Applications of graph theory to enzyme kinetics and protein folding kinetics. Steady and non-steady-state systems

Graphic methods have proved to be very useful in enzyme kinetics, as reflected in both raising the efficiency of performing calculations and aiding in the analysis of catalytic mechanisms. The kinetic relations among protein folding states are very similar to those between enzyme-catalyzed species. Therefore, it should be equally useful to provide a visually intuitive relation between kinetic calculations and folding mechanisms for protein folding kinetics, as manifested by the graphic rules in enzyme kinetics. It can actually be anticipated that, due to increasing interest in protein folding, the graphic method will become an important tool in folding kinetics as well. Based on the recent progress made in graphic methods of enzyme kinetics, in this review four graphic rules are summarized, which can be used to deal with protein folding systems as well as enzyme-catalyzed systems. Rules 1-3 are established for deriving the kinetic equations for steady-state processes and Rule 4 for those in the case of non-steady-state processes. In comparison with conventional graphic methods, which can only be applied to a steady-state system, the current rules have the following advantages: (1) Complicated and tedious calculations can be greatly simplified. (2) A lot of wasted labor can be turned away. (3) Final results can be double-checked by a formula provided in each of the graphic rules. (4) Transient kinetic systems can also be treated. The mathematical proof of Rules 1-4 is given in appendices A-D, respectively.     

Using stochastic roadmap simulation to predict experimental quantities in protein folding kinetics: folding rates and phi-values.

This paper presents a new method for studying protein folding kinetics. It uses the recently introduced Stochastic Roadmap Simulation (SRS) method to estimate the transition state ensemble (TSE) and predict the rates and the Phi-values for protein folding. The new method was tested on 16 proteins, whose rates and Phi-values have been determined experimentally. Comparison with experimental data shows that our method estimates the TSE much more accurately than an existing method based on dynamic programming. This improvement leads to better folding-rate predictions. We also compute the mean first passage time of the unfolded states and show that the computed values correlate with experimentally determined folding rates. The results on Phi-value predictions are mixed, possibly due to the simple energy model used in the tests. This is the first time that results obtained from SRS have been compared against a substantial amount of experimental data. The results further validate the SRS method and indicate its potential as a general tool for studying protein folding kinetics.     

Local folding coupled to RNA binding in the yeast ribosomal protein L30.

The ribosomal protein L30 from yeast Saccharomyces cerevisiae auto-regulates its own synthesis by binding to a structural element in both its pre-mRNA and its mRNA. The three-dimensional structures of L30 in the free (f L30) and the pre-mRNA bound (b L30) forms have been solved by nuclear magnetic resonance spectroscopy. Both protein structures contain four alternating alpha-helices and four beta-strands segments and adopt an overall topology that is an alphabetaalpha three-layer sandwich, representing a unique fold. Three loops on one end of the alphabetaalpha sandwich have been mapped as the RNA binding site on the basis of structural comparison, chemical shift perturbation and the inter-molecular nuclear Overhauser effects to the RNA. The structural and dynamic comparison of f L30 and b L30 reveals that local dynamics may play an important role in the RNA binding. The fourth helix in b L30 is longer than in f L30, and is stabilized by RNA binding. The exposed hydrophobic surface that is buried upon RNA binding may provide the energy necessary to drive secondary structure formation, and may account for the increased stability of b L30.     

Characterization of the critical state in protein folding. Effects of guanidine hydrochloride and specific Ca2+ binding on the folding kinetics of alpha-lactalbumin

The reversible unfolding and refolding kinetics of alpha-lactalbumin induced by concentration jump of guanidine hydrochloride were measured at pH 7.0 and 25 degrees C using tryptophan absorption at 292 nm, with varying concentrations of the denaturant and free Ca2+. The refolding reaction of alpha-lactalbumin from the fully unfolded (D) state occurs through the two stages: (1) instantaneous formation of a compact intermediate (the A state) that has a native-like secondary structure; (2) tight packing of the preformed secondary structure segments to lead finally to the native structure, this stage being the rate-determining step of the reaction and associated with acquisition of the specific structure necessary for strong Ca2+ binding. Under strongly native conditions, the observed kinetics of refolding is also complicated by the presence of a slow-folding species (10%) in the unfolded state. Considering these facts, the microscopic rate constants in folding and unfolding directions have been evaluated from the observed kinetics and from the equilibrium constants of the transitions among the native (N), A and D states. Close linear relationships have been found in the plots of the activation free energies, obtained from the microscopic rate constants, against the denaturant concentration. They are similar to the linear relationship between the free energy of unfolding and the denaturant concentration. It was demonstrated that the slope of the plots should be approximately proportional to a change in accessible surface area of the protein during the respective activation process, and that only a third of the difference in accessible surface area between A and N is buried in the critical activated state of folding. However, the selective effect of Ca2+ binding on the folding rate constant has been observed also, demonstrating that the specific Ca2+-binding substructure in the N state is already organized in the activated state. Thus, only a part of the protein molecule involving the Ca2+-binding region is organized in the activated state, with the other part of the molecule being left less organized, suggesting that the second stage of folding may be a sequential growing process of organized assemblage of the performed secondary structure segments.     

Role of cadmium in activating nuclear protein kinase C and the enzyme binding to nuclear protein.

Specific effects of cadmium on nuclear protein kinase C activity were found with 3T3/10T1/2 mouse fibroblast and rat liver nuclei. Treatment of the mouse fibroblasts in culture with 12-O-tetradecanoylphorbol-13-acetate resulted in the stimulation of nuclear protein kinase C activity in a "fixed" pool which is defined by its resistance to chelator extraction, whereas the chelator extractable enzyme activity, defined as the "labile" pool was unaffected. Cadmium was found to potentiate the effect of the phorbol ester, directed specifically to nuclei, since the particulate protein kinase C activity was not changed under similar treatment. In a reconstituted system consisting of rat liver nuclei and rat brain protein kinase C, cadmium stimulated the binding of the enzyme to a 105-kDa nuclear protein. The binding of a 105-kDa protein to protein kinase C is attributed strictly due to the cadmium effect, whereas a 50-kDa protein binding to protein kinase C was only enhanced by cadmium. We propose a mechanistic model, where cadmium substitutes zinc in the regulatory domain of protein kinase C rendering the putative protein-protein binding site exposed.     

Intestinal fatty acid binding protein: a specific residue in one turn appears to stabilize the native structure and be responsible for slow refolding.

The intestinal fatty acid binding protein is one of a class of proteins that are primarily beta-sheet and contain a large interior cavity into which ligands bind. A highly conserved region of the protein exists between two adjacent antiparallel strands (denoted as D and E in the structure) that are not within hydrogen bonding distance. A series of single, double, and triple mutations have been constructed in the turn between these two strands. In the wild-type protein, this region has the sequence Leu 64/Gly 65/Val 66. Replacing Leu 64 with either Ala or Gly decreases the stability and the midpoint of the denaturation curve somewhat, whereas mutations at Gly 65 affect the stability slightly, but the protein folds at a rate similar to wild-type and binds oleate. Val 66 appears not to play an important role in maintaining stability. All double or triple mutations that include mutation of Leu 64 result in a large and almost identical loss of stability from the wild-type. As an example of the triple mutants, we investigated the properties of the Leu 64 Ser/Gly 65 Ala/Val 66 Asn mutant. As measured by the change in intrinsic fluorescence, this mutant (and similar triple mutants lacking leucine at position 64) folds much more rapidly than wild-type. The mutant, and others that lack Leu 64, have far-UV CD spectra similar to wild-type, but a different near-UV CD spectrum. The folded form of the protein binds oleate, although less tightly than wild-type. Hydrogen/deuterium exchange studies using electrospray mass spectrometry indicate many more rapidly exchangeable amide protons in the Leu 64 Ser/Gly 65 Ala/Val 66 Asn mutant. We propose that there is a loss of defined structure in the region of the protein near the turn defined by the D and E strands and that the interaction of Leu 64 with other hydrophobic residues located nearby may be responsible for (1) the slow step in the refolding process and (2) the final stabilization of the structure. We suggest the possibility that this region of the protein may be involved in both an early and late step in refolding.     

Disulfide-dependent protein folding is linked to operation of the vitamin K cycle in the endoplasmic reticulum. A protein disulfide isomerase-VKORC1 redox enzyme complex appears to be responsible for vitamin K1 2,3-epoxide reduction

Gamma-carboxylation of vitamin K-dependent proteins is dependent on formation of reduced vitamin K1 (Vit.K1H2) in the endoplasmic reticulum (ER), where it works as an essential cofactor for gamma-carboxylase in post-translational gamma-carboxylation of vitamin K-dependent proteins. Vit.K1H2 is produced by the warfarin-sensitive enzyme vitamin K 2,3-epoxide reductase (VKOR) of the vitamin K cycle that has been shown to harbor a thioredoxin-like CXXC center involved in reduction of vitamin K1 2,3-epoxide (Vit.K>O). However, the cellular system providing electrons to the center is unknown. Here data are presented that demonstrate that reduction is linked to dithiol-dependent oxidative folding of proteins in the ER by protein disulfide isomerase (PDI). Oxidative folding of reduced RNase is shown to trigger reduction of Vit.K>O and gamma-carboxylation of the synthetic gamma-carboxylase peptide substrate FLEEL. In liver microsomes, reduced RNase-triggered gamma-carboxylation is inhibited by the PDI inhibitor bacitracin and also by small interfering RNA silencing of PDI in HEK 293 cells. Immunoprecipitation and two-dimensional SDS-PAGE of microsomal membrane proteins demonstrate the existence of a VKOR enzyme complex where PDI and VKORC1 appear to be tightly associated subunits. We propose that the PDI subunit of the complex provides electrons for reduction of the thioredoxin-like CXXC center in VKORC1. We can conclude that the energy required for gamma-carboxylation of proteins is provided by dithiol-dependent oxidative protein folding in the ER and thus is linked to de novo protein synthesis.     

Altered binding of the Cry1Ac toxin to larval membranes but not to the toxin-binding protein in Plodia interpunctella selected for resistance to different Bacillus thuringiensis isolates.

Immunoblotting and cytochemical procedures were used to determine whether toxin binding was altered in strains of the Indianmeal moth, Plodia interpunctella, selected for resistance to various strains of Bacillus thuringiensis. Each of these B. thuringiensis subspecies produces a mixture of protoxins, primarily Cry1 types, and the greatest insect resistance is to the Cry1A protoxins. In several cases, however, there was also resistance to toxins not present in the B. thuringiensis strains used for selection. The Cry1Ab and Cry1Ac toxins bound equally well over a range of toxin concentrations and times of incubation to a single protein of ca. 80-kDa in immunoblots of larval membrane extracts from all of the colonies. This binding protein is essential for toxicity since a mutant Cry1Ac toxin known to be defective in binding and thus less toxic bound poorly to the 80-kDa protein. This binding protein differed in size from the major aminopeptidase N antigens implicated in toxin binding in other insects. Binding of fluorescently labeled Cry1Ac or Cry1Ab toxin to larval sections was found at the tips of the brush border membrane prepared from the susceptible but not from any of the resistant P. interpunctella. Accessibility of a major Cry1A-binding protein appears to be altered in resistant larvae and could account for their broad resistance to several B. thuringiensis toxins.     

Lectin activity of the major surfactant protein (SP-A) may participate in, but is not required for, binding to rat type II cells.

Pulmonary surfactant is a complex mixture of lipids and proteins, of which surfactant protein A (SP-A) is the most abundant glycoprotein. The SP-A molecule has several distinct structural features that include a lectin-like domain, sharing structural features with other mammalian lectins. We have tested the hypothesis that lectin activity of the SP-A molecule is required for the binding to its receptor on the surface of alveolar Type II cells. By using colloidal gold immunocytochemistry in conjunction with electron microscopy, we evaluated the ability of mannosylated proteins to inhibit canine SP-A binding to rat Type II cells in vitro. After preincubation of SP-A with the mannosylated protein horse-radish peroxidase (HRP), SP-A was incubated with isolated filter-grown Type II cells. HRP did not alter the binding of SP-A to the Type II cell surface. Evidence that SP-A did bind to HRP was shown by coincident observation of gold-labeled SP-A and HRP precipitates. These results provide visual evidence that the lectin activity associated with SP-A is not required for its binding to receptor on rat alveolar Type II epithelial cells.     

The kinetics of the hydroxylation of procollagen by prolyl 4-hydroxylase. Proposal for a processive mechanism of binding of the dimeric hydroxylating enzyme in relation to the high kcat/Km ratio and a conformational requirement for hydroxylation of -X-Pro-Gly- sequences.

Prolyl 4-hydroxylase modifies only approx. 5% of the hydroxylatable prolyl residues in procollagen at a relatively high rate, after which the rate of further hydroxylation rapidly decreases. This suggests that the probability to exist in a defined hydroxylation-committed conformation differs between the numerous -X-Pro-Gly- sequences in the substrate. The enzyme reaction is characterized by the unusually high kcat/Km ratio of 3 x 10(9) M-1 s-1. To explain these kinetic features, an extremely high second-order rate constant for the association of enzyme and the subset of rapidly hydroxylated prolyl residues has to be assumed. A two-step mechanism is proposed in which diffusional constraints on the rate of association of prolyl 4-hydroxylase with hydroxylatable prolyl residues can be overcome. Upon encountering a random coil pro-alpha chain, the dimeric enzyme is first 'aspecifically' bound, followed by rapid transfers between different segments of the flexible peptide substrate via fast transitions between 'aspecific' single and double bound intermediate states. The rate of the second step, the productive (specific) binding of hydroxylation-committed -X-Pro-Gly- sequence to the active site, can be enhanced significantly by such an, in essence, 'one-dimensional' search. This processive mechanisms of binding does not necessarily imply many hydroxylation reactions during one encounter between enzyme and a peptide with several substrate sites as suggested previously in a slightly different model (De Waal, A. and De Jong, L. (1988) Biochemistry 27, 150-155).