Dynamic properties of network motifs contribute to biological network organization

Biological networks, such as those describing gene regulation, signal transduction, and neural synapses, are representations of large-scale dynamic systems. Discovery of organizing principles of biological networks can be enhanced by embracing the notion that there is a deep interplay between network structure and system dynamics. Recently, many structural characteristics of these non-random networks have been identified, but dynamical implications of the features have not been explored comprehensively. We demonstrate by exhaustive computational analysis that a dynamical property—stability or robustness to small perturbations—is highly correlated with the relative abundance of small subnetworks (network motifs) in several previously determined biological networks. We propose that robust dynamical stability is an influential property that can determine the non-random structure of biological networks.     

Conservation of high-flux backbone in alternate optimal and near-optimal flux distributions of metabolic networks

Constraint-based flux balance analysis (FBA) has proven successful in predicting the flux distribution of metabolic networks in diverse environmental conditions. FBA finds one of the alternate optimal solutions that maximizes the biomass production rate. Almaas et al. have shown that the flux distribution follows a power law, and it is possible to associate with most metabolites two reactions which maximally produce and consume a given metabolite, respectively. This observation led to the concept of high-flux backbone (HFB) in metabolic networks. In previous work, the HFB has been computed using a particular optima obtained using FBA. In this paper, we investigate the conservation of HFB of a particular solution for a given medium across different alternate optima and near-optima in metabolic networks of E. coli and S. cerevisiae. Using flux variability analysis (FVA), we propose a method to determine reactions that are guaranteed to be in HFB regardless of alternate solutions. We find that the HFB of a particular optima is largely conserved across alternate optima in E. coli, while it is only moderately conserved in S. cerevisiae. However, the HFB of a particular near-optima shows a large variation across alternate near-optima in both organisms. We show that the conserved set of reactions in HFB across alternate near-optima has a large overlap with essential reactions and reactions which are both uniquely consuming (UC) and uniquely producing (UP). Our findings suggest that the structure of the metabolic network admits a high degree of redundancy and plasticity in near-optimal flow patterns enhancing system robustness for a given environmental condition.     

Forward rate constants for receptor clusters. Variational methods for upper and lower bounds

We are interested in the effect of receptor clustering on k+, the diffusion-limited forward rate constant for the binding of a ligand to a cell surface receptor. Here we estimate the reduction in k+ when receptors are clustered in various configurations. We obtain two alternative expressions for the flux of ligands into receptors distributed on a surface. Next we show through a variational principle that these provide both upper and lower bounds on the flux when evaluated for trial concentration functions which satisfy only the boundary conditions of the Laplace equation. We use an analogy with electrostatics to calculate rigorous bounds within approx. 10% of the exact result for a variety of planar clusters of hemispherical receptor sites. We also obtain an exact result for the flux into a spheroidal receptor and use this result to obtain bounds on the flux into certain receptor clusters.     

Revisiting Robustness and Evolvability: Evolution in Weighted Genotype Spaces

Robustness and evolvability are highly intertwined properties of biological systems. The relationship between these properties determines how biological systems are able to withstand mutations and show variation in response to them. Computational studies have explored the relationship between these two properties using neutral networks of RNA sequences (genotype) and their secondary structures (phenotype) as a model system. However, these studies have assumed every mutation to a sequence to be equally likely; the differences in the likelihood of the occurrence of various mutations, and the consequence of probabilistic nature of the mutations in such a system have previously been ignored. Associating probabilities to mutations essentially results in the weighting of genotype space. We here perform a comparative analysis of weighted and unweighted neutral networks of RNA sequences, and subsequently explore the relationship between robustness and evolvability. We show that assuming an equal likelihood for all mutations (as in an unweighted network), underestimates robustness and overestimates evolvability of a system. In spite of discarding this assumption, we observe that a negative correlation between sequence (genotype) robustness and sequence evolvability persists, and also that structure (phenotype) robustness promotes structure evolvability, as observed in earlier studies using unweighted networks. We also study the effects of base composition bias on robustness and evolvability. Particularly, we explore the association between robustness and evolvability in a sequence space that is AU-rich – sequences with an AU content of 80% or higher, compared to a normal (unbiased) sequence space. We find that evolvability of both sequences and structures in an AU-rich space is lesser compared to the normal space, and robustness higher. We also observe that AU-rich populations evolving on neutral networks of phenotypes, can access less phenotypic variation compared to normal populations evolving on neutral networks.     

Defining control coefficients in non-ideal metabolic pathways

The extent to which an enzyme controls a flux has been defined as the effect on that flux of a small modulation of the activity of that enzyme divided by the magnitude of the modulation. We here show that in pathways with metabolic channelling or high enzyme concentrations and conserved moieties involving both enzymic and non-enzymic species, this definition is ambiguous; the magnitude of the corresponding flux control coefficient depends on how the enzyme activity is modulated. This is illustrated with two models of biochemically relevant pathways, one in which dynamic metabolite channelling plays a role, and one with a moiety-conserved cycle. To avoid such ambiguity, we view biochemical pathways in a more detailed manner, i.e., as a network of elemental steps. We define 'elemental control coefficients' in terms of the effect on a flux of an equal modulation of the forward and reverse rate constant of any such elemental step (which may correspond to transitions between enzyme states). This elemental control coefficient is independent of the method of modulation. We show how metabolic control analysis can proceed when formulated in terms of the elemental control coefficients and how the traditional control coefficients are related to these elemental control coefficients. An 'impact' control coefficient is defined which quantifies the effect of an activation of all elemental processes in which an enzyme is involved. It equals the sum of the corresponding elemental control coefficients. In ideal metabolic pathways this impact control coefficient reduces to the traditional flux control coefficient. Differences between the traditional control coefficients are indicative of non-ideality of a metabolic pathway, i.e. of channelling or high enzyme concentrations.     

Does Rubisco control the rate of photosynthesis and plant growth? An exercise in molecular ecophysiology

Experiments are described in which tobacco (Nicotiana tabacum L.) transformed with antisense rbcS to decrease expression of ribulose-1,5-bisphosphate carboxylase-oxygenase (Rubisco) was used to evaluate the contribution of Rubisco to the control of photosynthetic rate, and the impact of a changed rate of photosynthesis on whole plant composition, allocation and growth. (1) The concept of flux control coefficients is introduced. It is discussed how, with adequate precautions, a set of wild-type and transgenic plants with varying expression of an enzyme can be used to obtain experimental values for its flux control coefficient. (2) The flux control coefficient of Rubisco for photosynthesis depends on the short-term conditions. It increases in high light, or low CO₂. (3) When plants are grown under constant irradiance, the flux control coefficient in the growth conditions is low (<0.2) at irradiances of up to 1000μmol quanta m⁻² s⁻¹. In a natural irradiance regime exceeding 1500μmol quanta m⁻² s⁻² over several hours the flux coefficient rose to 0.8–0.9. It is concluded that plants are able to adjust the balance between Rubisco and the remainder of the photosynthetic machinery, and thereby avoid a one-sided limitation of photosynthesis by Rubisco over a wide range of ambient growth irradiance regimes. (4) When the plants were grown on limiting inorganic nitrogen, Rubisco had a higher flux control coefficient (0.5). It is proposed that, in many growth conditions, part of the investment in Rubisco may be viewed as a nitrogen store, albeit bringing additional marginal advantages with respect to photosynthetic rate and water use efficiency. (5) A change in the rate of photosynthesis did not automatically translate into a change in growth rate. Several factors are identified which contribute to this buffering of growth against a changed photosynthetic rate. (6) There is an alteration in whole plant allocation, resulting in an increase in the leaf area ratio. The increase is mainly due to a higher leaf water content, and not to changes in shoot/root allocation. This increased investment in whole plant leaf area partly counteracts the decreased efficiency of photosynthesis at the biochemical level. (7) Plants with decreased Rubisco have a lower intrinsic water use efficiency and contain high levels of inorganic cations and anions. It is proposed that these are a consequence of the increased rate of transpiration, and that the resulting osmotic potential might be a contributory factor to the increased water content and expansion of the leaves. (8) Starch accumulation in source leaves is decreased when unit leaf photosynthesis is reduced, allowing a more efficient use of the fixed carbon. (9) Decreased availability of carbohydrates leads to a down-regulation of nitrate assimilation, acting via a decrease in nitrate reductase activity.     

<Emphasis FontCategory="SansSerif">Investigation on changes of modularity and robustness by edge-removal mutations in signaling networks</Emphasis>

Background

Biological networks consisting of molecular components and interactions are represented by a graph model. There have been some studies based on that model to analyze a relationship between structural characteristics and dynamical behaviors in signaling network. However, little attention has been paid to changes of modularity and robustness in mutant networks.

Results

In this paper, we investigated the changes of modularity and robustness by edge-removal mutations in three signaling networks. We first observed that both the modularity and robustness increased on average in the mutant network by the edge-removal mutations. However, the modularity change was negatively correlated with the robustness change. This implies that it is unlikely that both the modularity and the robustness values simultaneously increase by the edge-removal mutations. Another interesting finding is that the modularity change was positively correlated with the degree, the number of feedback loops, and the edge betweenness of the removed edges whereas the robustness change was negatively correlated with them. We note that these results were consistently observed in randomly structure networks. Additionally, we identified two groups of genes which are incident to the highly-modularity-increasing and the highly-robustness-decreasing edges with respect to the edge-removal mutations, respectively, and observed that they are likely to be central by forming a connected component of a considerably large size. The gene-ontology enrichment of each of these gene groups was significantly different from the rest of genes. Finally, we showed that the highly-robustness-decreasing edges can be promising edgetic drug-targets, which validates the usefulness of our analysis.

Conclusions

Taken together, the analysis of changes of robustness and modularity against edge-removal mutations can be useful to unravel novel dynamical characteristics underlying in signaling networks.

     

Regulation of Rubisco activation in antisense plants of tobacco containing reduced levels of Rubisco activase

Following an increase in photon flux density (PFD), ribulose bisphosphate carboxylase/oxygenase (Rubisco) undergoes a slow activation which substantially limits the rate of photosynthesis. This activation process is mediated in part by Rubisco activase. Antisense DNA plants of tobacco were used to quantify the degree to which activase limits Rubisco activation. Reductions in leaf activase content caused proportional reductions in the rate of Rubisco activation following a PFD increase from 110 to 1200 micromol m(-2) sec(-1). This was the case for activase levels up to and slightly beyond normal wild-type activase levels. Activase therefore has a flux control coefficient of unity with respect to the Rubisco activation flux. Such a high control coefficient has rarely been measured for any metabolic system, and this is the highest control coefficient measured for an important photosynthetic flux. In contrast, the rate of Rubisco inactivation in leaves following a drop in PFD of 1200 to 110 micromol m(-2) sec(-1) was unchanged by a 60% reduction in activase levels. Despite the high degree of control that activase exerts over the rate of activation, and thus non-steady-state photosynthesis, it was shown that steady-state photosynthesis was largely unaffected by activase concentration until it was reduced below approximately 15% of the wild-type level. The significance of these results and their implications for published models of Rubisco activation are discussed.     

On the robustness of update schedules in Boolean networks

Deterministic Boolean networks have been used as models of gene regulation and other biological networks. One key element in these models is the update schedule, which indicates the order in which states are to be updated. We study the robustness of the dynamical behavior of a Boolean network with respect to different update schedules (synchronous, block-sequential, sequential), which can provide modelers with a better understanding of the consequences of changes in this aspect of the model. For a given Boolean network, we define equivalence classes of update schedules with the same dynamical behavior, introducing a labeled graph which helps to understand the dependence of the dynamics with respect to the update, and to identify interactions whose timing may be crucial for the presence of a particular attractor of the system. Several other results on the robustness of update schedules and of dynamical cycles with respect to update schedules are presented. Finally, we prove that our equivalence classes generalize those found in sequential dynamical systems.     

Analysis of structural robustness of metabolic networks

We study the structural robustness of metabolic networks on the basis of the concept of elementary flux modes. It is shown that the number of elementary modes itself is not an appropriate measure of structural robustness. Instead, we introduce three new robustness measures. These are based on the relative number of elementary modes remaining after the knockout of enzymes. We discuss the relevance of these measures with the help of simple examples, as well as with larger, realistic metabolic networks. Thereby we demonstrate quantitatively that the metabolism of Escherichia coli, which must be able to adapt to varying conditions, is more robust than the metabolism of the human erythrocyte, which lives under much more homeostatic conditions.     

Stoichiometric design of metabolic networks: multifunctionality,clusters, optimization,weak and strong robustness

Starting from a limited set of reactions describing changes in the carbon skeleton of biochemical compounds complete sets of metabolic networks are constructed. The networks are characterized by the number and types of participating reactions. Elementary networks are defined by the condition that a specific chemical conversion can be performed by a set of given reactions and that this ability will be lost by elimination of any of these reactions. Groups of networks are identified with respect to their ability to perform a certain number of metabolic conversions in an elementary way which are called the network’s functions. The number of the network functions defines the degree of multifunctionality. Transitions between networks and mutations of networks are defined by exchanges of single reactions. Different mutations exist such as gain or loss of function mutations and neutral mutations. Based on these mutations neighbourhood relations between networks are established which are described in a graph theoretical way. Basic properties of these graphs are determined such as diameter, connectedness, distance distribution of pairs of vertices. A concept is developed to quantify the robustness of networks against changes in their stoichiometry where we distinguish between strong and weak robustness. Evolutionary algorithms are applied to study the development of network populations under constant and time dependent environmental conditions. It is shown that the populations evolve toward clusters of networks performing a common function and which are closely neighboured. Under changing environmental conditions multifunctional networks prove to be optimal and will be selected.     

A new measure of the robustness of biochemical networks

MOTIVATION: The robustness of a biochemical network is defined as the tolerance of variations in kinetic parameters with respect to the maintenance of steady state. Robustness also plays an important role in the fail-safe mechanism in the evolutionary process of biochemical networks. The purposes of this paper are to use the synergism and saturation system (S-system) representation to describe a biochemical network and to develop a robustness measure of a biochemical network subject to variations in kinetic parameters. Since most biochemical networks in nature operate close to the steady state, we consider only the robustness measurement of a biochemical network at the steady state. RESULTS: We show that the upper bound of the tolerated parameter variations is related to the system matrix of a biochemical network at the steady state. Using this upper bound, we can calculate the tolerance (robustness) of a biochemical network without testing many parametric perturbations. We find that a biochemical network with a large tolerance can also better attenuate the effects of variations in rate parameters and environments. Compensatory parameter variations and network redundancy are found to be important mechanisms for the robustness of biochemical networks. Finally, four biochemical networks, such as a cascaded biochemical network, the glycolytic-glycogenolytic pathway in a perfused rat liver, the tricarboxylic acid cycle in Dictyostelium discoideum and the cAMP oscillation network in bacterial chemotaxis, are used to illustrate the usefulness of the proposed robustness measure.     

Metabolic design based on a coupled gene expression-metabolic network model of tryptophan production in Escherichia coli

The presumably high potential of a holistic design approach for complex biochemical reaction networks is exemplified here for the network of tryptophan biosynthesis from glucose, a system whose components have been investigated thoroughly before. A dynamic model that combines the behavior of the trp operon gene expression with the metabolic network of central carbon metabolism and tryptophan biosynthesis is investigated. This model is analyzed in terms of metabolic fluxes, metabolic control, and nonlinear optimization. We compare two models for a wild-type strain and another model for a tryptophan producer. An integrated optimization of the whole network leads to a significant increase in tryptophan production rate for all systems under study. This enhancement is well above the increase that can be achieved by an optimization of subsystems. A constant ratio of control coefficients on tryptophan synthesis rate has been identified for the models regarding or disregarding trp operon expression. Although we found some examples where flux control coefficients even contradict the trends of enzyme activity changes in an optimized profile, flux control can be used as an indication for enzymes that have to be taken into account in optimization.     

Robustness analysis of the Escherichia coli metabolic network

Genomic, biochemical, and strain-specific data can be assembled to define an in silico representation of the metabolic network for a select group of single cellular organisms. Flux-balance analysis and phenotypic phase planes derived therefrom have been developed and applied to analyze the metabolic capabilities and characteristics of Escherichia coli K-12. These analyses have shown the existence of seven essential reactions in the central metabolic pathways (glycolysis, pentose phosphate pathway, tricarboxylic acid cycle) for the growth in glucose minimal media. The corresponding seven gene products can be grouped into three categories: (1) pentose phosphate pathway genes, (2) three-carbon glycolytic genes, and (3) tricarboxylic acid cycle genes. Here we develop a procedure that calculates the sensitivity of optimal cellular growth to altered flux levels of these essential gene products. The results indicate that the E. coli metabolic network is robust with respect to the flux levels of these enzymes. The metabolic flux in the transketolase and the tricarboxylic acid cycle reactions can be reduced to 15% and 19%, respectively, of the optimal value without significantly influencing the optimal growth flux. The metabolic network also exhibited robustness with respect to the ribose-5-phosphate isomerase, and the ribose-5-phosephate isomerase flux was reduced to 28% of the optimal value without significantly effecting the optimal growth flux. The metabolic network exhibited limited robustness to the three-carbon glycolytic fluxes both increased and decreased. The development presented another dimension to the use of FBA to study the capabilities of metabolic networks.     

Recombination and the evolution of mutational robustness

Mutational robustness is the degree to which a phenotype, such as fitness, is resistant to mutational perturbations. Since most of these perturbations will tend to reduce fitness, robustness provides an immediate benefit for the mutated individual. However, robust systems decay due to the accumulation of deleterious mutations that would otherwise have been cleared by selection. This decay has received very little theoretical attention. At equilibrium, a population or asexual lineage is expected to have a mutation load that is invariant with respect to the selection coefficient of deleterious alleles, so the benefit of robustness (at the level of the population or asexual lineage) is temporary. However, previous work has shown that robustness can be favoured when robustness loci segregate independently of the mutating loci they act upon. We examine a simple two-locus model that allows for intermediate rates of recombination and inbreeding to show that increasing the effective recombination rate allows for the evolution of greater mutational robustness.     

Thermodynamic and kinetic control of ATP synthesis in yeast mitochondria: role of delta pH

ATP synthesis rate, measured as the variation in external Pi concentration, varied as a linear function of either delta mu H+ or delta Gpin, in such a manner that the delta Gpin/delta mu H+ ratio increased while VATP increased. We also observed a linear dependence of the flux control coefficient of the Pi carrier on delta pH. All the results presented can be explained by a relatively large delta pH drop when VATP increases.     

Rubidium and cesium fluxes in muscle as related to the membrane potential

The reduction of membrane potential in frog sartorius muscle produced by rubidium and cesium ions has been studied over a wide concentration range and compared with depolarization occasioned by potassium ions. The constant field theory of passive flux has been used to predict the potential changes observed. The potential data suggest certain permeability coefficient ratios and these are compared with ratios obtained from flux data using radioactive tracers. The agreement of the flux with the potential data is good if account is taken of the inhibition of potassium flux which occurs in the presence of rubidium and cesium ions. A high temperature dependence has been observed for cesium influx (Q₁₀ = 2.5) which is correlated with the observation that cesium ions depolarize very little at low temperatures. The observations suggest that cesium ions behave more like sodium ions at low temperatures and more like potassium ions at room temperature with respect to their effect on the muscle cell resting potential. The constant field theory of passive ion flux appears to be in general agreement with the experimental results observed if account is taken of the dependence of permeability coefficients on the concentrations of ions used and of possible interactions between the permeabilities of ions.     

Structural Controllability and Controlling Centrality of Temporal Networks

Temporal networks are such networks where nodes and interactions may appear and disappear at various time scales. With the evidence of ubiquity of temporal networks in our economy, nature and society, it''s urgent and significant to focus on its structural controllability as well as the corresponding characteristics, which nowadays is still an untouched topic. We develop graphic tools to study the structural controllability as well as its characteristics, identifying the intrinsic mechanism of the ability of individuals in controlling a dynamic and large-scale temporal network. Classifying temporal trees of a temporal network into different types, we give (both upper and lower) analytical bounds of the controlling centrality, which are verified by numerical simulations of both artificial and empirical temporal networks. We find that the positive relationship between aggregated degree and controlling centrality as well as the scale-free distribution of node''s controlling centrality are virtually independent of the time scale and types of datasets, meaning the inherent robustness and heterogeneity of the controlling centrality of nodes within temporal networks.     

Standard seasons

A renewed interest in climatic definitions of the seasons has surfaced in recent years. However, people usually associate biological phenomena and climate with the seasons. Standard seasons, defined as the periods of the year when the climate is within specified bounds, can be delimited in terms of biological phenomena if climatic thresholds are known for the groups of interest. By focusing attention on the time variation they illustrate climatic variation from a different perspective. An example of the type of information provided by standard seasons on the regional scale is given using human thermal standard seasons in the Pacific Basin. The latitudinal control of climate is indicated by the length of the hot and cold seasons. The moderation of maritime climates is seen in the shortness and slower rate of advance and retreat of the hot and cold seasons, and the extended length of the neutral (or spring) season. Standard seasons have widespread application in indicating the times of the year favorable or unfavorable for particular populations or activities. Bioclimatologists can contribute to season definition by expanding the range of information on climate-organism interaction on which seasonal thresholds are based and may find standard seasons useful in presenting the results of many types of investigations.