Mathematical Models Of Biological Systems Research Articles

Mathematical modeling in autoimmune diseases: from theory to clinical application.

The research & development (R&D) of novel therapeutic agents for the treatment of autoimmune diseases is challenged by highly complex pathogenesis and multiple etiologies of these conditions. The number of targeted therapies available on the market is limited, whereas the prevalence of autoimmune conditions in the global population continues to rise. Mathematical modeling of biological systems is an essential tool which may be applied in support of decision-making across R&D drug programs to improve the probability of success in the development of novel medicines. Over the past decades, multiple models of autoimmune diseases have been developed. Models differ in the spectra of quantitative data used in their development and mathematical methods, as well as in the level of "mechanistic granularity" chosen to describe the underlying biology. Yet, all models strive towards the same goal: to quantitatively describe various aspects of the immune response. The aim of this review was to conduct a systematic review and analysis of mathematical models of autoimmune diseases focused on the mechanistic description of the immune system, to consolidate existing quantitative knowledge on autoimmune processes, and to outline potential directions of interest for future model-based analyses. Following a systematic literature review, 38 models describing the onset, progression, and/or the effect of treatment in 13 systemic and organ-specific autoimmune conditions were identified, most models developed for inflammatory bowel disease, multiple sclerosis, and lupus (5 models each). ≥70% of the models were developed as nonlinear systems of ordinary differential equations, others - as partial differential equations, integro-differential equations, Boolean networks, or probabilistic models. Despite covering a relatively wide range of diseases, most models described the same components of the immune system, such as T-cell response, cytokine influence, or the involvement of macrophages in autoimmune processes. All models were thoroughly analyzed with an emphasis on assumptions, limitations, and their potential applications in the development of novel medicines.

Is Alzheimer's disease an individual-centered disease? Hypotheses from the atomic levels up to mathematical models for biological systems.

Is Alzheimer's disease an individual-centered disease? Hypotheses from the atomic levels up to mathematical models for biological systems.

Uncovering specific mechanisms across cell types in dynamical models.

Ordinary differential equations are frequently employed for mathematical modeling of biological systems. The identification of mechanisms that are specific to certain cell types is crucial for building useful models and to gain insights into the underlying biological processes. Regularization techniques have been proposed and applied to identify mechanisms specific to two cell types, e.g., healthy and cancer cells, including the LASSO (least absolute shrinkage and selection operator). However, when analyzing more than two cell types, these approaches are not consistent, and require the selection of a reference cell type, which can affect the results. To make the regularization approach applicable to identifying cell-type specific mechanisms in any number of cell types, we propose to incorporate the clustered LASSO into the framework of ordinary differential equation modeling by penalizing the pairwise differences of the logarithmized fold-change parameters encoding a specific mechanism in different cell types. The symmetry introduced by this approach renders the results independent of the reference cell type. We discuss the necessary adaptations of state-of-the-art numerical optimization techniques and the process of model selection for this method. We assess the performance with realistic biological models and synthetic data, and demonstrate that it outperforms existing approaches. Finally, we also exemplify its application to published biological models including experimental data, and link the results to independent biological measurements.

Structural identifiability of biomolecular controller motifs with and without flow measurements as model output.

Controller motifs are simple biomolecular reaction networks with negative feedback. They can explain how regulatory function is achieved and are often used as building blocks in mathematical models of biological systems. In this paper we perform an extensive investigation into structural identifiability of controller motifs, specifically the so-called basic and antithetic controller motifs. Structural identifiability analysis is a useful tool in the creation and evaluation of mathematical models: it can be used to ensure that model parameters can be determined uniquely and to examine which measurements are necessary for this purpose. This is especially useful for biological models where parameter estimation can be difficult due to limited availability of measureable outputs. Our aim with this work is to investigate how structural identifiability is affected by controller motif complexity and choice of measurements. To increase the number of potential outputs we propose two methods for including flow measurements and show how this affects structural identifiability in combination with, or in the absence of, concentration measurements. In our investigation, we analyze 128 different controller motif structures using a combination of flow and/or concentration measurements, giving a total of 3648 instances. Among all instances, 34% of the measurement combinations provided structural identifiability. Our main findings for the controller motifs include: i) a single measurement is insufficient for structural identifiability, ii) measurements related to different chemical species are necessary for structural identifiability. Applying these findings result in a reduced subset of 1568 instances, where 80% are structurally identifiable, and more complex/interconnected motifs appear easier to structurally identify. The model structures we have investigated are commonly used in models of biological systems, and our results demonstrate how different model structures and measurement combinations affect structural identifiability of controller motifs.

A New Visualization and Analysis Method for a Convolved Representation of Mass Computational Experiments with Biological Models

Modern computational biology makes widespread use of mathematical models of biological systems, in particular systems of ordinary differential equations, as well as models of dynamic systems described in other formalisms, such as agent-based models. Parameters are numerical values of quantities reflecting certain properties of a modeled system and affecting model solutions. At the same time, depending on parameter values, different dynamic regimes—stationary or oscillatory, established as a result of transient modes of various types—can be observed in the modeled system. Predicting changes in the solution dynamics type depending on changes in model parameters is an important scientific task. Nevertheless, this problem does not have an analytical solution for all formalisms in a general case. The routinely used method of performing a series of computational experiments, i.e., solving a series of direct problems with various sets of parameters followed by expert analysis of solution plots is labor-intensive with a large number of parameters and a decreasing step of the parametric grid. In this regard, the development of methods allowing the obtainment and analysis of information on a set of computational experiments in an aggregate form is relevant. This work is devoted to developing a method for the visualization and classification of various dynamic regimes of a model using a composition of the dynamic time warping (DTW-algorithm) and principal coordinates analysis (PCoA) methods. This method enables qualitative visualization of the results of the set of solutions of a mathematical model and the performance of the correspondence between the values of the model parameters and the type of dynamic regimes of its solutions. This method has been tested on the Lotka–Volterra model and artificial sets of various dynamics.

ASAS-NANP symposium: Mathematical Modeling in Animal Nutrition: The power of identifiability analysis for dynamic modeling in animal science:a practitioner approach.

Constructing dynamic mathematical models of biological systems requires estimating unknown parameters from available experimental data, usually using a statistical fitting procedure. This procedure is usually called parameter identification, parameter estimation, model fitting, or model calibration. In animal science, parameter identification is often performed without analytic considerations on the possibility of determining unique values of the model parameters. These analytical studies are related to the mathematical property of structural identifiability, which refers to the theoretical ability to recover unique values of the model parameters from the measures defined in an experimental setup and use the model structure as the sole basis. The structural identifiability analysis is a powerful tool for model construction because it informs whether the parameter identification problem is well-posed (i.e., the problem has a unique solution). Structural identifiability analysis is helpful to determine which actions (e.g., model reparameterization, choice of new data measurements, and change of the model structure) are needed to render the model parameters identifiable (when possible). The mathematical technicalities associated with structural identifiability analysis are very sophisticated. However, the development of dedicated, freely available software tools enables the application of identifiability analysis without needing to be an expert in mathematics and computer programming. We refer to such a non-expert user as a practitioner for hands-on purposes. However, a practitioner should be familiar with the model construction and software implementation process. In this paper, we propose to adopt a practitioner approach that takes advantage of available software tools to integrate identifiability analysis in the modeling practice in the animal science field. The application of structural identifiability implies switching our regard of the parameter identification problem as a downstream process (after data collection) to an upstream process (before data collection) where experiment design is applied to guarantee identifiability. This upstream approach will substantially improve the workflow of model construction toward robust and valuable models in animal science. Illustrative examples with different levels of complexity support our work. The source codes of the examples were provided for learning purposes and to promote open science practices.

Predictive Biodegradation of Multiple Toxic Pollutants in Bioreactors Treating Real Wastewater using ANN and GP

Industrial wastewater contains eco-toxic organic pollutants such as phenol, cyanide and xylenol, which are usually the most common toxic compounds from steel industries and their high concentrations negatively affects biological treatments through substrate inhibition. Optimizing the biodegradation of multiple substrates found in wastewater such as phenol, xylenol and cyanide under Aerobic (in presence of oxygen), Anaerobic (in absence of oxygen) and Anoxic (in presence of nitrates) biotreatment systems has been done in this study. To predict the performance and design control systems to meet the environmental regulatory standards, the complex non-linear behavior presented in the biological treatment requires an accurate model. Herein, by using Artificial intelligence (ANN, GA and GP), the mathematical modelling of biological reactors operating under batch and continuous systems treating phenol, xylenol and cyanide were attempted. Mathematical modelling of biological systems is complex due to lack of process knowledge and multiple governing factors thereby significantly increasing the research trends using data based driven models. To correlate the biological non-linear relationship and to achieve the minimum error goal for pollutant removal with an acceptable R2 value of 0.847, 0.892, 0.937 in the anoxic system for phenol, cyanide and xylenol removal, the ANN and GA were designed. These models represent, analyse and reliably make predictions of complex biological systems helping in controlling and enhancing the performance of the wastewater treatment plant. From these results, the better pollutant removal was done by the anoxic system when compared to aerobic and anaerobic systems especially for phenol and cyanide removal.

Perspectives on the Monod model in biological wastewater treatment

AbstractThe Monod model has been widely used in activated sludge and biofilm models to mathematically describe the kinetics of microbial growth. Mathematical models of biological systems can be distinguished according to their dimensionality as zero‐dimensional (0D), one‐dimensional (1D), two‐dimensional (2D) and three‐dimensional (3D). The Monod model is a 0D model, which does not take cell (aggregate) structure into account, yet has been used in different ways in activated sludge and biofilm modelling: In activated sludge modelling, it is used in its original sense (i.e. 0D), whilst in dimensional biofilm modelling, it is used to model intrinsic kinetics in the biofilm and thus in 1D, 2D and 3D. We believe that the latter is not the correct use of the Monod model, because both activated sludge and biofilm systems consist of similar cell aggregates that both feature concentration gradients. Diffusional mass transfer limitations are manifested by the values of kinetic parameters in the Monod model and should not be described explicitly by Fick's laws of diffusion. The 0D concept has been adopted in state‐of‐the‐art activated sludge models and more recently in 0D biofilm models. Measured kinetic parameters can be directly transferred to 0D models. The application of Fick's second law of diffusion in combination with the Monod model (i.e. the reaction–diffusion equation) therefore becomes irrelevant. In this manner, the Monod equation becomes a simple, universal approach to modelling that contains variable parameters. © 2022 Society of Chemical Industry (SCI).

A parsimonious model of blood glucose homeostasis.

The mathematical modelling of biological systems has historically followed one of two approaches: comprehensive and minimal. In comprehensive models, the involved biological pathways are modelled independently, then brought together as an ensemble of equations that represents the system being studied, most often in the form of a large system of coupled differential equations. This approach often contains a very large number of tuneable parameters (> 100) where each describes some physical or biochemical subproperty. As a result, such models scale very poorly when assimilation of real world data is needed. Furthermore, condensing model results into simple indicators is challenging, an important difficulty in scenarios where medical diagnosis is required. In this paper, we develop a minimal model of glucose homeostasis with the potential to yield diagnostics for pre-diabetes. We model glucose homeostasis as a closed control system containing a self-feedback mechanism that describes the collective effects of the physiological components involved. The model is analyzed as a planar dynamical system, then tested and verified using data collected with continuous glucose monitors (CGMs) from healthy individuals in four separate studies. We show that, although the model has only a small number (3) of tunable parameters, their distributions are consistent across subjects and studies both for hyperglycemic and for hypoglycemic episodes.

Implementation of a practical Markov chain Monte Carlo sampling algorithm in PyBioNetFit.

Bayesian inference in biological modeling commonly relies on Markov chain Monte Carlo (MCMC) sampling of a multidimensional and non-Gaussian posterior distribution that is not analytically tractable. Here, we present the implementation of a practical MCMC method in the open-source software package PyBioNetFit (PyBNF), which is designed to support parameterization of mathematical models for biological systems. The new MCMC method, am, incorporates an adaptive move proposal distribution. For warm starts, sampling can be initiated at a specified location in parameter space and with a multivariate Gaussian proposal distribution defined initially by a specified covariance matrix. Multiple chains can be generated in parallel using a computer cluster. We demonstrate that am can be used to successfully solve real-world Bayesian inference problems, including forecasting of new Coronavirus Disease 2019 case detection with Bayesian quantification of forecast uncertainty. PyBNF version 1.1.9, the first stable release with am, is available at PyPI and can be installed using the pip package-management system on platforms that have a working installation of Python 3. PyBNF relies on libRoadRunner and BioNetGen for simulations (e.g. numerical integration of ordinary differential equations defined in SBML or BNGL files) and Dask.Distributed for task scheduling on Linux computer clusters. The Python source code can be freely downloaded/cloned from GitHub and used and modified under terms of the BSD-3 license (https://github.com/lanl/pybnf). Online documentation covering installation/usage is available (https://pybnf.readthedocs.io/en/latest/). A tutorial video is available on YouTube (https://www.youtube.com/watch?v=2aRqpqFOiS4&t=63s). Supplementary data are available at Bioinformatics online.

Genome-scale Metabolic Model Guided Subtyping Lung Cancer towards Personalized Diagnosis

Mathematical modeling of biological systems are useful in (i) gaining better understanding about the physiological processes in an organism, (ii) simulating alternative scenarios, (iii) finding targets for improved performance within metabolic engineering context (iv) performing several functional analyses, e.g. identify drug targets (v) process scheduling within the context of industrial biotechnology etc. Increasing the predictive capability of these models is of common interest within systems biology studies which allows identification of more effective and personalized treatment strategies for complex metabolic diseases such as cancer by investigation of disease metabolism and providing correct subtyping and staging. By transforming gene-level information to flux/metabolite level information, current disease state can be analyzed and diagnosis of cancer subtype can be performed using a less invasive methods.In this study, subtyping and staging of lung cancer, that is one of the main causes of cancer related deaths, was performed by integrating publicly available RNAseq data of normal, lung adenoma and adenocarcinomas and lung squamous cell neoplasms to human genome scale metabolic model and classification of obtained flux distributions using linear support vector machine (SVM) classifications. Differential flux analysis and pathway enrichment methods showed that model adequately represented tumour metabolism. SVM classification accuracies were calculated as more than 99% for normal and cancer cells and 94% for adenomas and adenocarcinomas and squamous cell neoplasms, indicating high predictive capability of flux distributions.

A general fractional formulation and tracking control for immunogenic tumor dynamics

Mathematical modeling of biological systems is an important issue having significant effect on human beings. In this direction, the description of immune systems is an attractive topic as a result of its ability to detect and eradicate abnormal cells. Therefore, this manuscript aims to investigate the asymptotic behavior of immunogenic tumor dynamics based on a new fractional model constructed by the concept of general fractional operators. We discuss the stability and equilibrium points corresponding to the new model; then we modify the predictor–corrector method in general sense to implement the model and compare the associated numerical results with some real experimental data. As an achievement, the new model provides a degree of flexibility enabling us to adjust the complex dynamics of biological system under study. Consequently, the new general model and its solution method presented in this paper for the immunogenic tumor dynamics are new and comprise quite different information than the other kinds of classical and fractional equations. In addition to these, we implement a tracking control method in order to decrease the development of tumor‐cell population. The satisfaction of control purpose is confirmed by some simulation results since the controlled variables track the tumor‐free steady state in the whole realistic cases.

The applicability of mathematics in computational systems biology and its experimental relations

In 1966 Richard Levins argued that applications of mathematics to population biology faced various constraints which forced mathematical modelers to trade-off at least one of realism, precision, or generality in their approach. Much traditional mathematical modeling in biology has prioritized generality and precision in the place of realism through strategies of idealization and simplification. This has at times created tensions with experimental biologists. The past 20 years however has seen an explosion in mathematical modeling of biological systems with the rise of modern computational systems biology and many new collaborations between modelers and experimenters. In this paper I argue that many of these collaborations revolve around detail-driven modeling practices which in Levins’ terms trade-off generality for realism and precision. These practices apply mathematics by working from detailed accounts of biological systems, rather than from initially idealized or simplified representations. This is possible by virtue of modern computation. The form these practices take today suggest however Levins’ constraints on mathematical application no longer apply, transforming our understanding of what is possible with mathematics in biology. Further the engagement with realism and the ability to push realistic models in new directions aligns well with the epistemological and methodological views of experimenters, which helps explain their increased enthusiasm for biological modeling.

Charting a New Frontier Integrating Mathematical Modeling in Complex Biological Systems from Molecules to Ecosystems.

Advances in quantitative biology data collection and analysis across scales (molecular, cellular, organismal, and ecological) have transformed how we understand, categorize, and predict complex biological systems. This surge of quantitative data creates an opportunity to apply, develop, and evaluate mathematical models of biological systems and explore novel methods of analysis. Simultaneously, thanks to increased computational power, mathematicians, engineers and physical scientists have developed sophisticated models of biological systems at different scales. Novel modeling schemes can offer deeper understanding of principles in biology, but there is still a disconnect between modeling and experimental biology that limits our ability to fully realize the integration of mathematical modeling and biology. In this work, we explore the urgent need to expand the use of existing mathematical models across biological scales, develop models that are robust to biological heterogeneity, harness feedback loops within the iterative modeling process, and nurture a cultural shift towards interdisciplinary and cross-field interactions. Better integration of biological experimentation and robust mathematical modeling will transform our ability to understand and predict complex biological systems.

Countering reproducibility issues in mathematical models with software engineering techniques: A case study using a one-dimensional mathematical model of the atrioventricular node.

One should assume that in silico experiments in systems biology are less susceptible to reproducibility issues than their wet-lab counterparts, because they are free from natural biological variations and their environment can be fully controlled. However, recent studies show that only half of the published mathematical models of biological systems can be reproduced without substantial effort. In this article we examine the potential causes for failed or cumbersome reproductions in a case study of a one-dimensional mathematical model of the atrioventricular node, which took us four months to reproduce. The model demonstrates that even otherwise rigorous studies can be hard to reproduce due to missing information, errors in equations and parameters, a lack in available data files, non-executable code, missing or incomplete experiment protocols, and missing rationales behind equations. Many of these issues seem similar to problems that have been solved in software engineering using techniques such as unit testing, regression tests, continuous integration, version control, archival services, and a thorough modular design with extensive documentation. Applying these techniques, we reimplement the examined model using the modeling language Modelica. The resulting workflow is independent of the model and can be translated to SBML, CellML, and other languages. It guarantees methods reproducibility by executing automated tests in a virtual machine on a server that is physically separated from the development environment. Additionally, it facilitates results reproducibility, because the model is more understandable and because the complete model code, experiment protocols, and simulation data are published and can be accessed in the exact version that was used in this article. We found the additional design and documentation effort well justified, even just considering the immediate benefits during development such as easier and faster debugging, increased understandability of equations, and a reduced requirement for looking up details from the literature.

Anne Elizabeth Warner. 25 August 1940—16 May 2012

Anne Warner applied physiological techniques to developmental biology, elucidating the mechanisms of cell interaction and communication that pattern the early embryo. Through her determination and passion for science, she contributed crucial discoveries in the fields of muscle physiology, cellular differentiation and gap junction communication. She spent the majority of her career at University College London, which became her intellectual home and where she acquired a Royal Society Foulerton Research Professorship, becoming a highly respected and influential figure. In her work on gap junctions, Anne was the first to show that embryonic development and patterning required gap junctions, and that the restriction of junctional communication between cells played a key role in tissue differentiation. Anne excelled in her breadth of vision across research and its interdisciplinary possibilities. In 1998 she established the CoMPLEX Centre for systems biology at UCL, bringing her own group together with scientists from across the STEM subjects to build testable mathematical models of biological systems across multiple scales. Indefatigable in her capacity for leadership and committee work, she assumed an eclectic set of roles across a large span of research organizations and professional societies, and had a lifelong association with the Physiological Society. In 1984 she founded the Microelectrodes course at the Plymouth Marine Biology Laboratory, which has trained generations in the art of electrophysiology and still continues today. With uncompromisingly high standards, she inspired her mentees to be ambitious and fearless, and established postdoctoral fellowships to help the young scientists who followed after her.

Multiscale plant modeling: from genome to phenome and beyond.

Plants are complex organisms that adapt to changes in their environment using an array of regulatory mechanisms that span across multiple levels of biological organization. Due to this complexity, it is difficult to predict emergent properties using conventional approaches that focus on single levels of biology such as the genome, transcriptome, or metabolome. Mathematical models of biological systems have emerged as useful tools for exploring pathways and identifying gaps in our current knowledge of biological processes. Identification of emergent properties, however, requires their vertical integration across biological scales through multiscale modeling. Multiscale models that capture and predict these emergent properties will allow us to predict how plants will respond to a changing climate and explore strategies for plant engineering. In this review, we (1) summarize the recent developments in plant multiscale modeling; (2) examine multiscale models of microbial systems that offer insight to potential future directions for the modeling of plant systems; (3) discuss computational tools and resources for developing multiscale models; and (4) examine future directions of the field.

A Monte Carlo method to estimate cell population heterogeneity from cell snapshot data

Variation is characteristic of all living systems. Laboratory techniques such as flow cytometry can probe individual cells, and, after decades of experimentation, it is clear that even members of genetically identical cell populations can exhibit differences. To understand whether variation is biologically meaningful, it is essential to discern its source. Mathematical models of biological systems are tools that can be used to investigate causes of cell-to-cell variation. From mathematical analysis and simulation of these models, biological hypotheses can be posed and investigated, then parameter inference can determine which of these is compatible with experimental data. Data from laboratory experiments often consist of “snapshots” representing distributions of cellular properties at different points in time, rather than individual cell trajectories. These data are not straightforward to fit using hierarchical Bayesian methods, which require the number of cell population clusters to be chosen a priori. Nor are they amenable to standard nonlinear mixed effect methods, since a single observation per cell is typically too few to estimate parameter variability. Here, we introduce a computational sampling method named “Contour Monte Carlo” (CMC) for estimating mathematical model parameters from snapshot distributions, which is straightforward to implement and does not require that cells be assigned to predefined categories. The CMC algorithm fits to snapshot probability distributions rather than raw data, which means its computational burden does not, like existing approaches, increase with the number of cells observed. Our method is appropriate for underdetermined systems, where there are fewer distinct types of observations than parameters to be determined, and where observed variation is mostly due to variability in cellular processes rather than experimental measurement error. This may be the case for many systems due to continued improvements in resolution of laboratory techniques. In this paper, we apply our method to quantify cellular variation for three biological systems of interest and provide Julia code enabling others to use this method.

Acceleration of PDE-Based Biological Simulation Through the Development of Neural Network Metamodels.

Partial differential equation (PDE) mathematical models of biological systems and the simulation approaches used to solve them are widely used to test hypotheses and infer regulatory interactions based on optimization of the PDE model against the observed data. In this review, we discuss the ability of powerful machine learning methods to accelerate the parametric screening of biophysical informed- PDE systems. A major shortcoming in more broad adaptation of PDE-based models is the high computational complexity required to solve and optimize the models and it requires many simulations to traverse the very high-dimensional parameter spaces during model calibration and inference tasks. For instance, when scaling up to tens of millions of simulations for optimization and sensitivity analysis of the PDE models, compute times quickly extend from months to years for sufficient coverage to solve the problems. For many systems, this brute-force approach is simply not feasible. Recently, neural network metamodels have been shown to be an efficient way to accelerate PDE model calibration and here we look at the benefits and limitations in extending the PDE acceleration methods to improve optimization and sensitivity analysis. We use an example simulation to quantitatively and qualitatively show how neural network metamodels can be accurate and fast and demonstrate their potential for optimization of complex spatiotemporal problems in biology. We expect these approaches will be broadly applied to speed up scientific research and discovery in biology and other systems that can be described by complex PDE systems.

Dynamic Modeling of CHO Cell Metabolism Using the Hybrid Cybernetic Approach With a Novel Elementary Mode Analysis Strategy.

Chinese hamster ovary (CHO) cell culture has a major importance on the production of biopharmaceuticals, including recombinant therapeutic proteins such as monoclonal antibodies (MAb). Mathematical modeling of biological systems can successfully assess metabolism complexity while providing logical and systematic methods for relevant genetic target and culture parameter identification toward cell growth and productivity improvements. Most modeling approaches on CHO cells have been performed under stationary constraints, and only a few dynamic models have been presented on simplified reaction sets, due to substantial overparameterization problems. The hybrid cybernetic modeling (HCM) approach has been recently used to describe the dynamic behavior by incorporating regulation between different metabolic states by elementary mode participation control, with sets of equations evaluated by objective functions. However, as metabolic networks evaluated are constructed toward a genomic scale, and cell compartmentalization is considered, identification of the active set becomes more difficult as EM number exponentially grows. Thus, the development of robust approaches for EM active set selection and analysis with smaller computational requirements is required to impulse the use of cybernetic modeling on larger up to genome-scale networks. In this report, a novel elementary mode selection strategy, based on a polar representation of the convex solution space is presented and coupled to a cybernetic approach to model the dynamic physiologic and metabolic behavior of CHO-S cell cultures. The proposed Polar Space Yield Analysis (PSYA) was compared to other reported elementary mode selection approaches derived from Common Metabolic Objective Analysis (CMOA) used in Flux Balance Analysis (FBA), Yield Space Analysis (YSA), and Lumped Yield Space Analysis (LYSA). For this purpose, exponential growth phase dynamic metabolic models were calculated using kinetic rate equations based on previously modeled growth parameters. Finally, complete culture dynamic metabolic flux models were constructed using the HCM approach with selected elementary mode sets. The yield space elementary mode- and the polar space elementary mode- hybrid cybernetic models presented the best fits and performances. Also, a flux reaction perturbation prediction approach based on the polar yield solution space resulted useful for metabolic network flux distribution capability analysis and identification of potential genetic modifications targets.