Chemomechanical networks based on the nucleotide states of the two motor heads at site , with chemical transitions shown as solid blue lines.

<p>The motor can reach states at the neighbouring sites and through mechanical transitions (dashed lines). The motor’s step velocity can be calculated by periodically repeating the networks at site along the spatial coordinate. (a) Uni-cycle network for myosin consisting of the chemomechanical cycle . Dashed red lines show mechanical transitions along the filament coordinate , which emerge from the state TD into the forward and from state DT into the backward direction. Solid lines refer to chemical transitions. The arrows indicate the direction of the transition, and infrequent transitions are shown in grey. This uni-cycle network applies to forces below the stall force . (b) Three-cycle network introduced in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0055366#pone.0055366-Bierbaum1" target="_blank">[19]</a> that captures the myosin’s stepping properties for both sub- and superstall load forces. The network includes ADP release from the leading head and additional forward and backward mechanical transitions for forced stepping (dashed violet line), with the dots pointing into the direction of hydrolysis. In addition to the network cycle , two cycles and are present. While the enzymatic cycle contains only chemical transitions, the mechanical cycle consists only of the mechanical transition . Thus, a spatial displacement can arise by means of the network cycle (dashed red lines) or the mechanical cycle (dashed violet lines).</p

Cognate (blue), near-cognate (purple), and non-cognate tRNAs (white).

<p>for all sense codons in <i>E. coli</i> following the definitions of “cognate” and “near-cognate” as given in [<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0134994#pone.0134994.ref037" target="_blank">37</a>] and [<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0134994#pone.0134994.ref038" target="_blank">38</a>], respectively.</p

Protein Synthesis in <i>E. coli</i>: Dependence of Codon-Specific Elongation on tRNA Concentration and Codon Usage

<div><p>To synthesize a protein, a ribosome moves along a messenger RNA (mRNA), reads it codon by codon, and takes up the corresponding ternary complexes which consist of aminoacylated transfer RNAs (aa-tRNAs), elongation factor Tu (EF-Tu), and GTP. During this process of translation elongation, the ribosome proceeds with a codon-specific rate. Here, we present a general theoretical framework to calculate codon-specific elongation rates and error frequencies based on tRNA concentrations and codon usages. Our theory takes three important aspects of <i>in-vivo</i> translation elongation into account. First, non-cognate, near-cognate and cognate ternary complexes compete for the binding sites on the ribosomes. Second, the corresponding binding rates are determined by the concentrations of free ternary complexes, which must be distinguished from the total tRNA concentrations as measured <i>in vivo</i>. Third, for each tRNA species, the difference between total tRNA and ternary complex concentration depends on the codon usages of the corresponding cognate and near-cognate codons. Furthermore, we apply our theory to two alternative pathways for tRNA release from the ribosomal E site and show how the mechanism of tRNA release influences the concentrations of free ternary complexes and thus the codon-specific elongation rates. Using a recently introduced method to determine kinetic rates of <i>in-vivo</i> translation from <i>in-vitro</i> data, we compute elongation rates for all codons in <i>Escherichia coli</i>. We show that for some tRNA species only a few tRNA molecules are part of ternary complexes and, thus, available for the translating ribosomes. In addition, we find that codon-specific elongation rates strongly depend on the overall codon usage in the cell, which could be altered experimentally by overexpression of individual genes.</p></div

Dwell time distributions (a) for high assisting forces, for (b) and (c, d) for substall and superstall resisting forces, for [ATP], [ADP] and zero [P] with the data from [<b>4]</b>, <b>[7]</b>.

<p>The blue lines show obtained using the single cycle network for . In (c) the distribution of forward steps, (brown, dashed line) agrees with the data that does not exhibit rapid events as in . (d) Forced backward stepping for leads to a single exponential decay (dashed violet line) that arises through the mechanical transition in the network in Fig. 2(b).</p

(a) Typical stepping trajectory, i.e., spatial displacement as a function of time and (b) dwell time distribution of myosin V, adapted from [4].

<p>(a) In single-molecule experiments with a feedback loop, the data are monitored under constant external load. Hence, the distance between the bead monitoring the motor’s motion (upper gray trajectory, with the thin black line showing a filtered curve) and the trap center (black trajectory) remains constant. (b) Dwell time distribution of myosin V for saturating [ATP]. The solid line is a fit from Ref. <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0055366#pone.0055366-Rief1" target="_blank">[4]</a> that involves two exponential functions with decay rates 150/s and 12.5/s.</p

Dwell Time Distributions of the Molecular Motor Myosin V

<div><p>The dwell times between two successive steps of the two-headed molecular motor myosin V are governed by non-exponential distributions. These distributions have been determined experimentally for various control parameters such as nucleotide concentrations and external load force. First, we use a simplified network representation to determine the dwell time distributions of myosin V, with the associated dynamics described by a Markov process on networks with absorbing boundaries. Our approach provides a direct relation between the motor’s chemical kinetics and its stepping properties. In the absence of an external load, the theoretical distributions quantitatively agree with experimental findings for various nucleotide concentrations. Second, using a more complex branched network, which includes ADP release from the leading head, we are able to elucidate the motor’s gating effect. This effect is caused by an asymmetry in the chemical properties of the leading and the trailing head of the motor molecule. In the case of an external load acting on the motor, the corresponding dwell time distributions reveal details about the motor’s backsteps.</p> </div

(a) Root mean square deviation RMSD between the experimental data (green bars in Fig. 3 (c, d)) and the simulated dwell time distributions for the three-cycle network as a function of the gating parameter for [ATP] = 10 (crosses) and [ATP] = 2 (circles).

<p>A lower deviation indicates an improved agreement between the experimental values and the distributions that result from the three-cycle network. For both concentrations, the RMSD decreases with increasing , until it saturates for , as indicated by the dashed line. The fluctuations for large values of arise from the variance in the simulations. The solid lines serve as a guide to the eye. (b) In case of a variable ATP binding rate , the agreement between the simulated dwell time distributions (symbols) and the experimental data (green bars) is further improved. The agreement is optimal for (red crosses), and is significantly improved in contrast to the distribution based on the experimental value of (red circles). The inset shows the RMSD as a function of , illustrating the minimal deviation for </p

Motor velocity as a function of external load for the network formed by the cycle (lines) compared to experimental data (symbols) for varying [ATP].

<p>In the experiments, the concentrations of ADP and P are believed to be rather small. In the calculations, we consider the limit of [ADP] = [P] = 0.</p

Translation elongation cycle.

<p>The ribosome has three tRNA binding sites, the A, P, and E site. A ribosome that has just arrived at a new (green) codon of an mRNA (state “0”) has an empty A site, whereas the P site is occupied by a tRNA (here shown as small gray sphere) that is cognate or near-cognate to the preceding codon. Elongation factor EF-Tu (blue spheres), aa-tRNAs (green, orange, and purple small spheres), and GTP molecules (not shown) form ternary complexes. Free cognate, near-cognate and non-cognate ternary complexes bind to the ribosome with rates depending on their respective concentrations (green, orange, and purple arrows from state “0” to states “1”, “6”, and “11”, respectively). Since the initial binding is not codon-specific, all kinds of ternary complexes unbind again from the ribosome with the same dissociation rate. Alternatively, a cognate or near-cognate ternary complex can be recognized by the ribosome (dotted arrows from states “1” and “6” to states “4” and “9”, respectively) before the ternary complex is either completely released (arrows from states “4” and “9” to state “0”), brought back to the initial binding state (dotted arrows from states “4” and “9’ to states “1” and “6”, respectively), or its aa-tRNA is accommodated in the ribosomal A site (arrows from states “4” and ”9” to states “5” and “10”, respectively). Along with aa-tRNA accommodation, EF-Tu leaves the ribosome. The new A-site tRNA is then further processed and shifted to the P site, while the ribosome translocates to the next (purple) codon. The former P-site tRNA is now in the E site. Depending on the assumed pathway of tRNA release, the E-site tRNA either dissociates very rapidly from the ribosome (2-1-2 pathway), or stays until the next aa-tRNA has been accommodated in the ribosomal A site (2-3-2 pathway). The numerals correspond to the ribosomal states of the codon-specific Markov process introduced below.</p

Changing the codon usage of AAA for fixed ratios of the codon usages for all other codons.

<p>(A) The concentration of free Lys-tRNA^Lys ternary complexes decreases when the codon usage of one of its cognate codons, AAA, increases. (B) Elongation rates of codons AAA (solid line) and AAG (dashed line), both of which are cognate to tRNA^Lys. The solid and dashed lines coincide almost perfectly. (C) Near-cognate missense error frequencies for both codons (AAA: solid; AAG: dashed). Results are shown for both the 2-1-2 pathway (blue) and the 2-3-2 pathway (orange) of E-site tRNA release. The vertical dashed lines (black) indicate the wild type value 0.0467 of AAA codon usage, see <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0134994#pone.0134994.s008" target="_blank">S7 Table</a> in the Supporting Information.</p