Efficient Estimates of Surface Diffusion Parameters for Spatio-Temporally Resolved Virus Replication Dynamics.

Advanced methods of treatment are needed to fight the threats of virus-transmitted diseases and pandemics. Often, they are based on an improved biophysical understanding of virus replication strategies and processes in their host cells. For instance, an essential component of the replication of the hepatitis C virus (HCV) proceeds under the influence of nonstructural HCV proteins (NSPs) that are anchored to the endoplasmatic reticulum (ER), such as the NS5A protein. The diffusion of NSPs has been studied by in vitro fluorescence recovery after photobleaching (FRAP) experiments. The diffusive evolution of the concentration field of NSPs on the ER can be described by means of surface partial differential equations (sufPDEs). Previous work estimated the diffusion coefficient of the NS5A protein by minimizing the discrepancy between an extended set of sufPDE simulations and experimental FRAP time-series data. Here, we provide a scaling analysis of the sufPDEs that describe the diffusive evolution of the concentration field of NSPs on the ER. This analysis provides an estimate of the diffusion coefficient that is based only on the ratio of the membrane surface area in the FRAP region to its contour length. The quality of this estimate is explored by a comparison to numerical solutions of the sufPDE for a flat geometry and for ten different 3D embedded 2D ER grids that are derived from fluorescence z-stack data of the ER. Finally, we apply the new data analysis to the experimental FRAP time-series data analyzed in our previous paper, and we discuss the opportunities of the new approach.

Intracellular "In Silico Microscopes"-Comprehensive 3D Spatio-Temporal Virus Replication Model Simulations.

Despite their small and simple structure compared with their hosts, virus particles can cause severe harm and even mortality in highly evolved species such as humans. A comprehensive quantitative biophysical understanding of intracellular virus replication mechanisms could aid in preparing for future virus pandemics. By elucidating the relationship between the form and function of intracellular structures from the host cell and viral components, it is possible to identify possible targets for direct antiviral agents and potent vaccines. Biophysical investigations into the spatio-temporal dynamics of intracellular virus replication have thus far been limited. This study introduces a framework to enable simulations of these dynamics using partial differential equation (PDE) models, which are evaluated using advanced numerical mathematical methods on leading supercomputers. In particular, this study presents a model of the replication cycle of a specific RNA virus, the hepatitis C virus. The diffusion-reaction model mimics the interplay of the major components of the viral replication cycle, including non structural viral proteins, viral genomic RNA, and a generic host factor. Technically, surface partial differential equations (sufPDEs) are coupled on the 3D embedded 2D endoplasmic reticulum manifold with partial differential equations (PDEs) in the 3D membranous web and cytosol volume. The membranous web serves as a viral replication factory and is formed on the endoplasmic reticulum after infection and in the presence of nonstructural proteins. The coupled sufPDE/PDE model was evaluated using realistic cell geometries based on experimental data. The simulations incorporate the effects of non structural viral proteins, which are restricted to the endoplasmic reticulum surface, with effects appearing in the volume, such as host factor supply from the cytosol and membranous web dynamics. Because the spatial diffusion properties of genomic viral RNA are not yet fully understood, the model allows for viral RNA movement on the endoplasmic reticulum as well as within the cytosol. Visualizing the simulated intracellular viral replication dynamics provides insights similar to those obtained by microscopy, complementing data from in vitro/in vivo viral replication experiments. The output data demonstrate quantitative consistence with the experimental findings, prompting further advanced experimental studies to validate the model and refine our quantitative biophysical understanding.

Transdermal and lateral effective diffusivities for drug transport in stratum corneum from a microscopic anisotropic diffusion model.

This paper presents a computational model of molecular diffusion through the interfollicular stratum corneum. Specifically, it extends an earlier two-dimensional microscopic model for the permeability in two ways: (1) a microporous leakage pathway through the intercellular lipid lamellae allows slow permeation of highly hydrophilic permeants through the tissue; and (2) the model yields explicit predictions of both lateral (D‾‖sc) and transdermal (D‾⊥sc) effective (average, homogenized) diffusivities of solutes within the tissue. We present here the mathematical framework for the analysis and a comparison of the predictions with experimental data on desorption of both hydrophilic and lipophilic solutes from human stratum corneum in vitro. Diffusion in the lipid lamellae is found to make the effective diffusivity highly anisotropic, with the predicted ratio D‾‖sc/D‾⊥sc ranging from 34 to 39 for fully hydrated skin and 150 to more than 1000 for partially hydrated skin. The diffusivities and their ratio are in accord with both experimental data and the results of mathematical analyses performed by others.

Influence of T-Bar on Calcium Concentration Impacting Release Probability.

The relation of form and function, namely the impact of the synaptic anatomy on calcium dynamics in the presynaptic bouton, is a major challenge of present (computational) neuroscience at a cellular level. The Drosophila larval neuromuscular junction (NMJ) is a simple model system, which allows studying basic effects in a rather simple way. This synapse harbors several special structures. In particular, in opposite to standard vertebrate synapses, the presynaptic boutons are rather large, and they have several presynaptic zones. In these zones, different types of anatomical structures are present. Some of the zones bear a so-called T-bar, a particular anatomical structure. The geometric form of the T-bar resembles the shape of the letter "T" or a table with one leg. When an action potential arises, calcium influx is triggered. The probability of vesicle docking and neurotransmitter release is superlinearly proportional to the concentration of calcium close to the vesicular release site. It is tempting to assume that the T-bar causes some sort of calcium accumulation and hence triggers a higher release probability and thus enhances neurotransmitter exocytosis. In order to study this influence in a quantitative manner, we constructed a typical T-bar geometry and compared the calcium concentration close to the active zones (AZs). We compared the case of synapses with and without T-bars. Indeed, we found a substantial influence of the T-bar structure on the presynaptic calcium concentrations close to the AZs, indicating that this anatomical structure increases vesicle release probability. Therefore, our study reveals how the T-bar zone implies a strong relation between form and function. Our study answers the question of experimental studies (namely "Wichmann and Sigrist, Journal of neurogenetics 2010") concerning the sense of the anatomical structure of the T-bar.

A method to investigate the diffusion properties of nuclear calcium.

Modeling biophysical processes in general requires knowledge about underlying biological parameters. The quality of simulation results is strongly influenced by the accuracy of these parameters, hence the identification of parameter values that the model includes is a major part of simulating biophysical processes. In many cases, secondary data can be gathered by experimental setups, which are exploitable by mathematical inverse modeling techniques. Here we describe a method for parameter identification of diffusion properties of calcium in the nuclei of rat hippocampal neurons. The method is based on a Gauss-Newton method for solving a least-squares minimization problem and was formulated in such a way that it is ideally implementable in the simulation platform uG. Making use of independently published space- and time-dependent calcium imaging data, generated from laser-assisted calcium uncaging experiments, here we could identify the diffusion properties of nuclear calcium and were able to validate a previously published model that describes nuclear calcium dynamics as a diffusion process.

Synaptic activity induces dramatic changes in the geometry of the cell nucleus: interplay between nuclear structure, histone H3 phosphorylation, and nuclear calcium signaling.

Synaptic activity initiates many adaptive responses in neurons. Here we report a novel form of structural plasticity in dissociated hippocampal cultures and slice preparations. Using a recently developed algorithm for three-dimensional image reconstruction and quantitative measurements of cell organelles, we found that many nuclei from hippocampal neurons are highly infolded and form unequally sized nuclear compartments. Nuclear infoldings are dynamic structures, which can radically transform the geometry of the nucleus in response to neuronal activity. Action potential bursting causing synaptic NMDA receptor activation dramatically increases the number of infolded nuclei via a process that requires the ERK-MAP kinase pathway and new protein synthesis. In contrast, death-signaling pathways triggered by extrasynaptic NMDA receptors cause a rapid loss of nuclear infoldings. Compared with near-spherical nuclei, infolded nuclei have a larger surface and increased nuclear pore complex immunoreactivity. Nuclear calcium signals evoked by cytosolic calcium transients are larger in small nuclear compartments than in the large compartments of the same nucleus; moreover, small compartments are more efficient in temporally resolving calcium signals induced by trains of action potentials in the theta frequency range (5 Hz). Synaptic activity-induced phosphorylation of histone H3 on serine 10 was more robust in neurons with infolded nuclei compared with neurons with near-spherical nuclei, suggesting a functional link between nuclear geometry and transcriptional regulation. The translation of synaptic activity-induced signaling events into changes in nuclear geometry facilitates the relay of calcium signals to the nucleus, may lead to the formation of nuclear signaling microdomains, and could enhance signal-regulated transcription.

Diffusion simulation-based fiber tracking using time-of-arrival maps: a comparison with standard methods.

OBJECT: We propose a new tracking method based on time-of-arrival (TOA) maps derived from simulated diffusion processes. MATERIALS AND METHODS: The proposed diffusion simulation-based tracking consists of three steps that are successively evaluated on small overlapping sub-regions in a diffusion tensor field. First, the diffusion process is simulated for several time steps. Second, a TOA map is created to store simulation results for the individual time steps that are required for the tract reconstruction. Third, the fiber pathway is reconstructed on the TOA map and concatenated between neighboring sub-regions. This new approach is compared with probabilistic and streamline tracking. All methods are applied to synthetic phantom data for an easier evaluation of their fiber reconstruction quality. RESULTS: The comparison of the tracking results did show severe problems for the streamline approach in the reconstruction of crossing fibers, for example. The probabilistic method was able to resolve the crossing, but could not handle strong curvature. The new diffusion simulation-based tracking could reconstruct all problematic fiber constellations. CONCLUSION: The proposed diffusion simulation-based tracking method used the whole tensor information of a neighborhood of voxels and is, therefore, able to handle problematic tracking situations better than established methods.

The role of corneocytes in skin transport revised--a combined computational and experimental approach.

PURPOSE: To investigate mechanisms of compound-corneocyte interactions in a combined experimental and theoretical approach. MATERIALS AND METHODS: Experimental methods are presented to investigate compound-corneocyte interactions in terms of dissolution within water of hydration and protein binding and to quantify the extent of the concurrent mechanisms. Results are presented for three compounds: caffeine, flufenamic acid, and testosterone. Two compartmental stratum corneum models M1 and M2 are formulated based on experimentally determined input parameters describing the affinity to lipid, proteins and water. M1 features a homogeneous protein compartment and considers protein interactions only via intra-corneocyte water. In M2 the protein compartment is sub-divided into a cornified envelope compartment interacting with inter-cellular lipids and a keratin compartment interacting with water. RESULTS: For the non-protein binding caffeine the impact of the aqueous compartment on stratum corneum partitioning is overestimated but is successfully modeled after introducing a bound water fraction that is non-accessible for compound dissolution. For lipophilic, keratin binding compounds (flufenamic acid, testosterone) only M2 correctly predicts a concentration dependence of stratum corneum partition coefficients. CONCLUSIONS: Lipophilic and hydrophilic compounds interact with corneocytes. Interactions of lipophilic compounds are probably confined to the corneocyte surface. Interactions with intracellular keratin may be limited by their low aqueous solubility.

Advanced Hepatitis C Virus Replication PDE Models within a Realistic Intracellular Geometric Environment.

The hepatitis C virus (HCV) RNA replication cycle is a dynamic intracellular process occurring in three-dimensional space (3D), which is difficult both to capture experimentally and to visualize conceptually. HCV-generated replication factories are housed within virus-induced intracellular structures termed membranous webs (MW), which are derived from the Endoplasmatic Reticulum (ER). Recently, we published 3D spatiotemporal resolved diffusion⁻reaction models of the HCV RNA replication cycle by means of surface partial differential equation (sPDE) descriptions. We distinguished between the basic components of the HCV RNA replication cycle, namely HCV RNA, non-structural viral proteins (NSPs), and a host factor. In particular, we evaluated the sPDE models upon realistic reconstructed intracellular compartments (ER/MW). In this paper, we propose a significant extension of the model based upon two additional parameters: different aggregate states of HCV RNA and NSPs, and population dynamics inspired diffusion and reaction coefficients instead of multilinear ones. The combination of both aspects enables realistic modeling of viral replication at all scales. Specifically, we describe a replication complex state consisting of HCV RNA together with a defined amount of NSPs. As a result of the combination of spatial resolution and different aggregate states, the new model mimics a cis requirement for HCV RNA replication. We used heuristic parameters for our simulations, which were run only on a subsection of the ER. Nevertheless, this was sufficient to allow the fitting of core aspects of virus reproduction, at least qualitatively. Our findings should help stimulate new model approaches and experimental directions for virology.

Filtering, reconstruction, and measurement of the geometry of nuclei from hippocampal neurons based on confocal microscopy data.

The cell nucleus is often considered a spherical structure. However, the visualization of proteins associated with the nuclear envelope in rat hippocampal neurons indicates that the geometry of nuclei is far more complex. The shape of cell nuclei is likely to influence the nucleo-cytoplasmic exchange of macromolecules and ions, in particular calcium, a key regulator of neuronal gene expression. We developed a tool to retrieve the 3-D view of cell nuclei from laser scanning confocal microscopy data. By applying an inertia-based filter, based on a special structure detection mechanism, the signal-to-noise ratio of the image is enhanced, the signal is smoothed, gaps in the membrane are closed, while at the same time the geometric properties, such as diameters of the membrane, are preserved. After segmentation of the image data, the microscopy data are sufficiently processed to extract surface information of the membrane by creating an isosurface with a marching tetrahedra algorithm combined with a modified Dijkstra graph-search algorithm. All methods are tested on artificial data, as well as on real data, which are recorded with a laser scanning confocal microscope. Significant advantages of the inertia-based filter can be observed when comparing it to other state of the art nonlinear diffusion filters. An additional program is written to calculate surface and volume of cell nuclei. These results represent the first step toward establishing a geometry-based model of the-dynamics of cytoplasmic and nuclear calcium.

In-silico model of skin penetration based on experimentally determined input parameters. Part II: mathematical modelling of in-vitro diffusion experiments. Identification of critical input parameters.

This work describes a framework for in-silico modelling of in-vitro diffusion experiments illustrated in an accompanying paper [S. Hansen, A. Henning, A. Naegel, M. Heisig, G. Wittum, D. Neumann, K.-H. Kostka, J. Zbytovska, C.M. Lehr, U.F. Schaefer, In-silico model of skin penetration based on experimentally determined input parameters. Part I: experimental determination of partition and diffusion coefficients, Eur. J. Pharm. Biopharm. 68 (2008) 352-367 [corrected] A mathematical model of drug permeation through stratum corneum (SC) and viable epidermis/dermis is presented. The underlying geometry for the SC is of brick-and-mortar character, meaning that the corneocytes are completely embedded in the lipid phase. The geometry is extended by an additional compartment for the deeper skin layers (DSL). All phases are modelled with homogeneous diffusivity. Lipid-donor and SC-DSL partition coefficients are determined experimentally, while corneocyte-lipid and DSL-lipid partition coefficients are derived consistently with the model. Together with experimentally determined apparent lipid- and DSL-diffusion coefficients, these data serve as direct input for computational modelling of drug transport through the skin. The apparent corneocyte diffusivity is estimated based on an approximation, which uses the apparent SC- and lipid-diffusion coefficients as well as corneocyte-lipid partition coefficients. The quality of the model is evaluated by a comparison of concentration-SC-depth-profiles of the experiment with those of the simulation. Good agreements are obtained, and by an analysis of the underlying model, critical parameters of the models can be identified more easily.

In-silico model of skin penetration based on experimentally determined input parameters. Part I: experimental determination of partition and diffusion coefficients.

Mathematical modeling of skin transport is considered a valuable alternative of in-vitro and in-vivo investigations especially considering ethical and economical questions. Mechanistic diffusion models describe skin transport by solving Fick's 2nd law of diffusion in time and space; however models relying entirely on a consistent experimental data set are missing. For a two-dimensional model membrane consisting of a biphasic stratum corneum (SC) and a homogeneous epidermal/dermal compartment (DSL) methods are presented to determine all relevant input parameters. The data were generated for flufenamic acid (M(W) 281.24g/mol; logK(Oct/H2O) 4.8; pK(a) 3.9) and caffeine (M(W) 194.2g/mol; logK(Oct/H2O) -0.083; pK(a) 1.39) using female abdominal skin. K(lip/don) (lipid-donor partition coefficient) was determined in equilibration experiments with human SC lipids. K(cor/lip) (corneocyte-lipid) and K(DSL/lip) (DSL-lipid) were derived from easily available experimental data, i.e. K(SC/don) (SC-donor), K(lip/don) and K(SC/DSL) (SC-DSL) considering realistic volume fractions of the lipid and corneocyte phases. Lipid and DSL diffusion coefficients D(lip) and D(DSL) were calculated based on steady state flux. The corneocyte diffusion coefficient D(cor) is not accessible experimentally and needs to be estimated by simulation. Based on these results time-dependent stratum corneum concentration-depth profiles were simulated and compared to experimental profiles in an accompanying study.

Quantitative Analysis of Hepatitis C NS5A Viral Protein Dynamics on the ER Surface.

Exploring biophysical properties of virus-encoded components and their requirement for virus replication is an exciting new area of interdisciplinary virological research. To date, spatial resolution has only rarely been analyzed in computational/biophysical descriptions of virus replication dynamics. However, it is widely acknowledged that intracellular spatial dependence is a crucial component of virus life cycles. The hepatitis C virus-encoded NS5A protein is an endoplasmatic reticulum (ER)-anchored viral protein and an essential component of the virus replication machinery. Therefore, we simulate NS5A dynamics on realistic reconstructed, curved ER surfaces by means of surface partial differential equations (sPDE) upon unstructured grids. We match the in silico NS5A diffusion constant such that the NS5A sPDE simulation data reproduce experimental NS5A fluorescence recovery after photobleaching (FRAP) time series data. This parameter estimation yields the NS5A diffusion constant. Such parameters are needed for spatial models of HCV dynamics, which we are developing in parallel but remain qualitative at this stage. Thus, our present study likely provides the first quantitative biophysical description of the movement of a viral component. Our spatio-temporal resolved ansatz paves new ways for understanding intricate spatial-defined processes central to specfic aspects of virus life cycles.

3D Spatially Resolved Models of the Intracellular Dynamics of the Hepatitis C Genome Replication Cycle.

Mathematical models of virus dynamics have not previously acknowledged spatial resolution at the intracellular level despite substantial arguments that favor the consideration of intracellular spatial dependence. The replication of the hepatitis C virus (HCV) viral RNA (vRNA) occurs within special replication complexes formed from membranes derived from endoplasmatic reticulum (ER). These regions, termed membranous webs, are generated primarily through specific interactions between nonstructural virus-encoded proteins (NSPs) and host cellular factors. The NSPs are responsible for the replication of the vRNA and their movement is restricted to the ER surface. Therefore, in this study we developed fully spatio-temporal resolved models of the vRNA replication cycle of HCV. Our simulations are performed upon realistic reconstructed cell structures-namely the ER surface and the membranous webs-based on data derived from immunostained cells replicating HCV vRNA. We visualized 3D simulations that reproduced dynamics resulting from interplay of the different components of our models (vRNA, NSPs, and a host factor), and we present an evaluation of the concentrations for the components within different regions of the cell. Thus far, our model is restricted to an internal portion of a hepatocyte and is qualitative more than quantitative. For a quantitative adaption to complete cells, various additional parameters will have to be determined through further in vitro cell biology experiments, which can be stimulated by the results deccribed in the present study.

A three-dimensional mathematical model for the signal propagation on a neuron's membrane.

In order to be able to examine the extracellular potential's influence on network activity and to better understand dipole properties of the extracellular potential, we present and analyze a three-dimensional formulation of the cable equation which facilitates numeric simulations. When the neuron's intra- and extracellular space is assumed to be purely resistive (i.e., no free charges), the balance law of electric fluxes leads to the Laplace equation for the distribution of the intra- and extracellular potential. Moreover, the flux across the neuron's membrane is continuous. This observation already delivers the three dimensional cable equation. The coupling of the intra- and extracellular potential over the membrane is not trivial. Here, we present a continuous extension of the extracellular potential to the intracellular space and combine the resulting equation with the intracellular problem. This approach makes the system numerically accessible. On the basis of the assumed pure resistive intra- and extracellular spaces, we conclude that a cell's out-flux balances out completely. As a consequence neurons do not own any current monopoles. We present a rigorous analysis with spherical harmonics for the extracellular potential by approximating the neuron's geometry to a sphere. Furthermore, we show with first numeric simulations on idealized circumstances that the extracellular potential can have a decisive effect on network activity through ephaptic interactions.

A dynamic model for calculating methane emissions from digestate based on co-digestion of animal manure and biogas crops in full scale German biogas plants.

The focus of this work is the development of a model for the estimation of methane emissions for storage tanks of biogas plants. Those can be estimated depending on (i) hydraulic retention time in the digester, (ii) an arbitrary removal rate of the digestate from the storage tank and (iii) arbitrary temperature conditions in the storage tank. Furthermore, the model is capable of considering an arbitrary mixture of manure and crops in the input material. The model was validated by data from 21 full scale biogas plants in Germany digesting cow manure and crops. A realistic scenario for the removal rate and temperature conditions in the storage tank was then investigated and special emphasis was given to the effect of hydraulic retention time and proportion of crops in the mixture on the input VS methane yield from the digester and the storage tank.

Synaptic bouton properties are tuned to best fit the prevailing firing pattern.

The morphology of presynaptic specializations can vary greatly ranging from classical single-release-site boutons in the central nervous system to boutons of various sizes harboring multiple vesicle release sites. Multi-release-site boutons can be found in several neural contexts, for example at the neuromuscular junction (NMJ) of body wall muscles of Drosophila larvae. These NMJs are built by two motor neurons forming two types of glutamatergic multi-release-site boutons with two typical diameters. However, it is unknown why these distinct nerve terminal configurations are used on the same postsynaptic muscle fiber. To systematically dissect the biophysical properties of these boutons we developed a full three-dimensional model of such boutons, their release sites and transmitter-harboring vesicles and analyzed the local vesicle dynamics of various configurations during stimulation. Here we show that the rate of transmission of a bouton is primarily limited by diffusion-based vesicle movements and that the probability of vesicle release and the size of a bouton affect bouton-performance in distinct temporal domains allowing for an optimal transmission of the neural signals at different time scales. A comparison of our in silico simulations with in vivo recordings of the natural motor pattern of both neurons revealed that the bouton properties resemble a well-tuned cooperation of the parameters release probability and bouton size, enabling a reliable transmission of the prevailing firing-pattern at diffusion-limited boutons. Our findings indicate that the prevailing firing-pattern of a neuron may determine the physiological and morphological parameters required for its synaptic terminals.

Detailed modeling of skin penetration--an overview.

In recent years, the combination of computational modeling and experiments has become a useful tool that is proving increasingly powerful for explaining biological complexity. As computational power is increasing, scientists are able to explore ever more complex models in finer detail and to explain very complex real world data. This work provides an overview of one-, two- and three-dimensional diffusion models for penetration into mammalian skin. Besides diffusive transport this includes also binding of substances to skin proteins and metabolism. These models are based on partial differential equations that describe the spatial evolution of the transport process through the biological barrier skin. Furthermore, the work focuses on analytical and numerical techniques for this type of equations such as discretization schemes or homogenization (upscaling) techniques. Finally, the work compares different geometry models with respect to the permeability.

Mesophilic anaerobic co-digestion of cow manure and biogas crops in full scale German biogas plants: a model for calculating the effect of hydraulic retention time and VS crop proportion in the mixture on methane yield from digester and from digestate storage at different temperatures.

Data from 24 full scale biogas plants in Germany digesting cow manure and crops were evaluated. Special emphasis was given to the effect of hydraulic retention time HRT and proportion of crops in the mixture (VS basis) p(VS,Crops)(Inp) on the methane yield from the digester [Formula: see text] and the storage tank [Formula: see text] at 37 and 22°C. The evaluation has shown model parameters for maximal methane yield of manure and crops [Formula: see text] at 270 and 420 Lkg(-1), respectively. For example, at HRT of 60days, maximum methane yield result to 249 and 388 Lkg(-1) for a crop proportion in the input of 0.0 and 1.0, respectively. The calculation of [Formula: see text] considers first order reaction rates and a temperature term f(T). Hence, at any arbitrary temperature in the range of 12°C

Do two-phase biogas plants separate anaerobic digestion phases? - a mathematical model for the distribution of anaerobic digestion phases among reactor stages.

In this article a mathematical model is introduced, which estimates the distribution of the four anaerobic digestion phases (hydrolysis, acidogenesis, acetogenesis and methanogenesis) that occur among the leach bed reactor and the anaerobic filter of a biogas plant. It is shown that only the hydrolysis takes place in the first stage (leach bed reactor), while all other anaerobic digestion phases take place in both reactor stages. It turns out that, besides the usually measured raw materials of the acetogenesis and the methanogenesis phases (organic acids), it is also necessary to analyze the process liquid for raw materials of the acidogenesis phase, i.e., sugars, fatty acids, amino acids, etc. The introduced model can be used to monitor the inhibition of the anaerobic digestion phases in reactor stages and can, thus, help to improve the control system of biogas plants.