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.
Deep Dive into Discrete breathers in nonlinear network models of proteins.
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, d 0 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 k 2 is assumed to be the same for all interacting atom pairs [1]. As in previous ENM studies [8,28], we take R c =10 Å, and fix k 2 so that the low-frequency part of the linear spectrum match actual protein frequencies, as calculated through realistic force fields [10,11,12]. This gives k 2 = 5 kcal/mol/ Å2 , with the mass of each C α fixed to 110 a.m.u., that is, the average mass of aminoacid residues. Note that standard ENM corresponds to k 4 = 0, while in the present work k 4 = 5 kcal/mol/ Å4 . Proteins live and perform their functions immersed in water and exchange energy with the solvent through their sizable surface portion. In a previous paper we showed that complex energy relaxation patterns are observed as a result of the inhomogeneity of the coupling to the solvent of bulk and surface atoms [29]. In the presence of nonlinearity, boundary relaxation is known to drive a wide array of systems towards regions of phase space corresponding to localized modes that emerge spontaneously [30,31,32,33]. Thus, in order to study typical excitations of nonlinear origin in protein structures, it appears natural to perform a boundary cooling experiment. Our protocol is the following. After 50 psec of microcanonical molecular dynamics (MD) simulation performed at a temperature T eq , the protein is cooled down by adding a linear dissipation term to the force acting on surface atoms, that is, those belonging to amino-acids with more than 25 Å2 of solvent accessible surface area. This represents nearly 40% of the amino-acid residues, for all proteins considered in the present study. The viscous friction coefficient γ is set to 2 psec -1 , a typical value for protein atoms in a water environment [16]. Hereafter, the equilibration energies considered are in the range k B T eq = 2 -20 kcal/mol, that is, of the order of, e.g., the energy release of ATP hydrolysis. With such initial conditions, energy in the system remains high for a period of time long enough so that localization can occur.
In Fig. 1, we show the energy of dimeric citrate synthase (PDB code 1IXE) as a function of time, as well as the energy of two amino-acids of monomer A, Thr 208 and Ala 209. After t = 20 psec and a few large fluctuations, a DB centered at Thr 208 forms. At t = 200 psec, more than 80% of the total energy is located there. Note the slow decay of the total energy after t =100 psec and the periodic energy exchanges of Thr 208 with Ala 209, another among the few amino-acids involved in the DB. Note also that at t = 20 psec the energy of Thr 208 is higher than at t =0, that is, when the friction was turned
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