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The authors have declared that no competing interests exist.

Conceived and designed the experiments: PCW KYS. Performed the experiments: PCW KYS. Analyzed the data: PCW SCB JHDC KYS. Contributed reagents/materials/analysis tools: PCW. Wrote the paper: PCW SCB JHDC KYS.

Functional rearrangements in biomolecular assemblies result from diffusion across an underlying energy landscape. While bulk kinetic measurements rely on discrete state-like approximations to the energy landscape, single-molecule methods can project the free energy onto specific coordinates. With measures of the diffusion, one may establish a quantitative bridge between state-like kinetic measurements and the continuous energy landscape. We used an all-atom molecular dynamics simulation of the 70S ribosome (2.1 million atoms; 1.3 microseconds) to provide this bridge for specific conformational events associated with the process of tRNA translocation. Starting from a pre-translocation configuration, we identified sets of residues that collectively undergo rotary rearrangements implicated in ribosome function. Estimates of the diffusion coefficients along these collective coordinates for translocation were then used to interconvert between experimental rates and measures of the energy landscape. This analysis, in conjunction with previously reported experimental rates of translocation, provides an upper-bound estimate of the free-energy barriers associated with translocation. While this analysis was performed for a particular kinetic scheme of translocation, the quantitative framework is general and may be applied to energetic and kinetic descriptions that include any number of intermediates and transition states.

With a comprehensive understanding of the underlying energetics for a biomolecular machine, such as the ribosome, we may quantitatively identify the essential factors that determine function. In molecular-level machines, the energetic fluctuations arising from the surrounding solvent become comparable to the energetics that direct each process. In this regime, structural rearrangements may be described as diffusive movements across a free-energy landscape. By adopting this framework, it is possible to quantitatively relate theoretical predictions, computational results, experimental rates and biochemical data, which will allow for a self-consistent description to be obtained. To this end, we have used an explicit-solvent simulation (2.1 million atoms, 1.3 microseconds) to measure the diffusion along multiple structural coordinates that have been implicated in tRNA translocation in the ribosome. With the diffusion coefficients in hand, we demonstrate how they may be used to bridge theoretical and experimental descriptions of complex collective dynamics in a molecular machine.

Biological machines are ubiquitous in the cell and typically contain many molecules that include protein, RNA, and other cofactors. Each molecule provides a unique contribution to an assembly's energy landscape, which then governs the machine's function. Accordingly, quantifying the landscape's features and molecular origins may allow one to precisely manipulate the physical-chemical properties that dictate the biological dynamics. Despite the pressing need for a quantitative description of the energy landscapes that underpin function, most experimental techniques report on the rates of interconversion between states, where each state is a discretized approximation to an energetic basin. To bridge discrete and continuous descriptions for a molecular machine, such as the ribosome, it is necessary to quantify the diffusive properties of functionally-relevant collective rearrangements. The observed, or effective, diffusion of each component of a biomolecular complex is determined by the intrinsic diffusion of that component (free in solution) as well as the short-scale energetic roughness that is introduced by molecular interfaces

Short length-scale diffusion on a rough landscape

The ribosome has long been considered to function as a “thermal ratchet machine”

Explicit-solvent simulation of an E. coli ribosome (water molecules and ions not shown), colored by region: 23S/5S rRNA (gray), 16S rRNA (cyan), proteins (light blue), P/P tRNA (red) and A/A tRNA (yellow). During translocation, tRNA molecules adopt hybrid configurations (middle). Rotation of the 30S body (

A) rmsf, by residue, for the 23S (left) and 16S (right) rRNA. rmsf measures (See

A) Rotation coordinates

To quantify the energy landscape of translocation, it is necessary to identify coordinates that are able to accurately capture these motions (which encompass large-scale rotary movement of the subunits). Structural approaches, such as x-ray spectroscopy and cryo-electron microscopy (cryo-EM), provide snapshots of the ribosome that describe the average configurations of energetic minima. These structural models provide tremendous insights into the global architecture of biomolecular assemblies, though the models can not separate fluctuations that are due to movement of individual residues from many-residue collective rearrangements. For example, x-ray and cryo-EM models have shown that the 30S subunit rotates relative to the 50S subunit

To probe the energy landscape of 30S body and head rotation and tRNA displacements, we first identified reaction coordinates upon which to project the free energy. For an appropriate coordinate

To construct coordinates for 30S body and head rotation (

Analysis of the fluctuations of each subregion (50S, 30S-body and 30S-head) indicates that large sets of atoms within each subunit maintain their structural integrity while the core group undergoes displacements relative to other subunits. Accordingly, the configurations of the cores residues were analyzed for a variety of experimental structural models, in order to identify the vectors of rotation that define

With the rotation angles calculated for each simulated frame, we next asked if the dynamics in these coordinate spaces is diffusive, or not. We found that the rotation coordinates and tRNA coordinate exhibit diffusive behavior (

In our previous analysis of tRNA diffusion during accommodation

Effective diffusion coefficients describe the short length-scale energetic roughness

While we consider these values of the diffusion to be initial estimates, additional considerations suggest the presented values are reliable measures. First, the diffusive regime for each coordinate is reached at lag times (

Since the presented analysis indicates that

Using

Diffusion leads to free-energy barrier-crossing attempts and the barrier height determines the probability of successfully crossing

The ability to rigorously interconvert between the energy landscape and kinetics will be essential in order to unambiguously quantify the features of the biomolecular landscapes that underpin function. With knowledge of the diffusive properties, theoretical and experimental probes of the energy landscape may be directly compared to kinetic measurements, which will enable a comprehensive picture of the landscape to emerge. Thus, the diffusion provides a unifying foundation for understanding and interpreting all available data for a given biological process. In the presented study, we have made the first steps towards establishing such a framework for tRNA translocation in the ribosome. To do this, we probed the diffusive characteristics of subunit rotations and tRNA displacements, essential sub-processes that facilitate protein elongation in the cell. As computer hardware continues to increase in power, and new computational algorithms and models are developed, the current study will provide the context for understanding a gamut of biophysical measurements and predictions. Of particular interest are the detailed features of the underlying energy landscape, as well as the robustness of ribosome dynamics to external perturbations. Similar to macroscopic machines, by understanding the interplay between the moving parts of these systems, it may be possible to design strategies to exploit this knowledge and provide precise regulation of biomolecular dynamics in the cell. In such efforts, the presented approach for bridging kinetics and free-energies provides a way to systematically verify predictions about the landscape against experimental data. These tools allow us to integrate complementary information from experimental and computational techniques, which will be crucial when identifying the features of the energy landscapes that govern biological function.

The simulation is a direct continuation of our previous 100 ns explicit-solvent simulation of the ribosome

During translocation by the ribosome, there are multiple large-scale rotary motions that facilitate tRNA movement (

Start with a set of candidate residues for the 50S, 30S-body and 30S-head. Only rRNA residues were included in the candidate groups. Here, we refer to this set of residues as

Calculate the structural rmsf for all non-hydrogen atoms of each group

Remove residues from

Iteratively calculate the rmsf and remove residues from

Reduce the value of

Upon completion of this iterative-exclusion algorithm, there were 1353, 443 and 178 rRNA residues in the core groups of the 50S, 30S body and 30S head (

To define the coordinates for rotation (

With values of

As expected from the arguments of Kramers

Definitions of core residues. (top) Candidate residues to define the core regions of the 23S (orange), 16S head (red) and 16S body (yellow). All 23S residues were considered candidates with which to define the 23S core. However, the 16S head and 16S body were manually partitioned and analyzed separately. (bottom) From the first 1

(EPS)

Structural fluctuations estimated from crystallographic refinement. rmsf values were obtained from PDB entry 3F1F, via the relation

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Displacement-squared as a function of lag time. (left) The displacement squared of

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Uncertainty in measures of the diffusion coefficients. From the 1.3

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Functional form of

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Rates are robust to the functional form of the free-energy

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Drift in

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Displacement-squared for alternate rotation coordinates. The displacement squared of

(TIF)

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Supporting discussion and methodological details. Overview of elongation, details for

(PDF)

We would like to thank Dr. Jeffrey Noel (Rice University) for discussion regarding collective dynamics and reaction coordinates.