Discrete Breathers in Nonlinear Network Models of Proteins

Abstract

We introduce a topology-based nonlinear network model of protein dynamics with the aim of investigating the interplay of spatial disorder and nonlinearity. We show that spontaneous localization of energy occurs generically and is a site-dependent process. Localized modes of nonlinear origin form spontaneously in the stiffest parts of the structure and display site-dependent activation energies. Our results provide a straightforward way for understanding the recently discovered link between protein local stiffness and enzymatic activity. They strongly suggest that nonlinear phenomena may play an important role in enzyme function, allowing for energy storage during the catalytic process. The predictions of elastic network models (ENMs) of proteins [1,2,3,4] have proven useful in quantitatively describing amino-acid fluctuations at room temperature [1], often in good agreement with isotropic [2], as well as anisotropic measurements [5,6]. Moreover, it has been shown that a few low-frequency normal modes can provide fair insight on the large amplitude motions of proteins upon ligand binding [7,8,9], as previously noticed when more detailed models were considered [10,11,12], also by virtue of the robust character of the collective functional motions [13].However, low-frequency modes of proteins are known to be highly anharmonic [14,15], a property which has to be taken into account in order to understand energy storage and transfer within their structure as a consequence of ligand binding, chemical reaction, etc [16,17]. Indeed, there is growing experimental evidence that long-lived modes of nonlinear origin may exist in proteins [18,19]. Likewise, many theoretical studies have appeared suggesting that localized vibrations may play an active role in, e.g., enzyme catalysis [20]. These include topological excitations such as solitons [21] as well as discrete breathers (DBs) [22,23].The latter are nonlinear modes that emerge in many contexts as a result of both nonlinearity and discreteness [24]. Although their existence and stability properties are well understood in systems with translational invariance, much less is known of the subtle effects arising from the interplay of spatial disorder and anharmonicity [25,26,27]. For this purpose, in the present work we introduce the nonlinear network model (NNM). Our aim is to extend the simple scheme of ENMs, known to capture the topology-based features of protein dynamics [1,2,3], by adding anharmonic terms. Within the NNM framework, we show that spontaneous localization of energy can occur in protein-like systems and that its properties may be intuitively rationalized in the context of specific biological functions. In our model, the potential energy of a protein, E p , has the following form:where d ij is the distance between atoms i and j, d0 ij their distance in the structure under examination (as e.g. solved through X-ray crystallography) and R c is a cutoff that specifies the interacting pairs. As done in numerous studies, only C α atoms are taken into account [4] and...

Long-range energy transfer in proteins

“…Specifically, in the nonlinear network model (NNM) [33,39], the potential energy of a protein, E p , has the following form:…”

“…Indeed, their ability to pump energy from neighboring sites is a distinctive signature of DB self-excitation [32], e.g. observed as a consequence of surface cooling [33][34][35][36] or due to modulational instability of band-edge waves in nonlinear lattices [37,38]. Furthermore, within the frame of a nonlinear network model, we have shown that highly energetic DBs can form easily in proteins [33], and that they feature strongly site-modulated properties [39].…”

“…More precisely, we have shown that spatially localized bandedge normal modes (NM) can be continued from low energies to DB solutions centered at the same sites as the corresponding NMs (the NM sites). Note that the latter lie, as a rule, within the stiffest regions of a protein [33,39]. More generally, however, DBs display a gap in their excitation spectrum.…”

“…observed as a consequence of surface cooling [33][34][35][36] or due to modulational instability of band-edge waves in nonlinear lattices [37,38]. Furthermore, within the frame of a nonlinear network model, we have shown that highly energetic DBs can form easily in proteins [33], and that they feature strongly site-modulated properties [39]. More precisely, we have shown that spatially localized bandedge normal modes (NM) can be continued from low energies to DB solutions centered at the same sites as the corresponding NMs (the NM sites).…”

Energy thresholds of discrete breathers in thermal equilibrium and relaxation processes

“…In addition, when the thermalized nonlinear systems are cooling by boundary dissipations, long-lived discrete breathers can sometimes be observed after a long time relaxation 48,49,[51][52][53][54][55][56][57][58][59][60][61] .…”