BioMed Research International

Volume 2017, Article ID 2483264, 15 pages

https://doi.org/10.1155/2017/2483264

## Comparative Study of Elastic Network Model and Protein Contact Network for Protein Complexes: The Hemoglobin Case

^{1}Center for Systems Biology, School of Electronic and Information Engineering, Soochow University, Suzhou 215006, China^{2}Unit of Chemical-Physics Fundamentals in Chemical Engineering, Department of Engineering, Università Campus Bio-Medico di Roma, Rome, Italy^{3}Environment and Health Department, Istituto Superiore di Sanità, Viale Regina Elena 299, 00161 Roma, Italy

Correspondence should be addressed to Guang Hu; nc.ude.adus@gnauguh

Received 22 June 2016; Revised 17 November 2016; Accepted 20 December 2016; Published 22 January 2017

Academic Editor: Hesham H. Ali

Copyright © 2017 Guang Hu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The overall topology and interfacial interactions play key roles in understanding structural and functional principles of protein complexes. Elastic Network Model (ENM) and Protein Contact Network (PCN) are two widely used methods for high throughput investigation of structures and interactions within protein complexes. In this work, the comparative analysis of ENM and PCN relative to hemoglobin (Hb) was taken as case study. We examine four types of structural and dynamical paradigms, namely, conformational change between different states of Hbs, modular analysis, allosteric mechanisms studies, and interface characterization of an Hb. The comparative study shows that ENM has an advantage in studying dynamical properties and protein-protein interfaces, while PCN is better for describing protein structures quantitatively both from local and from global levels. We suggest that the integration of ENM and PCN would give a potential but powerful tool in structural systems biology.

#### 1. Introduction

Proteins rarely act alone: in the great majority of cases they perform a vast array of biological functions by forming functional complexes [1, 2]. The study of protein complexes not only elucidates the molecular mechanism of many diseases [3] but also provides structural information of protein-protein interactions [4]. With the increasing number of structural data, a lot of regularities have been found for protein complexes based on their topological structures [5]. However, the structural and assembly principles underlying protein complexes organization are not yet fully understood, which poses a great challenge in structural systems biology [6]. A well-studied example of protein complex is hemoglobin (Hb) tetramer, which contains two and two subunits as a dimer of dimer [7]. Hbs exist in three quaternary conformations: the low-affinity (deoxy, ) state and the high-affinity (oxy, ; carbonmonoxy, ) states. Hbs are never present in cells as monomers. Therefore, Hbs were considered as a sort of ‘obliged’ allosteric protein complexes and, even thanks to the great amount of both structural and physiological data, attracted a lot of attentions [8–10].

Network theory has become a versatile method to study structures and dynamics of biological systems [11–13]. As a dynamical-based method introduced by Tirion [14], Elastic Network Model (ENM) allows performing normal mode analysis at network level. Two mostly used ENM methods, Gaussian Network Model (GNM) and Anisotropic Network Model (ANM), were further proposed by Bahar and coworkers [15, 16]. ENM is an efficient computational tool to describe the essential vibrational dynamics encoded in the molecular topology [17–20]. It has been proved that the low-frequency modes of ENM are critical of collective motions [21], while the high-frequency modes can identify hot spots for protein-protein interactions [22].

The approach of Protein Contact Network (PCN) was proposed by Kannan and Vishveshwara [23] and now has become a new paradigm in protein ontology [24–28]. In a PCN, nodes correspond to , while edges exist if two amino acid residues (nodes) are close to each other under different cutoffs [29]. Based on this graphical representation, different topological parameters have been developed to describe protein structures and functions from both the global and the local prospective [30–32].

Both ENM and PCN offer computationally efficient tools to study the structure and function of protein complexes [33, 34], from predicting functionally important residues [35, 36], to characterize protein-protein interactions [37, 38] and allosteric communication paths [39, 40]. Of course, both models have strengths and weaknesses and their comparative study is needed.

In this paper, we have analyzed and compared four applications of ENM and PCN on Hb structures: conformational change characterization, modular analysis, allosteric mechanisms investigation, and interface characterization. Although there are several works reported on the ENM [41–43] and PCN [44, 45] studies of Hb independently, this work revisits Hb as case study and mainly focuses on the methodology comparison of ENM (specifically GNM and ANM) and PCN.

#### 2. Materials and Methods

##### 2.1. Data Sets

Hemoglobins (Hbs) have three states [7]. We select their structures for the ENM and PCN analysis, which are listed as follows: -Hb (PDB code: 2dn2), -Hb (PDB code: 2dn1), and -Hb (PDB code: 2dn3).

##### 2.2. Gaussian Network Model and Anisotropic Network Model

GNM [15] describes a protein as a network of connected by springs of uniform force constant if they are located within a cutoff distance (7 Å in this study). In GNM, the interaction potential for a protein of residues is [46]where and are the equilibrium and instantaneous distance between residues and , and is * Kirchhoff matrix*, which is written as follows:Then, square fluctuations are given byThe normal modes are extracted by eigenvalue decomposition: , where is the orthogonal matrix whose th column is th mode eigenvector. is the diagonal matrix of eigenvalues, . can be written in terms of the sum of the contribution of each mode as follows:Thus, the cross-correlation can be calculated byThe cross-correlation value ranges from −1 to 1: positive values mean that two residues have correlated motions, while the negative values mean that they have anticorrelated motions.

In ANM [16], the interaction potential for a protein of residues is [46]The motion of the ANM mode of proteins is determined by * Hessian matrix *, whose generic element is given as follows: where , , and represent the Cartesian components of residues and is the potential energy of the system. used here is 13 Å. Accordingly, ANMs provide the information not only about the amplitudes but also about the direction of residue fluctuations.

The similarity between two ANM modes, and , evaluated for proteins with two different conformations can be quantified in terms of inner product of their eigenvectors [39]; that is,The degree of overlap between th ANM modes and the experimentally observed conformation change of Hbs among different states is quantified by . Therefore, the cumulative overlap between and the directions spanned by subsets of ANM modes is calculated as follows:

The Markov model coupled with GNM was used for exploring the signal transductions of perturbations in proteins [47, 48]. The affinity matrix describes the interactions between residue pairs connected in GNM; its generic element is defined as follows:where is the number of atom-atom contacts between residues and based on a cutoff distance of 4 Å and is the number of side-chain atoms in residue . The density of contacts at each node is given byThe Markov transition matrix , whose element , determines the conditional probability of transmitting a signal from residue to residue in one time step. Accordingly, the hitting time for the transfer of a signal from residue to is given by [47]where is Kirchhoff matrix obtained by GNM. The average hit time for th residue is the average of over all starting points . The commute time is defined by the sum of the hitting times in both directions; that is, was defined as the corresponding distance, as the weight of the edge between node and in the network.

##### 2.3. Protein Contact Networks (PCNs)

Protein Contact Networks (PCNs) provide a coarse-grained representation of protein structure [49], based on coordinates from PDB files: network nodes are the residues, while links exist between nodes whose Euclidean distance (computed with respect to *α*-carbons) is within 4 to 8 Å, in order to account only for significant noncovalent intramolecular interactions [24, 50, 51].

After building up the network, it is possible to quantify its features through the adjacency matrix Ad, whose generic element is 1 if th and th nodes are connected by a link; otherwise it is 0.

The most basic descriptor is the* node degree*, defined for each node as the number of links involving the node itself:

Given a set of vertices , the* shortest path * between two nodes is the minimum number of edges connecting them (Figure 1). Its role is crucial since it has been demonstrated that the lower the network* average shortest path* (or* characteristic length*, computed as the average value over the whole number of node pairs), the higher the efficiency of signal transmission through the network [52]. In PCNs the average shortest path describes the protein attitude to allosteric regulation.