution and Stability of the G-matrix. Evolution. 62(10):2451--2461. doi:10.1111/j.1558-5646.2008.00472.x
[2] Arulampalam, M., Maskell, S., Gordon, N., and Clapp, T. (2002). Arulampalam, M, , Maskell, S., Gordon, N., and Clapp, T. A tutorial on particle filters for online nonlinear/non-Gaussian Bayesian tracking. IEEE Transactions on Signal Processing. 50(2):174--188. doi:10.1109/78.978374
[3] Caswell, H. (2001). Caswell, H, Matrix Population Models: Construction, Analysis, and Interpretation. Sinauer Associates Inc., second edition. .
[4] Darwin, C. (1859). Darwin, C, On the Origin of Species by Means of Natural Selection, or the Preservation of Favoured Races in the Struggle for Life. John Murray. doi:10.1017/cbo9780511694295
[5] Day, T. and Taylor, P.D. (1996). Day, T, and Taylor, P.D. Evolutionarily stable versus fitness maximizing life histories under frequency-dependent selection. Proceedings of the Royal Society of London. Series B: Biological Sciences. 263(1368):333--338. doi:10.1098/rspb.1996.0051
[6] Dembski, W.A. and Ruse, M. (2004). Dembski, W, A. and Ruse, M. Debating Design: From Darwin to DNA. Cambridge University Press. https://www.ebook.de/de/product/3437912/debating_design.html. .
[7] Engen, S. and Saether, B.-E. (2013). Engen, S, and Saether, B.-E. Evolution in Fluctuating Environments: Decomposing Selection into Additive Components of the Robertson-Price Equation. Evolution. 68(3):854--865. doi:10.1111/evo.12310
[8] Ergon, R. (2018). Ergon, R, The environmental zero-point problem in evolutionary reaction norm modeling. Ecology and Evolution. 8(8):4031--4041. doi:10.1002/ece3.3929
[9] Ergon, T. and Ergon, R. (2017). Ergon, T, and Ergon, R. When three traits make a line: evolution of phenotypic plasticity and genetic assimilation through linear reaction norms in stochastic environments. Journal of Evolutionary Biology. 30(3):486--500. doi:10.1111/jeb.13003
[10] Fisher, R.A. (1918). Fisher, R, A. The Correlation between Relatives on the Supposition of Mendelian Inheritance. Transactions of the Royal Society of Edinburgh. 52:399--433. .
[11] Gavrilets, S. and Scheiner, S.M. (1993). Gavrilets, S, and Scheiner, S.M. The genetics of phenotypic plasticity. V. Evolution of reaction norm shape. Journal of Evolutionary Biology, 1993. 6(1):31--48. doi:10.1046/j.1420-9101.1993.6010031.x
[12] Gavrilets, S. and Scheiner, S.M. (1993). Gavrilets, S, and Scheiner, S.M. The genetics of phenotypic plasticity. VI. Theoretical predictions for directional selection. Journal of Evolutionary Biology, 1993. 6(1):49--68. doi:10.1046/j.1420-9101.1993.6010049.x
[13] Gomulkiewicz, R. and Kirkpatrick, M. (1992). Gomulkiewicz, R, and Kirkpatrick, M. Quantitative Genetics and the Evolution of Reaction Norms. Evolution. 46(2):390. doi:10.2307/2409860
[14] Irwin, K.K. and Carter, P.A. (2013). Irwin, K, K. and Carter, P.A. Constraints on the evolution of function-valued traits: a study of growth inTribolium castaneum. Journal of Evolutionary Biology. 26(12):2633--2643. doi:10.1111/jeb.12257
[15] Johnson, R.A. and Wichern, D.W. (2008). Johnson, R, A. and Wichern, D.W. Applied Multivariate Statistical Analysis. Pearson, sixth edition. .
[16] Kalman, R.E. (1960). Kalman, R, E. A New Approach to Linear Filtering and Prediction Problems. Journal of Basic Engineering. 82(1):35. doi:10.1115/1.3662552
[17] Kingsolver, J.G., Gomulkiewicz, R., and Carter, P.A. (2001). Kingsolver, J, G., Gomulkiewicz, R., and Carter, P.A. Variation, selection and evolution of function-valued traits. Genetica. 112/113:87--104. doi:10.1023/a:1013323318612
[18] Kirkpatrick, M. and Heckman, N. (1989). Kirkpatrick, M, and Heckman, N. A quantitative genetic model for growth, shape, reaction norms, and other infinite-dimensional characters. Journal of Mathematical Biology. 27(4):429--450. doi:10.1007/bf00290638
[19] Kirkpatrick, M., Lofsvold, D., and Bulmer, M. (1990). Kirkpatrick, M, , Lofsvold, D., and Bulmer, M. Analysis of the Inheritance, Selection and Evolution of Growth Trajectories. Genetics. 124(2323560):979--993. https://www.ncbi.nlm.nih.gov/pmc/PMC1203988/. .
[20] Lande, R. (1979). Lande, R, Quantitative Genetic Analysis of Multivariate Evolution, Applied to Brain: Body Size Allometry. Evolution. 33(1):402. doi:10.2307/2407630
[21] Lande, R. (2009). Lande, R, Adaptation to an extraordinary environment by evolution of phenotypic plasticity and genetic assimilation. Journal of Evolutionary Biology. 22(7):1435--1446. doi:10.1111/j.1420-9101.2009.01754.x
[22] Lande, R. (2014). Lande, R, Evolution of phenotypic plasticity and environmental tolerance of a labile quantitative character in a fluctuating environment. Journal of Evolutionary Biology. 27(5):866--875. doi:10.1111/jeb.12360
[23] Lande, R. and Arnold, S.J. (1983). Lande, R, and Arnold, S.J. The Measurement of Selection on Correlated Characters. Evolution. 37(6):1210. doi:10.2307/2408842
[24] Lewis, F.L., Xie, L., and Popa, D. (2017). Lewis, F, L., Xie, L., and Popa, D. Optimal and Robust Estimation. CRC Press, second edition. doi:10.1201/9781315221656
[25] Ljung, L. (1999). Ljung, L, System Identification: Theory for the User. Prentice Hall, second edition. .
[26] Lush, J.L. (1937). Lush, J, L. Animal breeding plans. Ames, Ia., Collegiate Press, Inc.. Page 84. .
[27] McNamara, J.M., Barta, Z., Klaassen, M., and Bauer, S. (2011). McNamara, J, M., Barta, Z., Klaassen, M., and Bauer, S. Cues and the optimal timing of activities under environmental changes. Ecology Letters. 14(12):1183--1190. doi:10.1111/j.1461-0248.2011.01686.x
[28] Newman, K.B., Buckland, S.T., Morgan, B. J.T., King, R., Borchers, D.L., Cole, D.J., Besbeas, P., Gimenez, O., and Thomas, L. (2014). Newman, K, B., Buckland, S.T., Morgan, B. J.T., King, R., Borchers, D.L., Cole, D.J., Besbeas, P., Gimenez, O., and Thomas, L. Modelling Population Dynamics: Model Formulation, Fitting and Assessment using State-Space Methods. Springer New York. doi:10.1007/978-1-4939-0977-3
[29] Page, K.M. and Nowak, M.A. (2002). Page, K, M. and Nowak, M.A. Unifying Evolutionary Dynamics. Journal of Theoretical Biology. 219(1):93--98. doi:10.1006/jtbi.2002.3112
[30] Pigliucci, M. (2008). Pigliucci, M, Is evolvability evolvable? Nature Reviews Genetics. 9(1):75--82. doi:10.1038/nrg2278
[31] Pigliucci, M. and Finkelman, L. (2014). Pigliucci, M, and Finkelman, L. The Extended (Evolutionary) Synthesis Debate: Where Science Meets Philosophy. BioScience. 64(6):511--516. doi:10.1093/biosci/biu062
[32] Pigliucci, M. and Murren, C.J. (2003). Pigliucci, M, and Murren, C.J. Perspective: Genetic assimilation and a possible evolutionary paradox: can macroevolution sometimes be so fast as to pass us by? Evolution. 57(7):1455--1464. doi:10.1111/j.0014-3820.2003.tb00354.x
[33] Price, G.R. (1970). Price, G, R. Selection and covariance. Nature. 227(5257):520--521. doi:10.1038/227520a0
[34] Prigogine, N.G. (1977). Prigogine, N, G. Self-Organization in Nonequilibrium Systems: From Dissipative Structures to Order through Fluctuation. Wiley, NY. .
[35] Pross, A. and Pascal, R. (2013). Pross, A, and Pascal, R. The origin of life: what we know, what we can know and what we will never know. Open Biology. 3(3):120190--120190. doi:10.1098/rsob.120190
[36] Qin, S.J. (2006). Qin, S, J. An overview of subspace identification. Computers & Chemical Engineering. 30(10-12):1502--1513. doi:10.1016/j.compchemeng.2006.05.045
[37] Astrom, K.J. and Murray, R.M. (2008). Astrom, K, J. and Murray, R.M. Feedback Systems. An introduction for scientists and engineers. Princeton University Press. https://www.ebook.de/de/product/7106261/karl_johan_astrom_richard_m_murray_feedback_systems.html. .
[38] Rice, S.H. (2004). Rice, S, H. Evolutionary Theory: Mathematical and Conceptual Foundations. Oxford University Press. https://www.ebook.de/de/product/3268550/sean_h_rice_evolutionary_theory.html. .
[39] Rice, S.H. (2008). Rice, S, H. A stochastic version of the price equation reveals the interplay of deterministic and stochastic processes in evolution. BMC Evolutionary Biology. 8(1):262. doi:10.1186/1471-2148-8-262
[40] Robertson, A. (1966). Robertson, A, A mathematical model of the culling process in dairy cattle. Animal Production. 8(01):95--108. doi:10.1017/s0003356100037752
[41] Schlichting, C. and Pigliucci, M. (1998). Schlichting, C, and Pigliucci, M. Phenotypic Evolution: A Reaction Norm Perspective. Sinauer. .
[42] Steppan, S.J., Phillips, P.C., and Houle, D. (2002). Steppan, S, J., Phillips, P.C., and Houle, D. Comparative quantitative genetics: evolution of the G matrix. Trends in Ecology & Evolution. 17(7):320--327. doi:10.1016/s0169-5347(02)02505-3
[43] Tallman, G., Zhu, J., Mawson, B.T., Amodeo, G., Nouhi, Z., Levy, K., and Zeiger, E. (1997). Tallman, G, , Zhu, J., Mawson, B.T., Amodeo, G., Nouhi, Z., Levy, K., and Zeiger, E. Induction of CAM in Mesembryanthemum crystallinum Abolishes the Stomatal Response to Blue Light and Light-Dependent Zeaxanthin Formation in Guard Cell Chloroplasts. Plant and Cell Physiology. 38(3):236--242. doi:10.1093/oxfordjournals.pcp.a029158
[44] Turelli, M. (2017). Turelli, M, Commentary: Fisher's infinitesimal model: A story for the ages. Theoretical Population Biology. 118:46--49. doi:10.1016/j.tpb.2017.09.003
[45] Valena, S. and Moczek, A.P. (2012). Valena, S, and Moczek, A.P. Epigenetic Mechanisms Underlying Developmental Plasticity in Horned Beetles. Genetics Research International. 2012:1--14. doi:10.1155/2012/576303

  title={{Quantitative genetics state-space modeling of phenotypic plasticity and evolution}},
  author={Ergon, Rolf},
  journal={Modeling, Identification and Control},
  publisher={Norwegian Society of Automatic Control}