Irreversibility in unbranched pathways: preferred positions based on regulatory considerations

It has been observed experimentally that most unbranched biosynthetic pathways have irreversible reactions near their beginning, many times at the first step. If there were no functional reasons for this fact, then one would expect irreversible reactions to be equally distributed among all positions in such pathways. Since this is not the case, we have attempted to identify functional consequences of having an irreversible reaction early in the pathway. We systematically varied the position of the irreversible reaction in model pathways and compared the resulting systemic behavior according to several criteria for functional effectiveness, using the method of mathematically controlled comparisons. This technique minimizes extraneous differences in systemic behavior and identifies those that are fundamental. Our results show that a pathway with an irreversible reaction located at the first step, and with all other reactions reversible, is on average better than an otherwise equivalent pathway with all reactions reversible, which in turn is on average better than an otherwise equivalent pathway with an irreversible reaction located at any step other than the first. Pathways with an irreversible first reaction and low concentrations of intermediates (one of the primary criteria for functional effectiveness) exhibit the following profile when compared to fully reversible pathways: changes in the concentration of intermediates in response to changes in the level of initial substrate are equally low, the robustness of the intermediate concentrations and of the flux is similar, the margins of stability are similar, flux is more responsive to changes in demand for end product, intermediate concentrations are less responsive to changes in demand for end product, and transient times are shorter. These results provide a functional rationale for the positioning of irreversible reactions at the beginning of unbranched biosynthetic pathways.     

Mechanical disassembly of single virus particles reveals kinetic intermediates predicted by theory

New experimental approaches are required to detect the elusive transient intermediates predicted by simulations of virus assembly or disassembly. Here, an atomic force microscope (AFM) was used to mechanically induce partial disassembly of single icosahedral T=1 capsids and virions of the minute virus of mice. The kinetic intermediates formed were imaged by AFM. The results revealed that induced disassembly of single minute-virus-of-mice particles is frequently initiated by loss of one of the 20 equivalent capsomers (trimers of capsid protein subunits) leading to a stable, nearly complete particle that does not readily lose further capsomers. With lower frequency, a fairly stable, three-fourths-complete capsid lacking one pentamer of capsomers and a free, stable pentamer were obtained. The intermediates most frequently identified (capsids missing one capsomer, capsids missing one pentamer of capsomers, and free pentamers of capsomers) had been predicted in theoretical studies of reversible capsid assembly based on thermodynamic-kinetic models, molecular dynamics, or oligomerization energies. We conclude that mechanical manipulation and imaging of simple virus particles by AFM can be used to experimentally identify kinetic intermediates predicted by simulations of assembly or disassembly.     

A theoretical model successfully identifies features of hepatitis B virus capsid assembly.

The capsids of most spherical viruses are icosahedral, an arrangement of multiples of 60 subunits. Though it is a salient point in the life cycle of any virus, the physical chemistry of virus capsid assembly is poorly understood. We have developed general models of capsid assembly that describe the process in terms of a cascade of low order association reactions. The models predict sigmoidal assembly kinetics, where intermediates approach a low steady state concentration for the greater part of the reaction. Features of the overall reaction can be identified on the basis of the concentration dependence of assembly. In simulations, and on the basis of our understanding of the models, we find that nucleus size and the order of subsequent "elongation" reactions are reflected in the concentration dependence of the extent of the reaction and the rate of the fast phase, respectively. The reaction kinetics deduced for our models of virus assembly can be related to the assembly of any "spherical" polymer. Using light scattering and size exclusion chromatography, we observed polymerization of assembly domain dimers of hepatitis B virus (HBV) capsid protein. Empty capsids assemble at a rate that is a function of protein concentration and ionic strength. The kinetics of capsid formation were sigmoidal, where the rate of the fast phase had second-power concentration dependence. The extent of assembly had third-power concentration dependence. Simulations based on the models recapitulated the concentration dependences observed for HBV capsid assembly. These results strongly suggest that in vitro HBV assembly is nucleated by a trimer of dimers and proceeds by the addition of individual dimeric subunits. On the basis of this mechanism, we suggest that HBV capsid assembly could be an important target for antiviral therapeutics.     

Modeling capsid self-assembly: design and analysis

A series of simulations aimed at elucidating the self-assembly dynamics of spherical virus capsids is described. This little-understood phenomenon is a fascinating example of the complex processes that occur in the simplest of organisms. The fact that different viruses adopt similar structural forms is an indication of a common underlying design, motivating the use of simplified, low-resolution models in exploring the assembly process. Several versions of a molecular dynamics approach are described. Polyhedral shells of different sizes are involved, the assembly pathways are either irreversible or reversible and an explicit solvent is optionally included. Model design, simulation methodology and analysis techniques are discussed. The analysis focuses on the growth pathways and the nature of the intermediate states, properties that are hard to access experimentally. Among the key observations are that efficient growth proceeds by means of a cascade of highly reversible stages, and that while there are a large variety of possible partial assemblies, only a relatively small number of strongly bonded configurations are actually encountered.     

A practical kinetic model that considers endproduct inhibition in anaerobic digestion processes by including the equilibrium constant

The classical Michaelis-Menten model is widely used as the basis for modeling of a number of biological systems. As the model does not consider the inhibitory effect of endproducts that accumulate in virtually all bioprocesses, it is often modified to prevent the overestimation of reaction rates when products have accumulated. Traditional approaches of model modification use the inclusion of irreversible, competitive, and noncompetitive inhibition factors. This article demonstrates that these inhibition factors are insufficient to predict product inhibition of reactions that are close the dynamic equilibrium. All models investigated were found to violate thermodynamic laws as they predicted positive reaction rates for reactions that were endergonic due to high endproduct concentrations. For modeling of biological processes that operate close to the dynamic equilibrium (e.g., anaerobic processes), it is critical to prevent the prediction of positive reaction rates when the reaction has already reached the dynamic equilibrium. This can be achieved by using a reversible kinetic model. However, the major drawback of the reversible kinetic model is the large number of empirical parameters it requires. These parameters are difficult to determine and prone to experimental error. For this reason, the reversible model is not practical in the modeling of biological processes.This article uses the fundamentals of steady-state kinetics and thermodynamics to establish an equation for the reversible kinetic model that is of practical use in bio-process modeling. The behavior of this equilibrium-based model is compared with Michaelis-Menten-based models that use traditional inhibition factors. The equilibrium-based model did not require any empirical inhibition factor to correctly predict when reaction rates must be zero due to the free energy change being zero. For highly exergonic reactions, the equilibrium-based model did not deviate significantly from the Michaelis-Menten model, whereas, for reactions close to equilibrium, the reaction rate was mainly controlled by the quotient of mass action ratio (concentration of all products over concentration of all substrates) over the equilibrium constant K. This quotient is a measure of the displacement of the reaction from its equilibrium. As the new equation takes into account all of the substrates and products, it was able to predict the inhibitor effect of multiple endproducts. The model described is designed to be a useful basis for a number of different model applications where reaction conditions are close to equilibrium.     

Exploring the Paths of (Virus) Assembly

Assembly of viruses that have hundreds of subunits or folding of proteins that have hundreds of amino acids—complex biological reactions—are often spontaneous and rapid. Here, we examine the complete set of intermediates available for the assembly of a hypothetical viruslike particle and the connectivity between these intermediates in a graph-theory-inspired study. Using a build-up procedure, assuming ideal geometry, we enumerated the complete set of 2,423,313 species for formation of an icosahedron from 30 dimeric subunits. Stability of each n-subunit intermediate was defined by the number of contacts between subunits. The probability of forming an intermediate was based on the number of paths to it from its precedecessors. When defining population subsets predicted to have the greatest impact on assembly, both stability- and probability-based criteria select a small group of compact and degenerate species; ergo, only a few hundred intermediates make a measurable contribution to assembly. Though the number of possible intermediates grows combinatorially with the number of subunits in the capsid, the number of intermediates that make a significant contribution to the reaction grows by a much smaller function, a result that may contribute to our understanding of assembly and folding reactions.     

A reaction landscape identifies the intermediates critical for self-assembly of virus capsids and other polyhedral structures

The capsids of spherical viruses may contain from tens to hundreds of copies of the capsid protein(s). Despite their complexity, these particles assemble rapidly and with high fidelity. Subunit and capsid represent unique end states. However, the number of intermediate states in these reactions can be enormous-a situation analogous to the protein folding problem. Approaches to accurately model capsid assembly are still in their infancy. In this paper, we describe a sail-shaped reaction landscape, defined by the number of subunits in each species, the predicted prevalence of each species, and species stability. Prevalence can be calculated from the probability of synthesis of a given intermediate and correlates well with the appearance of intermediates in kinetics simulations. In these landscapes, we find that only those intermediates along the leading edge make a significant contribution to assembly. Although the total number of intermediates grows exponentially with capsid size, the number of leading-edge intermediates grows at a much slower rate. This result suggests that only a minute fraction of intermediates needs to be considered when describing capsid assembly.     

The Impact of Viral RNA on Assembly Pathway Selection

Many single-stranded RNA viruses self-assemble their protein containers around their genomes. The roles that the RNA plays in this assembly process have mostly been ignored, resulting in a protein-centric view of assembly that is unable to explain adequately the fidelity and speed of assembly in such viruses. Using bacteriophage MS2, we demonstrate here via a combination of mass spectrometry and kinetic modelling how viral RNA can bias assembly towards only a small number of the many possible assembly pathways, thus increasing assembly efficiency. Assembly reactions have been studied in vitro using phage coat protein dimers, the known building block of the T = 3 shell, and short RNA stem-loops based on the translational operator of the replicase cistron, a 19 nt fragment (TR). Mass spectrometry has unambiguously identified two on-pathway intermediates in such reactions that have stoichiometry consistent with formation of either a particle 3-fold or 5-fold axis. These imply that there are at least two sub-pathways to the final capsid. The flux through each pathway is controlled by the length of the RNA stem-loop triggering the assembly reaction and this effect can be understood in structural terms. The kinetics of intermediate formation have been studied and show steady-state concentrations for intermediates between starting materials and the T = 3 shell, consistent with an assembly process in which all the steps are in equilibrium. These data have been used to derive a kinetic model of the assembly reaction that in turn allows us to determine the dominant assembly pathways explicitly, and to estimate the effect of the RNA on the free energy of association between the assembling protein subunits. The results reveal that there are only a small number of dominant assembly pathways, which vary depending on the relative ratios of RNA and protein. These results suggest that the genomic RNA plays significant roles in defining the precise assembly sub-pathway followed to create the final capsid.     

Quantitative analysis of multi-component spherical virus assembly: scaffolding protein contributes to the global stability of phage P22 procapsids

Assembly of the hundreds of subunits required to form an icosahedral virus must proceed with exquisite fidelity, and is a paradigm for the self-organization of complex macromolecular structures. However, the mechanism for capsid assembly is not completely understood for any virus. Here we have investigated the in vitro assembly of phage P22 procapsids using a quantitative model specifically developed to analyze assembly of spherical viruses. Phage P22 procapsids are the product of the co-assembly of 420 molecules of coat protein and approximately 100-300 molecules of scaffolding protein. Scaffolding protein serves as an assembly chaperone and is not part of the final mature capsid, but is essential for proper procapsid assembly. Here we show that scaffolding protein also affects the thermodynamics of assembly, and for the first time this quantitative analysis has been performed on a virus composed of more than one type of protein subunit. Purified coat and scaffolding proteins were mixed in varying ratios in vitro to form procapsids. The reactions were allowed to reach equilibrium and the proportion of the input protein assembled into procapsids or remaining as free subunits was determined by size exclusion chromatography and SDS-PAGE. The results were used to calculate the free energy contributions for individual coat and scaffolding proteins. Each coat protein subunit was found to contribute -7.2(+/-0.1)kcal/mol and each scaffolding protein -6.1(+/-0.2)kcal/mol to the stability of the procapsid. Because each protein interacts with two or more neighbors, the pair-wise energies are even less. The weak protein interactions observed in the assembly of procapsids are likely important in the control of nucleation, since an increase in affinity between coat and scaffolding proteins can lead to kinetic traps caused by the formation of too many nuclei. In addition, we find that adjusting the molar ratio of scaffolding to coat protein can alter the assembly product. When the scaffolding protein concentration is low relative to coat protein, there is a correspondingly low yield of proper procapsids. When the relative concentration is very high, too many nuclei form, leading to kinetically trapped assembly intermediates.     

pH-dependence of intermediate steps of membrane fusion induced by the influenza fusion peptide

Membrane fusion mediated by the influenza-virus fusion protein is activated by low pH via a cascade of reactions. Some processes among them are irreversible, such as helix hairpin formation of the ectodomain, whereas others are reversible, such as exposure of the fusion peptide. Using this property, we attempted to dissect, in temporal order, different stages of the fusion reaction involving the fusion peptide by an acidic-neutral-acidic pH cycle. The fluorescence-quenching data indicated that both insertion depth and self-assembly are pH-reversible. In addition, lipid mixing assay was demonstrated to be arrested by neutral pH. By contrast, membrane leakage was shown to be irreversible with respect to pH. Our results, along with those from other studies on the pH-dependence of membrane fusion, are used to build a model for the virus-mediated fusion event from the perspective of pH-reversibility.     

General method of analysis of kinetic equations for multistep reversible mechanisms in the single-exponential regime: application to kinetics of open complex formation between Esigma70 RNA polymerase and lambdaP(R) promoter DNA

A novel analytical method based on the exact solution of equations of kinetics of unbranched first- and pseudofirst-order mechanisms is developed for application to the process of Esigma70 RNA polymerase (R)-lambdaPR promoter (P) open complex formation, which is described by the minimal three-step mechanism with two kinetically significant intermediates (I1, I2), [equation: see text], where the final product is an open complex RPo. The kinetics of reversible and irreversible association (pseudofirst order, [R] > [P]) to form long-lived complexes (RPo and I2) and the kinetics of dissociation of long-lived complexes both exhibit single exponential behavior. In this situation, the analytical method provides explicit expressions relating observed rate constants to the microscopic rate constants of mechanism steps without use of rapid equilibrium or steady-state approximations, and thereby provides a basis for interpreting the composite rate constants of association (ka), isomerization (ki), and dissociation (kd) obtained from experiment for this or any other sequential mechanism of any number of steps. In subsequent papers, we apply this formalism to analyze kinetic data obtained in the reversible and irreversible binding regimes of Esigma70 RNA polymerase (R)-lambdaP(R) promoter (P) open complex formation.     

A kinetic model for virus binding which involves release of cell-bound virus-receptor complexes

A general kinetic mechanism is presented for reversible binding of viruses to cells followed by an irreversible step that initiates the delivery of the viral genome. A novel feature is additional pathways for the release of both virus-occupied and unoccupied receptors from cells. Due to one simplifying assumption, it does not apply at low receptor densities. However, it is sufficiently general to be applicable to ligand binding and internalization for those systems in which ligand diffusion is rate limiting. Three different versions of the model fit the usual kinetic data for the binding of an eclipse mutant of bacteriophage phi X174 to Escherichia coli. However, in each case binding to cell-bound receptors is irreversible. Therefore, this explains the apparent failure of this system to obey the Law of Mass Action. One version of the model also predicts that the release rate of lipopolysaccharide receptors from the outer membrane may be significantly lowered when virus is bound to these receptors.     

Aminoacrylate intermediates in the reaction of Citrobacter freundii tyrosine phenol-lyase

Tyrosine phenol-lyase (TPL) from Citrobacter freundii is a pyridoxal 5'-phosphate (PLP)-dependent enzyme that catalyzes the reversible hydrolytic cleavage of l-Tyr to give phenol and ammonium pyruvate. The proposed reaction mechanism for TPL involves formation of an external aldimine of the substrate, followed by deprotonation of the alpha-carbon to give a quinonoid intermediate. Elimination of phenol then has been proposed to give an alpha-aminoacrylate Schiff base, which releases iminopyruvate that ultimately undergoes hydrolysis to yield ammonium pyruvate. Previous stopped-flow kinetic experiments have provided direct spectroscopic evidence for the formation of the external aldimine and quinonoid intermediates in the reactions of substrates and inhibitors; however, the predicted alpha-aminoacrylate intermediate has not been previously observed. We have found that 4-hydroxypyridine, a non-nucleophilic analogue of phenol, selectively binds and stabilizes aminoacrylate intermediates in reactions of TPL with S-alkyl-l-cysteines, l-tyrosine, and 3-fluoro-l-tyrosine. In the presence of 4-hydroxypyridine, a new absorption band at 338 nm, assigned to the alpha-aminoacrylate, is observed with these substrates. Formation of the 338 nm peaks is concomitant with the decay of the quinonoid intermediates, with good isosbestic points at approximately 365 nm. The value of the rate constant for aminoacrylate formation is similar to k(cat), suggesting that leaving group elimination is at least partially rate limiting in TPL reactions. In the reaction of S-ethyl-l-cysteine in the presence of 4-hydroxypyridine, a subsequent slow reaction of the alpha-aminoacrylate is observed, which may be due to iminopyruvate formation. Both l-tyrosine and 3-fluoro-l-tyrosine exhibit kinetic isotope effects of approximately 2-3 on alpha-aminoacrylate formation when the alpha-(2)H-labeled substrates are used, consistent with the previously reported internal return of the alpha-proton to the phenol product. These results are the first direct spectroscopic observation of alpha-aminoacrylate intermediates in the reactions of TPL.     

On the role of reversible denaturation (unfolding) in the irreversible thermal inactivation of enzymes

The contribution of the reversible thermal unfolding of an enzyme toward the overall irreversible thermoinactivation process has been examined both theoretically and experimentally. Using bovine pancreatic ribonuclease as a model, we have studied the effect of such variables as pH and salts both on the equilibrium constant of reversible denaturation and on the rate constant of the overall irreversible process. It has been demonstrated that at temperatures where a significant fraction of the enzyme molecules are in the native conformation, there is a correlation between the enzyme thermostabilities with respect to the reversible and irreversible inactivations: greater stability against the former is accompanied by greater stability against the latter. On the other hand, at very high temperatures (where essentially all of the enzyme molecules are unfolded), such a correlation does not exist. These findings are considered in terms of a kinetic model for irreversible enzyme thermoinactivation, and the implications of the derived relationship are discussed.     

Cryoenzymology of porcine pepsin

Physical and kinetic properties of porcine pepsin have been examined in aqueous methanol solvents at temperatures below ambient to seek evidence for covalent intermediates in the catalyzed hydrolysis of good substrates. It was first demonstrated that aqueous methanol cryosolvents have no significant deleterious effects upon this protein. The addition of methanol does lead to a drastic reduction in the midpoint of the thermal melting curve of pepsin. This could account for rate reductions previously observed in catalysis by this enzyme. This effect is lessened by the addition of active-site ligands including substrates and is fully reversible upon dilution into aqueous solution. Two substrates were chosen which have chromophoric groups on opposite sides of the scissile peptide bond. The UV spectral changes from hydrolysis of Pro-Thr-Glu-Phe-(NO2)Phe-Arg-Leu and the fluorescence spectral changes from hydrolysis of DNS-Ala-Ala-Phe-Phe-OP4P+-CH3 were studied at temperatures down to -60 degrees C. The resulting Arrhenius plots were linear in the region where pepsin exists in the native state with downward curvature exhibited at higher temperatures where the reversible denaturation occurs. No "burst" reactions were observed with either substrate. In addition, efforts at trapping intermediates by low-temperature denaturation and precipitation have provided no evidence for covalent intermediates on the reaction pathway. Although this evidence is negative, we cannot rule out the possibility of the formation of covalent intermediates following an initial rate-limiting step.     

Kinetic and thermodynamic characterization of the RNA guanylyltransferase reaction

An RNA guanylyltransferase activity is involved in the synthesis of the cap structure found at the 5' end of eukaryotic mRNAs. The RNA guanylyltransferase activity is a two-step ping-pong reaction in which the enzyme first reacts with GTP to produce the enzyme-GMP covalent intermediate with the concomitant release of pyrophosphate. In the second step of the reaction, the GMP moiety is then transferred to a diphosphorylated RNA. Both reactions were previously shown to be reversible. In this study, we report a biochemical and thermodynamic characterization of both steps of the reaction of the RNA guanylyltransferase from Paramecium bursaria Chlorella virus 1, the prototype of a family of viruses infecting green algae. Using a combination of real-time fluorescence spectroscopy, radioactive kinetic assays, and inhibition assays, the complete kinetic parameters of the RNA guanylyltransferase were determined. We produced a thermodynamic scheme for the progress of the reaction as a function of the energies involved in each step. We were able to demonstrate that the second step comprises the limiting steps for both the direct and reverse overall reactions. In both cases, the binding to the RNA substrates is the step requiring the highest energy and generating unstable intermediates that will promote the catalytic activites of the enzyme. This study reports the first thorough kinetic and thermodynamic characterization of the reaction catalyzed by an RNA capping enzyme.     

Phage P22 procapsids equilibrate with free coat protein subunits

Assembly of bacteriophage P22 procapsids has long served as a model for assembly of spherical viruses. Historically, assembly of viruses has been viewed as a non-equilibrium process. Recently alternative models have been developed that treat spherical virus assembly as an equilibrium process. Here we have investigated whether P22 procapsid assembly reactions achieve equilibrium or are irreversibly trapped. To assemble a procapsid-like particle in vitro, pure coat protein monomers are mixed with scaffolding protein. We show that free subunits can exchange with assembled structures, indicating that assembly is a reversible, equilibrium process. When empty procapsid shells (procapsids with the scaffolding protein stripped out) were diluted so that the concentration was below the dissociation constant ( approximately 5 microM) for coat protein monomers, free monomers were detected. The released monomers were assembly-competent; when NaCl was added to metastable partial capsids that were aged for an extended period, the released coat subunits were able to rapidly re-distribute from the partial capsids and form whole procapsids. Lastly, radioactive monomeric coat subunits were able to exchange with the subunits from empty procapsid shells. The data presented illustrate that coat protein monomers are able to dissociate from procapsids in an active state, that assembly of procapsids is consistent with reactions at equilibrium and that the reaction follows the law of mass action.