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

TL;DR: The Mathematical Analysis of the Becks et al. Experiments as discussed by the authors, which is the most closely related work to ours, can be found in Section 3.4.1.

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

Summary

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).

Figures

Figure 3.1. Plots of the four dimensionless variables w (nutrient), x (rods), y (cocci) and z (predators). Chaotic dynamics is evident.

Table 4.1. Values for Shannon’s measures of uncertainty (SMU) and the resulting redundancy (RD) values for the discrete probability distributions shown in Figures 4.1 through 4.4. The maximum possible SMU = (29)1/2 = 4.392.

Figure 2.3. 3-D system space plots of Equations (4) solution with an inflowing nutrient concentration given by Dn0 = [0.15(1 - 0.5sin(0.52t))]. A strange attractor is evident.

Table 1.1. Parameter values used in Equations (1.10) that yield chaotic dynamics with the dilution rate utilized in Becks et al. (2005).

Figure 1.6. Simulated nutrient and microbe dynamics that match the Becks et al. [2005, Fig. 1] results for D = 0.9/d (0.0375/hr).

Figure 3.2. A system-space plot of dimensionless rods vs. cocci vs. predators yields the usual 3-D section of the 4-D strange attractor.

Figure 1.1. Diagram of the Becks et al. (2005) chemostat system. A nutrient solution flowing left to right was consumed by two microbes (rods and cocci), with rods being stronger competitors for nutrient than cocci. A ciliate predator fed on both microbes, but preferred rods over cocci.

Figure 1.7. Resulting nutrient and microbe dynamics with D = 0.75/d (0.03125/h). Both the model and experiments produced classical steady states, but all microbes survived in the experiments while the rods died out in the model.

Figure 4.1. Discrete probability distribution for the dimensionless predator values shown in Figure 3.1. The domain of values was divided into 21 ranges of equal width 1.085E-1.

Table 2.2. New parameters in Equations (2.1), and a set of test values. Units of “S” are mg-1, units of the specific death rates “δ” are hr-1, and 0 ≤ EF ≤ 1 is dimensionless.

Table 1.2. Lyapunov exponents used to identify the presence of deterministic chaos based on the time series of concentrations shown in Figure 1.3 (Calculations were conducted using the R package “fractal” version: 2.0-1.)

Figure 3.5. Dimensionless plot of predator cells (y-axis) vs. rod cells (x-axis). Again, the relationship is not random in the classical sense.

Figure 4.2. Discrete probability distribution for the dimensionless nutrient values shown in Figure 3.1. The domain of values was divided into 21 ranges of equal width 3.6525E-2.

Table 2.3. Diagnostic nonlinear dynamics parameters used to identify the presence of deterministic chaos based on the time series of concentrations shown in Figures 2.2 and 2.3. (Note: Analysis using R software packages “FRACTAL”).

Figure 3.4. Dimensionless plot of predator cells (y-axis) vs. cocci cells (x-axis). Clearly, the relationship is not random in the classical sense.

Figure 1.4. Resulting nutrient and microbe kinetics with D = 0.45/d (0.01875/hr.), as was done in the Becks et al. (2005) experiments.

Figure 3.6. Dimensionless plot of Cocci cells (y-axis) vs. rods (x-axis). Again, the relationship is not random in the classical sense

Figure 4.3. Discrete probability distribution for the dimensionless cocci values shown in Figure 3.1. The domain of values was divided into 21 ranges of equal width 2.35E-2.

Figure 1.3. Irregular, non-periodic, dynamics of n(t), r(t), c(t) and p(t) associated with the system-space plot in Fig 1.2. By counting apparent concentration peaks, the mean frequency of the simulated data appears lower than that observed in Becks et al. [2005], but still within 50%. However, the mean model frequency can be selected by changing the dimensionless ratios of max specific growth and death rates to D (Chapter 3).

Table 2.1. Parameter values selected for an initial mathematical analysis of the Becks et al. experiments using the generalized Kot et al. Model (2.1). Units for “μ” are hr-1, for “K” are mg/cc, and “Y” are mg/mg. Microbial masses are held constant at the values given (mg) in Chapter 1.

Figure 2.2. Plots of nutrient and microbes as functions of time resulting from a solution of Equations (2.2), using parameter values specified in Tables 2.1 and 2.2, along with a sinusoidal, 12 hr. period, feeding rate. The only parameter difference with the solution shown in Figure 2.1 is the sinusoidal feeding rate.

Full text

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"
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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"
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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"
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!"#$%!&$'(
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
.$$Graham$et$al .$(2 0 0 7) $co n cl u d ed $th a t $“n itr ifica tio n $is $p ro n e$t o$c h a ot ic$
behavior$because$of$a$fragile$AOB-NOB$mutualism,”$i.e.,$interaction.$
$ Beninca$et$al.$(2008)$conducted$a$laboratory$experiment$over$a$period$$
$
$

Frequently asked questions

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. 

Q2. What future works have the authors mentioned in the paper "A broad exploration of coupled nonlinear dynamics in microbial systems motivated by chemostat experiments producing deterministic chaos" ?

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.