Optimization of kinetic proofreading: A general method for derivation of the constraint relations and an exploration of a specific case

We have previously presented a general cost-accuracy relationship for a broad class of kinetic proofreading mechanisms. In this paper we present a general matrix method, based upon classical enzyme kinetics, for the derivation of the constraint relation that characterizes specific proofreading mechanisms. For purpose of illustration we present the method in the context of a conventional Michaelis-Menten mechanism with side reactions. We then explore optimization of the general cost function under a variety of different constraints that may exist for such a mechanism. In this way we are able to contrast different perspectives on the optimization of enzyme design.     

Energy cost of proofreading to increase fidelity of transfer ribonucleic acid aminoacylation.

The paradox of relatively error free function in biological systems composed of relatively error prone components has recently come under intensive investigation. In the case of tRNA aminoacylation, aminoacyl-tRNA synthetases were discovered to have a separate function that allows misacylated molecules to be hydrolyzed more rapidly than correctly acylated molecules. This additional function of the synthetases provides a proofreading or verification mechanism that is believed to improve significantly the overall accuracy of tRNA aminoacylation. In this paper we provide an explicit relationship between the accuracy achieved by proofreading and the energy cost. Experimental data available in the literature are examined in light of this relationship. The following are the principal conclusions from our study: (1) high-accuracy proofreading of tRNA aminoacylation has a high energy cost, as much as 100 times greater than indications from early experimental work; (2) the minimum net error derived in previous theoretical studies is never actually reached; (3) mechanisms in which misacylation and subsequent proofreading occur on the surface of the same synthetase molecule achieve a much higher accuracy than mechanisms in which these functions occur on the surface of different synthetase molecules.     

Proofreading systems of multiple stages for improved accuracy of biological discrimination

The phenomenal accuracy of biological discrimination is due in many cases to specific proofreading mechanisms. We have previously developed a macroscopic theory of such mechanisms and applied it to the case of single-stage proofreading. In this article we apply the theory to systems with multiple stages of proofreading. A specific relationship between improved accuracy due to proofreading and the associated energy cost is given. This is a macroscopic relationship that must be satisfied regardless of the details of the underlying mechanisms. Five factors in the design of such systems are shown to influence their overall accuracy: (1) initial discrimination, (2) number of proofreading stages, (3) proofreading discrimination of each stage, (4) distribution of proofreading effort among the stages, and (5) total energy expended for proofreading. We show that there is an optimal distribution of proofreading effort that, for a given degree of accuracy, minimizes the energy cost of proofreading. We also provide a simple physical interpretation of this minimum condition. These results are used to examine proofreading in two experimental systems for which there is appropriate data available in the literature: the valyl-tRNA synthetase catalyzed misacylation of tRNA^Val with threonine and the isoleucyl-tRNA synthetase catalyzed misacylation of tRNA^Ile with valine. The correlation between the magnitude of a discrimination factor and the size of the corresponding enzymatic cavity is discussed.     

Energy considerations for kinetic proofreading in biosynthesis

An explicit model for kinetic proofreading in biosynthesis processes is treated in order to clarify the relation between the achieved accuracy, and the required energy which is provided by the hydrolysis of nucleoside triphosphates (atp, gtp). The displacement from equilibrium, which is restrictive for the discrimination, is explicitly taken into account. This means that the process is consistent with the principle of detailed balance. It proceeds in such a way that the molecules to be selected are associated together with nucleoside triphosphates to an enzyme complex. After an initial selection, hydrolysis takes place, whereafter there is a repeated testing in several steps. Our basic idea is that real proofreading systems are designed to use the energy of the triphosphate in the most efficient way to achieve a satisfactory accuracy, and therefore there should exist optimum kinetic properties of the system. A suitable formalism for finding such optimum situations is developed which yields new possibilities to interpret experimental data.     

Elucidation of T cell signalling models

A potential mechanism that allows T cells to reliably discriminate pMHC ligands involves an interplay between kinetic proofreading, negative feedback and a destruction of this negative feedback. We analyse a detailed model of these mechanisms which involves the TCR, SHP1 and ERK. We discover that the behaviour of pSHP1 negative feedback is of primary importance, and particularly the influence of a kinetic proofreading base negative feedback state on pSHP1 dynamics. The CD8 co-receptor is shown to benefit from a kinetic proofreading locking mechanism and is able to overcome pSHP1 negative influences to sensitise a T cell.     

Mechanisms of aminoacyl-tRNA synthetases: a critical consideration of recent results

During the last 10 years intensive and detailed studies on mechanisms and specificities of aminoacyl-tRNA synthetases have been carried out. Physical measurements, chemical modification of substrates, site-directed mutagenesis, and determination of kinetic parameters in misacylation reactions with noncognate amino acids have provided extensive knowledge which is now considered critically for its consistency. A common picture emerges: (1) The enzymes work with different catalytic cycles, kinetic constants, and specificities under different assay conditions. (2) Chemical modifications of substrates can have comparable influence on catalysis as can changes in assay conditions. (3) All enzymes show a specificity for the 2'- or 3'-position of the tRNA. (4) Hydrolytic proofreading is achieved in a pre- and a posttransfer process. In most cases pretransfer proofreading is the main step; posttransfer proofreading is often marginal. (5) Initial discrimination of substrates takes place in a two-step binding process. For some investigated enzymes, initial discrimination factors were found to depend on hydrophobic interaction and hydrogen bonds. (6) The overall recognition of amino acids is achieved in a process of at least four steps. At present, only a rough overall picture of aminoacyl-tRNA synthetase action can be given.     

DNA replication fidelity: kinetics and thermodynamics

Mechanisms that control the fidelity of DNA replication are discussed. Data are reviewed for 3 steps in a fidelity pathway: nucleotide insertion, exonucleolytic proofreading, and extension from matched and mismatched 3′-primer termini. Fidelity mechanisms that involve predominately K_m discrimination, V_max discrimination, or a combination of the two are analyzed in the context of a simple model for fidelity. Each fidelity step is divided into 2 components, thermodynamics and kinetic. The thermodynamic component, which relates to free-energy differences between right and wrong base pair, is associated with a K_m discrimination mechanism for polymerase. The kinetic component, which represents the enzyme's ability to select bases for insertion and excision to achieve fidelity greater than that availablek from base pairing free-energy differences, is associated with a V_max discrimination mechanism for polymerase. Currently available fidelity data for nucleotide insertion and primer extension in the absence of proofreading appears to have relatively large K_m and small V_max components. An important complication can arise when analyzing data from polymerases containing an associated 3′-exonuclease activity. In the presence of proofreading, a V_max discrimination mechanisms is likely to occur, but this may be the result of two K_m discrimination mechanisms acting serially, one for nucleotide insertion and other for excision. Possible relationships between base pairing free energy differences measured in aqueous solution and those defined within the polymerase active cleft are considered in the context of the enzyme's ability to exclude water, at least partially, from the vicinity of its active site.     

A kinetic view of clathrin assembly and endocytic cargo sorting

Specificity and sensitivity in biochemical reactions can be achieved through regulation of equilibrium binding affinity or through proofreading mechanisms that allow for the dissociation of unwanted intermediates. In this essay, we aim to provide our perspectives on how the concept of kinetic proofreading might apply in the context of cargo sorting in clathrin-mediated endocytosis.     

Isoleucyl-tRNA synthetase from Escherichia coli MRE 600: discrimination between isoleucine and valine with modulated accuracy

Discrimination between isoleucine and valine is achieved with different accuracies by isoleucyl-tRNA synthetase from E. coli MRE 600. The recognition process consists of two initial discrimination steps and a pretransfer and a posttransfer proofreading event. The overall discrimination factors D were determined from kcat and Km values observed in aminoacylation of tRNA(Ile)-C-C-A with isoleucine and valine. From aminoacylation of the modified tRNA species tRNA(Ile)-C-C-A(3'NH2) initial discrimination factors I1 and pretransfer proofreading factors II1 were calculated. Factors I1 were computed from ATP consumption and D1, the overall discrimination in aminoacylation of the modified tRNA; factors II1 were calculated as quotient of AMP formation rates. Initial discrimination factors I2 and posttransfer proofreading factors II2 were determined from AMP formation rates observed in aminoacylation of tRNA(Ile)-C-C-A. The observed overall discrimination varies up to a factor of about four according to conditions. Under standard assay conditions 72,000, under optimal conditions 144,000 correct aminoacyl-tRNAs are produced per one error while 1.1 or 1.7 ATPs are consumed. A comparison with isoleucyl-tRNA synthetase from yeast shows that both enzymes act principally with the same recognition mechanism, but the enzyme from E. coli MRE 600 exhibits higher specificity and lower energy dissipation and does not show such high variation of accuracy as observed with the enzyme from yeast.     

Proofreading, NTPases and translation: constraints on accurate biochemistry.

Two accurate individual reactions can work together as a more accurate overall mechanism. This is straightforward for a transient reaction, but the same accuracy in the steady state (termed proofreading) requires many preconditions. The preconditions for proofreading can be summarized as four statements, which allow experimental identification and subsequent confirmation of proofreading mechanisms.     

Reducing intrinsic biochemical noise in cells and its thermodynamic limit

In living cells, the specificity of biomolecular recognition can be amplified and the noise from non-specific interactions can be reduced at the expense of cellular free energy. This is the seminal idea in the Hopfield-Ninio theory of kinetic proofreading: The specificity is increased via cyclic network kinetics without altering molecular structures and equilibrium affinites. We show a thermodynamic limit of the specificity amplification with a given amount of available free energy. For a normal cell under physiological condition with sustained phosphorylation potential, this gives a factor of 10(10) as the upper bound in specificity amplification. We also study an optimal kinetic network design that is capable of approaching the thermodynamic limit.     

Kinetic proofreading of ligand-FcepsilonRI interactions may persist beyond LAT phosphorylation

Cells may discriminate among ligands with different dwell times for receptor binding through a mechanism called kinetic proofreading in which the formation of an activated receptor complex requires a progression of events that is aborted if the ligand dissociates before completion. This mechanism explains how, at equivalent levels of receptor occupancy, a rapidly dissociating ligand can be less effective than a more slowly dissociating analog at generating distal cellular responses. Simple mathematical models predict that kinetic proofreading is limited to the initial complex; once the signal passes to second messengers, the dwell time no longer regulates the signal. This suggests that an assay for kinetic proofreading might be used to determine which activation events occur within the initial signaling complex. In signaling through the high affinity IgE receptor FcepsilonRI, the transmembrane adaptor called linker for activation of T cells (LAT) is thought to nucleate a distinct secondary complex. Experiments in which the concentrations of two ligands with different dwell times are adjusted to equalize the level of LAT phosphorylation in rat basophilic leukemia 2H3 cells show that Erk2 phosphorylation, intracellular Ca(2+), and degranulation exhibit kinetic proofreading downstream of LAT phosphorylation. These results suggest that ligand-bound FcepsilonRI and LAT form a complex that is required for effective signal transmission.     

A new approach to DNA polymerase kinetics

Little is known of the detailed mechanisms of the polymerization reactions carried out by RNA and DNA polymerases. Besides technical reasons, there are mathematical difficulties not encountered in traditional enzymology. The product of the reaction after one polymerization step is also the substrate of the next step. A number of polymerases, isolated from various sources, have an exonuclease activity. The chain which is being synthesized may be either elongated or trimmed, and its growth has the character of a random walk. In this case, although the overall reaction scheme is more complex, the experiments are more informative, as every dNTP may be transformed into two distinct products: incorporated, or free dNMP.Having solved some of the mathematical difficulties of the random walk problem, we are able to propose a strategy for the study of the polymerization/excision kinetics. We measure the amount y(t) of nucleotide that is polymerized at time tand the amount x(t) of nucleoside monophosphate that has accumulated. When $\text{d}\text{y}\text{d}\text{x}$ is plotted against the concentration of dNTP, a curve is obtained with a characteristic shape, a straight line in a large number of cases. From there, kinetic constants can be estimated.The analysis is made in terms of four possible kinetic schemes. In the most elementary model there are only two rate constants, one for incorporation and one for excision. This model is a limiting case of all other models. The frayed-unfrayed model of Brutlag & Kornberg (1972), Hopfield's kinetic proofreading scheme (Hopfield, 1974), and the delayed-escape scheme (Ninio, 1975) are examined in detail, and we show how the kinetic experiments may in principle distinguish between the schemes. Our approach is illustrated with three experiments in which Escherichia coli DNA polymerase I acts on poly(dC), and poly(dT) · oligo(dA)₁₀.     

Nucleotide excision by E. coli DNA polymerase I in proofreading and non-proofreading modes

Escherichia coli DNA polymerase I exists in at least two distinct kinetic forms. When it binds to a template, the proofreading activity is usually switched off. As the enzyme progresses along the template, it becomes more and more competent for excision. This phenomenon introduces a link between fidelity and processivity. Processivity is best studied when the chain-length distributions of synthesized polymers are stationary. Even then, however, one cannot avoid multiple initiations on a given template by the same molecule of the enzyme. When synthesis is initiated with primers of lengths 15 or 20, a strange phenomenon is observed. It seems that the polymerase starts by hydrolyzing the primer down to a length of 7–10 nucleotides and only then starts to add nucleotides. It does so in a low-accuracy mode, suggesting that, while the exonuclease is clearly active, it does not contribute to proofreading. The warm-up of the proofreading function is therefore reinterpreted as a switch between two modes of behaviour: a mode 1 of low accuracy in which the 3′ → 5′ exonuclease, while active, is uncoupled from the polymerase and does not contribute to proofreading, and a mode 2 of high accuracy in which the exonuclease is kinetically linked to the polymerase activity.     

Nucleotide excision by E. coli DNA polymerase I in proofreading and non-proofreading modes

Escherichia coli DNA polymerase I exists in at least two distinct kinetic forms. When it binds to a template, the proofreading activity is usually switched off. As the enzyme progresses along the template, it becomes more and more competent for excision. This phenomenon introduces a link between fidelity and processivity. Processivity is best studied when the chain-length distributions of synthesized polymers are stationary. Even then, however, one cannot avoid multiple initiations on a given template by the same molecule of the enzyme. When synthesis is initiated with primers of lengths 15 or 20, a strange phenomenon is observed. It seems that the polymerase starts by hydrolyzing the primer down to a length of 7-10 nucleotides and only then starts to add nucleotides. It does so in a low-accuracy mode, suggesting that, while the exonuclease is clearly active, it does not contribute to proofreading. The warm-up of the proofreading function is therefore reinterpreted as a switch between two modes of behaviour: a mode 1 of low accuracy in which the 3'----5' exonuclease, while active, is uncoupled from the polymerase and does not contribute to proofreading, and a mode 2 of high accuracy in which the exonuclease is kinetically linked to the polymerase activity.     

Site dependent time optimization of protein synthesis with special regard to accuracy

The efficiency of protein synthesis is determined by its rate, accuracy, and energy consumption. With the energy consumption fixed, we optimize the system with respect to time and accuracy. Using an analytic model for a simple system and computer simulations for more complex systems, where also the possibility of errors is included, we demonstrate how different parts of the messenger RNA influence the protein production rate differently. The first part of the coding sequence is of major importance, since the availability of empty initiation sites is crucial, and queuing back to that region may interfere with initiation. The elongation rate at different positions depends on codon usage, on the concentrations of substrate and co-factors, and on the kinetic rate constants, including those of the proofreading branch(es). Ribosomal proofreading is a time consuming process and by allowing for more errors in the beginning of a protein, it is possible to increase the production rate of that protein. We calculate the mean translation time per functioning protein for various translation accuracies, and discuss the different strategies open to living cells.     

Thermodynamic constraints on kinetic proofreading in biosynthetic pathways.

We develop a quantitative theory of kinetic proofreading with an arbitrary number of checking steps after the hydrolysis of a nucleoside triphosphate. In particular, we investigate the relationship between the minimum dissipation of free energy required for a given error frequency in such systems. Several conclusions can be drawn from the present treatment: first, the ultimate accuracy of error correcting selective pathways is set by the displacement from equilibrium of the nucleoside triphosphates. Second, it is advantageous to achieve a desired accuracy at a small energy dissipation with several checking steps rather than a single one. This could explain antinomies in the amino acylation reaction as well as in mRNA translation, where small structural differences lead to large differences in flow rates between right and wrong substrates. Third, all checking steps should contribute equally to the accuracy, which implies a specific and symmetrical set of rate constants for the checking events on the enzyme.     

Integrated function of a kinetic proofreading mechanism: double-stage proofreading by isoleucyl-tRNA synthetase

Experimental measurements for isoleucyl-tRNA synthetase proofreading valyl-tRNAIle in Escherichia coli previously have been incorporated into the conventional Michaelis-Menten model for this system. This model was augmented to include two stages of proofreading--the aminoacyl adenylate and aminoacyl-tRNA stages--and used to predict the values of four additional rate constants that have been determined experimentally. The results suggest that two stages of conventional kinetic proofreading with binding sites designed for isoleucine (the "correct" substrate) are inconsistent with the experimental data, that a double-stage mechanism in which one stage (the "double-sieve") involves a binding site designed for valine (the "incorrect" substrate) and the other involves a binding site designed for isoleucine is consistent with all the experimental data, and that the experimental data are not sufficiently accurate to distinguish the stage at which the double-sieve mechanism operates in vivo. Furthermore, analysis of the model suggests that four parameters have the most questionable values and that experimental refinement of their estimates will be needed to determine which of the two stages involves the double-sieve mechanism.     

DNA polymerase alpha and models for proofreading.

Using a modified system to measure fidelity at an amber site in phi X174, we have employed DNA polymerase alpha to test different mechanisms for proofreading. DNA polymerase alpha does not exhibit the characteristics of "kinetic proofreading" seen with procaryotic polymerases. Polymerase alpha shows no evidence for a "next nucleotide" effect, and added deoxynucleoside monophosphates do not alter fidelity. Pyrophosphate, which increases error rates with a procaryotic polymerase, appears to weakly improve polymerase alpha fidelity. DNA polymerase alpha does exhibit a dramatic increase in error rate in the presence of a deoxycytidine thiotriphosphate (dCTP alpha S), but this enhanced mutagenesis also occurs under conditions where kinetic proofreading should be otherwise defeated. This particular effect with dCTP alpha S appears specific for DNA polymerase alpha and is not seen with the other polymerases tested.