Dynamic model for enzyme action

Background: Protein thermodynamic structure theory is an integrated approach to the study of protein dynamics and the mechanisms of enzyme catalysis. In this paper, a hypothesis arising from this theory is examined. The timescale of an enzymatic reaction (TER) gives a key to characterizing enzyme conformational changes. The aspects of timescale important in our approach are: (i) it is logically related to internal motions of the main chain of a protein; (ii) it sets the upper limit on the size or scope of protein conformational changes. Feature (i) is linked to the dynamic properties of enzyme-reactant complexes. Feature (ii) is linked to the dynamic sites of the main chain (promoting motion) involved in enzyme activity. Conclusion: Our analysis shows that a comprehensive understanding of enzymology can be established on the basis of protein thermodynamic structure theory.     

Chemical model of the enzyme system for hydroxylating cyclohexane

The kinetics of co-oxidation of cyclohexane and ferrous chloride by molecular oxygen in methanol in the temperature range 25–50°C has been studied. The total yield of the reaction products, cyclohexanol and cyclohexanone, equals about 25% of the reacted ferrous ion. A redox-active ligand, cysteine, added to the system FeCl₂-C₆H₁₂-O₂ increases product yield. The reaction involving FeCl₂ oxidation by oxygen in the presence of HCl is bimolecular. The effective value of the activation energy of reaction is 9.0 kcal/mol. In the absence of HCl the hydroxylating species is a complex [Fe³⁺O₂⁻], and in the presence of HCl this complex comes together with HO₂^• radicals. A scheme for co-oxidation of FeCl₂ and cyclohexane is proposed. Similarities and contrasts between various hydroxylating species in enzymatic and nonenzymatic hydroxylations are discussed.     

A microscopic model of enzyme kinetics.

Many in vivo enzymatic processes, such as those of the tissue factor pathway of blood coagulation, occur in environments with facilitated substrate delivery or enzymes bound to cellular or lipid surfaces, which are quite different from the ideal fluid environment for which the Michaelis-Menten equation was derived. To describe the kinetics of such reactions, we propose a microscopic model that focuses on the kinetics of a single-enzyme molecule. This model provides the foundation for macroscopic models of the system kinetics of reactions occurring in both ideal and nonideal environments. For ideal reaction systems, the corresponding macroscopic models thus derived are consistent with the Michaelis-Menten equation. It is shown that the apparent Km is in fact a function of the mechanism of substrate delivery and should be interpreted as the substrate level at which the enzyme vacancy time equals the residence time of ES-complexes; it is suggested that our microscopic model parameters characterize more accurately an enzyme and its catalytic efficiency than does the classical Km. This model can also be incorporated into computer simulations of more complex reactions as an alternative to explicit analytical formulation of a macroscopic model.     

A cellular automata model of enzyme kinetics

We have developed a cellular automata model of an enzyme reaction with a substrate in water. The model produces Michaelis-Menten kinetics with good Lineweaver-Burk plots. The variation in affinity parameters predicts that, in general, hydrophobic substrates are more reactive with enzymes, this attribute being more important than the relationship between enzyme and substrate. The ease of generation and the illustrative value of the model lead us to believe that cellular automata models have a useful role in the study of dynamic phenomena such as enzyme kinetics.     

A supramolecular enzyme model catalyzing the central cleavage of carotenoids

Several bis-beta-cyclodextrin porphyrins have been prepared as supramolecular receptors of carotenoids. The binding constants of carotenoids to receptors were determined by quenching the fluorescence of the porphyrins on hydrophobic binding of carotenoids within the cavities of cyclodextrins. K(a)=8.3 x 10(6) M(-1) was calculated for binding of beta,beta-carotene to bis-beta-cyclodextrin Zn porphyrin. The corresponding Ru complex catalyzes the central cleavage of carotenoids in the presence of tert-butyl hydroperoxide in a biphasic system.     

On a model and the kinetics of photo-enhanced enzyme reactions

The recently observed enhancement, by laser irradiation, of the specific activity of the enzyme chymotrypsin (which hydrolyses Benzoyl-L-tyrosine-ethyl ester) at low enzyme concentration is considered. The enhancement of the reaction rate is attributed to a coherently excited state of the enzyme molecule (activated through Raman scattering of the laser light) following a prediction due to Fröhlich. The model is described, the kinetics of the process is framed and the observed enzyme-concentration dependence of the specific activity is reproduced. Predictions of the model are delineated to urge verification of the main contentions through further experimentation.     

A model of enzyme adsorption and hydrolysis of microcrystalline cellulose with slow deactivation of the adsorbed enzyme

Reduction in the activity and the concentration of the adsorbed enzyme are noted in the experimental data. Two alternative mechanisms, inactivation of the adsorbed enzyme and mass transfer of the enzyme from the bulk solution to the solution within the cellulose fibril where the cellulase is assumed to be inactive, are used to represent the decline in activity. The decline in concentration of the adsorbed enzyme is represented by a modest product inhibition and, more importantly, the assumption that the concentration of the adsorption sites is proportional to the square of the remaining substrate concentration. Measurements of both adsorbed enzyme and product concentration over time are used in determining parameter values. The model is applied to a series of experiments having a 10-fold range of substrate concentration and to an experiment in which the product is removed continuously. For both deactivation mechanisms, a very good representation of product concentration (standard deviation 3.6%) is obtained over the full period (168 h) of hydrolysis; the representation of adsorbed enzyme is, however, less accurate (standard deviation 6.7-6.8%).     

Specific ligand-induced association of an enzyme. A new model of dissociating allosteric enzyme

The kinetic behavior of dissociative enzyme system of the type inactive monomer in equilibrium active dimer where dimeric form is stabilized by specific ligand (in particular by substrate) which is bound in the region of the contact of monomers has been analysed. It is assumed that the dissociation of dimer results in formation of monomers which retain the subsites for specific ligand binding. The shape of the dependences of enzyme reaction rate (v) on substrate concentration (S) has been characterized using the order of enzyme reaction rate with respect to substrate concentration: ns = d ln v/d ln [S]. When the substrate concentrations are low the dependences of v on [S] have S-shaped form (the maximum value of ns exceeds the unity) at the definite values of the parameters of the enzyme system. The value of ns approaches--2 at sufficiently high substrate concentrations (in the region where the substrate reveals the inhibitory effect due to blocking the association of inactive monomers into active dimer). The methods of calculation of the parameters of the dissociative enzyme system under discussion have been elaborated on the basis of the analysis of the experimental dependences of specific enzyme activity on enzyme concentration obtained at various fixed substrate concentrations.     

The seed germination model of enzyme catalysis.

The activity of enzymes and other biological macromolecules is often sensitively dependent on physiochemical context. Seed germination provides an analogy that helps to elicit the control and information processing capabilities of enzymatic networks. Like a seed, the enzyme takes a particular action (complexes with a specific substrate and catalyzes a specific reaction) when a specific set of milieu influences is satisfied. The context sensitivity, specificity and speed are enormously enhanced by the parallelism inherent in the electronic wave function (i.e. by the superposition of electronic states). This parallelism is converted to speedup through electronic-conformational interactions. The quantum speedup effect allows biological 'switches' to have qualitatively greater pattern recognition capabilities than electronic switches. Consequently the information processing and control capabilities of biomolecular systems exceed the capabilities obtainable from classical models and exceed the intuitive expectations that have developed through the study of such models.     

A simple model for a regulatory enzyme

A simple model for a regulatory enzyme is described which leads to relationships between the initial velocity of the catalysed reaction and the varied concentration of a substrate that are of the non-inflected or sigmoidal varieties without a maximum. The model assumes that the most relevant measure of protein configuration (itself determining the kinetic behaviour of the enzyme) is the apparent association constant, α_i, measured for the given fractional saturation of the ligand under investigation. It is further assumed that the original state of the protein in solution, α₀, is destabilized by an increment of energy, ΔG_p⁰, that is proportional to the fractional saturation of the enzyme by ligand so that the formation of a new configurational state, a,, can be represented by ?ΔG_p⁰ = RT ln $\text{α}{}_1\text{α}{}_0$. The rate or fractional saturation equation that can be derived from this model predicts both positive and negative cooperativity. Either equation can be transformed for linear representation, provided the maximum velocity or its equivalent maximum saturation is known, and estimates of α₀ and α_i (the apparent association constants at zero and complete saturation) can be obtained thereby. A procedure is also described by which an initial estimate of the maximum velocity or saturation can be improved. The model is tested by application to a range of data in the literature and it is shown to give fits to the data comparable in quality to those provided by the model of Monod, Wyman &; Changeux (1965).     

An "active enzyme" muscle model

A new model of skeletal muscle contraction is presented from a unified view of muscle physiology, chemical energetics and newly obtained experimental data concerning actomyosin ATPase in vitro.In this model an interaction between actin and myosin, involving two distinct active sites, is considered to be the essential elementary mechanism for muscle contractions. These two sites are located on myosin. One site, forming a myosin-ADP-P, complex, has stored energy derived from ATP splitting before the beginning of a contraction. Another site, forming a myosin-ATP complex, upon interacting with actin, catalyzes ATP hydrolysis, using a fraction of the stored energy. The hydrolysis at the latter site is responsible for tension development, while the stored energy is released to drive the contractile reaction between actin and myosin unidirectionally. (Thus, the two sites act co-operatively and they can be viewed as forming an active enzyme.)There has been a difficulty in explaining the shortening heat production with apparent lack of corresponding chemical change at the early stage of contraction. The active enzyme model accounts for the shortening heat as the irreversible release of the stored energy. The heat production appears to precede its corresponding ATP splitting for “refueling” which occurs after complete exhaustion of the stored energy, while the actomyosin ATP hydrolysis takes place proportionally to the work. At the macroscopic level, the model is compatible with Hill's tension-velocity and heat relation.     

Unstable electron pairing and the energy loan model of enzyme catalysis

A theory of enzyme catalysis is described which utilizes a thermodynamically consistent construction of a free energy diagram with different pathways for complex formation and decomposition. The switch to the decomposition pathway occurs when downward uncertainty and thermal fluctuations make possible a short-lived potential energy dominance in which parallel spin electrons are paired and thus free to drop below the energy floor normally maintained by the Pauli exclusion principle. Such pairing is possible if van der Waal's and other weak interactions holding the complex together impose confinement constraints on parallel spin electrons, thereby both increasing uncertainty fluctuations in their kinetic energy and weakly favoring a phase correlation in their motion (which can be interpreted in terms of an exchange of virtual particles). The paired configuration is highly unstable and thus energy released by pair falling is either immediately recaptured to re-establish a normal orbital structure or, if the pair persists long enough to produce a nuclear motion, recaptured at the end of this motion. In the latter case the release of energy can be thought of as an energy loan which finances the switch to the lower activation energy pathway without compromising an energy-balanced regeneration of the enzyme. The advantage is that the complex (because of its instability) has a real free energy which is lower than the free energy which would be assigned to it on the basis of its equilibrium concentration. This increases the specificity and speed of complex formation without decreasing the speed of decomposition. The theory predicts that the magnetic moment which marks the pair should accompany the nuclear (e.g. allosteric) motion and that the pair formation stage of the enzymatic process should have an anomalous temperature dependence. Variations of the model may be constructed to deal with a number of processes involving macromolecular motions, including sequential processes in catalysis, allosteric control, persistent molecular motions, self-assembly, energy transfer, channeled transport, and protection against inhibitors.     

A Michaelis-Menten-style model for the autocatalytic enzyme prostaglandin H synthase

Prostaglandin H synthase (PGHS) is an autocatalytic enzyme which plays a key role in the arachidonic acid metabolic pathway. PGHS mediates the formation of prostaglandin H₂, the precursor for a number of prostaglandins which are important in a wide variety of biological processes, including inflammation, blood clotting, renal function, and tumorigenesis. Here we present a Michaelis-Menten-style model for PGHS. A stability analysis determines when the reaction becomes self-sustaining, and can help explain the regulation of PGHS activity in vivo. We also consider a quasi-steady-state approximation (QSSA) for the model, and present conditions under which the QSSA is expected to be a good approximation. Applying the QSSA for this model can be useful in computationally intensive modeling endeavors involving PGHS.     

A novel pathway to enzyme deactivation: the cutinase model

Cutinase in aqueous solution at pH 4.5 deactivates following a parallel pathway. At 53 degrees C, 88% of the cutinase molecules are in the unfolded conformation, which can aggregate with a reaction order of 3 if the protein concentration is high (>/=12 microM). The aggregates show a sixfold increase in size as determined by dynamic light scattering. This aggregation process is the first phase observed during a deactivation experiment; however, after significant cutinase depletion and maturation of the aggregates, a first-order step starts to dominate and a second phase independent of the protein concentration is observed. Kinetic partitioning between aggregation and first-order irreversible changes of the unfolded conformation can occur during enzyme deactivation when the equilibrium between the native and the unfolded conformation is shifted and kept toward the unfolded conformation.     

Electrical and biochemical properties of an enzyme model of the sodium pump

The electrochemical properties of a widely accepted six-step reaction scheme for the Na+, K+-ATPase have been studied by computer simulation. Rate coefficients were chosen to fit the nonvectorial biochemical data for the isolated enzyme and a current-voltage (I-V) relation consistent with physiological observations was obtained with voltage dependence restricted to one (but not both) of the two translocational steps. The vectorial properties resulting from these choices were consistent with physiological activation of the electrogenic sodium pump by intracellular and extracellular sodium (Na+) and potassium (K+) ions. The model exhibited K+/K+ exchange but little Na+/Na+ exchange unless the energy available from the splitting of adenosine triphosphate (ATP) was reduced, mimicking the behavior seen in squid giant axon. The vectorial ionic activation curves were voltage dependent, resulting in large shifts in apparent Km's with depolarization. At potentials more negative than the equilibrium or reversal potential transport was greatly diminished unless the free energy of ATP splitting was reduced. While the pump reversal potential is at least 100 mV hyperpolarized relative to the resting potential of most cells, the voltage-dependent distribution of intermediate forms of the enzyme allows the possibility of considerable slope conductance of the pump I-V relation in the physiological range of membrane potentials. Some of the vectorial properties of an electrogenic sodium pump appear to be inescapable consequences of the nonvectorial properties of the isolated enzyme. Future application of this approach should allow rigorous quantitative testing of interpretative ideas concerning the mechanism and stoichiometry of the sodium pump.     

Kinetics of hog pancreas alpha-amylase; theoretical model of the dual-site enzyme

The model of the multiple attack mechanism has been proposed for the enzyme consisting of two hypothetical subunits or domains identical in structure and function. The theoretical coefficients of multiple attack: effectiveness and average number of unitary movements, have been computed for such a dual-site model of pancreatic alpha-amylase. Those coefficients affect the values of maximum velocity and Michaelis constant, respectively. Two possible manners of the subunit coupling were considered: symmetric and sequential. The dual-site model with a symmetric type of coupling appeared to be kinetically indistinguishable from the single-site model.