The equilibrium concept and density dependence tests What does it all mean?

Summary Tests of density dependent regulation of population size depend on the concept of equilibrium population size. Such an equilibrium is a purely theoretical construct whose existence in the field is debatable and whose value cannot be measured. An equilibrium is supposed to fluctuate in time, but the extent of the fluctuations relative to those of the population size is unknowable. It is impossible to separate a fluctuating population size from a fluctuating equilibrium value and from fluctuating deviations from an equilibrium value. Because it cannot be determined whether a given population size is above, at, or below equilibrium, the course of population size in unpredictable and density dependence tests cannot be expected to produce useful results. Stabilization tests may provide a more useful alternative.     

Steady-state head-to-tail polymerization of actin or microtubules. II. Two-state and three-state kinetic cycles

In a previous paper, bioenergetic aspects of head-to-tail polymerization for a two-state actin ATPase cycle were discussed. In section 2, here, the steady-state polymer length distribution for this case is derived. The distribution has the same mathematical form as at equilibrium, but the parameters are different. In section 3, both bioenergetic topics and the polymer length distribution are considered for the more general and realistic case of a three-state actin ATPase cycle. Again, the mathematical form of the steady-state distribution is the same as at equilibrium, but the parameters are more complicated. In section 4, the question is examined of how much the mean and variance of a polymer length distribution, obtained from a finite experimental sample of polymer (aggregate) molecules, would be expected to deviate from the true mean and variance (from an infinite sample). Also considered briefly in section 4 is the effect of hard polymer-polymer interactions on the equilibrium polymer length distribution, at finite polymer concentrations.     

Equilibrium and Nonequilibrium Models in Ecological Anthropology: An Evaluation of "Stability" in Maring Ecosystems in New Guinea

Three models pertaining to the stability of Maring ecosystems have been proposed. The first is the local stability model, in which a population seeks its own equilibrium state; the second is the regional stability model, in which each population is ultimately unstable, but populations persist somewhere in space; and the third is the disequilibrium model, in which neither stability nor population regulation is attained. In the disequilibrium model, exogenous factors prevent a population, which is moving toward some equilibrium state, from reaching it. The large number of quantitative anthropological and ecological studies in Highlands New Guinea has not shown clearly which of these three models best describes reality. Simulation of shifting agriculture in New Guinea shows that the Highlands systems are equilibrium-seeking, but have such limited recovery rates from disturbance that even small perturbations are sufficient to keep them from reaching equilibrium. When the influences of technological innovation, environmental change, and social-cultural evolution are taken into account, the disequilibrium model is the model of choice. These systems remain away from their stable equilibrium points most of the time, if those exist at all. Thus, New Guinea agroecosystems can be stable or unstable depending upon how stability is defined.     

Two locus models of selection and mutation within and among full-sib lines

Summary General models for continued full-sib mating with two diallelic autosomal loci taking account of linkage, mutation and selection within and among lines are considered. The problems are first approached by deriving the full probability transition matrix, taking account of linkage, mutation and within-line selection. Exact solutions to the equilibrium system are possible, but the computational effort is prohibitive, and this is exacerbated by the introduction of between-line selection. A second approach is based on decomposing the transition matrix into blocks whose properties suggest approximations that lead to a rapid iterative solution of the equilibrium system. Extensive numerical analysis of models of within-line selection and of combined within- and between-line selection were made. The results show that equilibrium values are essentially independent of the degree of linkage under models of within-line selection. This is because mutation plays a dominant role in determining equilibrium structure. Results from models of combined within- and between-line selection show that between-line selection has the dominant influence on gene frequency equilibrium. Both within-line and between-line selection produce appreciable linkage disequilibrium only when selection is disruptive. The results also suggest that much of the twolocus equilibrium structure can be predicted from a knowledge of single-locus equilibria.     

A quantum-dynamics study of the prototropic tautomerism of guanine and its contribution to spontaneous point mutations in Escherichia coli

High-level quantum-chemical and quantum-dynamics calculations are reported on the tautomerization equilibrium and rate constants of guanine and its complexes with one and two water molecules. The results are used to estimate the fraction of guanine present in the cell during DNA synthesis as the unwanted tautomer that forms an irregular base pair with thymine, thus giving rise to a spontaneous GC --> AT point mutation. A comparison of the estimated mutation frequency with the observed frequency in Escherichia coli is used to analyze two proposed mechanisms, differing in the extent of equilibration reached in the tautomerization reaction. In the absence of water, the equilibrium concentration of tautomeric forms is relatively large, but the barrier to their formation is high. If water is present, tautomeric forms are less favored, but water molecules may serve as efficient proton conduits causing rapid tautomerization. It is tentatively concluded that the mechanism in which a high tautomerization barrier keeps the tautomeric transformation far from a state of equilibrium is more likely than a mechanism in which water and/or polymerases produce a low equilibrium concentration of the tautomeric forms.     

Structure of stable protein folding intermediates by equilibrium phi-analysis: the apoflavodoxin thermal intermediate

Protein intermediates in equilibrium with native states may play important roles in protein dynamics but, in cases, can initiate harmful aggregation events. Investigating equilibrium protein intermediates is thus important for understanding protein behaviour (useful or pernicious) but it is hampered by difficulties in gathering structural information. We show here that the phi-analysis techniques developed to investigate transition states of protein folding can be extended to determine low-resolution three-dimensional structures of protein equilibrium intermediates. The analysis proposed is based solely on equilibrium data and is illustrated by determination of the structure of the apoflavodoxin thermal unfolding intermediate. In this conformation, a large part of the protein remains close to natively folded, but a 40 residue region is clearly unfolded. This structure is fully consistent with the NMR data gathered on an apoflavodoxin mutant designed specifically to stabilise the intermediate. The structure shows that the folded region of the intermediate is much larger than the proton slow-exchange core at 25 degrees C. It also reveals that the unfolded region is made of elements whose packing surface is more polar than average. In addition, it constitutes a useful guide to rationally stabilise the native state relative to the intermediate state, a far from trivial task.     

Sedimentation equilibrium analysis of protein interactions with global implicit mass conservation constraints and systematic noise decomposition

Sedimentation equilibrium is a powerful tool for the characterization of protein self-association and heterogeneous protein interactions. Frequently, it is applied in a configuration with relatively long solution columns and with equilibrium profiles being acquired sequentially at several rotor speeds. The present study proposes computational tools, implemented in the software SEDPHAT, for the global analysis of equilibrium data at multiple rotor speeds with multiple concentrations and multiple optical detection methods. The detailed global modeling of such equilibrium data can be a nontrivial computational problem. It was shown previously that mass conservation constraints can significantly improve and extend the analysis of heterogeneous protein interactions. Here, a method for using conservation of mass constraints for the macromolecular redistribution is proposed in which the effective loading concentrations are calculated from the sedimentation equilibrium profiles. The approach is similar to that described by Roark (Biophys. Chem. 5 (1976) 185-196), but its utility is extended by determining the bottom position of the solution columns from the macromolecular redistribution. For analyzing heterogeneous associations at multiple protein concentrations, additional constraints that relate the effective loading concentrations of the different components or their molar ratio in the global analysis are introduced. Equilibrium profiles at multiple rotor speeds also permit the algebraic determination of radial-dependent baseline profiles, which can govern interference optical ultracentrifugation data, but usually also occur, to a smaller extent, in absorbance optical data. Finally, the global analysis of equilibrium profiles at multiple rotor speeds with implicit mass conservation and computation of the bottom of the solution column provides an unbiased scale for determining molar mass distributions of noninteracting species. The properties of these tools are studied with theoretical and experimental data sets.     

Epistasis of quantitative trait loci under different gene action models

Modeling and detecting nonallelic (epistatic) effects at multiple quantitative trait loci (QTL) often assume that the study population is in zygotic equilibrium (i.e., genotypic frequencies at different loci are products of corresponding single-locus genotypic frequencies). However, zygotic associations can arise from physical linkages between different loci or from many evolutionary and demographic processes even for unlinked loci. We describe a new model that partitions the two-locus genotypic values in a zygotic disequilibrium population into equilibrium and residual portions. The residual portion is of course due to the presence of zygotic associations. The equilibrium portion has eight components including epistatic effects that can be defined under three commonly used equilibrium models, Cockerham's model, F2-metric, and F(infinity)-metric models. We evaluate our model along with these equilibrium models theoretically and empirically. While all the equilibrium models require zygotic equilibrium, Cockerham's model is the most general, allowing for Hardy-Weinberg disequilibrium and arbitrary gene frequencies at individual loci whereas F2-metric and F(infinity)-metric models require gene frequencies of one-half in a Hardy-Weinberg equilibrium population. In an F2 population with two unlinked loci, Cockerham's model is reduced to the F2-metric model and thus both have a desirable property of orthogonality among the genic effects; the genic effects under the F(infinity)-metric model are not orthogonal but they can be easily translated into those under the F2-metric model through a simple relation. Our model is reduced to these equilibrium models in the absence of zygotic associations. The results from our empirical analysis suggest that the residual genetic variance arising from zygotic associations can be substantial and may be an important source of bias in QTL mapping studies.     

Equilibrium shifts in enzyme reactions at high pressure

The effect of pressure on the equilibrium of a reaction was studied. Theoretical equilibrium constants and product concentrations have been calculated at elevated pressures. The theory is illustrated with an example of l-malate synthesis catalyzed by a fumarase. To study shifts in the equilibrium relatively low pressures can be applied (50–200 MPa), but our calculations show that for process optimisation much higher pressures (up to 1000 MPa) have to be used.

At these higher pressures, more stable enzymes are needed. We performed experiments with the hyperthermophilic β-glycosidase from Pyrococcus furiosus as a catalyst. Oligosaccharides were synthesized from glucose in an equilibrium reaction at pressures from 0.1 to 500 MPa. The enzyme remained active at 500 MPa. The equilibrium of the reaction was influenced by pressure and shifted towards the hydrolysis side, decreasing final oligosaccharide concentrations with increasing pressure. This pressure dependence of the final product concentration and the equilibrium constant could be described with a positive reaction volume of 2.4 mol/cm³.     

Exacerbating the Tragedy of the Commons: Private Inefficient Outcomes and Peer Effect in Experimental Games with Fishing Communities

Economic Experimental Games have shown that individuals make decisions that deviate down from the suboptimal Nash equilibrium. However, few studies have analyzed the case when deviation is above the Nash equilibrium. Extracting from above the Nash equilibrium is inefficient not only socially but also privately and it would exacerbate the tragedy of the commons. That would be the case of a race to the fish when stocks are becoming depleted or driver behavior on a highly congested road. The objective of this study is to analyze private inefficient extraction behavior in experimental games and to associate the type of player and the type of player group with such inefficient outcomes. To do this, we carried out economic experimental games with local coastal fishermen in Colombia, using a setting where the scarcity of the resource allows for an interior Nash equilibrium and inefficient over-extraction is possible. The state of the resource, the type of player and the composition of the group explain, in part, this inefficient behavior.     

Effect on the equilibrium between the Na+-form and the K+-form of the (Na+ + K+)-ATPase of modification of the enzyme with pyridoxal 5-phosphate

An increase in pH shifts the equilibrium between the K+-form and the Na+-form of the (Na+ + K+)-ATPase towards the Na+-form. pK for the proton effect on the equilibrium is decreased by modification of the enzyme with pyridoxal 5-phosphate. The reactivity of the enzyme towards pyridoxal 5-phosphate is increased by an increase in pH. Modification by pyridoxal 5-phosphate of epsilon-amino groups on lysine, which has a pK of about 8 with the enzyme in the K+-form and of about 7.4 in the Na+-form, shifts the equilibrium between E1Na+ and E2 towards E2, and the equilibrium between E2(K+occ) and E2 towards E2, but has no effect on the overall equilibrium between E1Na+ and E2(K+occ). An additional modification of epsilon-amino groups on lysine, which has a pK of 9.5-10 with the enzyme in the K+-form and of about 7.7 with the enzyme in the Na+-form, shifts the equilibrium between E2(K+occ) and E1Na+ towards E1Na+; this is due to a shift in the equilibrium between E2(K+occ) and E2 towards E2, but with no effect on the equilibrium between E1Na+ and E2. The results show that the transition from the K+-form to the Na+-form decreases the pK of lysine epsilon-amino groups on the enzyme, and that the protonation of these groups influences the equilibrium between the two conformations.     

Equilibrium of global amphibian species distributions with climate

A common assumption in bioclimatic envelope modeling is that species distributions are in equilibrium with contemporary climate. A number of studies have measured departures from equilibrium in species distributions in particular regions, but such investigations were never carried out for a complete lineage across its entire distribution. We measure departures of equilibrium with contemporary climate for the distributions of the world amphibian species. Specifically, we fitted bioclimatic envelopes for 5544 species using three presence-only models. We then measured the proportion of the modeled envelope that is currently occupied by the species, as a metric of equilibrium of species distributions with climate. The assumption was that the greater the difference between modeled bioclimatic envelope and the occupied distribution, the greater the likelihood that species distribution would not be at equilibrium with contemporary climate. On average, amphibians occupied 30% to 57% of their potential distributions. Although patterns differed across regions, there were no significant differences among lineages. Species in the Neotropic, Afrotropics, Indo-Malay, and Palaearctic occupied a smaller proportion of their potential distributions than species in the Nearctic, Madagascar, and Australasia. We acknowledge that our models underestimate non equilibrium, and discuss potential reasons for the observed patterns. From a modeling perspective our results support the view that at global scale bioclimatic envelope models might perform similarly across lineages but differently across regions.     

The role of guanine nucleotides in protein biosynthesis.

It is not possible to select one competing substrate molecule over another one when the selection system is at equilibrium. Therefore, it is convenient to view all substrate selections as transport phenomena. The requirement for a displacement from equilibrium to effect a substrate selection can be met in different ways. One particularly convenient way is to drive a nonselected substrate, such as GTP in protein synthesis, far from equilibrium. This allows the flux of selected substrate to be relatively slow, but effectively irreversible. Accordingly, the conventional view that GTP hydrolysis drives protein synthesis is amended. It is suggested that the regeneration GTP from GDP is the driving force for protein synthesis. Several different selection mechanisms are described in the context fof systems driven by displacements from equilibrium of the nonselected substrate. These are then evaluated in light of recent experimental results. The data argue against the relevance of proofreading mechanisms for aminoacyl-tRNA selection by the messenger RNA-programmed ribosome. Similarly, recent data suggesting that the translation of messenger RNA is not dependent on the presence of elongation factors and guanine nucleotides are reevaluated.     

Histone-DNA binding free energy cannot be measured in dilution-driven dissociation experiments

Despite decades of study on nucleosomes, there has been no experimental determination of the free energy of association between histones and DNA. Instead, only the relative free energy of association of the histone octamer for differing DNA sequences has been available. Recently, a method was developed based on quantitative analysis of nucleosome dissociation in dilution experiments that provides a simple practical measure of nucleosome stability. Solution conditions were found in which nucleosome dissociation driven by dilution fit well to a simple model involving a noncooperative nucleosome assembly/disassembly equilibrium, suggesting that this approach might allow absolute equilibrium affinity of the histone octamer for DNA to be measured. Here, we show that the nucleosome assembly/disassembly process is not strictly reversible in these solution conditions, implying that equilibrium affinities cannot be obtained from these measurements. Increases in [NaCl] or temperature, commonly employed to suppress kinetic bottlenecks in nucleosome assembly, lead to cooperative behavior that cannot be interpreted with the simple assembly/disassembly equilibrium model. We conclude that the dilution experiments provide useful measures of kinetic but not equilibrium stability. Kinetic stability is of practical importance: it may govern nucleosome function in vivo, and it may (but need not) parallel absolute thermodynamic stability.     

The curious case of Skokholm: equilibrium,non‐equilibrium and a phase shift in an island landbird assemblage

Previous work has indicated that the landbirds of Skokholm island (Wales) are not in equilibrium as defined in MacArthur–Wilson's classic theory of island biogeography. This study takes a new dataset with over six decades of data and investigates equilibrium on Skokholm using cluster analysis to identify periods of turnover stability. The attributes of the identified periods were investigated in relation to the MacArthur–Wilson model using analyses of change in numbers of species, S, from one year to the next and measures of variability in S quantified for each of the periods identified together with a consideration of the dynamics in the numbers of species by habitat groupings. Cluster analysis identified four main periods of which two middle periods appeared to be in equilibrium but with a phase shift in‐between. The first and last periods showed non‐equilibrium dynamics but plots of species by habitat groupings suggested that this was due to habitat changes going on at those times. This decadal long dataset indicates that the landbirds of Skokholm exhibit periods of both equilibrium and non‐equilibrium with the latter attributable to habitat change. The apparent phase shift in the equilibrium number of species was unexpected within the framework of island biogeographic theory and not easily explained using the current MacArthur–Wilson framework. There is a need to integrate the theory of island biogeography with more recent work on alternative stable states, tipping points, and phase (or regime) shifts, together with equilibrium and non‐equilibrium dynamics, into a single framework.     

Sibling competition stabilizes signalling resolution models of parent-offspring conflict

Young of altricial birds use conspicuous displays to solicit food from their parents. There is experimental evidence that the intensity of these displays is correlated with the level of food deprivation of young, and that parents respond to increased levels of solicitation by increasing the rate of food delivery to the nest. Game-theoretical models based on the handicap principle show that, when solicitation is costly, there is a signalling equilibrium at which there is a one-to-one correspondence between the condition of the young and the intensity of their display. Parents use this information to adjust their levels of investment on the current offspring. However, the models also have a non-signalling equilibrium, and computer simulations show that only the non-signalling equilibrium is stable. Here I show that when direct sibling competition is introduced into the model, in such a way that parents have control on the amount of food provided to the nest, but not on the way the food is allocated among siblings, the non-signalling equilibrium disappears and the signalling equilibrium becomes stable.     

Studies on inactivation of lipoprotein lipase: role of the dimer to monomer dissociation

Sedimentation equilibrium analysis demonstrated that preparations of bovine lipoprotein lipase contain a complex mixture of dimers and higher oligomers of enzyme protein. Enzyme activity profiles from sedimentation equilibrium as well as from gel filtration indicated that activity is associated almost exclusively with the dimer fraction. To explore if the enzyme could be dissociated into active monomers, 0.75 M guanidinium chloride was used. Sedimentation velocity measurements demonstrated that this treatment led to dissociation of the lipase protein into monomers. Concomitant with dissociation, there was an irreversible loss of catalytic activity and a moderate change in secondary structure as detected by circular dichroism. The rate of inactivation increased with decreasing concentrations of active lipase, but addition of inactive lipase protein did not slow down the inactivation. This indicates that reversible interactions between active species precede the irreversible loss of activity. The implication is that dissociation initially leads to a monomer form which is in reversible equilibrium with the active dimer, but which decays rapidly into an inactive form, and is therefore not detected as a stable component in the system.     

Discrete polymorphisms due to disruptive selection on a continuous trait--I: the one-locus case

We have investigated, numerically and analytically, long-term evolution under frequency-dependent disruptive selection of a continuous trait varying in a finite range and controlled by one diploid mendelian locus. We found that evolution converges towards a unique long-term equilibrium where only two extreme phenotypes are present with frequencies identical to those of the mixed strategy that would be the unique ESS of the game defined by the basic fitness function of the model. As long as this precise phenotypic composition is preserved, any genetic configuration of the polymorphism is equally acceptable (selectively neutral) at the equilibrium. Thus the number of alleles and their dominance pattern may vary considerably among different equilibrium populations. If genetic expression of the trait is variable but the amount of variability is genetically modifiable, disruptive selection, acting on such modifiers, produces a steady increase of expression variability before the equilibrium is attained. In this case a population at the long-term equilibrium might even be genetically monomorphic, with the phenotypic dimorphism resulting from purely random individual variation.     

Analysis of solvent-free ethyl oleate enzymatic synthesis at equilibrium conditions

This work reports experimental equilibrium data for the esterification of pure oleic acid and a fatty acid mixture with ethanol, using an immobilized Candida antarctica B lipase as catalyst. Reactions are performed in a solvent-free system, containing a mixture of substrates and different amounts of distilled water. According to the initial amount of water and the extent of the reaction, one or two liquid phases are present. Therefore, when the equilibrium is achieved, the liquid–liquid and chemical reaction equilibria have to be simultaneously satisfied.

Several reports dealing with enzymatic reactions performed in two-phase systems have found that the value of the reaction equilibrium constant calculated from overall experimental concentrations varies not only with temperature but also with substrate ratio and water content. Although this approach is a valuable way to explore equilibrium shifts in biphasic systems, it is limited to ideal systems with constant partition coefficients. The aim of this work is to consider the biphasic nature of the reactive mixture through a computational procedure that simultaneously takes into account liquid–liquid and reaction equilibria. This approach enables the determination of a classical temperature-dependent thermodynamic equilibrium constant, which accurately fits experimental equilibrium conversions over a wide range of operating conditions.     

Equilibrium and non‐equilibrium dynamics simultaneously operate in the Galápagos islands

Island biotas emerge from the interplay between colonisation, speciation and extinction and are often the scene of spectacular adaptive radiations. A common assumption is that insular diversity is at a dynamic equilibrium, but for remote islands, such as Hawaii or Galápagos, this idea remains untested. Here, we reconstruct the temporal accumulation of terrestrial bird species of the Galápagos using a novel phylogenetic method that estimates rates of biota assembly for an entire community. We show that species richness on the archipelago is in an ascending phase and does not tend towards equilibrium. The majority of the avifauna diversifies at a slow rate, without detectable ecological limits. However, Darwin's finches form an exception: they rapidly reach a carrying capacity and subsequently follow a coalescent‐like diversification process. Together, these results suggest that avian diversity of remote islands is rising, and challenge the mutual exclusivity of the non‐equilibrium and equilibrium ecological paradigms.