A theoretical derivation of the monod equation with a kinetics sense

Biodegradation of poly-3-hydroxybutyrate after soil inoculation with microbial consortium: Soil microbiome and plant responses to the changed environment

Biodegradable plastics play a vital role in addressing global plastics disposal challenges. Poly-3-hydroxybutyrate (P3HB) is a biodegradable bacterial intracellular storage polymer with substantial usage potential in agriculture. Poly-3-hydroxybutyrate and its degradation products are non-toxic; however, previous studies suggest that P3HB biodegradation negatively affects plant growth because the microorganisms compete with plants for nutrients. One possible solution to this issue could be inoculating soil with a consortium of plant growth-promoting and N-fixing microorganisms. To test this hypothesis, we conducted a pot experiment using lettuce (Lactuca sativa L. var. capitata L.) grown in soil amended with two doses (1 % and 5 % w/w) of P3HB and microbial inoculant (MI). We tested five experimental variations: P3HB 1 %, P3HB 1 % + MI, P3HB 5 %, P3HB 5 % + MI, and MI, to assess the impact of added microorganisms on plant growth and P3HB biodegradation. The efficient P3HB degradation, which was directly dependent on the amount of bioplastics added, was coupled with the preferential utilization of P3HB as a carbon (C) source. Due to the increased demand for nutrients in P3HB-amended soil by microbial degraders, respiration and enzyme activities were enhanced. This indicated an increased mineralisation of C as well as nitrogen (N), sulphur (S), and phosphorus (P). Microbial inoculation introduced specific bacterial taxa that further improved degradation efficiency and nutrient turnover (N, S, and P) in P3HB-amended soil. Notably, soil acidification related to P3HB was not the primary factor affecting plant growth inhibition. However, despite plant growth-promoting rhizobacteria and N2-fixing microorganisms originating from MI, plant biomass yield remained limited, suggesting that these microorganisms were not entirely successful in mitigating the growth inhibition caused by P3HB.

Disentangling the growth curve of microbial culture

Many researchers have studied the population dynamics of microbe of microbes as a typical example of population dynamics. The Monod equation, which mainly focuses on the growth and stationary phases, is used when plotting a growth curve. However, the growth potential in the late stage of culture has been overlooked. Previous studies considered the direct degradation of products to the limiting substrate. In this study, we considered microbial growth during the stationary phase, which enables us to describe the dynamics precisely. The microbes were divided into two populations: one grew by consuming the limiting substrate and the other degraded the products by metabolism. According to the numerical analysis of our model, microbes may choose one of two strategies: one consumes substrates and expands quickly, and the other grows slowly while cleaning up the environment in which they thrive. Furthermore, we found three types of microbial growth depending on their ability to detect metabolite accumulation. Using experimentally measured data, this model can estimate the dynamics of cell density, the substrates, and the metabolites used. The model's disentangling of growth curves offers novel interpretive possibilities for culture system dynamics.

Simulating degradation of organic compounds accounting for the growth of microorganisms (Monod kinetics) in a fully Lagrangian framework

Biologically mediated degradation of organic compounds in porous media is a complex mathematical problem, described by a non-linear differential equation. The organic compound gets in contact with the biomass, and an enzyme-catalysed reaction takes place. The net result is that part of the parent compound degrades into some daughter product, while some of the organic carbon is used for microbial growth. The rate of biomass growth in the presence of a limiting nutrient supply is usually modelled with the experimentally derived Monod equation, i.e., it is proportional to the actual existing biomass multiplied by a factor that is non-linear in terms of available organic matter. This non-linearity in the degradation equation implies a strong difficulty in directly implementing a numerical solution within a fully Lagrangian framework, and thus, numerical solutions have traditionally been sought in either an Eulerian, or else an Eulerian-Lagrangian framework. Here we pursue a fully Lagrangian solution to the problem. First, the Monod empirical equation is formulated as the outcome of a two-step reaction; while the approach is less general than other derivations existing in the literature based on a full understanding of the thermodynamics of the process, it allows two things: 1) providing some physical meaning to the actual parameters in the Monod equation, and more interestingly, 2) formulating a methodology for the solution of the degradation equation incorporating Monod kinetics by means of a particle tracking formulation. For the latter purpose, both reactants and biomass are represented by particles, and their location at any given time is represented by a kernel that accounts for the uncertainty in the actual physical location. By solving the reaction equation in a kernel framework, we can reproduce the Monod kinetics and, as a particular result in the case no biomass growth is allowed, the Michaelis-Menten kinetics. The methodology proposed is then successfully applied to reproduce two studies of microbially induced degradation of organic compounds in porous media, first, the observed kinetics of Pseudomonas putida F1 in batch reactors while growing on benzene, toluene and phenol, and second, the column study of carbon tetrachloride biodegradation by the denitrifying bacterium Pseudomonas Stutzeri KC.

Density-dependent effects are the main determinants of variation in growth dynamics between closely related bacterial strains

Although closely related, bacterial strains from the same species show significant diversity in their growth and death dynamics. Yet, our understanding of the relationship between the kinetic parameters that dictate these dynamics is still lacking. Here, we measured the growth and death dynamics of 11 strains of Escherichia coli originating from different hosts and show that the growth patterns are clustered into three major classes with typical growth rates, maximal fold change, and death rates. To infer the underlying phenotypic parameters that govern the dynamics, we developed a phenomenological mathematical model that accounts not only for growth rate and its dependence on resource availability, but also for death rates and density-dependent growth inhibition. We show that density-dependent growth is essential for capturing the variability in growth dynamics between the strains. Indeed, the main parameter determining the dynamics is the typical density at which they slow down their growth, rather than the maximal growth rate or death rate. Moreover, we show that the phenotypic landscape resides within a two-dimensional plane spanned by resource utilization efficiency, death rate, and density-dependent growth inhibition. In this phenotypic plane, we identify three clusters that correspond to the growth pattern classes. Overall, our results reveal the tradeoffs between growth parameters that constrain bacterial adaptation.

Simple Growth–Metabolism Relations Are Revealed by Conserved Patterns of Heat Flow from Cultured Microorganisms

Quantitative analyses of cell replication address the connection between metabolism and growth. Various growth models approximate time-dependent cell numbers in culture media, but physiological implications of the parametrizations are vague. In contrast, isothermal microcalorimetry (IMC) measures with unprecedented sensitivity the heat (enthalpy) release via chemical turnover in metabolizing cells. Hence, the metabolic activity can be studied independently of modeling the time-dependence of cell numbers. Unexpectedly, IMC traces of various origins exhibit conserved patterns when expressed in the enthalpy domain rather than the time domain, as exemplified by cultures of Lactococcus lactis (prokaryote), Trypanosoma congolese (protozoan) and non-growing Brassica napus (plant) cells. The data comply extraordinarily well with a dynamic Langmuir adsorption reaction model of nutrient uptake and catalytic turnover generalized here to the non-constancy of catalytic capacity. Formal relations to Michaelis–Menten kinetics and common analytical growth models are briefly discussed. The proposed formalism reproduces the “life span” of cultured microorganisms from exponential growth to metabolic decline by a succession of distinct metabolic phases following remarkably simple nutrient–metabolism relations. The analysis enables the development of advanced enzyme network models of unbalanced growth and has fundamental consequences for the derivation of toxicity measures and the transferability of metabolic activity data between laboratories.

Non-Linear Observer Design with Laguerre Polynomials

In this paper, a methodology for a non-linear system state estimation is demonstrated, exploiting the input and parameter observability. For this purpose, the initial system is transformed into the canonical observability form, and the function that aggregates the non-linear dynamics of the system, which may be unknown or difficult to be computed, is approximated by a linear combination of Laguerre polynomials. Hence, the system identification translates into the estimation of the parameters involved in the linear combination in order for the system to be observable. For the validation of the elaborated observer, we consider a biological model from the literature, investigating whether it is practically possible to infer its states, taking into account the new coordinates to design the appropriate observer of the system states. Through simulations, we investigate the parameter settings under which the new observer can identify the state of the system. More specifically, as the parameter θ increases, the system converges more quickly to the steady-state, decreasing the respective distance from the system’s initial state. As for the first state, the estimation error is in the order of 10−2 for θ=15, and assuming c0={0,1},c1=1. Under the same conditions, the estimation error of the system’s second state is in the order of 10−1, setting a performance difference of 10−1 in relation to the first state. The outcomes show that the proposed observer’s performance can be further improved by selecting even higher values of θ. Hence, the system is observable through the measurement output.

Self-Healing of Cementitious Materials via Bacteria: A Theoretical Study

Cracks on the surface of cementitious composites represent an entrance gate for harmful substances—particularly water—to devastate the bulk of material, which results in lower durability. Autogenous crack-sealing is a significantly limited mechanism due to a combination of the hydration process and calcite nucleation, and self-healing cementitious composites are a research area that require a great deal of scientific effort. In contrast to time-consuming experiments (e.g., only the preparation of an applicable bare concrete sample itself requires more than 28 days), appropriately selected mathematical models may assist in the deeper understanding of self-healing processes via bacteria. This paper presents theoretically oriented research dealing with the application of specific bacteria (B. pseudofirmus) capable of transforming available nutrients into calcite, allowing for the cracks on the surfaces of cementitious materials to be repaired. One of the principal objectives of this study is to analyze the sensitivity of the bacterial growth curves to the system parameters within the context of the logistic model in the Monod approach. Analytically calculated growth curves for various parameters (initial inoculation concentration, initial nutrition content, and metabolic activity of bacteria) are compared with experimental data. The proposed methodology may also be applied to analyze the growth of microorganisms of nonbacterial origin (e.g., molds, yeasts).

Limited Mechanistic Link Between the Monod Equation and Methanogen Growth: a Perspective from Metabolic Modeling

The Monod equation has been widely applied as the general rate law of microbial growth, but its applications are not always successful. By drawing on the frameworks of kinetic and stoichiometric metabolic models and metabolic control analysis, the modeling reported here simulated the growth kinetics of a methanogenic microorganism and illustrated that different enzymes and metabolites control growth rate to various extents and that their controls peak at either very low, intermediate, or very high substrate concentrations. In comparison, with a single term and two parameters, the Monod equation only approximately accounts for the controls of rate-determining enzymes and metabolites at very high and very low substrate concentrations, but neglects the enzymes and metabolites whose controls are most notable at intermediate concentrations. These findings support a limited link between the Monod equation and methanogen growth, and unify the competing views regarding enzyme roles in shaping growth kinetics. The results also preclude a mechanistic derivation of the Monod equation from methanogen metabolic networks and highlight a fundamental challenge in microbiology: single-term expressions may not be sufficient for accurate prediction of microbial growth. IMPORTANCE The Monod equation has been widely applied to predict the rate of microbial growth, but its application is not always successful. Using a novel metabolic modeling approach, we simulated the growth of a methanogen and uncovered a limited mechanistic link between the Monod equation and the methanogen's metabolic network. Specifically, the equation provides an approximation to the controls by rate-determining metabolites and enzymes at very low and very high substrate concentrations, but it is missing the remaining enzymes and metabolites whose controls are most notable at intermediate concentrations. These results support the Monod equation as a useful approximation of growth rates and highlight a fundamental challenge in microbial kinetics: single-term rate expressions may not be sufficient for accurate prediction of microbial growth.

Existing Empirical Kinetic Models in Biochemical Methane Potential (BMP) Testing, Their Selection and Numerical Solution

Biochemical Methane Potential (BMP) tests are a crucial part of feasibility studies to estimate energy recovery opportunities from organic wastes and wastewater. Despite the large number of publications dedicated to BMP testing and numerous attempts to standardize procedures, there is no “one size fits all” mathematical model to describe biomethane formation kinetic precisely. Importantly, the kinetics models are utilized for treatability estimation and modeling processes for the purpose of scale-up. A numerical computation approach is a widely used method to determine model coefficients, as a replacement for the previously used linearization approach. However, it requires more information for each model and some range of coefficients to iterate through. This study considers existing empirical models used to describe biomethane formation process in BMP testing, clarifies model nomenclature, presents equations usable for numerical computation of kinetic parameters as piece-wise defined functions, defines the limits for model coefficients, and collects and analyzes criteria to evaluate and compare model goodness of fit.