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A broad exploration of nonlinear dynamics in microbial systems motivated by chemostat experiments producing deterministic chaos.

26 Aug 2019-

Abstract................................................................................................................ 3 Chapter 1: Mathematical Analysis of the Becks et al. (1995) Experiments. ............. 4 1.

Topics: Chemostat (51%)

Summary (1 min read)

1.1. Introduction.

  • The 4 coupled dependent variables were concentrations of nutrient (mg/cc) and each of the 3 microbes (cells/cc).
  • Each set of data constituted a time series (concentrations at discrete times), and deterministic chaos was identified by using a computerized version of the analytical procedure developed by Rosenstein et al. (1993) for calculating the largest Lyapunov exponent (TISEAN package [Hegger et al., (1999)).

Y K m m Y K m and

  • So Equations (1.2) adapted to the Becks et al. supplemental experiments, admittedly in a non-unique way, may be written as Equations (1.10) after dividing through by the microbial masses (changed parameters in red.).
  • Now that the predators have been made to prefer rods over cocci with an increasing rod population, the rods are disadvantaged and would tend to die out.
  • These dimensionless equations may also serve as a basis for further study of a more abstract mathematical nature.
  • Further details concerning the mathematical nature of the introduced preference change are given in the “Supplemental Information” at the end of this chapter.

1.3. Results.

  • A set of parameters that produced chaotic dynamics is listed in Table 1.1 Based on parameter values selected in Kot et al. (1992) and value ranges given in Kravchenko et al. (2004), the Table 1.1 values appear reasonable in a physiological sense.
  • D was measured carefully in the experiments, so the authors decided to work only with those values.
  • For a D value of 0.9/d (0.0375/hr), both simulations and experiments produced a classical steady state with one microbe dying out.

1.4. Discussion and Conclusions.

  • Clearly, the developed model with the mix of measured and selected parameters and coupling functions is not capturing all of the experimental details, and this would be expected with a chaotic/classical model having many unmeasured parameters and microbe coupling functions.
  • There were several interesting parallels between the experimental and model results.
  • The authors therefore conclude that the availability of experimental results and a mathematical model, both producing classical and deterministic chaotic dynamics under similar conditions, is a useful first step that provides new insight that may lead to better understanding of complex phenomena in microbial systems and motivate further studies.
  • The statistical aspects of deterministic chaotic time series also have information measures, but classical steady or periodic states do not.
  • This leads one to consider biofilms (Benefield and Molz, 1985).

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Lawrence Berkeley National Laboratory
Recent Work
Title
A broad exploration of nonlinear dynamics in microbial systems motivated by chemostat
experiments producing deterministic chaos.
Permalink
https://escholarship.org/uc/item/9wr5396s
Authors
Molz, Fred
Faybishenko, Boris
Agarwal, Deborah
Publication Date
2022-08-05
Peer reviewed
eScholarship.org Powered by the California Digital Library
University of California

1"
"
A"Broad"Exploration"of"Coupled"Nonlinear"Dynamics"in"Microbial"
Systems"Motivated"by"Chemostat"Experiments"Producing"
Deterministic"Chaos."
""
Fred"Molz
1
,"Boris"Faybishenko
2
"and"Deborah"Agarwal
3"
1
Environmental"Engineering"&"Earth"Sciences"Dept.,"Clemson"University,"342"Computer"Court,"
Anderson,"SC""29625.""
2
Energy"Geosciences"Division,"Earth"and"Environmental"Sciences"Area,"Lawrence"Berkeley"
National"Laboratory,"Berkeley,"CA""94720""
3
Computer"Research"Division,"Lawrence"Berkeley"National"Laboratory,"Berkeley,"CA""94720""
"
"
LBNL Report Number LBNL-2001172
"
"
"
"
"
"
"
"
Acknowledgements"BF"and"DA"research"supported"by"the"U.S."DOE,"Office"of"
Science,"Office"of"Biological"and"Environmental"Research,"and"Office"of"Science,"
Office"of"Advanced"Scientific"Computing"under"the"DOE"Contract"No."DE-AC02-
05CH11231."FM"acknowledges"the"support"of"the"Clemson"University,"
Department"of"Environmental"Engineering"and"Earth"Sciences."
"

2"
"
"
"
Table"of"Contents."
ABSTRACT."................................................................................................................"3"
Chapter"1:"Mathematical"Analysis"of"the"Becks"et"al."(1995)"Experiments."............."4"
1.1."Introduction."..................................................................................................."4"
1.2."Mathematical"Model"Development.".............................................................."7"
1.3."Results."......................................................................................................"13"
1.4."Discussion"and"Conclusions."........................................................................."21"
1.5."References."..................................................................................................."25"
S1.""Supplemental"Information"Concerning"Predator"Preference"Change."........."27"
Chapter"2:"Further"Study"of"the"Becks"et"al."Equations."........................................."30"
2.1."Introduction."................................................................................................."30"
2.2."Another"Model"Generalization."...................................................................."31"
2.3."Results."........................................................................................................."33"
2.4."Conclusions.".................................................................................................."38"
2.5."References."..................................................................................................."38"
Chapter"3.""Dimensionless"Forms"for"Equations"(1.10)."........................................."40"
3.1."Introduction."................................................................................................."40"
3.2."Dimensionless"Formulation."........................................................................."40"
3.3."Example"Solution"to"the"Dimensionless"Equations."....................................."44"
3.4."Results"and"Discussion."................................................................................"44"
3.5."References."..................................................................................................."51"
Chapter"4:""How"Might"Information"Theory"Relate"to"Chaotic"Dynamics"in"
Biological"Systems?"................................................................................................"52"
4.1."Introduction."................................................................................................."52"
4.2."Interpretation"of"Shannon’s"Measure."........................................................."53"
4.3."The"concept"of"Redundancy."........................................................................"58"
4.4."What"About"Continuous"Probability"Densities?"..........................................."62"
4.5."Calculation"of"Chaotic"Information"Measures."............................................."64"
4.6."Summary"and"Future"Research"Suggestions."..............................................."70"
4.7."References."..................................................................................................."74"
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3"
"
!"#$%!&$'(
The$main$objective$of$this$report$is$to$develop$an$exploratory$mathematical$
analysis$motivated$by$the$Becks$et$al.$(2005)$experiments,$which$will$be$
presented$in$Chapter$1.$$Related$details$of$the$resulting$model$will$be$
presented$in$Chapter$2,$and$a$detailed$non-dimensionalization$will$follow$in$
Chapter$3.$$During$the$research$to$be$described,$a$potential$relationship$to$
Shannon$information$theory$was$realized,$and$this$will$be$developed$in$
Chapter$4.$$Several$avenues$for$future$research$are$suggested,$and$a$more$
mathematically-oriented$presentation$of$the$non-linear$dynamical$properties$
of$the$equations$developed$in$Chapter$1$may$be$found$in$Faybishenko$et$al.$
(2018).$$
"
"" "

4"
"
&)*+,-.(/0(1*,)-2*,34*5(!6*57838(9:(,)-("-4;8(-,(*5'(</==>?(@A+-.32-6,8'"
/'/'(B6 ,.9 C D4 ,396 ' (
$ Deterministic$chaotic$dynamics$in$biological$systems$has$not$received$as$
much$attention$as$that$in$electronic$or$fluid-mechanical$systems,$and$
mathematical$models$are$at$an$early$stage$of$development$(Faybishenko$and$
Molz$(2013).$$However,$this$has$started$to$change.$$Molz$and$Faybishenko$
(2013)$have$recently$concluded $that$three $pape rs$(Bec ks$et$al,$2005 ;$Graham $
et$al.$2007;$Beninca$et$al.,$2008),$using$experimental$studies$and$relevant$
mathematics,$may$provide$convincing$demonstrations$that$deterministic$
chaos$is$present$in$relatively$simple$biochemical$systems$of$an$ecological$
nature.$(See$Constantino$et$al.,$1997$for$additional$support.)$$For$example,$
Graham$et$al.$(2007)$reported$experimental$results$demonstrating$the$
phenomenon$of$chaotic$instability$in$biological$nitrification$in$a$controlled$
laboratory$environment.$$In$this$study,$the$aerob ic$biorea ctors$(ae rated $
containers$of$nutrient$solution$and$microbes)$were$filled$initially$with$a$
mixture$of$wastewater$from$a$treatment$plant$and$simulated$wastewater$
involving$a$mixt ur e$o f$m a n y$m ic rob e s.$T h e$m a in $v ar ia bl es $re cor d ed $a s$a $time$
series$were$total$bacteria,$ammonia-oxidizing$bacteria$(AOB),$nitrite-oxidizing$
bacteria$(NOB),$and$protozoa,$along$with$effluent$concentrations$of$nitrate,$
nitrite$and$total$ammonia.$$The$method$of$Rosenstein$et$al.$(1993)$was$used$to$
calculate$Lyapunov$exponents,$which$fell$roughly$in$a$range$from$0.05$to$$
0.2$d
-1
.$$Graham$et$al .$(2 0 0 7) $co n cl u d ed $th a t $“n itr ifica tio n $is $p ro n e$t o$c h a ot ic$
behavior$because$of$a$fragile$AOB-NOB$mutualism,”$i.e.,$interaction.$
$ Beninca$et$al.$(2008)$conducted$a$laboratory$experiment$over$a$period$$
$
$

Citations
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TL;DR: The mathematical model demonstrated that the occurrence of chaotic dynamics requires a predator, p, preference for r versus c to increase significantly with increases in r and c populations, leading to a modified version of the Monod kinetics model.
Abstract: Presented is a system of four ordinary differential equations and a mathematical analysis of microbiological experiments in a four-component chemostat-nutrient n, rods r, cocci c, and predators p. The analysis is consistent with the conclusion that previous experiments produced features of deterministic chaotic and classical dynamics depending on dilution rate. The surrogate model incorporates as much experimental detail as possible, but necessarily contains unmeasured parameters. The objective is to understand better the differences between model simulations and experimental results in complex microbial populations. The key methodology for simulation of chaotic dynamics, consistent with the measured dilution rate and microbial volume averages, was to cause the preference of p for r vs. c to vary with the r and c concentrations, to make r more competitive for nutrient than c, and to recycle some dying p biomass, leading to a modified version of the Monod kinetics model. Our mathematical model demonstrated that the occurrence of chaotic dynamics requires a predator, p, preference for r versus c to increase significantly with increases in r and c populations. Also included is a discussion of several generalizations of the existing model and a possible involvement of the minimum energy dissipation principle. This principle appears fundamental to thermodynamic systems including living systems. Several new experiments are suggested.

References
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Journal ArticleDOI
TL;DR: A new method for calculating the largest Lyapunov exponent from an experimental time series is presented that is fast, easy to implement, and robust to changes in the following quantities: embedding dimension, size of data set, reconstruction delay, and noise level.
Abstract: Detecting the presence of chaos in a dynamical system is an important problem that is solved by measuring the largest Lyapunov exponent. Lyapunov exponents quantify the exponential divergence of initially close state-space trajectories and estimate the amount of chaos in a system. We present a new method for calculating the largest Lyapunov exponent from an experimental time series. The method follows directly from the definition of the largest Lyapunov exponent and is accurate because it takes advantage of all the available data. We show that the algorithm is fast, easy to implement, and robust to changes in the following quantities: embedding dimension, size of data set, reconstruction delay, and noise level. Furthermore, one may use the algorithm to calculate simultaneously the correlation dimension. Thus, one sequence of computations will yield an estimate of both the level of chaos and the system complexity.

2,659 citations


"A broad exploration of nonlinear dy..." refers methods in this paper

  • ...The method of Rosenstein et al. (1993) was used to calculate Lyapunov exponents, which fell roughly in a range from 0.05 to 0.2 d-1....

    [...]

  • ...…set of data constituted a time series (concentrations at discrete times), and deterministic chaos was identified by using a computerized version of the analytical procedure developed by Rosenstein et al. (1993) for calculating the largest Lyapunov exponent (TISEAN package [Hegger et al., (1999))....

    [...]


Journal ArticleDOI
TL;DR: A variety of algorithms for data representation, prediction, noise reduction, dimension and Lyapunov estimation, and nonlinearity testing are discussed with particular emphasis on issues of implementation and choice of parameters.
Abstract: Nonlinear time series analysis is becoming a more and more reliable tool for the study of complicated dynamics from measurements. The concept of low-dimensional chaos has proven to be fruitful in the understanding of many complex phenomena despite the fact that very few natural systems have actually been found to be low dimensional deterministic in the sense of the theory. In order to evaluate the long term usefulness of the nonlinear time series approach as inspired by chaos theory, it will be important that the corresponding methods become more widely accessible. This paper, while not a proper review on nonlinear time series analysis, tries to make a contribution to this process by describing the actual implementation of the algorithms, and their proper usage. Most of the methods require the choice of certain parameters for each specific time series application. We will try to give guidance in this respect. The scope and selection of topics in this article, as well as the implementational choices that have been made, correspond to the contents of the software package TISEAN which is publicly available from this http URL . In fact, this paper can be seen as an extended manual for the TISEAN programs. It fills the gap between the technical documentation and the existing literature, providing the necessary entry points for a more thorough study of the theoretical background.

1,315 citations


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Abstract: We examine the thermodynamic evolution of various evolving systems, from primitive physical systems to complex living systems, and conclude that they involve similar processes which are phenomenological manifestations of the second law of thermodynamics. We take the reformulated second law of thermodynamics of Hatsopoulos and Keenan and Kestin and extend it to nonequilibrium regions, where nonequilibrium is described in terms of gradients maintaining systems at some distance away from equilibrium. The reformulated second law suggests that as systems are moved away from equilibrium they will take advantage of all available means to resist externally applied gradients. When highly ordered complex systems emerge, they develop and grow at the expense of increasing the disorder at higher levels in the system's hierarchy. We note that this behaviour appears universally in physical and chemical systems. We present a paradigm which provides for a thermodynamically consistent explanation of why there is life, including the origin of life, biological growth, the development of ecosystems, and patterns of biological evolution observed in the fossil record. We illustrate the use of this paradigm through a discussion of ecosystem development. We argue that as ecosystems grow and develop, they should increase their total dissipation, develop more complex structures with more energy flow, increase their cycling activity, develop greater diversity and generate more hierarchical levels, all to abet energy degradation. Species which survive in ecosystems are those that funnel energy into their own production and reproduction and contribute to autocatalytic processes which increase the total dissipation of the ecosystem. In short, ecosystems develop in ways which systematically increase their ability to degrade the incoming solar energy. We believe that our thermodynamic paradigm makes it possible for the study of ecosystems to be developed from a descriptive science to predictive science founded on the most basic principle of physics.

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"A broad exploration of nonlinear dy..." refers background in this paper

  • ...…in a biophysical/chemical sense, needs to be studied further, which is a truly interdisciplinary problem involving biochemistry, irreversible thermodynamics, nonlinear dynamics and perhaps Shannon information theory (Schneider and Kay, 1994; Ben-Naim, 2008; Feistel and Ebling, 2011; Sagawa, 2014)....

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TL;DR: The dynamic behavior observed for the virus population rules out the possibility that it is dominated by inactive species, and the viruses are suggested to be active members of the microbial food web as agents causing lysis in parts of the bacterial population, diverting part of theacterial production from the predatory food chain.
Abstract: Population sizes of algae, bacteria, heterotrophic flagellates, and viruses were observed through the 1989 spring diatom bloom in Raunefjorden in western Norway. The culmination of the diatom bloom was followed by a peak in the concentration of bacteria and an increase in the concentration of heterotrophic flagellates, a pattern consistent with the concept of a food chain from photosynthetically produced organic material, through bacteria, to bacterivorous flagellates. The concentration of viruses varied through the spring bloom from 5 x 10 in the prebloom situation to a maximum of 1.3 x 10 viruses ml 1 week after the peak of the diatom bloom. Coinciding with the collapse in the diatom bloom, a succession of bacteria and viruses was observed in the mucous layer surrounding dead or senescent diatoms, with an estimated maximum of 23% of the total virus population attached to the diatoms. The dynamic behavior observed for the virus population rules out the possibility that it is dominated by inactive species, and the viruses are suggested to be active members of the microbial food web as agents causing lysis in parts of the bacterial population, diverting part of the bacterial production from the predatory food chain.

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"A broad exploration of nonlinear dy..." refers methods in this paper

  • ...…the most abundant microbes are weakened leads to evolutionary dynamics of specific predator-prey pairs, which oscillate in time similar to those predicted by the classical Lotka-Volterra equations, and which are also called “Kill-the-Winner” systems (Bratbak et al., 1990;; Rohwer and Barott, 2012)....

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TL;DR: The first experimental demonstration of chaos in a long-term experiment with a complex food web, isolated from the Baltic Sea, demonstrates that species interactions in food webs can generate chaos, and implies that stability is not required for the persistence of complex food webs, and that the long- term prediction of species abundances can be fundamentally impossible.
Abstract: Mathematical models predict that species interactions such as competition and predation can generate chaos. However, experimental demonstrations of chaos in ecology are scarce, and have been limited to simple laboratory systems with a short duration and artificial species combinations. Here, we present the first experimental demonstration of chaos in a long-term experiment with a complex food web. Our food web was isolated from the Baltic Sea, and consisted of bacteria, several phytoplankton species, herbivorous and predatory zooplankton species, and detritivores. The food web was cultured in a laboratory mesocosm, and sampled twice a week for more than 2,300 days. Despite constant external conditions, the species abundances showed striking fluctuations over several orders of magnitude. These fluctuations displayed a variety of different periodicities, which could be attributed to different species interactions in the food web. The population dynamics were characterized by positive Lyapunov exponents of similar magnitude for each species. Predictability was limited to a time horizon of 15-30 days, only slightly longer than the local weather forecast. Hence, our results demonstrate that species interactions in food webs can generate chaos. This implies that stability is not required for the persistence of complex food webs, and that the long-term prediction of species abundances can be fundamentally impossible.

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"A broad exploration of nonlinear dy..." refers background in this paper

  • ...Beninca et al. (2008) conducted a laboratory experiment over a period of 6.3 years, which demonstrated chaotic dynamics in a plankton community in a water sample obtained from the Baltic Sea....

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  • ...Molz and Faybishenko (2013) have recently concluded that three papers (Becks et al, 2005; Graham et al. 2007; Beninca et al., 2008), using experimental studies and relevant mathematics, may provide convincing demonstrations that deterministic chaos is present in relatively simple biochemical…...

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Frequently Asked Questions (2)
Q1. What are the contributions mentioned in the paper "A broad exploration of coupled nonlinear dynamics in microbial systems motivated by chemostat experiments producing deterministic chaos" ?

Ben-Naim et al. this paper proposed Shannon 's measure of uncertainty ( SMU ) for deterministic chaotic dynamics. 

The potential relation of chaotic dynamics to the appropriate information flows, being careful of what “ information ” actually means in an ecological context, appears very interesting, and it should be considered as a prime area for future research.