Edge-based formulation of elastic network models

We present an edge-based framework for the study of geometric elastic network models to model mechanical interactions in physical systems. We use a formulation in the edge space, instead of the usual node-centric approach, to characterize edge fluctuations of geometric networks defined in $d$-dimensional space and define the edge mechanical embeddedness, an edge mechanical susceptibility measuring the force felt on each edge given a force applied on the whole system. We further show that this formulation can be directly related to the infinitesimal rigidity of the network, which additionally permits three- and four-center forces to be included in the network description. We exemplify the approach in protein systems, at both the residue and atomistic levels of description.

Figure 1

The extension of the spring can be written in terms of the displacements of the nodes [12].

Figure 2

Elastic response of PDK1 (PDB code: 3ORZ [34]). The top 2% of bonds by absolute extension are shown for three cases: (left) only the two-center interactions (bonds) are included in the network; (center) three-center (angle) interactions are added to the network and the top output extensions of bonds are shown; (right) dihedral angles are added and again only top output bond displacements are shown.

Figure 3

Infinitesimal rigidity of PDK1. Each cluster has a different color and “floppy” (nonrigid) atoms are shown in transparent gray: (left) only bonds included as constraints, leading to a single large cluster in blue with all other atoms floppy; (center) angle constraints added; (right) dihedral constraints also added leading to a large rigid cluster extending over the whole protein. The rigid clusters and floppy atoms are computed using the algorithm in Appendix pp5, which was introduced in Ref. [27].

Figure 4

(a) Structure of ADK from Escherichia coli (open conformation, PDB: 4AKE), with the lid and ${\mathrm{AMP}}_{\text{bind}}$ domains highlighted. (b) Closed (1AKE) and open (4AKE) forms of ADK showing that the main differences are in the lid and ${\mathrm{AMP}}_{\text{bind}}$ domains. (c) The distribution of average edge displacements computed for open ADK (4AKE). (d) The top 2% residue-residue interactions with highest displacements are concentrated in the lid and ${\mathrm{AMP}}_{\text{bind}}$ domains. Note that the viewpoint of the structures in (b) has been changed slightly relative to (a) and (d), so as to facilitate the visualization of the differences between the open and closed conformations.

Figure 5

Schematic for the derivation of the three-center (angle) interaction, where the two-center bond lengths $\left(ij\right),\left(jk\right)$ are kept constant and we compute the extension $i-k$ under those constraints.

Figure 6

Schematic for the derivation of the dihedral interaction: the three two-center bond lengths $\left(ij\right),\left(jk\right),\left(kl\right)$ remain constant, as well as the two three-center angle interactions $\left(ijk\right)$ and $\left(jkl\right)$ marked with the dashed lines. We compute the extension $i-l$ under those constraints.

Figure 7

Rigid cluster decomposition algorithm using infinitesimal rigidity [27]. For each trivial infinitesimal motion, such as the one in (a), the atoms are moved by a small distance along each $3N\times 1$ vector to a new position (b). A rigid tetrahedron of atoms is selected in the new position then in (c) this is moved back to its original position. Any atoms that also return to their original position at the same time (for all infinitesimal motions) are part of the same cluster. (d) The process is repeated until all atoms are clustered into rigid regions or are assigned as floppy.

Figure 8

The log absolute extension of interactions decreases linearly as a function of distance from the allosteric source site with slope $-0.142$ (correlation coefficient $=-0.603$, standard error $=0.0022$), i.e., the effect of the perturbation decays exponentially away from the allosteric site.