Dissipative structures and amplification of enantiomeric excess (an experimental point of view)

Various mathematical models have been proposed to account for the origin of chiral molecules in biological systems. Most of these models invoke non-linear phenomena, and are based on the general concept of dissipative structures. These theoretical models define the fundamental criteria which must be obeyed by the experimental systems that we have investigated. Our initial approach to this problem was an extensive search of the literature data in order to select a few systems or experimental situations which would satisfy the criteria defined by the theoretical models. For these reasons, we carried out a study of the possibility of stereospecific autocatalysis in the asymmetric polymerisation of benzofuran. Similarly, the formation of spatial dissipative structures by coupling of a transport process with an interfacial reaction was investigated as a simple experimental example of symmetry breaking.     

Probabilistic methods for addressing uncertainty and variability in biological models: application to a toxicokinetic model

Population variability and uncertainty are important features of biological systems that must be considered when developing mathematical models for these systems. In this paper we present probability-based parameter estimation methods that account for such variability and uncertainty. Theoretical results that establish well-posedness and stability for these methods are discussed. A probabilistic parameter estimation technique is then applied to a toxicokinetic model for trichloroethylene using several types of simulated data. Comparison with results obtained using a standard, deterministic parameter estimation method suggests that the probabilistic methods are better able to capture population variability and uncertainty in model parameters.     

Model building and model checking for biochemical processes

A central claim of computational systems biology is that, by drawing on mathematical approaches developed in the context of dynamic systems, kinetic analysis, computational theory and logic, it is possible to create powerful simulation, analysis, and reasoning tools for working biologists to decipher existing data, devise new experiments, and ultimately to understand functional properties of genomes, proteomes, cells, organs, and organisms. In this article, a novel computational tool is described that achieves many of the goals of this new discipline. The novelty of this system involves an automaton-based semantics of the temporal evolution of complex biochemical reactions starting from the representation given as a set of differential equations. The related tools also provide ability to qualitatively reason about the systems using a propositional temporal logic that can express an ordered sequence of events succinctly and unambiguously. The implementation of mathematical and computational models in the Simpathica and XSSYS systems is described briefly. Several example applications of these systems to cellular and biochemical processes are presented: the two most prominent are Leibler et al.'s repressilator (an artificial synthesized oscillatory network), and Curto-Voit-Sorribas-Cascante's purine metabolism reaction model.     

A framework for mapping,visualisation and automatic model creation of signal‐transduction networks

Intracellular signalling systems are highly complex. This complexity makes handling, analysis and visualisation of available knowledge a major challenge in current signalling research. Here, we present a novel framework for mapping signal‐transduction networks that avoids the combinatorial explosion by breaking down the network in reaction and contingency information. It provides two new visualisation methods and automatic export to mathematical models. We use this framework to compile the presently most comprehensive map of the yeast MAP kinase network. Our method improves previous strategies by combining (I) more concise mapping adapted to empirical data, (II) individual referencing for each piece of information, (III) visualisation without simplifications or added uncertainty, (IV) automatic visualisation in multiple formats, (V) automatic export to mathematical models and (VI) compatibility with established formats. The framework is supported by an open source software tool that facilitates integration of the three levels of network analysis: definition, visualisation and mathematical modelling. The framework is species independent and we expect that it will have wider impact in signalling research on any system.     

Biological nitrogen and phosphorus removal in membrane bioreactors: model development and parameter estimation

Membrane bioreactors (MBR) are being increasingly used for wastewater treatment. Mathematical modeling of MBR systems plays a key role in order to better explain their characteristics. Several MBR models have been presented in the literature focusing on different aspects: biological models, models which include soluble microbial products (SMP), physical models able to describe the membrane fouling and integrated models which couple the SMP models with the physical models. However, only a few integrated models have been developed which take into account the relationships between membrane fouling and biological processes. With respect to biological phosphorus removal in MBR systems, due to the complexity of the process, practical use of the models is still limited. There is a vast knowledge (and consequently vast amount of data) on nutrient removal for conventional-activated sludge systems but only limited information on phosphorus removal for MBRs. Calibration of these complex integrated models still remains the main bottleneck to their employment. The paper presents an integrated mathematical model able to simultaneously describe biological phosphorus removal, SMP formation/degradation and physical processes which also include the removal of organic matter. The model has been calibrated with data collected in a UCT-MBR pilot plant, located at the Palermo wastewater treatment plant, applying a modified version of a recently developed calibration protocol. The calibrated model provides acceptable correspondence with experimental data and can be considered a useful tool for MBR design and operation.     

A new model for biological pattern formation

Various non-equilibrium growth models have been used to explore the development of morphology in biological systems. Here we review a class of biological growth models which exhibit fractal structures and discuss the relationship of these models to a variety of other phenomena.     

农业废弃物养分循环利用技术模式评估模型的研究进展

农业废弃物的养分循环利用技术模式是实现农业循环经济的重要手段,其评估模型为优化养分循环利用技术提供了重要支撑。本文总结了农业废弃物养分循环技术模式评估框架、评估模型及评价指标、模型的数据源及其不确定性分析,以及模型应用尺度的研究进展。当前,常用于评估养分流动的模型主要是过程数学模型和产业生态学模型。过程数学模型和产业生态学模型在评估结果的可靠性和模拟尺度上存在较大差异,前者主要集中在实验室或中试规模,精度较高;后者可以实现从微观到宏观的多尺度模拟,数据的获取方式导致其具有较高的不确定性。最后,本文对农业废弃物养分循环利用技术评估模型的研究进行展望,提出为了在区域尺度上实现对农业生产系统废弃物资源化利用技术的准确评估,可以将过程数学模型与工业生态学模型相结合,建立可靠的模型框架和数据库,同时,在工厂、农场、村落、乡镇、区域等地理尺度进行模型拓展研究。     

Biology by numbers: mathematical modelling in developmental biology

In recent years, mathematical modelling of developmental processes has earned new respect. Not only have mathematical models been used to validate hypotheses made from experimental data, but designing and testing these models has led to testable experimental predictions. There are now impressive cases in which mathematical models have provided fresh insight into biological systems, by suggesting, for example, how connections between local interactions among system components relate to their wider biological effects. By examining three developmental processes and corresponding mathematical models, this Review addresses the potential of mathematical modelling to help understand development.     

The role of fiber-matrix interactions in a nonlinear fiber-reinforced strain energy model of tendon

The objective of this study was to develop a nonlinear and anisotropic three-dimensional mathematical model of tendon behavior in which the structural components of fibers, matrix, and fiber-matrix interactions are explicitly incorporated and to use this model to infer the contributions of these structures to tendon mechanical behavior. We hypothesized that this model would show that: (i) tendon mechanical behavior is not solely governed by the isotropic matrix and fiber stretch, but is also influenced by fiber-matrix interactions; and (ii) shear fiber-matrix interaction terms will better describe tendon mechanical behavior than bulk fiber-matrix interaction terms. Model versions that did and did not include fiber-matrix interaction terms were applied to experimental tendon stress-strain data in longitudinal and transverse orientations, and the R2 goodness-of-fit was evaluated. This study showed that models that included fiber-matrix interaction terms improved the fit to longitudinal data (R2(toe) = 0.88, R2(Lin) = 0.94) over models that only included isotropic matrix and fiber stretch terms (R2(Toe) = 0.36, R2(Lin) = 0.84). Shear fiber-matrix interaction terms proved to be responsible for the best fit to data and to contribute to stress-strain nonlinearity. The mathematical model of tendon behavior developed in this study showed that fiber-matrix interactions are an important contributor to tendon behavior The more complete characterization of mechanical behavior afforded by this mathematical model can lead to an improved understanding of structure-function relationships in soft tissues and, ultimately, to the development of tissue-engineered therapies for injury or degeneration.     

ChemChains: a platform for simulation and analysis of biochemical networks aimed to laboratory scientists

Background  

New mathematical models of complex biological structures and computer simulation software allow modelers to simulate and analyze biochemical systems in silico and form mathematical predictions. Due to this potential predictive ability, the use of these models and software has the possibility to compliment laboratory investigations and help refine, or even develop, new hypotheses. However, the existing mathematical modeling techniques and simulation tools are often difficult to use by laboratory biologists without training in high-level mathematics, limiting their use to trained modelers.     

Axial diffusion and wall permeability effects in perfused capillary-tissue structures

Attempts to experimentally examine oxygen supply and distribution in the isolated perfused heart and brain have renewed interest in mathematical models of artificially perfused capillary-tissue structures. The need to understand histograms of PO2 measurements from these isolated-perfused organ studies (modified Lagendorf preparations) has required that existing mathematical models and their boundary conditions be re-examined in the context of these experiments. A unifying system of equations and boundary conditions are examined here for the purpose of studying the effects of anisotropic diffusion, and capillary vessel wall permeability on both the capillary and tissue substrate supply. The mathematical models are explored for parameters of physiologic interest, and some comparisons are made with experimental determinations. The comparisons with data suggest an anisotropic transport of oxygen in the tissue that is unexplained by known physiologic mechanisms.     

Interpretation of nonlinear steady state enzyme kinetics—Cyclic and mathematical properties of cooperative,second-site and random pathway models

Nonlinear steady state kinetic patterns are frequently encountered in enzyme studies. Consequently, there is a need to develop procedures for systematically interpreting such data. This paper contributes to this development by identifying a common feature in nonlinear systems and by showing that quite different models commonly in use give very similar mathematical functions.Identical or similar cycles can result from quite different chemical events in enzyme mechanisms, cooperativity, second sites and random pathways. Such cycles can account for many of the observed nonlinear patterns, i.e., power functions, substrate activation and inhibition. Therefore nonlinear steady state kinetics generally requires the presence of a cycle(s) in the mechanism without specifying the underlying chemical events giving rise to that cycle(s).Rate equations for cooperative, second-site and random pathway models are derived and shown to yield virtually identical mathematical functions. Thus empirical equations composed of these functions can be used to represent nonlinear kinetic data without specifying the underlying chemical events.     

Characterization of amphiphilic secondary structures in neuropeptide Y through the design, synthesis, and study of model peptides

Neuropeptide Y (NPY) has the potential to form two amphiphilic secondary structures: a polyproline II-like helix in residues 1-8, and an alpha-helix in residues 13-32. NPY dimerizes in aqueous solution and forms stable monolayers at the air-water interface, suggesting that these amphiphilic conformations are stabilized at interfaces. Furthermore, the negative molar ellipticity of monomeric NPY at 222 nm (-8500 degree cm2/dmol), suggests that hydrophobic interactions with the NH2-terminal amphiphilic structure may stabilize the alpha-helix in residues 13-32 before it binds to cell surfaces, even at physiological concentrations. In order to investigate the role of these amphiphilic structures, five NPY models with multiple substitutions in positions 13-32 have been synthesized and studied. Our data demonstrate that the surfactant properties of NPY result from its potential to form amphiphilic secondary and tertiary structures and not from specific amino acid sequences in this region. However, specific residues on the hydrophilic face of the amphiphilic alpha-helix that have been substituted in the models appear to be required to reproduce the full potency of NPY in our pharmacological assays. A possible role for the amphiphilic structures in NPY in presenting such specific determinants to cell surface receptors in the correct conformation is suggested.     

Direct comparison of calculated hip joint contact forces with those measured using instrumented implants. An evaluation of a three-dimensional mathematical model of the lower limb

Characterisation of hip joint contact forces is essential for the definition of hip joint prosthesis design requirements. In vivo hip joint contact force measurements have been made using instrumented hip joint prostheses. However, to allow determination of the range of values of joint contact force and their directions relative to anatomical structures in a range of subject groups sufficient to form an agreed data base it is necessary to adopt a different approach without the use of an implanted transducer. The use of mathematical models of the lower limb to examine the forces in soft tissues and at the joints has provided valuable insight into internal loading conditions. Several authors have proposed mathematical musculo-skeletal models. However, there have been only limited attempts at validation of these models. It is possible to use the results of in vivo force measurements from instrumented prostheses to validate the results calculated using the mathematical models. In this study two subjects with instrumented hip joint prostheses were studied. Forces at the hip joints were calculated using a three-dimensional model of the leg. Walking at slow, normal and fast speeds (0.97-2.01m/s), weight transfer from two to one leg and back again, and sit to stand were studied. Direct comparisons were made between the 'gold standard' measured hip joint contact forces and the calculated forces. There was general agreement between the calculated and measured forces in both pattern and magnitude. There were, however, discrepancies. Reasons for these differences in results are discussed and possible model developments suggested.     

A structurally relevant coarse-grained model for cholesterol

Detailed atomistic computer simulations are now widely used to study biological membranes, including increasingly mixed lipid systems that involve, for example, cholesterol, which is a key membrane lipid. Typically, simulations of these systems start from a preassembled bilayer because the timescale on which self-assembly occurs in mixed lipid systems is beyond the practical abilities of fully atomistic simulations. To overcome this limitation and study bilayer self-assembly, coarse-grained models have been developed. Although there are several coarse-grained models for cholesterol reported in the literature, these generally fail to account explicitly for the unique molecular features of cholesterol that relate to its function and role as a membrane lipid. In this work, we propose a new coarse-grained model for cholesterol that retains the molecule's unique features and, as a result, can be used to study crystalline structures of cholesterol. In the development of the model, two levels of coarse-graining are explored and the importance of retaining key molecular features in the coarse-grained model that are relevant to structural properties is investigated.     

An optimization framework of biological dynamical systems

Different biological dynamics are often described by different mathematical equations. On the other hand, some mathematical models describe many biological dynamics universally. Here, we focus on three biological dynamics: the Lotka-Volterra equation, the Hopfield neural networks, and the replicator equation. We describe these three dynamical models using a single optimization framework, which is constructed with employing the Riemannian geometry. Then, we show that the optimization structures of these dynamics are identical, and the differences among the three dynamics are only in the constraints of the optimization. From this perspective, we discuss the unified view for biological dynamics. We also discuss the plausible categorizations, the fundamental nature, and the efficient modeling of the biological dynamics, which arise from the optimization perspective of the dynamical systems.     

A methodology for performing global uncertainty and sensitivity analysis in systems biology

Accuracy of results from mathematical and computer models of biological systems is often complicated by the presence of uncertainties in experimental data that are used to estimate parameter values. Current mathematical modeling approaches typically use either single-parameter or local sensitivity analyses. However, these methods do not accurately assess uncertainty and sensitivity in the system as, by default, they hold all other parameters fixed at baseline values. Using techniques described within we demonstrate how a multi-dimensional parameter space can be studied globally so all uncertainties can be identified. Further, uncertainty and sensitivity analysis techniques can help to identify and ultimately control uncertainties. In this work we develop methods for applying existing analytical tools to perform analyses on a variety of mathematical and computer models. We compare two specific types of global sensitivity analysis indexes that have proven to be among the most robust and efficient. Through familiar and new examples of mathematical and computer models, we provide a complete methodology for performing these analyses, in both deterministic and stochastic settings, and propose novel techniques to handle problems encountered during these types of analyses.     

A cometabilic kinetics model incorporating enzyme inhbition, inactivation, and recovery: I. Model development, analysis, and testing

Cometabolic biodegradation prcesses are important for bioremediation of hazardous waste sites. However, these proceeses are not well understood and have not been modeled thoroughly. Traditional Michaelis-Menten kinetics models often are used, but toxic effects and bacterial responses to toxicity may cause changes in enzyme levels, rendering such models inappropriate. In this article, a conceptual and mathematical model of cometabolic enzyme kinetics i described. Model derivation is based on enzyme/growth-substrate/nongrowth-substrate interaction and incorporates enzyme inhibition (caused by the presence of a cometabolic compound), inactivation (resulting from toxicity of a cometabolic product), and recovery (associated with bacterial synthesis of new enbzyme in response to inactivation). The mathematical model consists of a system of two, nonlinear ordinary differential equations that can be solved implicitly using numerical methods, providing estimates of model parameters. Model analysis shows that growth substraate adn nongrowth substrate oxidation rates are related by a dimensionless constant. Reliability of tehy model solution prcedure is verifiedl by abnalyzing data ses, containing random error, from simulated experimentss with trichhloroethyylene (TCE) degradation by ammonia-oxidizing bacterialunder various conditions. Estimation of the recovery rate contant is deterimined to be sensitive to intial TCE concentration. Model assumptions are evaluated in a companion article using data from TCE degradation experiments with amoniaoxidizing bacteria. (c) 1995 John Wiley & Sons, Inc.     

Highly accurate sigmoidal fitting of real-time PCR data by introducing a parameter for asymmetry

Background  

Fitting four-parameter sigmoidal models is one of the methods established in the analysis of quantitative real-time PCR (qPCR) data. We had observed that these models are not optimal in the fitting outcome due to the inherent constraint of symmetry around the point of inflection. Thus, we found it necessary to employ a mathematical algorithm that circumvents this problem and which utilizes an additional parameter for accommodating asymmetrical structures in sigmoidal qPCR data.