Edited by: Anke Marianne Herrmann, Swedish University of Agricultural Sciences, Sweden
Reviewed by: Mustafa Yucel, Middle East Technical University, Turkey; John Senko, University of Akron, United States
This article was submitted to Microbiological Chemistry and Geomicrobiology, a section of the journal Frontiers in Environmental Science
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Determining how microbial communities organize and function at the ecosystem level is essential to understanding and predicting how they will respond to environmental change. Mathematical models can be used to describe these communities, but properly representing all the biological interactions in extremely diverse natural microbial ecosystems in a mathematical model is challenging. We examine a complementary approach based on the maximum entropy production (MEP) principle, which proposes that systems with many degrees of freedom will likely organize to maximize the rate of free energy dissipation. In this study, we develop an MEP model to describe biogeochemistry observed in Siders Pond, a phosphate limited meromictic system located in Falmouth, MA that exhibits steep chemical gradients due to density-driven stratification that supports anaerobic photosynthesis as well as microbial communities that catalyze redox cycles involving O, N, S, Fe, and Mn. The MEP model uses a metabolic network to represent microbial redox reactions, where biomass allocation and reaction rates are determined by solving an optimization problem that maximizes entropy production over time, and a 1D vertical profile constrained by an advection-dispersion-reaction model. We introduce a new approach for modeling phototrophy and explicitly represent oxygenic photoautotrophs, photoheterotrophs and anoxygenic photoautotrophs. The metabolic network also includes reactions for aerobic organoheterotrophic bacteria, sulfate reducing bacteria, sulfide oxidizing bacteria and aerobic and anaerobic grazers. Model results were compared to observations of biogeochemical constituents collected over a 24 h period at 8 depths at a single 15 m deep station in Siders Pond. Maximizing entropy production over long (3 day) intervals produced results more similar to field observations than short (0.25 day) interval optimizations, which support the importance of temporal strategies for maximizing entropy production over time. Furthermore, we found that entropy production must be maximized locally instead of globally where energy potentials are degraded quickly by abiotic processes, such as light absorption by water. This combination of field observations and modeling results indicate that natural microbial systems can be modeled by using the maximum entropy production principle applied over time and space using many fewer parameters than conventional models.
Mass and energy flow associated with the growth and predation of bacteria, archaea and eukaryotes in microbial food webs, coupled with abiotic reactions and transport processes, define biogeochemical cycles that occur in ecosystems ranging in size from less than a liter (Marino et al.,
Understanding how ecosystems function at the systems level has a long tradition in theoretical ecology (Chapman et al.,
In this study we develop an MEP-based model to predict microbial biogeochemistry in a meromictic pond located in Falmouth, MA (Siders Pond) that includes metabolic processes for phytoplankton, green sulfur bacteria, aerobic organoheterotrophic bacteria, sulfate reducing bacteria, sulfide oxidizing bacteria, photoheterotrophs and aerobic and anaerobic predators (Figure
Schematic of catalysts and associated reactions used in the MEP model for Siders Pond. Functional groups include:
Our interest in investigating the applicability of MEP theory for describing microbial communities has, in addition to the basic science question, an applied objective. Standard models used to describe microbial biogeochemistry contain a large number of poorly constrained parameters, such as maximum growth rates, substrate affinity constants, growth efficiencies, prey preferences, substrate inhibition constant, and others. Consequently, data are required to determine parameter values, but since there is almost always more parameters than constraining observations, it is not difficult to obtain good agreement between model output and observations. Consequently, a good model fit does not equate to an understanding of the underlying mechanisms in a system. As a consequence, such models usually do poorly when extrapolated to different systems (Vallino,
A challenging and unresolved aspect of MEP principle involves the temporal and spatial scales over which it applies (Dewar et al.,
This section describes Siders Pond sample collection procedures and sample analyses followed by the development of the MEP model and associated 1D transport model for Siders Pond.
Siders Pond is a small coastal meromictic kettle hole (volume: 10^{6} m^{3}; area: 13.4 ha; maximum depth: 15 m) that receives approximately 1 × 10^{6} m^{3} of fresh and 0.15 × 10^{6} m^{3} of saltwater each year (Caraco,
Samples were collected from Siders Pond, Falmouth, MA over a 24 h period starting at 6:45 on Jun 25th and ending at 7:37 on Jun 26th, 2015 from a single station located within the deepest basin of the pond (41.548212°N, 70.622412°W). A total of 7 casts were conducted over the 24 hr period, and each cast sampled 8 depths to generate a 2D sampling grid designed for comparison to model outputs (Figure
Layout of the Siders Pond 2D (
Unless otherwise noted, all samples were filtered as described above and stored in 20 mL acid-washed, high-density polyethylene scintillation vials (Fisher Scientific). Samples were preserved for later analysis as follows. Inorganic phosphate: 15 mL samples were amended with 20 μL of 5 N HCl and placed on dry ice. Dissolved inorganic carbon (DIC) and sulfate: samples were collected in 12 mL exetainers (Labco, UK) by overfilling bottles from the bottom up, and then capped without bubbles and placed on ice. Hydrogen sulfide: while filling exetainers, 25 μL of sample was pipette transferred to 6 mL of 2% zinc acetate in a 20 mL scintillation vial and place on ice. Dissolved organic carbon (DOC): 25 mL of sample was collected in previously ashed 30 mL glass vials to which 40 μL of 5 N HCl was added then stored on ice. Particulate organic carbon (POC): approximately 300 mL of sample was passed through new, ashed, 25 mm GF/F filter then stored in a plastic Petri dish on dry ice.
Samples were analyzed at the Ecosystems laboratory, MBL as follows. Phosphate: samples were stored at −20°C then later analyzed following the spectrophotometric method of Murphy and Riley (
The equations used to model biogeochemistry in Siders Pond are provided in detail in the
Names of state variables and associated symbols.
Salt | Phytoplankton | ||
Dissolved oxygen | Green sulfur bacteria | ||
Dissolved inorganic carbon | Aerobic predators | ||
Inorganic phosphate | Anaerobic predators | ||
Sulfate | Aerobic organoheterotrophic bacteria | ||
Hydrogen sulfide | Sulfate reducing bacteria | ||
Phytoplankton carbohydrates | Photoheterotrophs | ||
Green sulfur bacteria carbohydrates | Sulfide oxidizing bacteria | ||
Labile organic carbon | |||
Refractory organic carbon | |||
Refractory organic phosphate |
Our approach views a complex microbial community as a collection of catalysts (denoted with the special symbol
Reactions associated with the 8 biological catalysts,
The catalyst
respectively, where the stoichiometric coefficients,
The reaction efficiency variable, ε_{SRB}, is
Gibbs free energies of reaction, Δ_{r}
where
In this version of the MEP model we introduce catalysts associated with phototrophic growth, specifically phytoplankton (Phy),
The carbon fixation reaction for phytoplankton is given by,
where
where
and the Gibbs free energy of reaction for CO_{2} fixation defined by Equation (5) is given by,
In this formulation, ε_{Phy} governs the efficiency for the conversation of electromagnetic potential into chemical potential. If 100% of photosynthetic active radiation is converted to chemical potential, no entropy is produced, and the free energy of reaction for Equation (5) is zero, so the reaction will not proceed due to thermodynamic constraints. However, as ε_{Phy} is decreased, some photons are dissipated as heat, which drives the reaction forward, and all photons are dissipated as heat when ε_{Phy} = 0, but no catalyst is synthesized.
The reaction rate for CO_{2} fixation depends on the rate of photon capture, which is given by,
where
where
The reaction for photoheterotrophs (PH) differs slightly from that above. In this case only one reaction is used (
Siders Pond is horizontally well mixed, so an advection-dispersion-reaction (ADR) model that includes particle sinking was used to approximate vertical transport of the 19 state variables (Figure
The primary external drivers in the model are temperature, pH and photosynthetic active radiation. Surface irradiance was based on a model of solar zenith angle (Brock,
Neumann boundary conditions were used for state variables at the pond's surface and gas transport for O_{2}, CO_{2}, and H_{2}S across the air-water interface was accounted for using a stagnant-film model. Robin boundary conditions were used for the bottom boundary based on the flux of material entering the boundary associated with the intrusion of seawater diluted with groundwater. In addition, aerobic and anaerobic decomposition of sinking organic matter from the water column contributed to a sink for O_{2} and H_{2}SO_{4}, and a source for H_{3}PO_{4}, H_{2}CO_{3}, and H_{2}S to the overlying water.
Entropy production occurs when an energy potential is destroyed and dissipated as heat to the environment, but not when the potential is converted to another potential. For example, a flame converting chemical potential into heat or light being absorbed by water both result in maximum entropy production; these are irreversible processes and the Gibbs free energy is destroyed. On the other hand, entropy is not produced if electromagnetic potential is converted reversibly into chemical potential, but thermodynamic theory requires that reversible reactions must proceed infinitely slowly. In the model, as the reaction efficiency for phytoplankton, ε_{Phy}, approaches 1, electromagnetic potential is converted to chemical potential without entropy production, but the thermodynamic force,
Instantaneous entropy production per unit volume,
Average entropy production,
The value of △
Entropy production associated with the dissipation of electromagnetic radiation is readily calculated from the Gibbs free energy for photons, Equation (6), and photon capture rate, Equation (9). The photon free energy can be dissipated as heat along two pathways: (1) interception by water or (2) particles, such as bacteria and grazers, as well as by the non-photosynthetic components of phytoplankton. Photons intercepted by the photosynthetic machinery of phototrophs can be either dissipated as heat, if ε_{j} = 0, or all energy potential can be transferred to chemical potential, if ε_{j} = 1, but typically it is a combination of both, so that 0 < ε_{j} < 1. Consequently, total entropy production,
The stoichiometry, thermodynamics and kinetics of the 28 reactions that comprise the metabolic network (Table
If the MEP model developed here were cast as a conventional biogeochemistry model, there would be 89 biological parameters associated with growth kinetics. Instead, there are 28 optimal control (OC) variables (8 ε_{j} and 20 Ω_{i, j}), but only one biological parameter, because the OC variables are determined by maximizing entropy production described above. The sole biological parameter, Δ
To determine model performance, 2D linear interpolation was used to extract values of model state variables at times and depths corresponding to those taken for observations. Root mean squared errors were then calculated between interpolated model outputs and observations to quantify model skill (Fitzpatrick,
All solutions presented here are from local entropy maximization at 10 depths (0, 1, 2, 3, 4, 6, 8, 10, 12, and 15 m). We investigated two optimization time intervals, △
Solutions obtained from the short (0.25 day) and long (3 day) interval optimizations are compared to biogeochemical observations from Siders Pond that were collected to form a 2D sample grid on Jun 25th and 26th, 2015 (Figure
Root mean squared errors between model predictions and observations for the short (SIO,
0.25 | 484. | 32.9 | 6050. | 2500. | 3650. | 388. | |||
3.00 | 212. | 4.88 | 198. |
Contour plots of photosynthetic active radiation (PAR)
Simulations of substrate concentrations for autotrophs, namely inorganic phosphate and dissolved inorganic carbon (DIC), show qualitative agreement to observations for both SIO and LIO solutions (Figure
Contour plots of inorganic phosphate
Simulations of hydrogen sulfide show a chemocline at approximately 9 and 10 m for the SIO and LIO solutions, respectively, which are slightly deeper than the H_{2}S chemocline observed in Siders Pond at about 8 m (Figures
Contour plots of hydrogen sulfide
The last of the observations are dissolved organic carbon (DOC) and particulate organic carbon (POC) concentrations (Figure
Contour plots of dissolved organic carbon (DOC)
Since the MEP optimization model generates a large number of outputs, this section highlights some of those outputs to contrast the simulations based on the short (0.25 d) interval optimization (SIO) to that from the long (3 d) interval optimization (LIO). Consider entropy production, which is the variable that is being sequentially maximized over a 0.25 d interval (SIO) or a 3 d interval (LIO) at 10 different depths (Figure
Contour plots of simulated total entropy production,
The different contributors to total entropy production, namely by reaction,
An analysis of phytoplankton (Phy) growth by the SIO and LIO simulations (Figure
Contour plots of phytoplankton concentration,
Instead of producing phytoplankton, the SIO solution produces more green sulfur bacteria (GSB), which reach a maximum concentration of 735 mmol m^{−3}, compare to only 23 mmol m^{−3} in the LIO solution (Figures
Contour plots of green sulfur bacteria concentration,
Contour plots of photoheterotroph concentration,
Bacterial densities are similar in the two simulations (Figures
Another significant difference between the SIO and LIO simulations is a greater importance of predation,
Contour plots of grazer concentration,
An interesting result that derives from the focus on dissipating energy potentials rather than on growing organisms is the importance of chemotrophs on dissipating electromagnetic potential. For instance, aerobic organoheterotrophic bacteria,
Contour plots of entropy production by bacteria associated with light absorption by bacterial biomass,
The two primary objectives of this study were to demonstrate that (1) a model based on free energy dissipation can reasonably describe microbial community organization and function with relatively few parameters and (2) that microbial systems operate collectively over characteristic timescales that are likely longer than what common wisdom would suggest. The secondary objectives were to demonstrate how the model can be used in systems with spatial dimensions and to extend the approach to include phototrophs. While improvements could be made with explicit data assimilation (Edwards et al.,
Temporal strategies, such as circadian clocks (Wolf and Arkin,
One of the main questions that arise in applying MEP is what is the appropriate timescale over which biology organizes? That is, in the current implementation of the model, what is an appropriate value for △
Strategies that coordinate function over space also lead to greater free energy dissipation (Vallino,
Other areas that could improve the MEP modeling approach are as follows. Our metabolic network (Figure
Our results demonstrate that models based on MEP can reasonably simulate how microbial communities organize and function in Siders Pond over time and space while using a minimum of adjustable parameters. The improved qualitative and quantitative agreement between model predictions and observations using long (3 day, LIO) versus short (0.25 day, SIO) interval optimization supports the hypothesis that biological systems maximize entropy production over long time scales. The modeling presented here extends the MEP approach to include an explicit spatial dimension, and new metabolic reactions were introduced to model phototrophs and entropy production associated with the destruction of electromagnetic potential. By considering the dissipation of both chemical and electromagnetic potentials, the MEP model shows that heterotrophs, such as bacteria, dissipate far more free energy in the photic zone by passive light absorption than by chemical respiration. Short interval optimization results in higher grazing rates and turnover of organic carbon, as well as rapid (bang-bang) changes in the reaction control variables, while long interval optimization supports higher phytoplankton growth and standing stocks near the surface of the pond. We also found that maximization of local entropy production, as opposed to global entropy production, must be used for energy potentials that are quickly dissipated by abiotic processes, such as light absorption by water and particles. Taken together, results validate our general conjecture that biological systems evolve and organize to maximize entropy production over the greatest possible spatial and temporal scales, while abiotic processes maximize instantaneous and local entropy production.
Both JV and JH contributed equally to project concepts and design, and both conducted the 24 h sampling of Siders Pond. Preliminary results from metatranscriptomics and metagenomics from JH were used to improve development of the metabolic network. JV developed the model and wrote first draft of the manuscript with editing and inputs during the process from JH. Both authors contributed to manuscript revision, read and approved the submitted version.
The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
We thank Petra Byl for assistance in field work preparation and sampling of Siders Pond, as well as insightful conversations about the project. We also thank Jane Tucker for dissolved inorganic carbon analyses, Suzanne Thomas for hydrogen sulfide analyses, Sam Kelsey for dissolved organic carbon analyses, Kate Morkeski for sulfate, phosphate and particulate organic carbon analyses, and Rich McHorney, Leslie Murphy, Emily Reddington, and Gretta Serres for assistance with Siders Pond sampling logistics. We also thank the Semester in Environmental Science Program at MBL for use of their Hydrolab and Jon Boat used for sampling, and a special thank you to Tom Gregg who allowed us to access Siders Pond through his property as well as for providing logistical support.
The Supplementary Material for this article can be found online at: