The combination of transformed and constrained Gibbs energies

Gibbs free energy is the thermodynamic potential representing the fundamental equation at constant temperature, pressure, and molar amounts. Transformed Gibbs energies are important for biochemical systems because the local concentrations within cell compartments cannot yet be determined accurately. The method of Constrained Gibbs Energies adds kinetic reaction extent limitations to the internal constraints of the system thus extending the range of applicability of equilibrium thermodynamics from predefined constraints to dynamic constraints, e.g., adding time-dependent constraints of irreversible chemical change. In this article, the implementation and use of Transformed Gibbs Energies in the Gibbs energy minimization framework is demonstrated with educational examples. The combined method has the advantage of being able to calculate transient thermodynamic properties during dynamic simulation.     

Analysis and Application of an Equilibrium Model forin vitroBioassay Systems with Three Components: Receptor, Hormone and Hormone-Binding-Protein

A simple theoretical framework is presented for bioassay studies using three componentin vitrosystems. An equilibrium model is used to derive equations useful for predicting changes in biological response after addition of hormone-binding-protein or as a consequence of increased hormone affinity. Sets of possible solutions for receptor occupancy and binding protein occupancy are found for typical values of receptor and binding protein affinity constants. Unique equilibrium solutions are dictated by the initial condition of total hormone concentration. According to the occupancy theory of drug action, increasing the affinity of a hormone for its receptor will result in a proportional increase in biological potency. However, the three component model predicts that the magnitude of increase in biological potency will be a small fraction of the proportional increase in affinity. With typical initial conditions a two-fold increase in hormone affinity for its receptor is predicted to result in only a 33% increase in biological response. Under the same conditions an 11-fold increase in hormone affinity for receptor would be needed to produce a two-fold increase in biological potency. Some currently used bioassay systems may be unrecognized three component systems and gross errors in biopotency estimates will result if the effect of binding protein is not calculated. An algorithm derived from the three component model is used to predict changes in biological response after addition of binding protein toin vitrosystems. The algorithm is tested by application to a published data set from an experimental study in anin vitrosystem (Limet al., 1990,Endocrinology127,1287–1291). Predicted changes show good agreement (within 8%) with experimental observations.     

The steady state kinetics of some biological systems: I

The steady state kinetics of some typical catalytic systems of biological importance have been formulated. The conditions for the existence of a maximum or limiting velocity are examined and discussed. In particular it is shown that the limiting velocity for a given component is simply the rate expression for a given number of steps of theoverall process; from the general condition for a limiting velocity these steps may be specified. The stringency of the conditions which must be imposed upon the steady state solution in order that it may be assumed that one or more steps are essentially at equilibrium is pointed out. The application of the general method to coupled or branched systems and to cyclic systems is briefly discussed. This work was done while the author was a Senior Research Fellow of the National Institute of Health.     

The contribution of proximity and orientation to catalytic reaction rates

We present calculations of the possible magnitude of propinquity which has been proposed to play an important role in enzymic catalysis. The effect has been evaluated as in the past by calculating the ratio of bimolecular to intramolecular reaction rates. The ratio is estimated for intramolecular catalysis in rigid systems as well as for systems with five and six rotatable bonds. Our method differs from others mainly in the way cyclization has been treated. The reaction rate (in all cases) is proportional to the probability that the reactive units have the appropriate spatial and orientational positioning for reaction. This probability is obtained by evaluating a distance distribution function within the spatial and angular intervals to which the units are constrained after reaction. For the bimolecular case we have made the usual assumption that the distribution function is uniform. For the intramolecular reaction, neither the spatial nor the angular part of the distribution function is uniform. The pertinent parameters in this case are the bond lengths and angles, and the statistical weight matrices describing torsional rotation. The difficulty in obtaining analytical expressions for the distribution function is circumvented by using Monte Carlo methods. It is argued that the spatial contribution to rate accelerations in rigid systems can be as high as 10⁷ M, depending upon the size of the volume to which the reactive units are constrained after reaction. The limitation on the smallest physically reasonable volume is estimated from considerations of energy requirements and vibrational amplitudes. Accelerations by five- and six-membered ring cyclizations were estimated at 10³ M, the six-membered ring exhibiting the smaller rate enhancement.     

A stochastic model for a closed biochemical system at equilibrium

This paper presents a stochastic model, a theoretical multi-variate probability function describing concentrations of reactants in a closed biochemical system at equilibrium. The theory applies to the complete range of biochemical systems from single enzyme reactions to combinations of reactions to complete pathways. Prior to examining the general system, probability functions are derived for the following systems as examples: a reaction with a competitive inhibitor, a bisubstrate reaction using the ping-pong mechanism and a series of two mono-substrate reactions. The theory of Markov processes is used to derive the probability functions for each of the example systems and then for the general system which includes the example systems as special cases. The probability function for any appropriate biochemical system proves to be the product of independent Poisson probabilities conditioned on the conservation equations. Finally, the implications of the theory are briefly discussed and possible extensions proposed.     

Product-Form Stationary Distributions for Deficiency Zero Chemical Reaction Networks

We consider stochastically modeled chemical reaction systems with mass-action kinetics and prove that a product-form stationary distribution exists for each closed, irreducible subset of the state space if an analogous deterministically modeled system with mass-action kinetics admits a complex balanced equilibrium. Feinberg’s deficiency zero theorem then implies that such a distribution exists so long as the corresponding chemical network is weakly reversible and has a deficiency of zero. The main parameter of the stationary distribution for the stochastically modeled system is a complex balanced equilibrium value for the corresponding deterministically modeled system. We also generalize our main result to some non-mass-action kinetics.     

Reaction equilibrium for lipase-catalyzed condensation in organic solvent systems

Lipase-catalyzed condensation in an organic solvent is useful for the syntheses of esters. To reasonably design and optimize the reaction conditions, knowledge of the reaction equilibrium is required. The interaction of water with other reactants and the quantitative predictions for adsorption of water by a desiccant are discussed. The solvent effects on the reaction equilibrium are also elucidated in mixtures of nitrile and tert-alcohol.     

Fluctuation analysis in small chemically reacting systems

A theory of noise fluctuations is developed which is applicable to systems of any size in which unimolecular or bimolecular reactions are occurring. The main difference between small and large reacting systems is that in the former the probability of finding a particle in a particular state does not obey a Gaussian distribution, but satisfies a distribution which reflects the mechanism of the chemical reaction. This difference is reflected in the main result of the theory: an autocorrelation function that is expressible as a sum of exponentials, the amplitudes of which are explicit functions of the moments of the distribution. Thus, by using small systems, the autocorrelation function,in principle, allows the elucidation of reaction mechanisms. Numerical simulations indicate that for reacting systems having ten or fewer particles, the deviation of the autocorrelation function from a single exponential should be easily detectable, and that estimates of the first four moments of the distribution should be possible. Accurate inference of the distribution, however, will require further mathematical and experimental advances.     

Non-planckian equilibrium radiation of plasma-like media

Consideration of equilibrium radiation of plasma-like media shows that the spectral distribution of such radiation differs from that of Planckian equilibrium radiation (blackbody radiation). The physical reason for this difference consists in the impossibility of propagation of photons with the dispersion law ω = ck in systems of charged particles. The thermodynamics of equilibrium electromagnetic radiation in plasma is also considered. It is shown that the difference of the thermodynamic properties of such radiation from those of Planckian radiation is characterized by the parameter a = ℏΩ _p /T. This difference is especially pronounced in plasma media in which a ≥ 1. Applications of the results obtained to plasmas of metals (first of all, liquid metals in which charged particles have no distant order) and to the plasma model of the early Universe are discussed.     

Computer-aided solvent screening for biocatalysis

A computer-aided solvent screening methodology is described and tested for biocatalytic systems composed of enzyme, essential water and substrates/products dissolved in a solvent medium, without cells. The methodology is computationally simple, using group contribution methods for calculating constrained properties related to chemical reaction equilibrium, substrate and product solubility, water solubility, boiling points, toxicity and others. Two examples are provided, covering the screening of solvents for lipase-catalyzed transesterification of octanol and inulin with vinyl laurate. Esterification of acrylic acid with octanol is also addressed. Solvents are screened and candidates identified, confirming existing experimental results. Although the examples involve lipases, the method is quite general, so there seems to be no preclusion against application to other biocatalysts.     

Metabolic engineering in silico

This review briefs on the main directions in the field of mathematical modeling of metabolic processes aimed at a rational design of genetically modified organisms. The class of generalized Hill functions is described, and their application to modeling of nonlinear processes in Escherichia coli metabolic systems is illustrated by several examples. A model for the pyrimidine biosynthesis in E. coli, taking into account the nonlinear effects of a negative allosteric regulation of enzyme activities involved in the control of the subsequent stages by the end products of synthesis, is considered. It has been shown that the model displays its own continuous oscillation mode of functioning with a period of approximately 50 min, which is close to the duration of E. coli cell cycle. The need in considering the nonlinear effects in the models as essential elements in the function of metabolic systems far from equilibrium is discussed.     

Co-operativity in monomeric enzymes

It has been known for at least 20 years that monomeric enzymes can in principle show kinetic behaviour similar in appearance to the binding of ligands to oligomeric proteins in which there are co-operative interactions between multiple binding sites. However, the initial lack of experimental examples of kinetic co-operativity suggested that in nature co-operativity always arose from interactions between binding sites. Now, however, several examples are known, most of which cannot be explained in terms of multiple binding sites on one polypeptide chain. All current theoretical models for monomeric co-operativity postulate that it arises from the presence in the mechanism of parallel pathways for substrate binding that are slow compared with the possible rate of the catalytic reaction. Rapid removal of the intermediates produced in the slow steps prevents them from approaching equilibrium and allows the appearance of kinetic properties that would not be possible in systems at equilibrium.     

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.     

Fluorescence correlation spectroscopy. I. Conceptual basis and theory

We describe a method for determining chemical kinetic constants and diffusion coefficients by measuring the rates of decay of spontaneous concentration fluctuations. The equilibrium of the system is not disturbed during the measurement. We measure the number of molecules of a specified type in a defined open volume as a function of time and compute the time course of the deviations from the thermodynamic mean concentration. The method is based on the principle that the rates of decay of spontaneous microscopic fluctuations are determined by the same phenomenological rate coefficients as those of macroscopic departures from equilibrium which result from external perturbations. Hence, an analysis of fluctuations yields the same chemical rate constants and diffusion coefficients as are measured by conventional procedures. In practice the number of the specified molecules is measured by a property such as absorbance or fluorescence which is specific and sensitive to chemical change. The sample volume is defined by a light beam which traverses the cell. As the molecules appear in or disappear from the light beam, either due to diffusion or chemical reaction, their concentration fluctuations give rise to corresponding fluctuations of the intensity of absorbed or emitted light. This paper presents the theory needed to derive chemical rate constants and diffusion coefficients from these fluctuations in light intensity. The theory is applied to three examples of general interest: pure diffusion in the absence of chemical reaction; the binding of a small rapidly diffusing ligand to a larger slowly diffusing macromolecule; and a unimolecular isomerization. The method should be especially useful in studying highly cooperative systems, relatively noncooperative systems with intermediate states closely spaced in free energy, small systems, and systems not readily subject to perturbations of state.     

A global stability criterion for simple control loops

Summary A method for studying biological control loops has been developed, which suffices to prove global stability for the Goodwin equations when the Hill coefficient is equal to 1. This holds for arbitrary reaction constants, even if time delays are included in the system. For a generalized class of repfessible systems, including the Goodwin Equations for >1, the method gives a sufficient condition for global stability, in terms of solutions of an algebraic equation in a single variable. When the criterion is not satisfied, the same equation gives bounds on any possible limit cycles. The method also shows that inducible systems with a unique equilibrium are globally stable. The system of equations studied allows each reaction rate equation to be non-linear, and to include a time delay.     

An exact test for Hardy-Weinberg and multiple alleles

Algorithms for generating the exact distribution of a finite sample drawn from a population in Hardy-Weinberg equilibrium are given for multiple alleles. The finite sampling distribution is derived analogously to Fisher's 2 X 2 exact distribution and is equivalent to Levene's conditional finite sampling distribution for Hardy-Weinberg populations. The algorithms presented are fast computationally and allow for quick alternatives to standard methods requiring corrections and approximations. Computation time is on the order of a few seconds for three-allele examples and up to 2 minutes for four-allele examples on an IBM 3081 machine.     

Metabolism and motility in prebiotic structures

Easily accessible, primitive chemical structures produced by self-assembly of hydrophobic substances into oil droplets may result in self-moving agents able to sense their environment and move to avoid equilibrium. These structures would constitute very primitive examples of life on the Earth, even more primitive than simple bilayer vesicle structures. A few examples of simple chemical systems are presented that self-organize to produce oil droplets capable of movement, environment remodelling and primitive chemotaxis. These chemical agents are powered by an internal chemical reaction based on the hydrolysis of an oleic anhydride precursor or on the hydrolysis of hydrogen cyanide (HCN) polymer, a plausible prebiotic chemistry. Results are presented on both the behaviour of such droplets and the surface-active properties of HCN polymer products. Such motile agents would be capable of finding resources while escaping equilibrium and sustaining themselves through an internal metabolism, thus providing a working chemical model for a possible origin of life.     

The effects of intraspecific competition and stabilizing selection on a polygenic trait

The equilibrium properties of an additive multilocus model of a quantitative trait under frequency- and density-dependent selection are investigated. Two opposing evolutionary forces are assumed to act: (i) stabilizing selection on the trait, which favors genotypes with an intermediate phenotype, and (ii) intraspecific competition mediated by that trait, which favors genotypes whose effect on the trait deviates most from that of the prevailing genotypes. Accordingly, fitnesses of genotypes have a frequency-independent component describing stabilizing selection and a frequency- and density-dependent component modeling competition. We study how the equilibrium structure, in particular, number, degree of polymorphism, and genetic variance of stable equilibria, is affected by the strength of frequency dependence, and what role the number of loci, the amount of recombination, and the demographic parameters play. To this end, we employ a statistical and numerical approach, complemented by analytical results, and explore how the equilibrium properties averaged over a large number of genetic systems with a given number of loci and average amount of recombination depend on the ecological and demographic parameters. We identify two parameter regions with a transitory region in between, in which the equilibrium properties of genetic systems are distinctively different. These regions depend on the strength of frequency dependence relative to pure stabilizing selection and on the demographic parameters, but not on the number of loci or the amount of recombination. We further study the shape of the fitness function observed at equilibrium and the extent to which the dynamics in this model are adaptive, and we present examples of equilibrium distributions of genotypic values under strong frequency dependence. Consequences for the maintenance of genetic variation, the detection of disruptive selection, and models of sympatric speciation are discussed.