On the existence of thermodynamically stable rigid solids

Abstract

Customarily, crystalline solids are defined to be rigid since they resist changes of shape determined by their boundaries. However, rigid solids cannot exist in the thermodynamic limit where boundaries become irrelevant. Particles in the solid may rearrange to adjust to shape changes eliminating stress without destroying crystalline order. Rigidity is therefore valid only in the metastable state that emerges because these particle rearrangements in response to a deformation, or strain, are associated with slow collective processes. Here, we show that a thermodynamic collective variable may be used to quantify particle rearrangements that occur as a solid is deformed at zero strain rate. Advanced Monte Carlo simulation techniques are then used to obtain the equilibrium free energy as a function of this variable. Our results lead to a unique view on rigidity: While at zero strain a rigid crystal coexists with one that responds to infinitesimal strain by rearranging particles and expelling stress, at finite strain the rigid crystal is metastable, associated with a free energy barrier that decreases with increasing strain. The rigid phase becomes thermodynamically stable when an external field, which penalizes particle rearrangements, is switched on. This produces a line of first-order phase transitions in the field-strain plane that intersects the origin. Failure of a solid once strained beyond its elastic limit is associated with kinetic decay processes of the metastable rigid crystal deformed with a finite strain rate. These processes can be understood in quantitative detail using our computed phase diagram as reference.

On identifying collective displacements in apo-proteins that reveal eventual binding pathways

“…It is often unclear how such collective variable may be defined. Drawing from ideas which were first demonstrated in crystalline solids [27][28][29] we have defined a new collective variable, NAP, which quantifies non-a ne thermally excited displacements of proteins. Firstly, we show that the susceptibility of di↵erent locations of a protein to non-a ne displacements can be used as an indicator to determine regions which are important for binding events.…”

“…In solids, non-a ne displacements are obtained by systematically projecting out 27 trivial homogeneous deformations of atoms, capturing only those displacements responsible for irreversible plastic events. 28,29 The NAP, is the squared sum of the mean amplitudes of these displacements. In proteins NAP behaves as a collective coordinate responsible for revealing binding pathways and also for determining possible allosteric couplings between spatially distant residues.…”

“…The current work o↵ers a practical middle ground between expensive Molecular Dynamics simulation of protein-ligand recognition processes and inaccurate docking based approach by employing an idea from materials physics, which correctly accounts for deformations in protein structure relevant for binding. We use a formalism, introduced earlier to analyse atomic displacements in solids, [27][28][29] to predict the protein locations which are potentially susceptible towards ligand recognition. We build on our work under the basic premises that all external and internal stresses result in two types of mutually orthogonal deformations namely those which are "a ne" i.e.…”

Non-affine displacements encode collective conformational fluctuations in proteins

“…For example, in crystals, these non-affine modes with the largest eigenvalues correspond to precursors to crystal defects 16,17 and are implicated in plastic events and loss of rigidity. 26 In proteins too, they correspond to local conformational changes which are important for ligand binding and allostery. 23 One may also characterize regions in the protein which are particularly amenable to non-affine displacements using the local NAP susceptibility defined as (NAP(i) − NAP(i) ) 2 .…”

Exploring the link between crystal defects and nonaffine displacement fluctuations

“…The affine displacements couple to local stresses (and torques) while the non-affine component of the displacement couples to a new "non-affine" field [15,21,22]. Enhancing non-affine fluctuations by increasing temperature, applying large strains or the nonaffine field leads to the creation of defects [15,18,22]. Indeed, atomic fluctuations that act as precursors to the formation of defects such as dislocation dipoles have been shown to be the most prevalent, though not the sole, non-affine displacement even within a small oscillation, harmonic, approximation [15,18].…”