Modeling the effect of codon translation rates on co-translational protein folding mechanisms of arbitrary complexity

Abstract

In a cell, the folding of a protein molecule into tertiary structure can begin while it is synthesized by the ribosome. The rate at which individual amino acids are incorporated into the elongating nascent chain has been shown to affect the likelihood that proteins will populate their folded state, indicating that co-translational protein folding is a far from equilibrium process. Developing a theoretical framework to accurately describe this process is, therefore, crucial for advancing our understanding of how proteins acquire their functional conformation in living cells. Current state-of-the-art computational approaches, such as molecular dynamics simulations, are very demanding in terms of the required computer resources, making the simulation of co-translational protein folding difficult. Here, we overcome this limitation by introducing an efficient approach that predicts the effects that variable codon translation rates have on co-translational folding pathways. Our approach is based on Markov chains. By using as an input a relatively small number of molecular dynamics simulations, it allows for the computation of the probability that a nascent protein is in any state as a function of the translation rate of individual codons along a mRNA's open reading frame. Due to its computational efficiency and favorable scalability with the complexity of the folding mechanism, this approach could enable proteome-wide computational studies of the influence of translation dynamics on co-translational folding.

FIG. 1.

Co-translational protein folding. (a) The ribosome translates codons contained in the open reading frame of a mRNA molecule into a nascent protein. (b) Starting from the 5′ end (codon 1), the ribosome uni-directionally slides (large gray arrow) along the mRNA molecule and converts the genomic information in the ORF into a nascent protein (blue), which emerges through a channel known as the ribosome exit tunnel. (c) At a given nascent chain length L, the nascent chain has the potential to form tertiary structure; such states may include folded, intermediate, and unfolded conformations. The arrows indicate that these states may be able to interconvert at the given chain length.

FIG. 2.

A free-energy surface of the MIT domain in bulk solution near its melting temperature and implied time scales from the Markov analysis. (a) The free-energy contour surface is projected along the order parameters Q₁₂ and Q₁₂₋₃ at 320 K. The free energy at a given point on this surface is calculated as −RTln[P(Q₁₂, Q₁₂₋₃)], where R is the universal gas constant, T is the temperature K, and P(Q₁₂, Q₁₂₋₃) is the probability of the simulation sampling a point (Q₁₂, Q_12-3) during the simulations. The energy scale, on the right, is in units of kcal/mol. The three states (U, I, and F) can be seen as blue basins in this surface. (b) The largest implied time scales as a function of the lag time, Δt, for nascent chain lengths of 69 (circles), 75 (squares), 85 (diamonds), 95 (upward triangles), 105 (downward triangles), and 115 residues (stars). Lines are to guide the eye.

FIG. 3.

A parallel co-translational folding reaction scheme with (a) rates and (b) elementary transition probabilities indicated. Assuming state S₁ corresponds to the folded state, then a domain that folds via this mechanism can take parallel pathways to the folded state, either directly from S₂ or S₃. At length L, these three states can reversibly and directly interconvert with one another with rates $k_{i,j}^{\left(L\right)}$ and elementary transition probabilities $t_{i,j}^{\left(L\right)}$. Addition of a residue to the nascent chain shifts the system irreversibly from length L to length L + 1 with rate $k_A^{\left(L\right)}$ and elementary reaction probability $a_i^{\left(L\right)}$ that state $S_i^{\left(L\right)}$ transitions to state $S_i^{(L+1)}$ after one step on this reaction network.

FIG. 4.

Equation (10) accurately predicts the effect of codon translation rates on the probability of populating different states during co-translational folding in coarse-grained Langevin dynamics simulations. (a) The 77-residue MIT domain consists of three helices and was fused to the N-terminus of an unstructured 43-residue polyglycine linker. (b) The domain forms a helix bundle in the folded state. (c) The synthesis of this nascent chain was simulated using a coarse-grained model, a simulation structure of which is shown in which the intermediate is present. (d) The populations of unfolded, intermediate, and folded states are shown, respectively, in black, red, and green at different translation rates. The predictions from Eq. (10) are shown as solid lines (their width corresponds to the 68% confidence interval). The predictions were made at $k_A^{\left(L\right)}$ rates equal to 0.01*$k_F^{\left(\mathit{bulk}\right)}$ ((d), top panel), 0.1*$k_F^{\left(\mathit{bulk}\right)}$ ((d), middle panel), and $k_F^{\left(\mathit{bulk}\right)}$ ((d), bottom panel). The continuous translation results from the coarse-grained model are shown as symbols at the various $k_A^{\left(L\right)}$ values; error bars correspond to the standard error about the mean (computed from 300 independent trajectories at each nascent chain length).