Reduction of a biochemical model with preservation of its basic dynamic properties

The complexity of full-scale metabolic models is a major obstacle for their effective use in computational systems biology. The aim of model reduction is to circumvent this problem by eliminating parts of a model that are unimportant for the properties of interest. The choice of reduction method is influenced both by the type of model complexity and by the objective of the reduction; therefore, no single method is superior in all cases. In this study we present a comparative study of two different methods applied to a 20D model of yeast glycolytic oscillations. Our objective is to obtain biochemically meaningful reduced models, which reproduce the dynamic properties of the 20D model. The first method uses lumping and subsequent constrained parameter optimization. The second method is a novel approach that eliminates variables not essential for the dynamics. The applications of the two methods result in models of eight (lumping), six (elimination) and three (lumping followed by elimination) dimensions. All models have similar dynamic properties and pin-point the same interactions as being crucial for generation of the oscillations. The advantage of the novel method is that it is algorithmic, and does not require input in the form of biochemical knowledge. The lumping approach, however, is better at preserving biochemical properties, as we show through extensive analyses of the models.     

Has Large-Scale Named-Entity Network Analysis Been Resting on a Flawed Assumption?

The assumption that a name uniquely identifies an entity introduces two types of errors: splitting treats one entity as two or more (because of name variants); lumping treats multiple entities as if they were one (because of shared names). Here we investigate the extent to which splitting and lumping affect commonly-used measures of large-scale named-entity networks within two disambiguated bibliographic datasets: one for co-author names in biomedicine (PubMed, 2003–2007); the other for co-inventor names in U.S. patents (USPTO, 2003–2007). In both cases, we find that splitting has relatively little effect, whereas lumping has a dramatic effect on network measures. For example, in the biomedical co-authorship network, lumping (based on last name and both initials) drives several measures down: the global clustering coefficient by a factor of 4 (from 0.265 to 0.066); degree assortativity by a factor of ∼13 (from 0.763 to 0.06); and average shortest path by a factor of 1.3 (from 5.9 to 4.5). These results can be explained in part by the fact that lumping artificially creates many intransitive relationships and high-degree vertices. This effect of lumping is much less dramatic but persists with measures that give less weight to high-degree vertices, such as the mean local clustering coefficient and log-based degree assortativity. Furthermore, the log-log distribution of collaborator counts follows a much straighter line (power law) with splitting and lumping errors than without, particularly at the low and the high counts. This suggests that part of the power law often observed for collaborator counts in science and technology reflects an artifact: name ambiguity.     

Lumped and population stochastic models of skeletal muscle: Implications and predictions

Lumped models of skeletal muscle have been assumed a) in the design of experiments and the interpretation of experimental findings, b) in theoretical studies. In this paper, a population model that takes into account the differing properties and separate (independent) activation of motor units is presented as the most appropriate for muscle. A realistic (for muscle) transformation, populationlumped model, resulting in the lumping of motor unit neural signals or system responses, is proposed. On this basis, the possibility of modelling muscle as a single system is examined; and the consequences of treating muscle as a lumped system, in experiments or theoretical studies, are discussed. Also, the advantages of lumping, in models of muscle, are reviewed. Predictions of a computer population model, together with actual recordings from a hand muscle, are used to confirm the results of the analysis.     

Inner filter effects and their interferences in the interpretation of culture fluorescence

Inner filter effects and their interferences in the measurement and interpretation of culture fluorescence are discussed. An approximate light intensity model for a typical open-ended culture fluorescence measuring device is developed for calculating the fluorescence response of a component of interest in a general three component solution. The model is tested using well de fined synthetic fluorescent systems. The model is then extended for correlating culture fluorescence with cell density and metabolic state of microbial cultures based on a lumping approximation. The extended model has been utilized to derive culture fluorescence parameters of yeast culture at three distinct metabolic states.     

The problem of time unit in Leslie’s population model

The conditions that will allow the lumping together of several age classes in the Leslie model are investigated. We show that if the lumping is to be valid for all population distributions, then the parameters of the model must be periodic. Lumping is valid when the population is in equilibrium, but equilibrium should be tested before the model is lumped.     

Exact epidemic models on graphs using graph-automorphism driven lumping

The dynamics of disease transmission strongly depends on the properties of the population contact network. Pair-approximation models and individual-based network simulation have been used extensively to model contact networks with non-trivial properties. In this paper, using a continuous time Markov chain, we start from the exact formulation of a simple epidemic model on an arbitrary contact network and rigorously derive and prove some known results that were previously mainly justified based on some biological hypotheses. The main result of the paper is the illustration of the link between graph automorphisms and the process of lumping whereby the number of equations in a system of linear differential equations can be significantly reduced. The main advantage of lumping is that the simplified lumped system is not an approximation of the original system but rather an exact version of this. For a special class of graphs, we show how the lumped system can be obtained by using graph automorphisms. Finally, we discuss the advantages and possible applications of exact epidemic models and lumping.     

Development of models of active ion transport for whole-cell modelling: cardiac sodium-potassium pump as a case study

This study presents a method for the reduction of biophysically-based kinetic models for the active transport of ions. A lumping scheme is presented which exploits the differences in timescales associated with fast and slow transitions between model states, while maintaining the thermodynamic properties of the model. The goal of this approach is to contribute to modelling of the effects of disturbances to metabolism, associated with ischaemic heart disease, on cardiac cell function.

The approach is illustrated for the sodium-potassium pump in the myocyte. The lumping scheme is applied to produce a 4-state representation from the detailed 15-state model of Läuger and Apell, Eur. Biophys. J. 13 (1986) 309, for which the principles of free energy transduction are used to link the free energy released from ATP hydrolysis (ΔG_ATP) to the transition rates between states of the model. An iterative minimisation algorithm is implemented to determine the transition rate parameters based on the model fit to experimental data. Finally, the relationship between ΔG_ATP and pump cycling direction is investigated and compared with recent experimental findings.     

How objective are genera in euascomycetes?

Intra- and intergeneric distances derived from maximum-likelihood phylogenetic trees inferred from 254 nuclear ITS rDNA sequences were examined for seven families of euascomycetes, representing five classes. The intra- and intergeneric distances were well separated in most cases, but the distances varied between families. The analysis of the distance distributions provides a powerful tool for identifying certain taxa with highly deviating distances and thus cases of excessive lumping or splitting. Some cases of lumping and splitting found in different families are briefly discussed. The results of the analysis show that the generic concepts differ between the families. The consequences for nomenclature are discussed and a method abandoning binomial nomenclature while keeping the style of species names is recommended to ensure nomenclatural stability.     

Methods of Model Reduction for Large-Scale Biological Systems: A Survey of Current Methods and Trends

Complex models of biochemical reaction systems have become increasingly common in the systems biology literature. The complexity of such models can present a number of obstacles for their practical use, often making problems difficult to intuit or computationally intractable. Methods of model reduction can be employed to alleviate the issue of complexity by seeking to eliminate those portions of a reaction network that have little or no effect upon the outcomes of interest, hence yielding simplified systems that retain an accurate predictive capacity. This review paper seeks to provide a brief overview of a range of such methods and their application in the context of biochemical reaction network models. To achieve this, we provide a brief mathematical account of the main methods including timescale exploitation approaches, reduction via sensitivity analysis, optimisation methods, lumping, and singular value decomposition-based approaches. Methods are reviewed in the context of large-scale systems biology type models, and future areas of research are briefly discussed.     

Combining sources in stable isotope mixing models: alternative methods

Stable isotope mixing models are often used to quantify source contributions to a mixture. Examples include pollution source identification; trophic web studies; analysis of water sources for soils, plants; or water bodies, and many others. A common problem is having too many sources to allow a unique solution. We discuss two alternative procedures for addressing this problem. One option is a priori to combine sources with similar signatures so the number of sources is small enough to provide a unique solution. Aggregation should be considered only when isotopic signatures of clustered sources are not significantly different, and sources are related so the combined source group has some functional significance. For example, in a food web analysis, lumping several species within a trophic guild allows more interpretable results than lumping disparate food sources, even if they have similar isotopic signatures. One result of combining mixing model sources is increased uncertainty of the combined end-member isotopic signatures and consequently the source contribution estimates; this effect can be quantified using the IsoError model (). As an alternative to lumping sources before a mixing analysis, the IsoSource mixing model () can be used to find all feasible solutions of source contributions consistent with isotopic mass balance. While ranges of feasible contributions for each individual source can often be quite broad, contributions from functionally related groups of sources can be summed a posteriori, producing a range of solutions for the aggregate source that may be considerably narrower. A paleohuman dietary analysis example illustrates this method, which involves a terrestrial meat food source, a combination of three terrestrial plant foods, and a combination of three marine foods. In this case, a posteriori aggregation of sources allowed strong conclusions about temporal shifts in marine versus terrestrial diets that would not have otherwise been discerned.     

Metabolic control analysis for large changes: extension to variable elasticity coefficients

Modular approaches are powerful systems biology strategies to deal with complexity. They consist in lumping conceptually all that is irrelevant to the problem under study, leaving explicit the portions of interest. Modular (or top-down) metabolic control analysis is a theoretical and experimental approach to study the sensitivity properties of complex metabolic systems. Initially, it was conceived for infinitesimal changes but, recently, it started to be developed for large metabolic changes. A central result of this approach is that the systemic properties, represented by control coefficients, can be expressed as a function of the properties of isolated modules, the elasticity coefficients. Here we extend the theory for large changes to the case that the elasticity coefficients depend on the extent of the change. The novel theory is used to analyse experimental data related to the control of glycolytic flux in Escherichia coli. Our analysis shows that the pattern of control for large changes is quantitatively and qualitatively different from the one obtained applying the infinitesimal treatment.     

Silent endocrine tumors

Some tumors of hormonal organs are clinically active, while others are not. The silent tumors may be discovered by accident or because of effects due to their increase in size. From a simple steady state analysis of hormonal feedback systems follows that hormonal cell multiplication does not significantly influence the systems steady state behaviour (hence the clinical silence). —Exceptions to this rule occur in three situations: when the gain of the system is low; when the growth concerns cells with isolated sensor or reference functions; or because of the growth of autonomous cells. In many biological systems the dangerous situation of clamping to low levels upon sensor cell multiplication has been prevented by lumping, such as the combination of sensor and comparator functions into sensor-comparator cells.     

Variable expression of an autosomal dominant syndrome: (BBB syndrome or G syndrome)

We report a family in which Opitz-Frias G syndrome is expressed across 4 generations. The propositus displays hypertelorism, low grade hypospadias, cleft palate and lips and cleft larynx, making the diagnosis of G syndrome very likely. A cousin of his mother discloses similar clefts, vulviform hypospadias, anal imperforation and mental retardation. His clinical appearance fits perfectly the diagnosis of BBB syndrome. A nephew shows ambiguous genitalia and hypertelorism. Authors suggest the lumping of the BBB and the G syndrome.     

Muscle mass in musculoskeletal models

Most current models of musculoskeletal dynamics lump a muscle's mass with its body segment, and then simulate the dynamics of these body segments connected by joints. As shown here, this popular approach leads to errors in the system's inertia matrix and hence in all aspects of the dynamics. Two simplified mathematical models were created to capture the relevant features of monoarticular and biarticular muscles, and the errors were analyzed. The models were also applied to two physiological examples: the triceps surae muscles that plantar flex the human ankle and the biceps femoris posterior muscle of the rat hind limb. The analysis of errors due to lumping showed that these errors can be large. Although the errors can be reduced in some postures, they cannot be easily eliminated in models that use segment lumping. Some options for addressing these errors are discussed.     

Mapping functional similarity of predators on the basis of trait similarities

Theoretical and empirical studies in community ecology often simplify their study system by lumping together species on the basis of trait similarities (e.g., their taxonomy, resource or microhabitat usage) and then assume species sharing similar traits are functionally similar. To date, no study has directly tested whether species more similar with respect to any of these traits are more similar in their functional effects on population or ecosystem processes. In this study, we examined the association between traits and functional effects of six different aquatic predatory vertebrates. We demonstrated that functional similarity across multiple response variables was not correlated with trait similarity, but differences in trait values were associated with significantly different effects on individual response variables. The exact relationship between species traits and functional effect of predators, however, was different for each response variable. Using traits to predict functional similarity among species may be useful when considering individual response variables, but only if it is known which traits have the greatest influence on the response variable of interest. It is doubtful that any one scheme will predict the functional similarity of species across a diverse array of response variables because each response will likely be strongly influenced by different traits.     

Reduced modeling of signal transduction – a modular approach

Background  

Combinatorial complexity is a challenging problem in detailed and mechanistic mathematical modeling of signal transduction. This subject has been discussed intensively and a lot of progress has been made within the last few years. A software tool (BioNetGen) was developed which allows an automatic rule-based set-up of mechanistic model equations. In many cases these models can be reduced by an exact domain-oriented lumping technique. However, the resulting models can still consist of a very large number of differential equations.     

Metabolic network simulation using logical loop algorithm and Jacobian matrix

A novel method to accomplish efficient numerical simulation of metabolic networks for flux analysis was developed. The only inputs required are the set of stoichiometric balances and the atom mapping matrices of all components of the reaction network. The latter are used to automatically calculate isotopomer mapping matrices. Using the symbolic toolbox of MATLAB the analytical solution of the stoichiometric balance equation system, isotopomer balances and the analytical Jacobian matrix of the total set of stoichiometric and isotopomer balances are created automatically. The number of variables in the isotopomer distribution equation system is significantly reduced applying modified isotopomer mapping matrices. These allow lumping of several consecutive isotopomer reactions into a single one. The solution of the complete system of equations is improved by implementing an iterative logical loop algorithm and using the analytical Jacobian matrix. This new method provided quick and robust convergence to the root of such equation systems in all cases tested. The method was applied to a network of lysine producing Corynebacterium glutamicum. The resulting equation system with the dimension of 546 x 546 was directly derived from 12 isotopomer balance equations. The results obtained yielded identical labeling patterns for metabolites as compared to the relaxation method.