Here the weighted CSN was studied with the 2D hydrophobic-hydrophilic (HP) square lattice model proposed by Dill [35]. The main point of that model is that the interactions between hydrophobic amino acids are the major driving force in protein folding. There are many advantages to using the HP model. First of all, complete enumeration of the sequence space and the conformation space is possible for sequences of given length. In our work, all sequences have a length of 13. Moreover, the definition of the interaction potential of the conformation is simple. In the present study, the interaction energy E of a conformation was defined as the number of topological contacts between adjacent hydrophobic amino acids that were not neighbors in the sequence, denoted by E_HH = −1, E_PP = E_HP = 0. It is common practice in the HP model to simply regard E as an “energy”, and neglect the fact that it is more correctly a free energy [35,36].

First, the conformation space was mapped onto an unweighted network. All allowed conformations of a chain, which has unique ground-state conformation, were enumerated. Conformation of the chain was evolved by the conventional Monte Carlo elementary moves [37]: the end rotation, the corner flips and the crankshaft move. As shown in Figure 1(a), the end monomer can be rotated either 90° (conformation c1 switches to conformation c2) or 180° (conformation c1 switches to conformation c3); conformation c3 can switch to c4 by single corner flips; and conformation c4 translates to conformation c5 by the “crankshaft” move, which can move the two monomers at the same time. All moves must obey the excluded volume criteria: no lattice sites can be doubly occupied. Then, the conformation space can be mapped to an unweighted network. Each conformation of the chain is identified as a vertex of the network. A link is created between two vertices if the two corresponding conformations translate through a single elementary Monte Carlo move. As seen in Figure 1(b), since the conformation c1 can be switched to the conformation c2 by a single Monte Carlo move, so there is a link between them. However, the conformation c2 can not be switched to the conformation c4 by a single Monte Carlo move, so that there is no link between them.

Second, the energy weight was added to the network. According to the interaction energy of the conformation, the Boltzmann factor of each conformation exp[−E / k_BT] can be calculated, where E and k_B are the energy of the conformation and the Boltzmann constant, respectively, and T is the absolute temperature. We set k_B = 1 and T = 1 in this work. The energy weight W_i of the vertex i in the networks, represents the importance of i-th conformation in the conformation space, which can be defined as following: assuming that the vertex i connects with vertices j, k, h, the sum of the Boltzmann factors of the vertices j, k, h is treated as the weight of the vertex i. Obviously, the numerical value of the weight of the vertex i is a real number. Therefore, a vertex with the weight w means the numerical value of the vertex’s energy weight falls into the half open interval [w-1, w).

Finally, we chose HP sequences to construct the weighted CSNs. As to the HP model, the relationship of the conformation–sequence has been studied thoroughly [38], and the report that the features of protein folding are weakly dependent on the chain length [39,40] and the properties of protein folding tested by using 2D lattice model and 3D lattice model are similar [36,37] may provide confidence of the validity of using short chain models to study the folding problem. Here we started from a set of 2¹³ HP sequences. Through searching the complete conformations of each sequence, a set of 309 sequences with unique lowest-energy state was found from the 2¹³ HP sequences. A natural protein sequence has a unique global minimum of free energy which is well separated in energy from other misfolded states. In a lattice HP model, the protein-like folds are associated with sequences that have a minimal number of lowest-energy states [41,42]. And the sequences with only one global ground state candidate are good sequences [43]. The native structures of these sequences have well stability and designability [19,43]. We randomly chose three short sequences from these 309 sequences in order to exhibit the scale-free property of the energy-weighted CSNs. The first sequence was (HHPPPHPHPPHPH), and the other two sequences were (HHHHHPHHPPPHP) and (HHHPPPHHPHHHP).

To uncover the relationship between the folding dynamics and the power-law property of the weighted CSN, the set of 309 sequences, as mentioned above, with unique lowest-energy state was applied. Then according to the method of the construction of the complex network above, we constructed 309 weighted complex networks. It is found out that all of these weighted networks show power-law tail properties, thus we can obtain all the scaling exponent γ s of the weighted CSNs.

Additionally, a known parameter, Z score (Z = Δ/Γ), was introduced in this article. In the Z score expression, Δ denotes the average energy difference between the ground state and all the other states and Γ is the standard deviation of the energy spectrum, which can be expressed as follows [19]: (1) Δ = 1 N C ∑ α > 0 ( E α − E 0 ) , (2) Γ 2 = 1 N C ∑ α > 0 E α 2 − ( 1 N C ∑ α > 0 E α ) 2 ,where E_a (α > 0) represents the energies of the excited conformation, and E₀ is the lowest state energy, and N_C is the number of excited compact conformations. According to the Equations (1) and (2), we can get all values of the Δ/Γ of the conformation spaces generated by the sequences that have unique ground-state. Then the relationship between the exponent scale γ and Z score can be studied.

In this article, a multistep greedy algorithm (MSG) in combination with a local refinement procedure named “vertex mover” (VM) [31,32] were applied to detect the module structure of the weighted CSNs. The MSG-VM algorithm is an agglomerative hierarchical clustering method and is based on the optimization of the modularity function that is defined as: (3) Q = ∑ i = 1 N C [ I ( i ) L − ( d i 2 L ) 2 ] ,with I(i) the weights of all edges linking pairs of vertices in community i, d_i the sum over all degrees of vertices in module i, L the total weight of all edges, and N_C the number of community. At the first step of the algorithm, each vertex in the network is considered as a community. Then the process of the algorithm involves finding the changes in Q that would result from the amalgamation of communities, selecting the largest of them. The MSG algorithm is to allow the merging of l (l > 1) pairs of communities at each iteration step, developing the algorithm that can only merge a pair of communities at each iteration step [30].

The algorithm works as follows: i. Start with the modularity change matrix ΔQ, whose initial value is Q 0   =   − ∑ i = 0 N d i 2 4 L 2 (the means of d_i and L are the same in the expression (3). N is the number of vertices.); ii. Change the initial values of ΔQ according to Δ Q i j   =   I L   −   d i d j 2 L 2 with I the weight of the edges connection the vertices i and j, d_x the degree of vertex x = i, j, and L the total edge weight; iii. Select the elements with among highest l values, then join the corresponding communities, and update the matrix ΔQ_ij; iv. Repeat the third step until getting the best values of modularity.

In the MSG-VM algorithms, the value of l and the weight of each edge of the network should be set. Here, an empirical formula was applied [32] for the choice of the step width l, which can be expressed as follows: (4) l = α L ,where L is the total weight of the edges and α = 0.25 [32]. On the other hand, we defined the weight of the edge in the network as follows: assuming there is an edge e between two nodes n₁ and n₂, the product of the Boltzmann factors of the two nodes n₁ and n₂ is defined as the weight of edge e, which is given by: (5) W e = exp [ − ( E n 1 + E n 2 ) / k B T ] ,where W_e is the weight of the edge e; E_n1 and E_n2 are the energies of the conformations which correspond to nodes n₁ and n_2, respectively; k_B is the Boltzmann constant; T is the absolute temperature. We set k_B = 1 and T = 1 in this work.

For a complex network, the widely studied characteristic is the degree. Degree of a vertex is the total number of its connections. This quantity is also called “connectivity”. The degree of a vertex is a local quantity, and the total distribution of vertex degrees often determines some important global characteristics of the network. As to different kinds of networks, the degree distributions follow different forms. The degree distribution of a scale-free network is of a fat-tailed form [44,45]. A good approximation of the degree distribution of a random network is a binomial distribution, which can be replaced by Poission distribution for large number of vertices [4]. In this paper, each vertex in the network was endowed with an energy weight, so the weight is a basic quantity of a vertex, and the weight distribution is an important quality of the weighted network.

Figure 2 shows the connectivity distribution of the unweighted CSN. Obviously, the connectivity distribution agrees well with the Poission distribution, thus the topology of the unweighted network is relatively homogeneous, with most of nodes having approximately the same number of the edges. The result is consistent with the previous work [15]. In Figures 3(a)–(c), it is found that the weight distributions of the CSNs show a well defined fat-tail property. The continuous lines correspond the straight-line fit ~ w ^−γ on the tail of the weight distribution, where w is the energy weight of the vertex, and γ is the scaling exponent. Furthermore, we also investigated the small-world property of the weighted CSNs.

Using the edge weight defined as (5), we firstly calculated the average shortest distance l_w of the weighted CSN and compared it with that of random network l_random ~ ln(n)/ln(〈k〉) [4], where 〈k〉 is average degree of the CSN, and n is the number of the nodes in the CSN. Then we randomly chose 10 weighted CSNs from the whole 309 weighted CSNs as the case study. The results shows that the l_w of these weighted CSNs are all around 60 (data not shown), which is similar with the previous work [15]. The results also indicate that the l_w of the weighted CSNs is larger than l_random and is of the same magnitude of the l_random. In our model, we observed l_random ≈ 10. Secondly, we estimated the clustering coefficient C_w of the weighted CSNs, obtaining that C w C random: 10², where C_random ≈ 〈k〉/n is the clustering coefficient of random network. This result indicates that C_w is much larger than C_random. In summary, the above results suggest that the weighted CSNs have scale-free and small-world properties [2].

The result indicates that the energy weight is a key factor for the appearance of the power-law property of the CSN. To understand the meaning of the energy weight, we should consider the connectivity of the conformations and more details about the lattice model. It is well known that the lattice model is a simplified model of a polymer. In the lattice model, polymers including proteins can be treated as linear strings of beads, and the details of the structures are not accurately represented. This simplified approximation is based on the following hypothesis: when the real chain conformations are under good solvent conditions or in “theta”[46] (where conformations are highly expanded), the structural details such as side chain are subsumed under the property of chain stiffness [47], in other words, some of the degrees of freedom of the structures are “frozen”, but an actual polymer chain is flexible in the three-dimension Euclidean space. For a given stable conformation, due to the influence of the structure details such as the side chain, there are small conformational fluctuations around the stable conformation [47–51]. As to the lattice model, it means that for a given lattice conformation, there are many tiny difference conformations fluctuating around this conformation. In other words, there is an ensemble of subconformations belonging to the given lattice conformation. Further, the Boltzmann factor of a conformation incarnates the relative quantity of the subconformations of the ensemble [16,52]. On the other hand, the properties of some dynamical processes, such as protein folding, are determined not by the spectrum but also by the connectivity of the conformations [53]. If the energy barrier between the two conformations is too high, then we can say that the two conformations are not “dynamically connected” [8]. If a conformation is not dynamically connected to other low energy structures, then it would be kinetically inaccessible in spite of its low energy [8,54]. Therefore, assuming that the vertex i connects with vertices j, k, h, the weight of vertex i, as mentioned above, may reflects the number of all possible potential dynamically connected subconformations connecting to the conformation that corresponds to the vertex i. In other words, the weight of a node in the CSN represents the number of possible potential links connecting the node, implying the importance of the node in the CSN.

When the interaction energy of the monomers in the chain is not taken into account, all conformations are of the same free energy, i.e. unit free energy. It means that each conformation has the same weight in the conformation space. Therefore, the topology of the CSN is simply determined by the connectivity of the conformations. In this case, the degree distribution of the network is binomial distribution and significantly deviates from the power-law form, as shown in Figure 2. In terms of the free energy landscape, the free energy surface is flat [5] and the power-low property disappears.

The power-law distribution indicates that there are a few “hubs” which have large connectivity and a mass of vertices with a relatively small number of links in the network [3]. In the weighted network of conformation space, the power-law property indicates that a few conformations play the role as “hubs” in the dynamic process of complex systems, such as protein folding. The large weight of a conformation shows that this conformation has relatively more connections and the energies of its connective conformations are low. According to the definition of the energy weight, the large weight of a conformation implies that there are abundant possible routes to access it.

Figure 4 shows the relation between γ and Δ/Γ, in which there is high correlation between γ and Δ/Γ. It has been reported that Z score correlates significantly with the folding rates [55,56] and the sequences with a large ratio Δ/Γ fold fast [19,57,58]. Therefore, the large value of γ implies the fast folding.

This result can be understood by the free-energy landscape view. According to the mathematical knowledge of the power-law function [59], the larger scaling exponent indicates that the value of the function decays to zero more quickly for the relatively large independent variable. For example, for the power-law function f (x) = x ^−β (β > 0), the larger value of β indicates that the value of f (x) inclines to zero more quickly when x → +∞. This property can be applied to analyze to the weight distribution of the weighted CSN. As expressed above, the weight distribution of the weighted CSN is a function, which gives the possibility that a randomly selected node has a definite weight. This function is of power-law property and γ is its scaling exponent. So, the larger scaling exponent γ indicates the less possibility finding the nodes with large weight. In other words, for the larger γ, there is relatively less number of nodes which have large weight. On the other hand, according to the physical meaning of the weight of the nodes, the large weight of a node means that this node has dense connections around it. Therefore, a lesser number of nodes with large weight in the network signify there are fewer crowded groups in the network. In terms of free-energy landscape, this scenario implies that the free-energy surface has a small number of deep valleys and high barriers between them [5,13]. So, the larger γ indicates that the free-energy landscape is smoother [8].

It has been shown [19,24] that the ratio Δ/Γ can be used to measure how much a sequence is “protein-like”, in other words, it is a good criteria for the thermodynamic stability or the stability of against mutation of the sequences. Simultaneously, the similar perspective has been reflected recently in the research of biomolecular binding [60] and cellular networks [61,62] in which the ratio of the energy gap versus roughness of the underlying energy landscape, which in essential is equivalent to the ratio Δ/Γ, has been recognized as an optimization criteria for the specificity of binding in binding energy landscape or the global thermodynamic stability (or robustness) of the network. The relationship between Δ/Γ and γ shown in Figure (4) suggests that the scaling exponent of the power-law of the networks should be considered carefully as a valuable topological parameter for the thermodynamic stability (or the robustness) of the CSN, which potentially may be good news for the design of artificial networks.

This result is also helpful for protein design, whose goal is to identify amino acid sequences that can fold well and lead to a given structure [63,64]. It has been shown [65,66] that for the Z score, which was always defined to have negative value in those papers, minimization is equivalent to maximizing the energy gap between mis-folded or unfolded conformations and the native state of the proteins, and such maximization results in stable and fast-folding proteins. Optimization of the Z score thus provides a quantitative method for the problem of protein design and has been widely applied [52,67,68]. Therefore, considering the result shown in Figure (4), the scaling exponent γ may provide an alternative optimization criterion for designing protein-like sequences.

Figures 5 and 6 show the relationship between the scaling exponent γ s and the modularity Qs and the relationship between the ratio Δ/Γ and the modularity Qs, respectively. The values of modularity Qs are obtained by the MSG-VM algorithm [31,32], and the parameters which are needed in the algorithm can be calculated through the Equations (4) and (5). It can be found from both Figure 5 and Figure 6 that Q is more than 0.4 for most weighted CSNs. In addition, as shown in Figure 5, the γ s and the Qs correlate well with a linear inverse relationship on average. A similar relationship between the ratio Δ/Γ and the modularity Qs can be found in Figure 6.

The value of Q will be zero for the randomized network [30]. Nonzero values represent deviations from randomness, and in practice it is found that a value of Q above 0.3 is a good criterion for a distinct modularity structure [30]. Therefore, the result that most values of Qs are more than 0.4 implies that the majority of the weighted CSNs have significant community structures. This indicates the overall topology of the CSN is heterogeneous [69] with subsets of vertices within which vertex-vertex connections are dense, but there are only scant connections between the subsets.

The inverse proportion relationships shown in Figures 5 and 6 may have some interesting implications concerning the modularity mechanism [70] and the topology of the CSNs. In some biological networks, the modularity structure seems to increase the stability, robustness, and flexibility [33,71] of the networks, and recent theory suggests that modularity can be enhanced when the environmental changes over time [72]. From Figure 6, we can find that the larger value of Q on average corresponds to the relative smaller value of Δ/Γ. In other words, the sequences that fold relatively fast appear to avoid forming significant community structures in the conformation space generated by the folding process. When addressed in terms of the free energy landscape, the scenario may be more easily understood. In the view of free energy landscape, the behavior of protein folding can be described by a free energy landscape that looks like a funnel [8,9], where there are many different depth minima and various height barriers among them on the coarse surface [8]. In this language, the surface of the energy landscape can be divided into many basin areas, with the deeper minima having large basins of attraction [5,73,74]. Inside a basin and among the basins between which the height of the barriers is small, there are many dynamical accessible routes among the conformations [8]. On contrast, if the barriers between the minima are large, there are a small number of pathways among them. This scene reveals the original source of the modularity structure in the weighted CSNs. Moreover, under the evolution pressure, the sequences of protein are selected to have relatively smooth energy landscape on which there are a small quantity of deep minima and high barriers in order to fold quickly [8,57], in other words, the sequences which fold relatively rapidly avoid to form conspicuous modularity structures in the weighted CSNs.

To obtain the values of the modularity Q in the article, we should calculate the weight of the edges in the network, which describes in some sense how the nodes closely connected. We defined the weight of edges according to expression (5). A possible explanation can help us to understand the meaning of the weight of the edges. As pointed out above, the Boltzmann factor of a lattice conformation depicts the relative quantity of the group of the subconformations around the conformation. When two conformations connect in the CSN, it means there are two potential groups connected to each other and the weight of an edge linking two nodes represents the quantity of the possible accessible routes between the corresponding two groups of the subconformations. The larger value of the product of the Boltzmann factor of the two nodes connected by an edge means the more potential accessible routes between them, thus may carefully consider that these two nodes more closely connected.

Finally, several tasks remain to be done in the future. First of all, one may consider the conformation space generated by a three-dimensional (3D) lattice chain and a more reasonable weighting method. In addition, more details about relationship between the topology of the CSN and protein dynamics could be studied to obtain more profound insights into the protein dynamics and the topology of the CSN. Besides, the farther modular analysis of CSN will help us to comprehend expressly the rugged energy landscape. For example, through modular analysis, one may find out more information about the micro-structure on the surface of the energy landscape, such as “local minima” [69]. Finally, one may also consider the rate of transition [75] between the conformations as the weight of edge and the directed network [1] in the modularity analysis in the future.