Universality of fold-encoded localized vibrations in enzymes

Abstract

Enzymes speed up biochemical reactions at the core of life by as much as 15 orders of magnitude. Yet, despite considerable advances, the fine dynamical determinants at the microscopic level of their catalytic proficiency are still elusive. In this work, we use a powerful mathematical approach to show that rate-promoting vibrations in the picosecond range, specifically encoded in the 3D protein structure, are localized vibrations optimally coupled to the chemical reaction coordinates at the active site. Remarkably, our theory also exposes an hithertho unknown deep connection between the unique localization fingerprint and a distinct partition of the 3D fold into independent, foldspanning subdomains that govern long-range communication. The universality of these features is demonstrated on a pool of more than 900 enzyme structures, comprising a total of more than 10,000 experimentally annotated catalytic sites. Our theory provides a unified microscopic rationale for the subtle structure-dynamics-function link in proteins.

Figure 1

Coarse grained model. (A) All-atom view of the L-Lactate Dehydrogenase dimer (LDH, PDB id: 1I0Z). (B) Elastic Network Model deduced from the tertiary structure. (C) Sparsity pattern of the 3N × 3N force constant matrix ${ℍ}_{ij}^{\alpha \beta }$ used to compute the localization landscape. Each non-vanishing term is represented by a blue spot. In our case of uniform spring constant and sharp cutoff coupling, this matrix is a direct representation of the connectivity pattern among residues.

Figure 2

Comparison between Normal Modes and LL computation for human L-Lactate Dehydrogenase dimer (LDH, PDB id: 1I0Z). (A) Wave localization is visualized by plotting the 10 highest-frequency normal modes. Their frequencies range from 94.2 to 99.3 cm⁻¹ (2.82–2.98 THz). (B) The LL (u) is drawn along the protein backbone. Catalytic sites, shown explicitly alongside the LL, clearly lie very close to the LL maxima, corresponding to the sites of highest localization (hot spots).

Figure 3

3D LL for human L-Lactate Dehydrogenase dimer. The 3D LL is shown by color-coding the 3D coarse-grained scaffold according to the amplitudes of the LL depicted in Fig. 2B. We observe here that peaks (hot spots) of the localization landscape that appear distant when plotted along the backbone chain are in fact found around the same spatial location (the red spots). Wave localization is thus predicted to occur within two distinct 3D domains lying at the center of the molecule: these domains host the catalytic activity.

Figure 4

Compressive motions and localization sites in the L-Lactate Dehydrogenase dimer (LDH). The displacement amplitude (top graph) associated with the vibrational eigenvector #10 (frequency 94.17 cm⁻¹, 2.82 THz) is localized along the reaction coordinate residues VAL-31, GLY-32, MET-33, as predicted by the LL (function U, middle graph). The computation of the local compression factor (see Supplementary Information) clearly shows that these localized modes are compression modes.

Figure 5

A rate-promoting vibration in the L-Lactate Dehydrogenase dimer (LDH). Localization landscape color-coded on the coarse-grained structure with a close-up of the compression field corresponding to the vibrational eigenvector #10 (frequency 94.17 cm⁻¹, 2.82 THz) along the reaction coordinate: residues VAL-31, GLY-32, MET-33 compress towards ARG 106. The localized eigenmode #10 corresponds to the rate-promoting vibrations found in ref..

Figure 6

Localization and functional domains. (A) Partitioning of the molecule obtained from the LL. On the landscape plotted on the backbone chain, one selects the 4 highest local maxima (marked by a spike on the color bar) separated by the 4 lowest local minima (marked by the dotted lines). In the LL theory, each domain can be seen as a local harmonic oscillator, weakly coupled to the others. (B) The partitioning of the LDH obtained in frame A is here plotted on the tertiary structure, exhibits distinct spatial domains.

Figure 7

Domains in other enzymes. The partitioning procedure is illustrated for four enzymes. The clustering of the enzymes into vibrationally independent subregions is a general feature.

Figure 8

Proximity score for 10,566 annotated catalytic sites (933 enzymes) from the catalytic site atlas, gauging the match between a functional site and a main localization hot spot. Frame A: The relative distance is scored by taking the shortest distance between catalytic sites and the main localization hot spots, divided by the chain length. Frame B: Main histogram. 95% of active sites are found at one of the highest localization spots with an error smaller that 0.2% of the enzyme length along the chain. Inset: Size distribution.