Skip to main content
SpringerLink
Log in

Criticality, adaptability and early-warning signals in time series in a discrete quasispecies model

  • Research Article
  • Published:
Frontiers in Biology

Abstract

Complex systems from different fields of knowledge often do not allow a mathematical description or modeling, because of their intricate structure composed of numerous interacting components. As an alternative approach, it is possible to study the way in which observables associated with the system fluctuate in time. These time series may provide valuable information about the underlying dynamics. It has been suggested that complex dynamic systems, ranging from ecosystems to financial markets and the climate, produce generic early-warning signals at the “tipping points,” where they announce a sudden shift toward a different dynamical regime, such as a population extinction, a systemic market crash, or abrupt shifts in the weather. On the other hand, the framework of Self-Organized Criticality (SOC), suggests that some complex systems, such as life itself, may spontaneously converge toward a critical point. As a particular example, the quasispecies model suggests that RNA viruses self-organize their mutation rate near the error-catastrophe threshold, where robustness and evolvability are balanced in such a way that survival is optimized. In this paper, we study the time series associated to a classical discrete quasispecies model for different mutation rates, and identify early-warning signals for critical mutation rates near the error-catastrophe threshold, such as irregularities in the kurtosis and a significant increase in the autocorrelation range, reminiscent of 1/f noise. In the present context, we find that the early-warning signals, rather than broadcasting the collapse of the system, are the fingerprint of survival optimization.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Addison P (1997). Fractals and chaos: An illustrated course. Bristol and Philadelphia: Institute of Publishing

    Book  Google Scholar 

  • Aldana M, Balleza E, Kauffman S, Resendiz O (2007). Robustness and evolvability in genetic regulatory networks. J Theor Biol, 245(3): 433–448

    Article  PubMed  Google Scholar 

  • Anderson J P, Daifuku R, Loeb L A (2004). Viral error catastrophe by mutagenic nucleosides. Annu Rev Microbiol, 58(1): 183–205

    Article  PubMed  CAS  Google Scholar 

  • Azhar M A, Gopala K (1992). Clustering Poisson process and burst noise. Jpn J Appl Phys, 31(Part 1, No. 2A): 391–394

    Article  Google Scholar 

  • Bak P (1996). How Nature works: the science of self-organized criticality. New York: Springer-Verlag.

    Google Scholar 

  • Bak P, Tang C, Wiesenfeld K (1987). Self-organized criticality: An explanation of the 1/f noise. Phys Rev Lett, 59(4): 381–384

    Article  PubMed  Google Scholar 

  • Ballerini M, Cabibbo N, Candelier R, Cavagna A, Cisbani E, Giardina I, Lecomte V, Orlandi A, Parisi G, Procaccini A, Viale M, Zdravkovic V (2008). Interaction ruling animal collective behavior depends on topological rather than metric distance: evidence from a field study. Proc Natl Acad Sci USA, 105(4): 1232–1237

    Article  PubMed  CAS  Google Scholar 

  • Bedeian A G, Mossholder K W (2000). On the use of the coefficient of variation as a measure of diversity. Organ Res Methods, 3(3): 285–297

    Article  Google Scholar 

  • Berglund N, Gentz B (2002). Metastability in simple climate models: pathwise analysis of slowly driven Langevin equations. Stoch Dyn, 2(03): 327–356

    Article  Google Scholar 

  • Biebricher C K, Eigen M (2005). The error threshold. Virus Res, 107(2): 117–127

    Article  PubMed  CAS  Google Scholar 

  • Biggs R, Carpenter S R, Brock W A (2009). Turning back from the brink: detecting an impending regime shift in time to avert it. Proc Natl Acad Sci USA, 106(3): 826–831

    Article  PubMed  CAS  Google Scholar 

  • Boettiger C, Hastings A (2013). Tipping points: From patterns to predictions. Nature, 493(7431): 157–158

    PubMed  CAS  Google Scholar 

  • Bull J J, Meyers L A, Lachmann M (2005), Quasispecies made simple. Plos Comput Biol, 1:e61 (0450–0460).

    CAS  Google Scholar 

  • Carpenter S R, Brock W A (2006). Rising variance: a leading indicator of ecological transition. Ecol Lett, 9(3): 311–318

    Article  PubMed  CAS  Google Scholar 

  • Carpenter S R, Brock W A (2010). Early warnings of regime shifts in spatial dynamics using the discrete Fourier transform. Ecosphere, 1(5): 10

    Article  Google Scholar 

  • Carpenter S R, Cole J J, Pace M L, Batt R, Brock W A, Cline T, Coloso J, Hodgson J R, Kitchell J F, Seekell D A, Smith L, Weidel B (2011). Early warnings of regime shifts: a whole-ecosystem experiment. Science, 332(6033): 1079–1082

    Article  PubMed  CAS  Google Scholar 

  • Cavagna A, Cimarelli A, Giardina I, Parisi G, Santagati R, Stefanini F, Viale M (2010). Scale-free correlations in starling flocks. Proc Natl Acad Sci USA, 107(26): 11865–11870

    Article  PubMed  CAS  Google Scholar 

  • Crotty S, Cameron C E, Andino R (2001). RNA virus error catastrophe: direct molecular test by using ribavirin. Proc Natl Acad Sci USA, 98(12): 6895–6900

    Article  PubMed  CAS  Google Scholar 

  • DeCarlo L T (1997). On the meaning and use of kurtosis. Psychol Methods, 2(3): 292–307

    Article  Google Scholar 

  • Doane D P, Seward L E (2011). Measuring skewness: A forgotten statistic? J Stat Educ, 19: 1–18

    Google Scholar 

  • Drake J M, Griffen B D (2010). Early warning signals of extinction in deteriorating environments. Nature, 467(7314): 456–459

    Article  PubMed  CAS  Google Scholar 

  • Eigen M (2002). Error catastrophe and antiviral strategy. Proc Natl Acad Sci USA, 99(21): 13374–13376

    Article  PubMed  CAS  Google Scholar 

  • Flandrin P (1989). On the spectrum of fractional brownian motion. IEEE Trans Inf Theory, 35(1): 197–199

    Article  Google Scholar 

  • Fossion R, Landa E, Stránský P, Velázquez V, López Vieyra J C, Garduño, García D, Frank A (2010). Scale invariance as a symmetry in physical and biological systems: Listening to photons, bubbles and heartbeats. In: Benet L, Hess P O, Torres J M, Wolf K B, editors, Symmetries in Nature: Symposium in Memoriam Marcos Moshinsky (Cuernavaca, Mexico, 7–14 august), New York: AIP Conf. Proc., volume 1323: 74–90

    Google Scholar 

  • Guttal V, Jayaprakash C (2008). Changing skewness: an early warning signal of regime shifts in ecosystems. Ecol Lett, 11(5): 450–460

    Article  PubMed  Google Scholar 

  • Halley J M (1996). Ecology, evolution and 1/f-noise. Trends Ecol Evol, 11(1): 33–37

    Article  PubMed  CAS  Google Scholar 

  • Halley J M, Inchausti P (2004). The increasing importance of 1/f-noises as models of ecological variability. Fluct Noise Lett, 4(02): R1–R6

    Article  Google Scholar 

  • Hausdorff J M, Peng C K (1996). Multiscaled randomness: A possible source of 1/f noise in biology. Phys Rev E Stat Phys Plasmas Fluids Relat Interdiscip Topics, 54(2): 2154–2157

    Article  PubMed  CAS  Google Scholar 

  • Hausdorff J M, Zemany L, Peng C K, Goldberger A L (1999). Maturation of gait dynamics: stride-to-stride variability and its temporal organization in children. J Appl Physiol, 86(3): 1040–1047

    PubMed  CAS  Google Scholar 

  • Jonsson C B, Milligan B G, Arterburn J B (2005). Potential importance of error catastrophe to the development of antiviral strategies for hantaviruses. Virus Res, 107(2): 195–205

    Article  PubMed  CAS  Google Scholar 

  • Kauffman S (1995). At home in the universe: The search for the laws of self-organization and complexity. New York, Oxford: Oxford University Press

    Google Scholar 

  • Keshner M S (1982). 1/f noise. Proc IEEE, 70(3): 212–218

    Article  Google Scholar 

  • Kleinen T, Held H, Petschel-Held G (2003). The potential role of spectral properties in detecting thresholds in the earth system: application to the thermohaline circulation. Ocean Dyn, 53(2): 53–63

    Article  Google Scholar 

  • Landa E, Morales I O, Fossion R, Stránský P, Velázquez V, Vieyra J C, Frank A (2011). Criticality and long-range correlations in time series in classical and quantum systems. Phys Rev E Stat Nonlin Soft Matter Phys, 84(1 Pt 2): 016224

    Article  PubMed  CAS  Google Scholar 

  • Lauring A S, Andino R (2010). Quasispecies theory and the behavior of RNA viruses. PLoS Pathog, 6(7): e1001005

    Article  PubMed  Google Scholar 

  • Lenski R E, Barrick J E, Ofria C (2006). Balancing robustness and evolvability. PLoS Biol, 4(12): e428

    Article  PubMed  Google Scholar 

  • Livina V N, Lenton T M (2007). A modified method for detecting incipient bifurcations in a dynamical system. Geophys Res Lett, 34(3): L03712

    Article  Google Scholar 

  • Manneville P (1980). Intermittency self-similarity and 1/f spectrum in dissipative dynamical systems. J Phys (Paris), 41(11): 1235–1243

    Article  Google Scholar 

  • Miramontes O (1995). Order-disorder transitions in the behavior of ant societies. Complexity, 1: 50–60

    Google Scholar 

  • Peng C K, Mietus J, Hausdorff JM, Havlin S, Stanley H E, Goldberger A L, (1993). Long-range anticorrelations and non-Gaussian behavior of the heartbeat. Phys Rev Lett, 70(9): 1343–1346

    Article  Google Scholar 

  • Procaccia I, Schuster H (1983). Functional renormalization-group theory of universal 1/f noise in dynamical systems. Phys Rev A, 28(2): 1210–1212

    Article  Google Scholar 

  • Rosenstein M T, Collins J J, Luca C J D (1993). A practical method for calculating largest Lyapunov exponents from small data sets. Physica D, 65(1–2): 117–134

    Article  Google Scholar 

  • Scheffer M, Bascompte J, Brock WA, Brovkin V, Carpenter S R, Dakos V, Held H, van Nes E H, Rietkerk M, Sugihara G (2009). Earlywarning signals for critical transitions. Nature, 461(7260): 53–59

    Article  PubMed  CAS  Google Scholar 

  • Scheffer M, Carpenter S, Foley J A, Folke C, Walker B (2001). Catastrophic shifts in ecosystems. Nature, 413(6856): 591–596

    Article  PubMed  CAS  Google Scholar 

  • Scheffer M, Carpenter S R, Lenton T M, Bascompte J, Brock W, Dakos V, van de Koppel J, van de Leemput I A, Levin S A, van Nes E H, Pascual M, Vandermeer J (2012). Anticipating critical transitions. Science, 338(6105): 344–348

    Article  PubMed  CAS  Google Scholar 

  • Schuster H, Just W (2005) Deterministic chaos: an introduction. Weinheim: Wiley-VCH Verlag GmbH & Co. KGaA, 4th edition.

    Book  Google Scholar 

  • Solé R V (2003). Phase transitions in unstable cancer cell populations. Eur Phys J B, 35: 117–123

    Article  Google Scholar 

  • Solé R V, Ferrer R, González-García I, Quer J, Domingo E (1999b). Red queen dynamics, competition and critical points in a model of RNA virus quasispecies. J Theor Biol, 198(1): 47–59

    Article  PubMed  Google Scholar 

  • Solé R V, Manrubia S C, Benton M, Kauffman S, Bak P (1999a). Criticality and scaling in evolutionary ecology. Trends Ecol Evol, 14(4): 156–160

    Article  PubMed  Google Scholar 

  • Solé R V, Miramontes O, Goodwin B C (1993a). Oscillations and chaos in ant societies. J Theor Biol, 161(3): 343–357

    Article  Google Scholar 

  • Solé R V, Miramontes O, Goodwin B C (1993b) Emergent behaviour in insect societies: Global oscillations, chaos and computation. In: Haken H, Mikhailov A, editors, Interdisciplinary Approaches To Nonlinear Complex Systems, Springer Series in Synergetics. Berlin Heidelberg: Springer-Verlag, volume 62:pp, 77–88

    Chapter  Google Scholar 

  • Wakeley J (2006). Coalescent theory: An introduction. Greenwood Village, Colorado: Roberts & Co

    Google Scholar 

  • Wilke C O, Wang J L, Ofria C, Lenski R E, Adami C (2001). Evolution of digital organisms at high mutation rates leads to survival of the flattest. Nature, 412(6844): 331–333

    Article  PubMed  CAS  Google Scholar 

  • Williams G P (1997). Chaos theory tamed. Washington D.C.: Joseph Henry Press

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Instituto Nacional de Geriatría, Periférico Sur No. 2767, Col. San Jerónimo Lídice, Del. Magdalena Contreras, 10200, México D.F., Mexico

    R. Fossion

  2. Centro de Ciencias de la Complejidad (C3), Universidad Nacional Autónoma de México, 04510, México D.F., Mexico

    R. Fossion, D. A. Hartasánchez & A. Frank

  3. Institut de Biologia Evolutiva (CSIC-Universitat Pompeu Fabra), 08003, Barcelona, Catalonia, Spain

    D. A. Hartasánchez

  4. Systems Biology Group, Instituto Nacional de Medicina Genomica (INMEGEN), Mexico, Mexico

    O. Resendis-Antonio

  5. Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México, 04510, México D.F., Mexico

    A. Frank

Authors
  1. R. Fossion
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. D. A. Hartasánchez
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. O. Resendis-Antonio
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. A. Frank
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to R. Fossion.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Fossion, R., Hartasánchez, D.A., Resendis-Antonio, O. et al. Criticality, adaptability and early-warning signals in time series in a discrete quasispecies model. Front. Biol. 8, 247–259 (2013). https://doi.org/10.1007/s11515-013-1256-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11515-013-1256-0

Keywords

Search

Navigation