Constraints and missing reactions in the urea cycle.

The stoichiometric relations in a series of biochemical reactions are summarized by a stoichiometric number matrix (with a column for each reaction) and a conservation matrix (with a row for each constraint). These two matrices for a series or cycle of biochemical reactions are related because the columns of the stoichiometric number matrix are in the null space of the conservation matrix, and the rows of the transpose of the conservation matrix are in the null space of the transpose of the stoichiometric number matrix. The conservation matrix for a system of biochemical reactions is of interest because it shows the nature of the constraints in addition to the conservation of atoms and groups. Constraints beyond those for the conservation of atoms and groups indicate "missing reactions" that do not occur because the enzymes involved couple reactions that could occur and still conserve atoms and groups. The interpretation of conservation matrices and stoichiometric matrices for a reaction system is complicated by the fact that they are not unique. However, their row-reduced forms are unique, as are their dimensions, which represent the number of reactants and number of independent reactions. Two matrices that look different contain the same information if they have the same row-reduced form. The urea cycle, which involves five enzyme-catalyzed reactions, and its net reaction are discussed in terms of the linear constraints produced by enzyme catalysis. A procedure to obtain a set of conservation equations that will yield the correct net reaction is described.     

Metabolic analysis in drug design

Biotechnology is often presented as if progress in the past two decades represented a major success, but the reality is quite different. For example, ten major classes of antibiotics were discovered between 1935 and 1963, but after 1963 there has been just one, the oxazolidones. To illustrate the possibilities of doing better by taking account of the real behaviour of metabolic systems, we can examine how one might modify the activity of an enzyme in the cell (for example by genetic manipulation, or by the action of an inhibitor, etc.) to satisfy a technological aim. For example, if the objective is to eliminate a pest, one might suppose that the effect of an inhibitor could be to depress an essential flux to a level insufficient for life, or to raise the concentration of an intermediate to a toxic level. The former may seem the more obvious, but the latter is easier to achieve in practice, and there are some excellent examples of industrial products that work in that way, such as the herbicide 'Roundup' and antimalarials of the quinine class. A study of glycolysis in the parasite Trypanosoma brucei (which causes African sleeping sickness) indicates that for this approach to work the selected target enzyme must have a substrate with a concentration that is not limited by stoichiometric constraints. That is not necessarily easy to find in a complicated system, and typically needs the metabolic network to be analysed in the computer.     

Ab initio prediction of thermodynamically feasible reaction directions from biochemical network stoichiometry

Analysis of the stoichiometric structure of metabolic networks provides insights into the relationships between structure, function, and regulation of metabolic systems. Based on knowledge of only reaction stoichiometry, certain aspects of network functionality and robustness can be predicted. Current theories focus on breaking a metabolic network down into non-decomposable pathways able to operate in steady state. The physics underlying these theories is based on mass balance and the laws of thermodynamics. However, due to the inherent nonlinearity of the thermodynamic constraints on metabolic fluxes, computational analysis of large-scale biochemical systems can be expensive. In this study, it is shown how the feasible reaction directions may be determined by either computing the allowable ranges under the mass-balance and thermodynamic constraints or by analyzing the stoichiometric structure of the network. The computed reaction directions translate into a set of linear constraints necessary for thermodynamic feasibility. This set of necessary linear constraints is shown to be sufficient to guarantee feasibility in certain cases, thus translating the nonlinear thermodynamic constraints to linear. We show that for a reaction network of 44 internal reactions representing energy metabolism, the computed linear inequality constraints represent necessary and sufficient conditions for thermodynamic feasibility.     

Effects of budget constraints on conservation network design for biodiversity and ecosystem services

Limited budgets and budget cuts hamper the development of effective biodiversity conservation networks. Optimizing the spatial configuration of conservation networks given such budget constraints remains challenging. Systematic conservation planning addresses this challenge. Systematic conservation planning can integrate both biodiversity and ecosystem services as conservation targets, and hence address the challenge to operationalize ecosystem services as an anthropocentric argument for conservation. We create two conservation scenarios to expand the current conservation network in the Dutch province of Limburg. One scenario focuses on biodiversity only and the other integrates biodiversity and ecosystem services. We varied conservation budgets in these scenarios and used the software Marxan to assess differences in the resulting network configurations. In addition, we tested the network’s cost-effectiveness by allocating a conservation budget either in one or in multiple steps. We included twenty-nine biodiversity surrogates and five ecosystem services. The inclusion of ecosystem services to expand Limburg’s conservation network only moderately changed prioritized areas, compared to only conserving biodiversity. Network expansion in a single time-step is more efficient in terms of compactness and cost-effectiveness than implementing it in multiple time-steps. Therefore, to cost-effectively plan conservation networks, the full budget should ideally be available before the plans are implemented. We show that including ecosystem services to cost-effectively expand conservation networks can simultaneously encourage biodiversity conservation and stimulate the protection of conservation-compatible ecosystem services.     

Actinomycin D complexes with oligonucleotides as models for the binding of the drug to DNA. Paramagnetic induced relaxation experiments on drug-nucleic acid complexes.

Mn(II) ions have been used as a paramagnetic probe to investigate the geometry of drug-oligonucleotide complexes. Nuclear magnetic resonance and electron spin resonance experiments show that Mn(II) ions bind approximately two orders of magnitude stronger to the 5'-terminal phosphate group than to the 3'-5' phosphodiester linkage of deoxydinucleotides. By using mixtures of nucleotides in which only one nucleotide contains a terminal phosphate group, the location of the Mn(II) ion in the drug-nucleotide-Mn(II) complexes may be preselected. The paramagnetic induced relaxation of the nuclear spin systems in these complexes has been used to investigate the geometry of these complexes. These data confirm that actinomycin D is able to recognize and preferentially bind guanine (as opposed to adenine) nucleotides in the quinoid portion of the phenoxazone ring, while both adenine and guanine will bind to the benzenoid portion of the phenoxazone ring. These results suggest that stacking forces are primarily responsible for the general requirement of a guanine base when actinomycin D binds to DNA.     

Thermodynamic Constraints in Kinetic Modeling: Thermodynamic‐Kinetic Modeling in Comparison to Other Approaches

Kinetic models of reaction networks may easily violate the laws of thermodynamics and the principle of detailed balance. In large network models, the constraints that are imposed by these laws are particularly difficult to address. This hinders modeling of biochemical reaction networks. Thermodynamic‐kinetic modeling is a method that provides a thermodynamically sound and formally appealing way for deriving dynamic model equations of reaction systems. State variables of this approach are thermokinetic potentials that describe the ability of compounds to drive a reaction. A compound has a parameter called capacity, which is the ratio of its concentration and thermokinetic potential. A reaction is described by its resistance which is the ratio of the thermokinetic driving force and flux. In these aspects, the formalism is similar to the modeling formalism for electrical networks and an analogous graphical representation is possible. The thermodynamic‐kinetic modeling formalism is equivalent to the traditional kinetic modeling formalism with the exception that it is not possible to build thermodynamically infeasible models. Here, the thermodynamic‐kinetic modeling formalism is reviewed, compared to other approaches, and some of its advantages are worked out. In contrast to other approaches, thermodynamic‐kinetic modeling does not rely on an explicit enumeration of stoichiometric cycles. It is capable of describing rate laws far from equilibrium. Further, the parameterization by capacities and resistances is particularly intuitive and powerful.     

Trade-off between Multiple Constraints Enables Simultaneous Formation of Modules and Hubs in Neural Systems

The formation of the complex network architecture of neural systems is subject to multiple structural and functional constraints. Two obvious but apparently contradictory constraints are low wiring cost and high processing efficiency, characterized by short overall wiring length and a small average number of processing steps, respectively. Growing evidence shows that neural networks are results from a trade-off between physical cost and functional value of the topology. However, the relationship between these competing constraints and complex topology is not well understood quantitatively. We explored this relationship systematically by reconstructing two known neural networks, Macaque cortical connectivity and C. elegans neuronal connections, from combinatory optimization of wiring cost and processing efficiency constraints, using a control parameter , and comparing the reconstructed networks to the real networks. We found that in both neural systems, the reconstructed networks derived from the two constraints can reveal some important relations between the spatial layout of nodes and the topological connectivity, and match several properties of the real networks. The reconstructed and real networks had a similar modular organization in a broad range of , resulting from spatial clustering of network nodes. Hubs emerged due to the competition of the two constraints, and their positions were close to, and partly coincided, with the real hubs in a range of values. The degree of nodes was correlated with the density of nodes in their spatial neighborhood in both reconstructed and real networks. Generally, the rebuilt network matched a significant portion of real links, especially short-distant ones. These findings provide clear evidence to support the hypothesis of trade-off between multiple constraints on brain networks. The two constraints of wiring cost and processing efficiency, however, cannot explain all salient features in the real networks. The discrepancy suggests that there are further relevant factors that are not yet captured here.     

Kinetic constraints for formation of steady states in biochemical networks

The constraint-based analysis has emerged as a useful tool for analysis of biochemical networks. This work introduces the concept of kinetic constraints. It is shown that maximal reaction rates are appropriate constraints only for isolated enzymatic reactions. For biochemical networks, it is revealed that constraints for formation of a steady state require specific relationships between maximal reaction rates of all enzymes. The constraints for a branched network are significantly different from those for a cyclic network. Moreover, the constraints do not require Michaelis-Menten constants for most enzymes, and they only require the constants for the enzymes at the branching or cyclic point. Reversibility of reactions at system boundary or branching point may significantly impact on kinetic constraints. When enzymes are regulated, regulations may impose severe kinetic constraints for the formation of steady states. As the complexity of a network increases, kinetic constraints become more severe. In addition, it is demonstrated that kinetic constraints for networks with co-regulation can be analyzed using the approach. In general, co-regulation enhances the constraints and therefore larger fluctuations in fluxes can be accommodated in the networks with co-regulation. As a first example of the application, we derive the kinetic constraints for an actual network that describes sucrose accumulation in the sugar cane culm, and confirm their validity using numerical simulations.     

Calculability analysis in underdetermined metabolic networks illustrated by a model of the central metabolism in purple nonsulfur bacteria

Metabolite balancing has turned out to be a powerful computational tool in metabolic engineering. However, the linear equation systems occurring in this analysis are often underdetermined. If it is difficult or impossible to find the missing constraints, it is nevertheless feasible in some cases to determine the values of a subset of the unknown rates. Here, a procedure for finding out which reaction rates can be uniquely calculated in underdetermined metabolic networks and computing these rates is given. The method is based on the null space to the stoichiometry matrix corresponding to the reactions with unknown rates. It is shown that this method is considerably easier to handle than an algorithm given previously (Van der Heijden et al., 1994a). Furthermore, a useful elementary representation of the null space is presented which is closely related with the elementary flux modes. This unique representation is central to a more general approach to observability/calculability analysis. In particular, it allows one to find, in an easy way, those sets of measurable rates that enable a calculation of a certain unknown rate. Besides, rates which are never calculable by metabolite balancing may be easily detected by this method. The applicability of these methods is illustrated by a model of the central metabolism in purple nonsulfur bacteria. The photoheterotrophic growth of these representatives of anoxygenic photosynthetic bacteria is stoichiometrically analyzed. Interesting metabolic constraints caused by the necessary balancing of NADPH can be detected in a highly underdetermined system. This is, to our knowledge, the first application of stoichiometric analysis to the metabolic network in this bacteria group using metabolite balancing techniques. A new software tool, the FluxAnalyzer, is introduced. It allows quantitative and structural analysis of metabolic networks in a graphical user interface.     

Combining flux and energy balance analysis to model large-scale biochemical networks

Stoichiometric Network Theory is a constraints-based, optimization approach for quantitative analysis of the phenotypes of large-scale biochemical networks that avoids the use of detailed kinetics. This approach uses the reaction stoichiometric matrix in conjunction with constraints provided by flux balance and energy balance to guarantee mass conserved and thermodynamically allowable predictions. However, the flux and energy balance constraints have not been effectively applied simultaneously on the genome scale because optimization under the combined constraints is non-linear. In this paper, a sequential quadratic programming algorithm that solves the non-linear optimization problem is introduced. A simple example and the system of fermentation in Saccharomyces cerevisiae are used to illustrate the new method. The algorithm allows the use of non-linear objective functions. As a result, we suggest a novel optimization with respect to the heat dissipation rate of a system. We also emphasize the importance of incorporating interactions between a model network and its surroundings.     

Identifying All Moiety Conservation Laws in Genome-Scale Metabolic Networks

The stoichiometry of a metabolic network gives rise to a set of conservation laws for the aggregate level of specific pools of metabolites, which, on one hand, pose dynamical constraints that cross-link the variations of metabolite concentrations and, on the other, provide key insight into a cell''s metabolic production capabilities. When the conserved quantity identifies with a chemical moiety, extracting all such conservation laws from the stoichiometry amounts to finding all non-negative integer solutions of a linear system, a programming problem known to be NP-hard. We present an efficient strategy to compute the complete set of integer conservation laws of a genome-scale stoichiometric matrix, also providing a certificate for correctness and maximality of the solution. Our method is deployed for the analysis of moiety conservation relationships in two large-scale reconstructions of the metabolism of the bacterium E. coli, in six tissue-specific human metabolic networks, and, finally, in the human reactome as a whole, revealing that bacterial metabolism could be evolutionarily designed to cover broader production spectra than human metabolism. Convergence to the full set of moiety conservation laws in each case is achieved in extremely reduced computing times. In addition, we uncover a scaling relation that links the size of the independent pool basis to the number of metabolites, for which we present an analytical explanation.     

A base-triple structural domain in RNA.

An oligonucleotide modeled on a proposed base-triple domain of the Tetrahymena group I intron has been characterized by NMR. The oligonucleotide contains two double-helix regions with adjacent single-stranded nucleotides. The NMR data show that the two helices stack coaxially, although the rotation between the two helices is approximately twice as large as the rotation between normal base pairs. The rotation between the two helices allows the single-stranded nucleotides to form U.U.G and A.G.C base triples in the minor groove. The A.G.C base triple contains a hydrogen bond between the adenine N1 and a 2'-hydroxyl in the minor groove of the G.C pair. A similar hydrogen bond between an adenine and a 2'-hydroxyl in transfer RNA suggests that this could be a recurring tertiary interaction in RNA.     

Assessment of the metabolic capabilities of Haemophilus influenzae Rd through a genome-scale pathway analysis

The annotated full DNA sequence is becoming available for a growing number of organisms. This information along with additional biochemical and strain-specific data can be used to define metabolic genotypes and reconstruct cellular metabolic networks. The first free-living organism for which the entire genomic sequence was established was Haemophilus influenzae. Its metabolic network is reconstructed herein and contains 461 reactions operating on 367 intracellular and 84 extracellular metabolites. With the metabolic reaction network established, it becomes necessary to determine its underlying pathway structure as defined by the set of extreme pathways. The H. influenzae metabolic network was subdivided into six subsystems and the extreme pathways determined for each subsystem based on stoichiometric, thermodynamic, and systems-specific constraints. Positive linear combinations of these pathways can be taken to determine the extreme pathways for the complete system. Since these pathways span the capabilities of the full system, they could be used to address a number of important physiological questions. First, they were used to reconcile and curate the sequence annotation by identifying reactions whose function was not supported in any of the extreme pathways. Second, they were used to predict gene products that should be co-regulated and perhaps co-expressed. Third, they were used to determine the composition of the minimal substrate requirements needed to support the production of 51 required metabolic products such as amino acids, nucleotides, phospholipids, etc. Fourth, sets of critical gene deletions from core metabolism were determined in the presence of the minimal substrate conditions and in more complete conditions reflecting the environmental niche of H. influenzae in the human host. In the former case, 11 genes were determined to be critical while six remained critical under the latter conditions. This study represents an important milestone in theoretical biology, namely the establishment of the first extreme pathway structure of a whole genome.     

Biological network comparison using graphlet degree distribution

MOTIVATION: Analogous to biological sequence comparison, comparing cellular networks is an important problem that could provide insight into biological understanding and therapeutics. For technical reasons, comparing large networks is computationally infeasible, and thus heuristics, such as the degree distribution, clustering coefficient, diameter, and relative graphlet frequency distribution have been sought. It is easy to demonstrate that two networks are different by simply showing a short list of properties in which they differ. It is much harder to show that two networks are similar, as it requires demonstrating their similarity in all of their exponentially many properties. Clearly, it is computationally prohibitive to analyze all network properties, but the larger the number of constraints we impose in determining network similarity, the more likely it is that the networks will truly be similar. RESULTS: We introduce a new systematic measure of a network's local structure that imposes a large number of similarity constraints on networks being compared. In particular, we generalize the degree distribution, which measures the number of nodes 'touching' k edges, into distributions measuring the number of nodes 'touching' k graphlets, where graphlets are small connected non-isomorphic subgraphs of a large network. Our new measure of network local structure consists of 73 graphlet degree distributions of graphlets with 2-5 nodes, but it is easily extendible to a greater number of constraints (i.e. graphlets), if necessary, and the extensions are limited only by the available CPU. Furthermore, we show a way to combine the 73 graphlet degree distributions into a network 'agreement' measure which is a number between 0 and 1, where 1 means that networks have identical distributions and 0 means that they are far apart. Based on this new network agreement measure, we show that almost all of the 14 eukaryotic PPI networks, including human, resulting from various high-throughput experimental techniques, as well as from curated databases, are better modeled by geometric random graphs than by Erd?s-Rény, random scale-free, or Barabási-Albert scale-free networks. AVAILABILITY: Software executables are available upon request.     

Modularity and reliability in the organization of organisms

An organism persists only if it satisfies internal and external constraints. Within the organism networks of processes meet the constraints. In such networks a principle of matching often obtains: the pattern of coupling among processes matches the correlation among constraints. That is, a module—a cluster of coupled processes—meets a constraint. Dissociable modules meet dissociàble constraints. A hierarchy of modules meets a hierarchy of constraints. We have inquired whether such matching is predicted by an optimality criterion in a simple example. We find that in an ensemble of networks with unreliable processes, the networks that meet the constraints with highest reliability obey the principle of matching. The difference in reliability between modular and nonmodular networks that meet the same constraints is a function of the probability of success per process. Our results suggest that this difference is maximal at a probability of success that increases monotonically with the number of processes in the network.     

Characterization of purine nucleotide metabolism in primary rat cardiomyocyte cultures

Primary rat cardiomyocyte cultures were utilized as a model for the study of purine nucleotide metabolism in the heart muscle, especially in connection with the mechanisms operating for the conservation of adenine nucleotides. The cultures exhibited capacity to produce purine nucleotides from nonpurine molecules (de novo synthesis), as well as from preformed purines (salvage synthesis). The conversion of adenosine to AMP, catalyzed by adenosine kinase, appears to be the most important physiological salvage pathway of adenine nucleotide synthesis in the cardiomyocytes. The study of the metabolic fate of IMP formed from [14C]formate or [14C]hypoxanthine and that of AMP formed from [14C]adenine or [14C]adenosine revealed that in the cardiomyocyte the main flow in the nucleotide interconversion pathways is from IMP to AMP, whereas the flux from AMP to IMP appeared to be markedly slower. Following synthesis from labeled precursors by either de novo or salvage pathways, most of the radioactivity in purine nucleotides accumulated in adenine nucleotides, and only a small proportion of it resided in IMP. The results suggest that the main pathway of AMP degradation in the cardiomyocyte proceeds through adenosine rather than through IMP. About 90% of the total radioactivity in purines effluxed from the cells during de novo synthesis from [14C]formate or following prelabeling of adenine nucleotides with [14C]adenine were found to reside in hypoxanthine. The activities in cell extracts of AMP 5'-nucleotidase and IMP 5'-nucleotidase, which catalyze nucleotide degradation, and of AMP deaminase, a key enzyme in the purine nucleotide cycle, were low. The nucleotidase activity resembles, and that of the AMP deaminase contrasts the respective enzyme activities in extracts of cultured skeletal-muscle myotubes. The results indicate that in the cardiomyocyte, in contrast to the myotube, the main mechanism operating for conservation of nucleotides is prompt phosphorylation of AMP, rather than operation of the purine nucleotide cycle. The primary cardiomyocyte cultures are a plausible model for the study of purine nucleotide metabolism in the heart muscle.     

Basic concepts and principles of stoichiometric modeling of metabolic networks

Metabolic networks supply the energy and building blocks for cell growth and maintenance. Cells continuously rewire their metabolic networks in response to changes in environmental conditions to sustain fitness. Studies of the systemic properties of metabolic networks give insight into metabolic plasticity and robustness, and the ability of organisms to cope with different environments. Constraint-based stoichiometric modeling of metabolic networks has become an indispensable tool for such studies. Herein, we review the basic theoretical underpinnings of constraint-based stoichiometric modeling of metabolic networks. Basic concepts, such as stoichiometry, chemical moiety conservation, flux modes, flux balance analysis, and flux solution spaces, are explained with simple, illustrative examples. We emphasize the mathematical definitions and their network topological interpretations.     

Metabolism of adenine nucleotides in thymocyte cytoplasm and subcellular fractions after treatment with concanavalin A, papaverine and adenosine

Studies with rat thymocytes labeled with [14C]adenine and fractionated by digitonin treatment revealed that the cytoplasm of these cells contains about 60% of the total adenine nucleotide pool with a higher ATP/ADP ratio and metabolic activity as compared with the structural components. The incorporation of [14C]adenine and [14C]adenosine into thymocyte adenine nucleotides results in predominant labeling of cytoplasmic ATP, in which the specific radioactivity of this nucleoside triphosphate is two and three times as high as in subcellular structures. Concanavalin A decreases the ATP level in thymocytes without changing its specific radioactivity. This compound does not influence the total content and amount labeled adenine nucleotides in the structural fraction. Papaverine accelerates the catabolism of ATP, mainly in thymocyte cytoplasm and, in a lesser degree, in its structural fraction. In each fraction the papaverine-induced catabolism of ATP is localized in the compartment which is more intensively labeled with [14C]adenine than the whole fractionation ATP pool. Adenosine markedly accelerates adenine nucleotide catabolism in the cytoplasmic and structural fractions of thymocytes; however, only in the first one of them this acceleration is due to ATP elevation. Papaverine and adenosine do not directly influence either the content or specific radioactivity of adenine nucleotides of the structural fraction isolated from [14C]adenine-labeled thymocytes.