Proteins are the basic building block of nearly all biological functions. All the protein consists of primary amino acid sequence that translates the tertiary structure of the protein, and ultimately describes the overall protein’s function. According to the central protein dogma, the amino acid sequence determines the protein structure and functional relationship. Protein structure and function is commonly correlated but usually, proteins are dynamic in nature, and their inner motions among the molecule facilitate their function. In fact, the protein’s dynamics is the intrinsic interface between its structure and function [1]. However, only the local dynamical property of an enzyme active sites are not involved in the functional and catalytic competent state essential to perform the proteins biological functions. Indeed, the overall protein’s dynamics is integral for its whole biological functions. Numerous studies and investigations affirm the impacts of protein dynamics on enzyme catalysis, substrate binding, the proliferation of allosteric signal, and the interaction between the proteins [2-8]. Further, it has been investigated that the dynamics of amino acids that are not directly engaged in catalysis are also significant for the other functions of enzyme [9]. Protein dynamism is the result of time-dependent change and evolution in the structure of a protein which involves fluctuations of equilibrium controlling the biological function [10]. By deforming the inner structure of the protein can successfully determine these time-dependent fluctuations. The mechanistic features of the protein determine the structural flexibility at all levels, from the static fluctuations of residues chains to the reorientation of huge rigid bodies like protein domains. Formerly numerous studies have been focused on the qualitative descriptions of very less number of static structures, but there is wide evidence that the structure of a protein is correlated to the dynamics and it may be evolutionarily and structurally conserved. Thus, it’s a need for analyzing protein dynamics in a systematic, efficient and organize quantitative manner. Despite the fact that the number of dynamical experimental data is overgrowing day by day e.g. from the investigation of the time dimension in X-ray and NMR studies, the knowledge gained from them is still incomplete. A number of computational approaches have been adapted to counterpart the experimental structural biology data and furnish dynamical models of the biomolecules. Molecular Dynamics (MD) simulations are extensively used to simulate the protein dynamics on time scale of up to the microsecond. Computationally it’s laborious and cost-effective to analyze the data by MD simulation and the time scales essential to large sample motions such as domain reorientation is not easily accomplished. Comparatively Normal Mode Analysis (NMA) using Elastic Network Models (ENM) is an economical and efficient method for characterizing and predicting slow motions of proteins. The productivity and performance of ENM methodology also makes them more suitable for automated comparative analysis among multiple protein structures. NMA has been extensively used to analyze the latent protein flexibility [11-14]. NMA modeling the atoms motions in a protein as a pair harmonic oscillator and combine each mode of motion to an oscillation frequency. Lower frequencies of different modes correspond to overall or domain motions and have been investigated to coordinate well to functionally related protein dynamics [15-18]. In addition it has been shown to obtain results in precise harmony with molecular dynamics simulations when characterizing the cumulative proteins motions [14,19-23]. In Elastic Network Models, the protein is shown as a spring’s network, grabbing the density of particles and the interaction strength between them. Inter-atomic interactions illustrated by a simple harmonic potential and the protein is commonly modeled with a small number of atoms, mostly single bead per residue residing at the C-alpha position. This granularity of the model is suitable to reduce the dimensionality, motivated by the approximate independence of overall-residue motion from the side-chain dynamics in globular proteins [24]. In this study, we apply the single mode analysis of NMA approach to our protein FnCel5A (PDB ID: 3rjy) to investigate the flexibility and motions of its structure (Figure 1). This analysis is helpful in providing information, how the structure of FnCel5A moves and how its flexibility is conserved over its structure.

Figure 1: Structure of FnCel5A: The substrates (in green) and the key residues (in orange) of active site are represented by sticks. The yellow dots represent the distance between the atoms of key residues and substrates.

Protein sequence and structure

An FnCel5A protein was selected from the online Protein Data Bank with PDB ID (3Rjy) [25] based upon structural data availability. From the previous study, we know that that the selected protein is a homopolymer, therefore, single polypeptide chain (chain A) was chosen to analyze the protein sequence and structure, as well as protein dynamics.

Pymol analysis

The FnCel5A structure was analyzed by using Pymol software to find out the highest B factor regions and was compared with the deformation energies and other information’s of the Normal Mode Analysis.

Calculation of normal modes

Hinsen et al., developed the WEBnm@ server using the Elastic Network Model (ENM) with the C-alpha force field [24], accessible in the Molecular Modeling Toolkit (MMTK) [26]. Each residue is indicated by a mass at the position of its Cα atoms.

The PDB file 3.rjy was submitted to the online WEBnm@ server [27,28] for single NMA analysis using elastic network model to analyze the FnCel5A structure to find out the highest B-factors and other flexible regions by calculating the deformation energies, eigenvaluse atomic displacement and the fluctuation between C-alpha atoms.

Deformation energies

Deformation energies shown in Table 1 and eigenvalues shown in Figure 2 showing the energy associated with each mode and is inversely related to the amplitude of the motion described by the corresponding modes. The protein structure moves along all the normal modes at once. A mode can only be evaluated in isolation if it is energetically well separated from other modes. The PDB file 3.rjy contains atoms that have no alpha-carbon, or not a standard amino acid, therefore those atoms were ignored. The eigenvectors and eigenvalues of the mass-weighted matrix of the second derivative of U(r) correspond to normal modes and the frequency squares for each normal mode, respectively. In WEBnm @ uses a molecular modeling toolbox and uses the coarse granular ENM to compute normal low frequency modes as shown in Figure 3. As shown in Figure 2 the B-factors that corresponds to high deformation energy are the residues: 55-62, 88-105, 211-217, 236-275, 328-332, which comprise the surface loops of the FnCel5A protein. Both the normal mode analysis and Pymole study correlate with each other.

Table 1: Values of the deformation energy for the lowest-frequency non-trivial modes (modes 7 to 20).

Figure 2: B-factor representation of FnCel5A: Regions with high B-factor are located in long loops (from pale green to deep blue).

Figure 3: The eigenvalues for the 50 lowest-frequency non-trivial modes.

Atomic Displacement

The Figure 4 showing the displacement square of each C-alpha atom (for modes 7 to 12) normalized so that the sum of all residues is equal to 100. The highest values correspond to the most displaced regions. Clusters of peaks on the plots identify significantly displaced regions, while isolated peaks may reflect local flexibility.

Figure 4: First column: residue identifier, second column: normalized squared displacements.

Fluctuations

Figure 5 describe the fluctuation flexibility of the Cα atoms of the enzyme. The residue numbers are indicated on the x-axis of the graph and the amplitude of the fluctuations on the y-axis. The graph shows normalized squared fluctuations of each Cα atom and is computed by using the default WEBnm @. The fluctuation square of each C-alpha atom (calculated from 200 lowest non-trivial modes), was normalized so that the sum of all residues become equal to 100.

Figure 5: First column: residue identifier. Second column: normalized squared atomic fluctuations.

Overlap between normal modes and difference vector

The Figure 6A and B showing the squared inner product between the difference vector and the normal modes identifies which modes contribute most to the structural difference.

Figure 6: (a) Cluster peak overlap plot and (b) Normal overlap plot.

Correlation Matrix Result

The correlation matrix shows the correlated movement of the C-alphas atoms in the FnCel5A protein. Both axes denote the C-alphas atoms of the protein in sequential order, so that each cell in the plot shows the coupling of two residues in the protein on a range from -1 (anti-correlated) via 0 (uncorrelated) to 1 (correlated). The measure expresses the expected inner production of atomic displacement, so both correlated and anti-correlated motion could be interpreted as coupled motions. Selected, significant correlation matrix values are mapped onto the input structure for a 3-D interpretation of the patterns seen on the heat map. The correlation matrix is searched in all four directions for an uninterrupted network of correlated residues that fulfill the following: First, a score threshold is set such that that it falls in a percentile between 95 and 99.9 (depending on the input structure size) of the distribution of the absolute scores in the correlation matrix. The positive correlation scores, thus, range from this threshold up to one, while the negative correlation scores range from the negative of the threshold down to -1. Then, pairs of correlated residues that have a distance below 0.8 nm are excluded. The resulting lists of residues form a group that is connected in sequence, and by regions of strong correlations that go beyond local contacts. These groups may consist of only a pair of residues many residues that are connected by sticks. There is a weak correlation in atomic fluctuation (red color regions in Figure 7), indicates that residues in this regions fluctuate simultaneously in a correlated fashion.

Figure 7: First row: residue identifier, First column: residue identifier. Other cells: Inter-residue correlations.

The dynamic simulation is critical for understanding macromolecular function with biologically relevant sizes and time scales but this approach is cost-effective and time consuming as compared to Normal Mode Analysis. In this study, normal mode analysis (NMA) was used to analyze the complex dynamic function of a macromolecule FnCel5A as a set of simple harmonic oscillation vibrating around a fixed equilibrium conformation. This approach, was developed from classical mechanics, and was first used to find out the dynamical behavior of small biological systems about thirty years ago. During this era, abundant of evidence have been combined to validate NMA as a powerful tool for simulating macromolecular motions at expanding length scales. Recently, NMA, with coarse representations, has become an effective alternative to molecular dynamics simulations to study the slow and large-scale movements of macromolecular machines. Interesting insights into these systems can be obtained very quickly with NMA to characterize their flexibility, to predict the directions of their collective conformational changes, or to help in the evaluation of experimental structural data. In protein crystallography, Atomic displacement parameters (ADPs, also known as thermal or B-factors) have given increasing significance as they contribute enough information on the flexibility of side-chain and main-chain atoms when anisotropically [29,30] or isotropically [31] maintained. The ADPs also play an important role to determine the accuracy of atomic position and its coordinates [32,33]. It can also contribute statistically by analyzing various structural results [34]. The best predictions of polypeptide chain flexibility were based on analyzes of ADPs in known structural kits [35-37]. In this study the eigenvalue and ADPs showing significant effect on FnCel5A structure. Unfortunately, ADP averages and variances are often significantly different for different protein structure determinations, since absolute ADP values do not depend only on various physical phenomena such as true oscillations for stable conformations or static, dynamic, and lattice distortions, but also depend on structural differences in refinement methods and levels [38].

In recent years, great interest in NMA-inspired approaches has been observed as they provide a unique and biologically relevant analytical solution to the equilibrium dynamics of biomolecules. The single-mode analysis calculates the lowest frequency modes and normal proteins and offers different types of calculations to analyze these modes. The single-mode analysis of FnCel5A shows that the three-dimensional structures of this protein contain the information necessary to describe its functional movements. Most of the global or collective motion modes predicted by NMA are insensitive to the details of model and energy parameters, and rather depend on the topology of contacts between equilibrium residues; this justifies the widespread use of the more efficient particle size EN model described here. This information on FnCel5A can be used in conjunction with experimental studies to resolve the supramolecular dynamics and longtime scales of the FnCel5A structure, which may not be accessible with conventional simulation approaches.

The author enthusiastically acknowledges the Microbial Metabolism Laboratory, School of Life Sciences and Biotechnology, Shanghai Jiao Tong University Shanghai P.R China, for the provision of facilities to carry out this research. The author has not got any special technical and financial support from any institute or funding agency to perform this research.