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Title: Deducing the Kinetics of Protein Synthesis In Vivo from the Transition Rates Measured In Vitro}
Article:
Abstract:
The molecular machinery of life relies on complex multistep processes that involve numerous individual transitions, such as molecular association and dissociation steps, chemical reactions, and mechanical movements. The corresponding transition rates can be typically measured in vitro but not in vivo. Here, we develop a general method to deduce the in-vivo rates from their in-vitro values. The method has two basic components. First, we introduce the kinetic distance, a new concept by which we can quantitatively compare the kinetics of a multistep process in different environments. The kinetic distance depends logarithmically on the transition rates and can be interpreted in terms of the underlying free energy barriers. Second, we minimize the kinetic distance between the in-vitro and the in-vivo process, imposing the constraint that the deduced rates reproduce a known global property such as the overall in-vivo speed. In order to demonstrate the predictive power of our method, we apply it to protein synthesis by ribosomes, a key process of gene expression. We describe the latter process by a codon-specific Markov model with three reaction pathways, corresponding to the initial binding of cognate, near-cognate, and non-cognate tRNA, for which we determine all individual transition rates in vitro. We then predict the in-vivo rates by the constrained minimization procedure and validate these rates by three independent sets of in-vivo data, obtained for codon-dependent translation speeds, codon-specific translation dynamics, and missense error frequencies. In all cases, we find good agreement between theory and experiment without adjusting any fit parameter. The deduced in-vivo rates lead to smaller error frequencies than the known in-vitro rates, primarily by an improved initial selection of tRNA. The method introduced here is relatively simple from a computational point of view and can be applied to any biomolecular process, for which we have detailed information about the in-vitro kinetics.
Introduction:
Life is based on the continuous synthesis, modification, and degradation of proteins and other macromolecules. These processes are performed by complex biomolecular machines that bind their ligands and transform them into product molecules. Examples are provided by the transcription of DNA by RNA polymerases, the translation of mRNA by ribosomes, or the degradation of proteins by proteasomes. Each of these processes involves several steps: the binding of the ligand molecules, chemical reactions catalyzed at the active sites, as well as specific conformational changes and directed mechanical movements of parts of the molecular machinery. In principle, the kinetics of such multistep processes can be understood in terms of the individual transitions and the associated transition rates, a well-established approach both for enzyme kinetics[1]–[3] and for free energy transduction by molecular motors[4],[5]. In practice, the values of the individual transition rates can be typically measured in vitro but not in vivo, and the in-vitro rates depend on the composition of the buffer. Because the cytosol represents a rather complex buffer, it is difficult to assess whether a certain in-vitro assay provides a reliable description of the process in vivo. One important tool that is missing for such an assessment is a simple measure by which we can quantitatively compare the kinetics of a multistep process in different environments. Here, we develop a general method that provides such a measure and allows the deduction of the in-vivo rates from their in-vitro values. Our method has two basic components. First, we introduce the ‘kinetic distance’, i. e., a distance metric for the kinetics, by which we can describe the similarity or dissimilarity of multistep processes in vitro and in vivo in a quantitative manner. The kinetic distance depends logarithmically on the rates and has an intuitive interpretation in terms of the associated free energy barriers. Second, we minimize the kinetic distance between the in-vitro and in-vivo processes, imposing the constraint that the deduced rates reproduce a known global property such as the overall in-vivo speed. Computationally, this constraint defines a hypersurface in the multi-dimensional space of transition rates. In order to demonstrate the predictive power of our method, we apply it to the elongation cycle of protein synthesis, a key process of gene expression. In all living cells, proteins are synthesized by ribosomes, which translate the codon sequences of mRNA into peptide chains of proteins. During the elongation cycle of this process, the ribosome translates one codon after another by binding a ternary complex consisting of aminoacyl-tRNA( aa-tRNA), elongation factor Tu( EF-Tu) and GTP. The amino acid is transferred from the tRNA to the nascent peptide chain, and the ribosome moves to the next codon with the help of elongation factor G( EF-G) allowing for the next elongation cycle. Translation elongation involves several individual states with rapid transitions between them[6],[7]. The different states have been studied by a variety of experimental techniques: chemical probing methods[8], pre-steady state kinetics[9]–[15], electron microscopy[16]–[19], X-ray crystallography[20]–[22], and single molecule methods[23],[24]. The kinetic measurements in vitro provided values for the individual transition rates but, so far, it has not been possible to measure the corresponding rates in the cell. The different states and transitions of the elongation cycle are schematically shown in Fig. 1. When the ribosome dwells at a certain codon and binds a ternary complex, the tRNA within this complex can be cognate, near-cognate, or non-cognate to the codon, which implies that the elongation cycle contains three different reaction pathways corresponding to the three branches in Fig. 1. During each round of elongation, the ribosome typically explores all three pathways in order to select a cognate tRNA and to reject the near-cognate and non-cognate ones. The individual rates of these pathways were measured in vitro at 20°C and/or 37°C using the ribosomes and translation factors from Escherichia coli[6],[13],[25]. Here, we combine these results with new data on the overall elongation rates in vitro to first derive a complete set of individual in-vitro rates at both temperatures. We then minimize the kinetic distance between the in-vitro and in-vivo processes, taking into account two known properties of the in-vivo process: the overall elongation rates[26],[27] and the tRNA concentrations[28], both of which have been measured in E. coli for different growth conditions. The in-vivo rates of the elongation cycle obtained in this way are then validated by three independent sets of in-vivo data[29]–[31]. First, we compute codon-specific elongation rates and show that these rates correlate well with relative translation rates as obtained experimentally by[29]. Second, we predict the time-dependent incorporation of radioactively labeled amino acids into proteins as studied in vivo by[30]. The time course of synthesis obtained theoretically is in excellent agreement with the experimental data. Third, using the same in-vivo rates, we also compute the missense error frequency and obtain good agreement with the experimental results of[31]. In all three cases, our computations do not involve any fit parameter and, thus, directly validate the derived set of in-vivo rates.
Discussion:
The theoretical approach described here involves two novel concepts. First, we introduced the kinetic distance to provide a quantitative measure for the similarity of the in-vitro and in-vivo kinetics. This distance has an intuitive interpretation in terms of the free energy barriers that govern the individual transition rates along the reaction pathways, and provides a genuine metric in the mathematical sense. Second, we constructed a constrained minimization procedure in order to deduce the unknown in-vivo values of the individual transition rates from their known in-vitro values. It is instructive to compare our approach with flux control or sensitivity analysis, a widely used method for multistep reaction pathways[3],[35]–[37], which has also been applied to protein synthesis[38]. The latter method explores the local vicinity of a given kinetics and describes the linear response of the overall flux to small changes in the individual transition rates in terms of flux control or sensitivity coefficients. In contrast, the theoretical approach introduced here is not restricted to the linear response regime but explores the space of transition rates in a global manner via an extremum principle( Fig. 2B). Furthermore, both the coordinate transformation from the individual transition rates to the single barrier shifts and the constraint arising from the global in-vivo property make our approach highly nonlinear. When we applied our computational method to translation elongation by ribosomes, we obtained predictions for the individual in-vivo rates that could be validated by three independent sets of data for codon-dependent translation speeds, codon-specific translation dynamics and missense error frequencies of protein synthesis. In all cases, we found good agreement between theory and experiment without adjusting any fit parameter. Even for the largest growth condition of 2. 5 dbl/h, most of the deduced in-vivo rates are similar to the measured in-vitro rates( Fig. 5B) but three in-vivo rates are significantly increased compared to their in-vitro values: the rejection rate for near-cognates, the dissociation rate after initial binding, and the recognition rate for cognate and near-cognate ternary complexes. The largest difference is found for the rejection rate, which is increased in vivo by a factor of 3. 9, while the dissociation rate and the recognition rate are increased by a factor 3. 3 and 2. 2, respectively. For all transition rates of the elongation cycle, we find that the deviations between the in-vivo and in-vitro rates correspond to relatively small shifts of the corresponding free energy barriers( Fig. 5A). In fact, all single barrier shifts are predicted to be smaller than. Because the cytosol represents a rather complex buffer, such small changes in the free energy barriers can be easily envisaged, arising, e. g., from changes in the hydrogen bond networks around the ribosome or from changes in the flexibility of some parts of this complex. On the other hand, our results also show that the high-fidelity buffer at 37°C, used here and developed by[25] represents a good approximation to the cytosol as far as the ribosomal kinetics is concerned, in contrast to earlier estimates in Ref.[39]. The free energy barriers considered here could be studied by Molecular Dynamics simulations. The latter method has been recently applied to explore the free energy landscape of tRNA translocation through the ribosome[40],[41]. From such simulations, one can estimate the attempt frequencies for barrier crossing which are difficult to determine by other computational methods. In principle, these simulation techniques could also be used to investigate how the energy landscape changes as one varies the ambient buffer conditions in the simulations. Even though the predicted shifts of the free energy barriers are relatively small, the associated changes of the transition rates have an interesting consequence for the relative importance of initial selection and proofreading for the error frequency of protein synthesis. For the codon-specific Markov process depicted in Fig. 3, the efficiency of initial selection and proofreading are described by the coefficients and, respectively. The in-vivo value of the initial selection coefficient is increased by a factor of 7. 7 compared to the corresponding in-vitro value whereas the proofreading coefficient is increased by a factor of 2. 9. The combination of improved initial selection and proofreading leads to a reduction of the in-vivo error frequency by a factor of 6. 7, a reduction that is primarily achieved by the improved initial selection of the bound ternary complexes. In the present study, the codon-dependence of the elongation cycle arose from the initial binding rates that depend on the concentrations of cognate, near-cognate, and non-cognate tRNA, because we used the same transition rates along the reaction pathways for all cognate as well as for all near-cognate tRNAs. Thus, the values of the rates,,…of the cognate branch represent average values, obtained by averaging over all cognate tRNAs of all codons, and likewise for the internal rates,,…of the near-cognate branch. In vitro, the decoding rates of different cognate codons were observed to be rather similar[14],[25] whereas the GTPase activation rate was found to vary between 0. 06/s and 1. 3/s for different near-cognate codons of tRNA[25]. Likewise, recent in-vivo experiments provided evidence that the error frequency on 4 out of 14 near-cognate codons of tRNA is much higher than on the remaining 10 near-cognate codons[42]. Theoretically, it is straightforward to include codon-specific decoding and processing rates. Experimentally, it is, however, quite challenging to determine these rates in vitro for all codons and tRNA species. Our theory for protein synthesis by ribosomes can be extended in a variety of ways. For example, one could study how the overall elongation rate or the missense error frequency vary with changes in the overall ternary complex composition or as a function of individual ternary complex concentrations. Likewise, one may investigate how changes in internal transition rates arising, e. g., from protein or rRNA mutagenesis, affect the speed and accuracy of translation elongation. The computational method developed here to deduce the in-vivo from the in-vitro rates is relatively simple and can be applied, in general, to any multistep process or Markov model, for which one can estimate the in-vitro rates. Simple examples are provided by the folding and unfolding of proteins, the catalytic activity of enzymes with one active site, or the motility of molecular motors. More complex examples are transcription by RNA polymerase, protein refolding by chaperones, or protein degradation by proteases. Our method can also be applied to the large number of biochemical processes that have been studied by flux control or sensitivity analysis. Furthermore, the similarity measure provided by the kinetic distance could be useful in the context of systems biology, where the importance of detailed kinetics has been recently emphasized[43]. One important target in systems biology is to standardize the experimental data for such networks. Using the kinetic distance introduced here, one could, in fact, compare the kinetic data obtained by different groups in a systematic and quantitative manner.
The proverb ‘life is motion’ also applies to the molecular scale . Indeed , if we looked into any living cell with molecular resolution , we would observe a large variety of highly dynamic processes . One particularly striking aspect of these dynamics is that all macromolecules within the cell are continuously synthesized , modified , and degraded by complex biomolecular machines . These ‘nanorobots’ follow intricate reaction pathways that form networks of molecular transitions or transformation steps . Each of these steps is stochastic and takes , on average , a certain amount of time . A fundamentally important question is how these individual step times or the corresponding transition rates determine the overall speed of the process in the cell . This question is difficult to answer , however , because the step times can only be measured in vitro but not in vivo . Here , we develop a general computational method by which one can deduce the individual step times in vivo from their in-vitro values . In order to demonstrate the predictive power of our method , we apply it to protein synthesis by ribosomes , a key process of gene expression , and validate the deduced step times by three independent sets of in-vivo data .