Reading the Secrets of Biological Fluctuations

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I explore tools and applications of the van Kampen expansion of the stochastic master equation in several areas of population biology. I highlight fluctuation-dissipation and …

I explore tools and applications of the van Kampen expansion of the stochastic master equation in several areas of population biology. I highlight fluctuation-dissipation and fluctuation-enhancement regimes, using information from noise to help model choice, and examples where noise changes the macroscopic dynamics of a system.

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  • 1. Noisy Biology Fluctuation Regimes Model Choice Macroscopic Phenomena Fluctuation Dominance Reading the Secrets of Biological Fluctuations Carl Boettiger UC Davis June 27, 2008 Carl Boettiger, UC Davis Biological Fluctuations 1/21
  • 2. Noisy Biology Fluctuation Regimes Model Choice Macroscopic Phenomena Fluctuation Dominance 1 Noisy Biology 2 Fluctuation Regimes 3 Model Choice 4 Macroscopic Phenomena 5 Fluctuation Dominance Carl Boettiger, UC Davis Biological Fluctuations 2/21
  • 3. Noisy Biology Fluctuation Regimes Model Choice Macroscopic Phenomena Fluctuation Dominance Why study fluctuations? Biology is noisy and we want to understand it. Stochasticity can drive phenomena we would miss in deterministic models. Fluctuations hold the key to deeper biological understanding? Grenfell et al. (1998) Nature Carl Boettiger, UC Davis Biological Fluctuations 3/21
  • 4. Noisy Biology Fluctuation Regimes Model Choice Macroscopic Phenomena Fluctuation Dominance Variables at the Macroscopic and Individual Levels Deterministic models describe macroscopic behavior Individual based model are described by transition rates between states – a Markov process Macroscopic variable φ is independent of details of system (intensive), i.e. population density Individual-based variable n depends on system size (extensive), i.e. population number In a given area Ω with a macroscopic density φ, we’d expect to find n = φΩ on average, which is more accurate with larger Ω. Carl Boettiger, UC Davis Biological Fluctuations 4/21
  • 5. Noisy Biology Fluctuation Regimes Model Choice Macroscopic Phenomena Fluctuation Dominance Theory of Fluctuations Markov process Linear Noise Approximation 1/2 ⇒ = n = Ωφ+Ω ξ=⇒ Fundamental Equations dφ = α1,0 (φ)+α1,0 (φ)σ 2 (1) dt dσ 2 = 2α1,0 (φ)σ 2 + α2,0 (φ) (2) dt α1,0 (φ) = b(φ) − d(φ), α2,0 = b(φ) + d(φ) Carl Boettiger, UC Davis Biological Fluctuations 5/21
  • 6. Noisy Biology Fluctuation Regimes Model Choice Macroscopic Phenomena Fluctuation Dominance 1 Noisy Biology 2 Fluctuation Regimes 3 Model Choice 4 Macroscopic Phenomena 5 Fluctuation Dominance Carl Boettiger, UC Davis Biological Fluctuations 6/21
  • 7. Noisy Biology Fluctuation Regimes Model Choice Macroscopic Phenomena Fluctuation Dominance Distinct Fluctuation Regimes dn n “ n” n =c 1− −e dt | N {z N } |{z}N bn dn dσ 2 = 2α1,0 (φ)σ 2 + α2,0 (φ) dt Carl Boettiger, UC Davis Biological Fluctuations 7/21
  • 8. Noisy Biology Fluctuation Regimes Model Choice Macroscopic Phenomena Fluctuation Dominance Near Equilibrium: Fluctuation Dissipation Regime In the dissipation regime, fluctuations exponentially relax to the equilibrium level b(n)+d(n) σ2 = ˆ 2[d (n)−b (n)] N = 1000, e = 0.2, c=1 e ˆ ˜ n = N 1 − c = 800 ˆ σ 2 = N c = 200 ˆ e Dots are simulation averages, lines are theoretical prediction Carl Boettiger, UC Davis Biological Fluctuations 8/21
  • 9. Noisy Biology Fluctuation Regimes Model Choice Macroscopic Phenomena Fluctuation Dominance Fluctuation Enhancement With an initial condition starting deep in the enhancement regime, fluctuations grow exponentially. At N = 400, dissipation takes over and fluctuations return to the same equilibrium as before. Carl Boettiger, UC Davis Biological Fluctuations 9/21
  • 10. Noisy Biology Fluctuation Regimes Model Choice Macroscopic Phenomena Fluctuation Dominance 1 Noisy Biology 2 Fluctuation Regimes 3 Model Choice 4 Macroscopic Phenomena 5 Fluctuation Dominance Carl Boettiger, UC Davis Biological Fluctuations 10/21
  • 11. Noisy Biology Fluctuation Regimes Model Choice Macroscopic Phenomena Fluctuation Dominance Which model best describes this data? dn n “ n” n =c 1− −e dt | N {z N } |{z} N bn dn dn rn2 = |{z} − rn dt K bn |{z} dn . . . and why does it matter? Carl Boettiger, UC Davis Biological Fluctuations 11/21
  • 12. Noisy Biology Fluctuation Regimes Model Choice Macroscopic Phenomena Fluctuation Dominance Using the Information Hidden in the Fluctuations 1 Independently parameterize birth & death rates, see which is density dependent Predicted fluctuations 2 Works with single realization at equilibrium 3 With replicates: The dynamic equations can determine functions b(n) and d(n) 4 Uses more information to inform model choice 5 Can discount weights of points from high-variance regions when model-fitting Carl Boettiger, UC Davis Biological Fluctuations 12/21
  • 13. Noisy Biology Fluctuation Regimes Model Choice Macroscopic Phenomena Fluctuation Dominance 1 Noisy Biology 2 Fluctuation Regimes 3 Model Choice 4 Macroscopic Phenomena 5 Fluctuation Dominance Carl Boettiger, UC Davis Biological Fluctuations 13/21
  • 14. Noisy Biology Fluctuation Regimes Model Choice Macroscopic Phenomena Fluctuation Dominance Stochastic Corrections: Deflation and Inflation dφ α1,0 (φ) < 0 =⇒ Fluctuations = α1,0 (φ)+α1,0 (φ)σ 2 dt suppress the average relative to the deterministic approximation. Our theory accurately predicts the extent of this effect. Recall α2,0 = bn + dn controls the magnitude of this effect. Ecological and evolutionary consequences for when variability is favorable? Carl Boettiger, UC Davis Biological Fluctuations 14/21
  • 15. Noisy Biology Fluctuation Regimes Model Choice Macroscopic Phenomena Fluctuation Dominance Fluctuation Phenomena: Deflation dn n “ n” n =c 1− −e dt | N {z N } |{z}N bn dn dφ = α1,0 (φ)+α1,0 (φ)σ 2 dt Carl Boettiger, UC Davis Biological Fluctuations 15/21
  • 16. Noisy Biology Fluctuation Regimes Model Choice Macroscopic Phenomena Fluctuation Dominance 1 Noisy Biology 2 Fluctuation Regimes 3 Model Choice 4 Macroscopic Phenomena 5 Fluctuation Dominance Carl Boettiger, UC Davis Biological Fluctuations 16/21
  • 17. Noisy Biology Fluctuation Regimes Model Choice Macroscopic Phenomena Fluctuation Dominance Fluctuation Dominance Far from equilibrium, enhancement can expand the fluctuations until they reach the macroscopic scale. Variance equation fails dramatically Mean trajectory need not follow the deterministic trajectory Bimodal distribution of trajectories can emerge Conjecture: occurs when neighborhood exists for which α1,0 ≈ 0 and α1,0 ≈ 0 Carl Boettiger, UC Davis Biological Fluctuations 17/21
  • 18. Noisy Biology Fluctuation Regimes Model Choice Macroscopic Phenomena Fluctuation Dominance Breakdown of the approximation Carl Boettiger, UC Davis Biological Fluctuations 18/21
  • 19. Noisy Biology Fluctuation Regimes Model Choice Macroscopic Phenomena Fluctuation Dominance Breakdown of the Canonical Equation of Adaptive Dynamics Carl Boettiger, UC Davis Biological Fluctuations 19/21
  • 20. Noisy Biology Fluctuation Regimes Model Choice Macroscopic Phenomena Fluctuation Dominance Further Topics This approach can be applied to a variety of stochastic processes in biology. . . The multivariate theory: multiple species or age structured populations. Predicts covariances as well. Macroevolutionary theory: inferring speciation and extinction rates from phylogenetic trees Adaptive dynamics: quantifying uncertainty in the canonical equation, correcting for fluctuations. Carl Boettiger, UC Davis Biological Fluctuations 20/21
  • 21. Noisy Biology Fluctuation Regimes Model Choice Macroscopic Phenomena Fluctuation Dominance Acknowledgments Advisors & Advice Alan Hastings Joshua Weitz Many here for helpful discussions! Funding DOE CSGF UC Davis Population Biology Graduate Group Carl Boettiger, UC Davis Biological Fluctuations 21/21