Nanoscale Systems MMTA • ISSN: 2299-3290
Review Article • DOI: 10.2478/nsmmt-2013-0003 • NanoMMTA • Vol. 2 • 2013 • 30-48
Nanonetworks: The graph theory framework for
modeling nanoscale systems
Abstract
Nanonetwork is defined as a mathematical model of nanosize objects with biological, physical and chemical attributes,
which are interconnected within certain dynamical process.
To demonstrate the potentials of this modeling approach for
quantitative study of complexity at nanoscale, in this survey,
we consider three kinds of nanonetworks: Genes of a yeast
are connected by weighted links corresponding to their coexpression along the cell cycle; Gold nanoparticles, arranged on
a substrate, are linked via quantum tunneling junctions which
enable single-electron conduction; A network of similar profiles of force–distance curves consists of sequences of states
of a molecular complex from HIV–1 virus observed in repeated single-molecule force spectroscopy experiments. The
graph-theory analysis of these systems reveals their organizational principles, quantifies the relation between the function of nanostructured materials and their architecture, and
helps understand the character of fluctuations at nanoscale.
Jelena Živković1,2∗ , Bosiljka Tadić1†
1 Department of theoretical physics; Jožef Stefan Institute,
Box 3000, SI-1001 Ljubljana Slovenia
2 Scanning Probe Microscopy Group, Institute for Molecules
and Materials, Radboud University,
Nijmegen, The Netherlands, EU
Received 12 November 2012
Revised 4 January 2013
Accepted in revised form 24 January 2013
Keywords
Conducting nanoparticle films • Genetic networks of yeast •
Single-molecule force spectroscipy data • Graph theory • Network community detection• Bionanosystems • Viral RNA • Cell
• Complex systems
PACS: 62.23.St, 82.37.Rs, 87.18.Cf, 89.75.-k, 89.75.Hc
MSC: 5Cxx, 90B10, 90C35, 92E10, 05C62
© Versita sp. z o.o.
1.
Introduction
The multiscale structure of materials and devices composed of many nanosize objects are actual examples of complex
systems, in that they possess emergent properties at a higher scale [1–8]. Controlled architecture of nanomaterials
that can perform a distinct function makes the right basis for their applications in nanotechnology. On the other hand,
specifically these features of the nanostructured systems, together with the enhanced fluctuations at nanoscale [9], impose
tremendous challenges for theoretical and numerical modeling and require the development of new multidisciplinary
approaches [10].
In recent years, progress in the modeling of complex systems have been made by suitable mapping onto mathematical
graphs (networks) and use of the graph theory methods. The basic idea consists of identifying the elements of the
graphs—the nodes and the edges (links) connecting the nodes, out of the available features of the system considered.
It usually occurs that, in the network representing a complex system, both nodes and edges have other attributes that
need to be taken into account. This leads to discovery of new types of graphs and increases demands for the adequate
mathematical concepts. In this way, a theory of complex networks develops as a new branch of science [11, 12]. On the
∗
†
E-mail: [email protected]
E-mail: [email protected] (Corresponding author)
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Nanonetworks: The graph theory for nanosystems
other hand, the quantitative analysis of the underlying complex systems was enabled by use of graph theory methods
[13, 14]. In particular, these approaches led to a noticeable advancement in bioinformatics [15, 16] and econophysics
[17], and contributed to developing the quantitative analysis of social systems [18–20].
Recently several problems related to the nanosize systems have been tackled by using the complex networks theory.
In particular, these are related to
• Coping with enhanced fluctuations at nanoscale, e.g., in the single-molecule force spectroscopy [21]: The method
was shown to be sensitive to the nature of fluctuations (beyond the problem of the signal-to-noise ratio) in
the kinetics of the molecular bond and capable of separating the force–distance curves from different molecular
systems.
• Discovering hidden structures in cooperative functioning of genes in a cell: Despite recent improvements in
experimental methods, which increased dramatically the amount of information regarding individual genes of a
genome, their functional principle at a larger scale is still to be understood. In recent years, the network approach
was used to elicit such functional structures from high-throughput expression data [22–28].
• Quantifying complex ordering of nanoobjects, e.g., in self-assembly processes of nanoparticles, nanorods, and
nano-bio heterostructures: Consequently, the role that the structure plays in the physical and chemical properties
of the assembly has been studied quantitatively [29–36].
In the latter case, the geometrical ordering of nanoobjects into a complex structure can be directly mapped onto a
network. Whereas, in the former two cases, the use of networks is more formal, as it will be described below. The
term nanoparticle network has already become familiar in the literature [29], usually indicating an irregular structure
of interacting nanoparticles [32, 37–39] or nanoparticles linked with molecular systems [1, 31, 40–42]. On the other
hand, the genetic and biochemical networks [43] and the networks related to the force-distance curve fluctuations [21]
are mathematical constructs that help to pinpoint relevant relationships among the elements of the system, which are for
this purpose recognized as nodes of a network. Other types of nanonetworks involving bionanosystems in real space are
arrays of molecular machines, which are designed to expand capabilities of a single machine to perform communication,
computation or sensing tasks at molecular level [44, 45]. Moreover, the biological concepts and principles of molecular
systems are being used to design robust technology networks [46].
The aim of this survey is to build the unifying concept of nanonetworks, performed at three distinct levels in the
analysis of nanosystems pointed out above and to elucidate potential benefits of the graph theory approach. In the
investigated systems, our attention will be on the dynamics of nanosystems which is fundamentally associated with
their descriptions by complex networks. The elements of network theory needed for this exposition are introduced and
illustrated by utilizing ’archetypal’ examples from gene coexpression networks. Furthermore, in section 3 we present
nanoparticle networks, which are used for the characterization of complex nanomaterials. With few examples from
nanoparticle films we explain two viewpoints of the nanonetwork approach, i.e., modeling the arrangement of nanoparticles
in a film, and modeling the dynamics of single-electron tunneling conduction through such structures. Subsequently,
in section 4 we discuss in greater detail how the formal theory of nanonetworks can be implemented in the evaluation
of data from the single-molecule force spectroscopy experiments. For this goal, distinct sets of data are used from the
dynamic force spectroscopy experiments of peptide–RNA interaction in HIV–1 virus, obtained in different experimental
setups and conditions. Besides, by carrying out further analysis of the data within the nanonetwork approach, we reach
some new conclusions which confirm and improve the earlier results.
Recalling that a number of competent reviews exists in the field of complex networks and genetic networks as well
as maintaining the bibliography manageable, we do not provide with exhaustive referencing, but present some textbooks
and recent reviews where additional references can be found. Likewise, a vast literature is available regarding the
applicability of nanoparticle networks in technological designs. In agreement with our objectives, in this survey we
quote the references in which the underlying association of the function of nanoparticle assemblies with their network
representations is pointed out.
2.
Network features primer
A prominent feature of all complex systems is the emergence of functional properties in an ensemble. Consequently,
the networks representing such systems and their dynamics frequently manifest complex architecture with hidden topological structures. In contrast to familiar random graphs [13], the mathematical theory of complex networks is still
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developing. Distinct classes of complex networks have been recognized in applications from natural systems and processes to social dynamics [11, 12, 19, 47] and numerical algorithms devised for their analysis. Here, we summarize those
network characteristics that appear pertinent to analysis of nanosize systems and their assemblies.
2.1.
Useful topology measures of nanonetworks
Different topology measures can be defined and determined in complex networks at local scale (the properties of
nodes and edges), mesoscopic (smaller or larger groups of nodes, nodes neighborhoods, functional motifs and modules),
and global network level (paths, cycles, spanning trees). The standard topology analysis is further adapted, having in
mind that the elements of the nanonetworks have additional physical, chemical or biological attributes, e.g., charge or
dipole moment of the nanoparticles, and information about processes in which they take part. Specifically, consider
a network of N nodes and L links, which is given by the adjacency matrix Cij . In the case of binary links, Cij = 1
if the edge ij exists and Cij = 0 if the nodes are not connected. In the weighted networks, the matrix element Wij
represents a multiplicity or otherwise determined weight of a link. Different types of links among nodes are indicated
in Fig. 1a,b. Some typical nanoobjects and their relations that can be considered as nodes and links of a nanonetwork
are shown schematically. The following groups of topology measures can be defined and used to characterize various
nanonetworks:
• Centrality measures of a node i determine the position of that node in its neighborhood and in the network as
a whole [13, 48]. Specifically, the node degree is determined by the number of edges which connect the node
P
P
with its neighbors. In a directed network the number of incoming qin
qout
= j Cij links
i =
i
j Cji and outgoing
P
is defined. In the case of weighted networks, node strength can be defined as `i = j (Wij + Wji ) to take into
account the widths of all node’s connections. Two global topology measures determine positioning of a node in
the entire network. In particular: closeness centrality is determined as the inverse of the sum of all shortest paths
from that node to all other nodes in the network; betweenness centrality is the fraction σikj of shortest paths σij
P
between all pairs (i, j) of nodes in the network that pass through that node, Bk = i,j6=k σikj /σij . An analogous
centrality is often defined for a link in the network.
i
• Eigenvector centrality is defined as i-th component of the eigenvector Vmax
belonging to the largest eigenvalue
λmax of the adjacency matrix of the network. It gives another measure of “importance” of node i, which is relevant
for diffusion and synchronization processes on the network. The eigenvalue spectral analysis of the network, i.e.,
computing the spectrum of eigenvalues and associated eigenvectors of the adjacency matrix (or another operator),
gives additional information of network topology related to the dynamics on it. Specifically, the normalized
p
Laplacian operator for the symmetrical network with weighted links Wij , is defined [48] as L = δij − Wij / `i `j .
It has the eigenvalue spectrum limited [49] to the range λi ∈ [0, 2], which is suitable for comparing networks of
different sizes. Estrada index, which is often used for the characterization of the topological structure of molecules
P
P
[50–52], is also defined on the basis of graph spectral theory, i.e., EE(G) = i Ei /N where Ei = k |Vki |2 eλk and
k = 1, 2, · · · N enumerates the eigenvalues of the adjacency matrix of the network.
• Modularity Q > 0 indicates network’s inhomogeneity at mesoscopic scale, i.e., that a partition of the network
into subgraphs (modules) of different sizes is applicable. The nodes inside a module are assumed to be connected
by higher density (or strength) of links compared with the links between different modules [48]. Apart from their
topological features and size, the network modules, e.g., in the genetic networks mentioned above, often contain
nodes which share a specific function. Therefore, finding suitable modular structure of a network is of crucial
importance for understanding its function. It should be noted that devision of a network into separate modules is
an optimization problem. Thus the quality Q of the partition can be defined. For instance,
Q=
m
qi qj 1 XX
Cij −
;
2L r=1 i,j∈r
2L
(1)
for a devision of a network of L links into m modules. Here Cij represents the network adjacency matrix, qi and
qj are degrees of the nodes i and j, and the sum i, j ∈ r runs over all pairs of nodes within the rth module. A
large number of methods and algorithms have been developed for detecting the network’s modular structure [53].
Their efficiency depends on the overall network structure and size. In this work we make use of the eigenvalue
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spectral analysis method described in [49]. The method is based on the property of the eigenvectors belonging
to the sub-largest eigenvalues of the adjacency matrix (or smallest nonzero eigenvalues of the related normalized
Laplacian operator) to localize on the subgraphs. This method is efficient when the graphs are not too large and
have strong clustering and weighted links, as it is the case with the correlation networks. In addition, we use the
maximum-modularity algorithm [54], which is more suitable to analyze large graphs.
• Node neighborhood structures are topology measures that can be defined at the level of nearest or extended
neighborhood of a node. Such measures are interesting for network sequencing and graph alignment [16, 59]
methods, by which nodes with similar environment and possibly similar function are found on the same network,
or groups of nodes (functional modules) detected across different networks.
• Network flows are properties of links, e.g., they measure the amount of traffic along a link [14]. In the nanonetworks
studied in this review, the dynamical flow can be defined in relation to the process on the network. Other attributes
of the network links can be used, emanating from a construction principle, for instance, the weight, the degree of
correlation among the adjacent nodes, and the betweenness centrality (topological flow). The most central links
are determined, and paths or cycles of most central links can be considered. The maximum-flow spanning tree
is a tree representation of the entire network, on which each node is connected to the tree by its maximum-flow
link. Hence, it represents a union of the most central paths on the network. An example is shown in Fig. 1c.
A large number of network features can be computed within graph visualization packages as, for instance in Pajek
[55], Cytoscape [56] and Gephi [57]. For more specific analysis, the appropriate program codes need to be developed.
It should be stressed that the network structure often reflects certain organizational principle of the system. Thus, the
networks of complex topology occur in relation to functional nanosystems, see for instance the genetic networks in Refs.
[24, 58].
Fig 1.
(Color online) A: Elements of a nanonetwork are nodes and links (edges, arcs) between them. They may have different origin and physical,
chemical or biological attributes. B: Three types of colloidal interactions and occurrence of tunneling junctions among two types of nanoobjects are indicated schematically. C: Example of the maximum flow spanning tree in the network of genes co-expressed along the cell-cycle
of yeast. Red, yellow, green and blue colors of nodes indicate maximum expression of a gene in the cell cycle phases G1, S, G2, and M,
respectively.
2.2.
Mapping digital signal correlations to networks: gene coexpression network of yeast
This useful mathematical procedure is demonstrated here by using the gene co-expression network of yeast Saccharomyces cerevisiae [28]. In general, study of genetic networks (for recent review see [23, 24, 60]) is a very active area
of research at the interface of bioinformatics and natural sciences that builds on advances of detailed biology analysis
of genes and high-throughput measurements. Note that different types of networks that are related to the functioning
of genes in a genome have been studied in the literature. In particular, for the yeast Saccharomyces cerevisiae one
can study: gene coexpression networks, which are based on system-scale expression data; gene regulatory network
[61, 62], which is constructed from biological information about the transcription factors, i.e., proteins synthesized by
certain genes or gene groups to further regulate expressions of other genes; and protein–protein interaction networks,
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Fig 2.
Time series of the expressions of the selected genes, measured in 17 instances along the cell-cycle (left). The correlation matrix of the
normalized gene expressions (right). The indexes are ordered by gene’s maximum expression in ascending time (or phase of the cell cycle).
PPI, [58, 63], representing bio-chemical interactions among different proteins, which themselves are produced from their
genetic blueprints. As mentioned above, we are interested in dynamical features of gene cooperation, hence we use the
expression data measured along the cell cycle of yeast [64].
The gene expression is a cellular process by which the information stored in DNA is extracted in order to synthesize
proteins that carry out specified functions in the cell. It is a part of the whole set of processes commonly known as gene
regulation, which control flow of information from DNA to proteins and in the opposite direction, carrying the demands
what proteins in given conditions need to be produced by the related genes [65]. Which genes are coexpressed in a
given moment of the lifetime of a cell? This partial information gives some insight into these complex processes of gene
cooperation [66]. Here, we show how the graph theory methods can be employed to identify groups of coexpressed
genes along the cell-cycle of yeast Saccharomyces cerevisiae from the gene expression data measured at several time
intervals [28]. The goal here is to demonstrate the general mathematical procedure, which we will also use in section
4 in a different context. The conclusion reached in Ref. [28] is that different groups of coexpressed genes along the cell
cycle of yeast appear to share same physical location in the cell. These findings are here discussed in view of research
of the protein interaction network of yeast [58, 63, 66].
The genome-wide expression data of yeast Saccharomyces cerevisiae, have been measured at 10 minutes intervals
along two consecutive cell cycles [64]. The data can be represented as a set of time series fi (k), for k = 1, 2, · · · 17
indicating separate measurement, i.e., time point, and the index i = 1, 2 · · · N indicates a given gene in the genome of
yeast. The expressions of the genes selected for our analysis are shown in Fig. 2(left). Note that first 9 points complete
the first cell cycle [64]. Following Ref. [28], from the whole genome we select those genes that are expressed over an
extended period (two or more cell cycle phases). The selection criterion is that their expression fluctuates (constantly
expressed housekeeping genes are not considered) but stays over the average expression on the genome. The selected
set contains 604 genes. For further analysis the expressions in each measurement (instance of time in the cell cycle)
are first normalized with the average expression in that measurement point, then the correlation matrix is computed.
Different procedures exist for reconstructing the dynamical system from the time series [26, 67]. Here, we follow
the procedure used in [26, 28]. First step is to compute the gene–gene correlations as Pearson’s correlations of the time
series. The resulting matrix elements Cij are given by
P
Cij =
k [fi (k)
− hfi i][fj (k) − hfj i]
.
σi σj
(2)
In general, k is given by a discrete set of measured values, and σi , σj stand for the standard deviation of the time series
(digital signal) fi (k) and fj (k). In the present case, k = 1, 2, · · · 17 is the index of time intervals along approximately two
cell cycles of yeast, and fi (k) is the expression level of gene i at time instance k. Note that the Pearson’s correlations,
normalized by the standard deviations of the correlated signals, can have values in the range Cij ∈ [−1, +1] with the
negative values indicating anti-correlations, while correlations correspond to the positive values. For the purpose of this
work we are interested in strong positive correlations. In some situations, strong negative correlations are also interesting
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Nanonetworks: The graph theory for nanosystems
[26]. The matrix is mapped to the range [0, 1] so that perfect anti-correlation is mapped to zero, and then normalization
of the correlation matrix is performed in order to enhance positively correlated signals against the background. Similar
procedure was described in Ref. [26]. For the normalization, we used the affinity transformation method [68, 69], where
the element is enhanced as Cij → Fij Cij if the rows i and j correlate with the rest of the matrix in a similar way, which
gives the meta-correlation Fij ∼ 1, and diminished otherwise. The factor Fij is also computed as a Pearson’s coefficient
of the matrix elements in the row i and row j, which are rearranged as follows: {Cij , Ci1 , ..., Ci,j−1 , Ci,j+1 , . . . , CiN } and
{Cji , Cj1 , ..., Cj,i−1 , Cj,i+1 , . . . , CjN } and the diagonal elements are removed. In Fig. 2(right) the correlation matrix of the
selected 604 genes is shown, considering the normalized matrix elements which exceed the threshold W0 = 0.6. The
genes in the matrix are sorted according to the cell cycle phases (G1,S,G2,M) in which their maximum expression is
found.
In the next step, we find groups of genes that are strongly correlated to each other. Note that in the matrix representation, such groups can be visualized as blocks along the diagonal. In the equivalent graph (network) representation,
as shown in Fig. 3, they can be visually tracked as dense blobs in the network. As mentioned above, different methods
can be used to detect such densely connected areas of the graph [53]. Note that the advanced graph visualization
packages mentioned above also have built in some algorithms for detecting graph modularity. Applying the eigenvalue
spectral analysis method to the correlation matrix of Fig. 2, four groups of genes co-expressed over the threshold 0.6 are
identified in Ref. [28]. By identifying the nodes (genes) in each subgraph and further inspection of their biological data
in MIPS database [70], it was found that the majority of the genes within the identified groups are, in fact, co-localized
in one or more cellular components, i.e., cytoplasm, mitochondria, nucleus, etc.
Fig 3.
(Color online) Modular structure of the yeast co-expression network of genes correlated over the threshold 0.8. Separately shown is a
close-up view of one of the modules with the ORF labels of the genes.
Here, we perform an additional analysis of the co-expression gene network by first shifting the threshold to higher
value 0.8 and using the maximum-modularity method in standard resolution. Five modules are found as shown in Fig. 3.
While the partial overlap in the eigenvectors does not allow full separation of the branches [28], the maximum-modularity
algorithm always gives a complete partition of the graph, which is a subject of the applied resolution. As a result, we
obtain the complete lists of genes within these modules. They are identified by their open-reading-frame (ORF) labels,
which allows search for further information in the biological databases and the related literature. Some results of such
a search are summarized in Table 1, where we show cross-listing of the genes in these modules with the genes in three
largest protein complexes. Our findings are in line with recent research of the protein-protein interaction network of
yeast [58, 63], where the co-localization of the identified protein complexes in the cell components is found, and the
co-expression of the genes within large protein complexes [66]. The Table 1 shows how different modules found by our
analysis, containing the genes which are strongly co-expressed along entire cell-cycle, are participating in two largest
protein complexes—Cytoplasmic ribosome and Mitochondrial ribosome complex and in the New complex reported in
Ref. [63].
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Table 1.
Cross-listing of the genes in four of the modules from Fig. 3 with two largest protein complexes of yeastCytoplasmic ribosome complex
and Mitochondrial ribosome complex, and New complex from Ref. [63].
Module142 Module133 Module121 Module100
Cytopl.
YGR027C
YGL147C
ribosome YGR118W YLR367W
YER074W YLL045C
YBR048W YLR344W
complex
Mitoch.
YKL180W
YKL138C
YBL041W
YOL038W YNL081C
ribosome YMR188C YKR006C
YOR259C
complex YMR188C YBL038W
YML009C
YNR037C
YMR125W
New
YBL026W
complex YOR159C
YLL036C
3.
Modeling complex nanostructures by nanonetworks
In the literature the term nanoparticle network has been used sometimes to suggest irregular ordering of a nanoparticle assembly [1, 3, 5, 34, 71, 72]. Here, we argue that mapping the nanoparticle structure onto a mathematical graph
(network) is a giant step ahead in modeling and quantitative analysis of the nanoparticle assemblies and their functional
properties [29]. First, by employing the graph theory methods one can determine quantitative measures, which characterize the ordering of a nanoparticle assembly. Furthermore, mapping onto a graph is a necessary step in modeling
functional properties of the assembly, i.e., by implementing physical processes on such structures and quantifying the
role that specific structural elements paly in these processes [8, 73, 74]. We briefly demonstrate both of these aspects of
the nanoparticle networks in the case of metallic nanoparticle films, where single-electron tunneling (SET) conduction
occurs under Coulomb blockade conditions [35–38]. Interested readers should consult detailed analysis in the review
[29] and the references therein.
3.1.
Quantifying structure of nanoparticle assemblies
In contrast to the genetic networks discussed above, the nanoparticle networks visualize mutual relations of the
nanoparticles or colloids in a physical space. Well studied examples are gold nanoparticle films—arrays of gold
nanoparticles on a substrate, which are interesting for nanoelectronics applications due to their SET conduction properties [5, 34, 35, 75]. Another interesting system with similar SET conduction mechanisms is chemically derived graphene
oxide sheet, a disorder analogue of graphene, presumably containing arrays of the graphene quantum dots which are
separated by insulating regions [38, 76]. A straightforward way for characterizing such structures by mathematical graphs
is as follows: each nanoparticle (colloid, quantum dot) represents a node while a link indicates certain association among
them, i.e., either simple geometrical proximity on the substrate [35], or even a chemical link between colloids [33, 77, 78].
For example, to study SET processes on such structures, two nanoparticles are “linked” in the network only when their
mutual distance is such that allows for the electron tunneling between them [29, 35]. Hence, each link indicates the
tunneling junction in the nanoparticle arary. Having constructed the adjacency matrix of the graph, the structure of the
assembly can be characterized in detail by graph theory methods.
How does a given complex structure, i.e., similar to the one obtained by the self-assembly process in the laboratory,
emerge and how it can be tuned? Modeling of the assembly processes in which structures with certain topological features
arise, and thus identifying the parameters relevant for their appearance, is of primary importance in mathematical
modeling of the nanoparticle networks. For the illustration, in Fig. 4 are shown two network models of nanoparticle
films, which are grown by the cell aggregation algorithm of Ref. [79] with different parameters. In the algorithm, to match
some experimentally observed nanoparticle films [36, 75], the hexagonal or nearly hexagonal cells are attached to the
graph (initially one cell graph). In the growth process the average degree inside the graph is kept constant < qi >= 3.
Furthermore, the constraints of planarity is systematically observed, i.e., the 5-cliques, K5 , and the minimal non-planar
graphs with six nodes, K3,3 , are not allowed. The parameter ν, the inverse “chemical potential” for attachment of a loop
along a nesting area, controls the branching of the graph. For ν = 0 branched structure appears, as the one in Fig. 4a
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Fig 4.
(Color online) Two examples of planar graphs grown by cell-aggregation algorithm of Ref. [79]: (a) Fixed size of 6-nodes cell (µ2 = 0) and
large chemical potential for cell attachment (ν = 0.0) resulting in the branched structure. The widths of the links indicate topological flow;
(b) Small irregularity of cell sizes (µ2 = 0.5) and reduced chemical potential for cell attachment (ν = 1) yielding the plane-filling structure.
The widths of links indicate current flow when the extended electrode with positive voltage is in the top-left corner, while the negative voltage
electrode in the bottom-right corner area. Color codes from pale green to red indicate increased centrality of nodes on both graphs. Black:
unvisited nodes in the SET process.
while for a finite ν, the graph has a tendency to fill the plane, Fig. 4b. The probability that a cell departs from pure
hexagon is controlled by the parameter µ2 , that is the second moment of a distribution of cell size.
3.2.
Quantifying structure–function interdependency
The property of a nanoparticle network which is most relevant for the SET conduction is the distribution of
betweenness centrality of the links. Note that due to the constant number of links per node in the above examples, the
betweenness centrality of links and nodes on this type of networks is similar. It is depicted in Figs. 4 by the widths of the
links and by the color of the nodes (red color indicated most central nodes). As the figure shows, a variety of topological
paths is found in the case of branched structure, although the elementary cells are all equal. Several “highways”, the
routs with large topological centrality, are visible. However, the paths inside the small homogeneous area in the central
part of the same network are almost equal.
For a given network, part of the topological flow that plays a role in the conduction process is, of course, dependent
on the actual position of the electrodes. A situation shown in Fig. 4b corresponds to the electrodes positioned along
a part of the network nodes in the top-left corner (high voltage electrode) and the bottom-right corner (zero voltage
electrode). Consequently, the shown betweenness centrality of the nodes and the links in that network is the dynamical
centrality [29, 80]. It represents the number of single electron tunnelings between the nodes, simulated within the
capacitively-coupled system model [35, 36, 80–82].
Fig 5.
(Color online) Visualization of the conducting paths in single-electron tunnelings under applied voltage through a nanoparticle network with
large topological defects. The simulated data are from Ref. [35].
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The network mapping of an actual structure of nanoparticle assembly, apart from characterizing its structure, is a
necessity in the modeling of physical processes on such structure. The adjacency matrix Cij of the nanoparticle network
directly appears in the interaction energy of the assembly, e.g., in modeling the spin reversal processes on complex
geometry [8, 73]. Similarly, in the metallic nanoparticle films modeled as capacitively-coupled systems with long-range
interactions, the capacitance matrix Mij = CCij , where C stands for the capacity of a nanoparticle, appears in the
electrostatic energy of the system [29]. In this way, the geometry of the assembly is directly coupled to the physical
process that it supports. Moreover, the graph visualization techniques offer a possibility of inspecting the details of the
process on that graph. For instance, the SET on nanoparticle films is a collective dynamical percolation process, which
occurs at threshold voltage VT . It is characterized by a nonlinear current–voltage characteristic I(V ) ∼ (V /VT − 1)ζ .
The measured values of the exponent ζ strongly correlate with the structure of the film [35]. The visualization of the
conduction paths [35], an example is shown in Fig. 5, gives an insight into the course of the tunneling processes through
the film. Therefore, it helps understand the fluctuations in the current [77], which can be measured at the position of the
low-voltage electrode. Both experiments and simulations [29, 35] suggest that large nonlinearity exponents in the range
ζ ∈ [3.0, 4.1] are compatible with topologically inhomogeneous structures, i.e., the structures with broad distributions
of betweenness centrality of the links. In these cases, the river-like structure of the conduction paths enhances the
cooperative flow of electrons. This yields potentially larger bursts of current [29, 77] compared to regular or weakly
irregular arrays, which are characterized with paths of simple geometry and smaller exponents ζ ∈ [1.0, 2.8]. Recently,
similar conclusions were reached in the experiments with the disordered arrays of graphene quantum dots [38].
4.
Evaluation of the single-molecule force spectroscopy data by network modeling
In the above studied nanoparticle networks as well as in the genetic networks, we deal with many nanosize systems,
which are interacting in either physical or virtual space. In contrast, the application of the network modeling to evaluation
of the dynamic force spectroscopy data considers different states of the same molecule (or molecular complex) in the
phase space.
A typical setup for single-molecule force spectroscopy with manipulation techniques based on atomic force microscopy (AFM) is schematically shown in Fig. 6a. The molecules of interest are brought into proximity of each other
which allows them to form a chemical bond. Subsequently, they are pulled apart by pulling apparatus, i.e., AFM tip,
which results in bond rupture. During each pulling event, molecules change their position and the internal states. In
the experiment, each pulling event is stored as a digital signal Force–Distance (FD) curve. Some examples of FD
curves are shown in Fig. 6b. Therefore, the profile of a FD curve contains information about the whole history of states
of a molecular system during one pulling event. Typically, a large number of repeated pulling events is performed to
improve statistics. The shape of each FD curve may depend on the number of factors, including the local environmental
fluctuations (these experiments are performed in liquid), pulling velocity, motion of the linkers by which the molecules
are immobilized to the pulling apparatus, occurrence of unspecific binding, multiple binding, and more. Recently we
have devised a method based on graph theory approach that attempts to separate the FD curves depending on the
very nature of these processes [21]. Following Ref. [21], below we describe the method in detail and present some new
developments.
The guiding idea of this approach is that such information about the binding processes is embedded in the similarity
between the FD profiles. The similarity between FD curves can be quantified by Pearson’s correlations, formally in the
same way as discussed in section 2.2 for the gene expressions. Thus, the first step is to map the dataset of empirical
FD curves onto a correlation matrix.
4.1.
Description of the experimental data and network mapping
To demonstrate how the method works, we use experimental data of RRE–Rev complex from HIV–1 virus [83, 84],
which plays an important role in the late phase of viral replication in the host cell. Rev peptide binds to it’s viral RNA
target Rev Responsive Element (RRE), a 351 nt long sequence which is located in the Env gene. We first create the
correlation matrix which contains all pairs of FD curves considered. Specifically, we use N = 1188 FD curves gathered
in different pulling experiments containing the data from RNA–peptide interaction (988 curves, RRE–Rev complex from
HIV–1) and ssDNA–ssDNA (200 curves), described in Ref. [21]. The curves were pre-processed in the following way:
We removed the contact part which mainly carries information on the cantilever. Further we applied an offset to shift
the interaction free part of the curves to the baseline, and only the part where interactions are expected was kept, i.e.,
approximately first 200 nm.
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The curves for the correlation matrix were selected from several experiments with different experimental setups. For
this purpose, the following experimental setups are realized: two setups with a different number of flexible linkers, which
couple molecules to the surface and the cantilever, are used in combination with different pulling velocities v and with
modified or absent target molecule. Two-linker setup (2t): both molecules were coupled to the cantilever/gold surface
via PEG (Polyethylene Glycol) spacers, see Fig. 6a; One-linker setup (1t): one molecule was coupled directly to the
gold surface and the other to the cantilever via the PEG spacer. The PEG linkers of length ∼ 40 nm were used. In
addition, we consider 200 FD curves from the one-linker setup but without target molecule, where the Rev peptide may
bind to the gold surface, and 188 FD curves from the two-linker setup in the presence of Neomycin, which can block
the binding sites on the target molecule. The groups of FD curves from different measurements appear as Blocks I–VI
in the correlation matrix shown in Fig. 7, also described in the top part of the Table 2.
Fig 6.
(a) Schematic representation of a typical experimental setup: RRE is immobilized to gold surface via PEG linker. The Rev peptide is
immobilized to the Si N cantilever using the same type of linkers. (b) Examples of Force–Distance curves obtained in one experiemnt [83].
In the correlation matrix, the first block (I) consists of 200 curves from measurements of the RNA–peptide interaction
in the setup with two linkers and low pulling velocity 581 nm/s. The second block (II) contains 188 binding curves of
RNA–peptide in the presence of Neomycin, in the same setup with two linkers and velocity 581nm/s. In the third block
(III) there are 200 curves from RNA–peptide interaction but with one-linker and high pulling velocity 2540 nm/s. The
fourth block (IV) contains 200 curves measured on ssDNA–ssDNA interaction (with 30 complementary bp), in the setup
with two linkers and very low pulling velocity 218nm/s. The fifth block (V) has 200 curves of RNA–peptide interaction,
with one linker and medium velocity 1160 nm/s, and finally, the block (VI) consists of 200 curves of peptide–Au (gold
surface) interaction, with one linker and velocity 1160 nm/s.
The overlay of all FD curves from the first block is shown in Fig. 7(left). A variety of the break points, visible as
deeps at different distances and forces, suggests large inhomogeneity of the profiles in this block. Indeed, by inspecting
the correlation matrix, the curves from block-I are not strongly correlated among each other, but they correlate with
different intensity with the blocks II, IV, V and VI. Notably, the blocks III and VI have more coherent structure, leading
to stronger correlations inside the block while also some cross-block correlations among them and the block V are
significant.
4.2.
Data selection and information from the selected groups of DF curves
In Ref. [21] the groups of curves that exhibit a strong similarity across different blocks have been found. In particular,
considering the correlation matrix Wij , i, j = 1, 2, · · · N of FD curves as a weighted undirected graph, and using the
eigenvalue spectral analysis of the Laplacian operator related to the matrix Wij , four communities of similar curves were
identified. Two large groups, termed G2 and G3 , contain FD curves from different blocks, as indicated in the Table 2, will
be further discussed in detail. It is remarkable, that the largest group G2 contains most of the curves from RNA–peptide
interactions and none of the ssDNA–ssDNA interaction curves, which are all contained in G3 . The histograms of rupture
forces and distances at break-points of all curves in these two groups are shown in Fig. 8.
The largest group G2 contains 579 curves all related to RNA–peptide interactions from different blocks and peptide–
Au interactions from block VI. For further separation of these FD curves in G2 two methods are applied: the eigenvalue
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Jelena Živković, Bosiljka Tadić
Fig 7.
Overlay of FD curves from block-I (left), and the correlation matrix of FD curves from all blocks I to VI, indicated along the top line, with the
threshold 0.5 (right).
Table 2.
Number of FD curves from different experimental blocks that can be identified by the eigenvalue spectral algorithm in Ref. [21] as
belonging to two major groups G2 and G3 . Modules G2− Mk , k = 0, 1, 2, 3 in the group G2 here identified by maximum-modularity
algorithm. Description: RR (Rev-RNA); R-R+Neo (Rev-RNA+Neomycin); Au-gold surface; 1t, 2t (one, two linkers), h.v. (high velocity);
m.v. (medium velocity); l.v. (low velocity); < F > average disruption force in a group. Original data from Ref. [21].
I
II
III
V
VI
Identity
1–200
201–388
389–588 589–788
789–988
989–1188
1t,high.v 2t,low.v
1t,medium.v 1t,medium.v [pN]
Description 2t,low.v 2t,low.v
Interaction R-R
G2
51
R-R+Neo R-R
28
177
G3
60
68
0
G2− M0
21
6
22
G2− M2
3
1
G2− M1
1
G2− M3
0
IV
hF i
Block
ssDNA-ssDNA R-R
0
135
181
Rev-Au
148
141±13
1
7
34±2
0
33
68
151±7
0
0
21
76
119±5
0
97
0
76
4
180±6
0
27
0
23
0
140±8
spectral method [21] and the modular partition of the graph at an increased threshold 0.9. The resulting graph partition
is shown in Fig. 9.
Applying the eigenvalue spectral analysis for the subgraphs, it was found [21] that the group G2 splits into three
subgroups. They contain similar FD curves from blocks V and VI (subgroup g1 ), from block III (subgroup g2 ) and mixed
blocks I and II (subgroup g3 ). Similarity of the curves in these subgroups is demonstrated in Fig. 10, where the first 20
curves from each group are depicted. The average rupture forces < FD > computed from the histograms of the selected
subsets appear to be different [21], in particular, G2− g1: 66 ± 6, G2− g2: 135 ± 8, and G2− g3: 38 ± 4 (values are in
pN). Moreover, these curves exhibit similar rupture patterns. The patterns are shown as contour plots in Fig. 11 (top
row). Similar analysis were applied to the groups G3 [21], leading to the rupture patterns in Fig. 11 (bottom row).
Further separation of the curves in G2 over different blocks is achieved by graph-theory methods applied to the
graph in Fig. 9, where the correlation threshold in substantially increased. The complete graph representing correlations
within the groups G2 above the threshold 0.9 is further partitioned into subgraphs by the maximum-modularity algorithm.
Note that the number of nodes which remained connected at this correlation level is reduced from 579 to 467. Among
these correlated nodes, four modules are found, M0 , M2 and M1 , M3 , which are indicated by different colors of nodes in
Fig. 9. The number of FD curves in each module is listed in the Table 2. More importantly, the original IDs of each
curve in a module are preserved so that the topologically selected groups of curves can be separated from the original
dataset.
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Nanonetworks: The graph theory for nanosystems
Fig 8.
Histograms of the rupture forces and the rupture distances in groups G2 and G3 of the FD curves separated by the eigenvalue spectral
method. Data for G2 are shifted to the left to avoid overlap.
Fig 9.
(Color online) Networks of FD curves from the previously selected group G2 with higher similarity (correlation threshold 0.9 applied). The
subgroups, marked by different colors, are found by maximum-modularity method: G2− M0 and G2 − M2 are shown in nuances of pink while
cyan and red color indicate the modules G2− M1 and G2− M3 , respectively. Right figure shows one of the modules G2− M0 , which contains
some FD curves from block VI (Rev-Au interaction) and most of the misplaced curves from other blocks, possibly “missing-target” events,
cf. Table 2.
The conclusion based on the modular separation of G2 curves is as follows, cf. Table 2. The modules G2− M1 and
G2− M3 contain the FD curves from blocks III and V, both of which are about RNA–peptide binding, measured at different
velocities in the same setup with one linker. Thus, this similarity is not surprising. (Note that block I also contains
RNA–peptide interaction but at lower velocity and setup with two linkers, which resulted in much weaker correlations
than the applied threshold.) Consequently, these curves can be further analysed to extract the bond strength and bond
kinetics using the appropriate approaches [84]. Here, we construct two histograms of the “good” curves (G2− M1 and
G2− M3 ) in blocks III and V, cf. Fig. 12. The average rupture forces from these histograms are also given in the Table 2.
However, the curves from the blocks III and V which belong to modules G2− M0 and G2− M2 correlate strongly with the
block VI. Thus, they might contain information about binding of the peptide to the surface, rather than to RNA target.
Therefore, 22 curves in block III and 33+21 curves in block V are considered as missed binding events and should be
disregarded in the analysis of RNA–peptide bond. Another set of 68+76 curves from two modules G2− M0 and G2− M2
which belong to the block VI, however, contain genuine information about binding of the Rev peptide to the Au surface
and could be further analysed to determine the strength of that bond. It should be also noted that a larger degree of
similarity was detected between the FD curves in blocks V and VI, corresponding to the same pulling velocity, than
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Jelena Živković, Bosiljka Tadić
Fig 10.
(Color online) FD curves from a large group G2 with lower similarity (correlation threshold 0.6) separated by the eigenvalue spectral method
in three subgroups g1 , g2 and g3 . The first 20 curves from each subgroup are shown. They belong to blocks V and IV, block III, and blocks
I and II, respectively.
between blocks III and VI, where different velocities are used. The rupture patterns of the remaining (good) FD curves
from the block III and the block V, shown in Fig. 12, represent the same binding process at different velocity. The average
rupture forces in these blocks are found as 207 ± 8 and 145 ± 7 pN, for higher and lower velocity, respectively. These
histograms are the starting point for the analysis of the kinetics of the Rev–RNA bond [84]. Such analysis is performed
within the reaction-rate theory framework [85], which is left out of this review.
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Nanonetworks: The graph theory for nanosystems
Fig 11.
5.
Contour plots of the rupture events in (top row) three subgroups of the group G2 and (bottom row) of group G3 [Data from supportive
information of [21]].
Summary, Discussion and Conclusions
Summary and Discussion. In mathematical modeling of nanosize systems, the unifying concept of nanonetworks
powered by graph theory methods, can be applied at three different levels. To demonstrate it in this survey, we have
used a selected set of previously published works. The results, revisited in view of recent developments, also led to
several new conclusions which enhance previous findings. Specifically, the nanonetworks can be envisaged as graphs
with
• Nanosize objects linked in physical space: Chemically linked colloidal particles are straightforward example [77].
Another situation, as discussed in section 3, represents a conducting nanoparticle network, with an assembly
of identical gold nanoparticles on a substrate. The particles are fixed in space and connected by virtual links,
indicating the quantum tunneling junctions, which depend on the type of particles and the distance between them.
Defining the adjacency matrix of such a nanonetwork has two immediate consequences for modeling the system.
First, the structure of the assembly can be quantified. More importantly, the interaction energy of the assembly
can be handled in mathematically rigorous manner, also facilitating the numerical implementation of the SET
processes. Simulations of the SET on different nanoparticle networks [29] gave new insight into the cooperative
conduction mechanisms.
• Nanosize objects linked in a virtual space: The networks of genes are a typical example of this sort, where genes
(with their full biological identity) are considered as network nodes while the links can represent different types
of associations among them, depending on their function in the cell. The genetic networks and related proteinprotein interaction networks are extensively studied in recent years [23, 24]. For completeness and illustration
purposes, in section 2.2 we discussed the coexpression network of the genes of yeast, which are expressed along
the entire cell cycle. Patterns of gene coexpression along the cell cycle of yeast, resulting from the present
network analysis, are in line with recent research of the protein complexes identified on the related PPI network
of yeast [58, 63].
• Sequences of states of a single nanosize object in its phase space: This formal application of the network theory
methods to manage with enhanced fluctuations at nanoscale has been introduced recently for the evaluation of
single-molecule force spectroscopy data in Ref. [21]. It is still to gain its importance in the field. Sensitivity of
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Fig 12.
Top: Histograms of the rupture forces computed from the FD curves in the modules M1 and M3 . Middle and bottom: Rupture patterns of
“good” FD curves belonging to the block III and V, respectively.
the method to select FD curves with similar profiles is demonstrated on several sets of force spectroscopy data
of molecular binding between the peptide and viral RNA (well known RNA–Rev peptide complex from HIV–1
virus). The data are measured in different setups and driving speeds and experimental modifications including
the situation where the target molecule is absent. Here, we have extended the analysis of the data. Specifically,
by increasing the correlation threshold we were able to separate strongly correlated profiles using maximum· NanoMMTA · Vol. 2 · 2013 · 30-48·
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Nanonetworks: The graph theory for nanosystems
modularity methods. The results for the strength of the RNA–peptide bond obtained from newly selected datasets
are in agreement with Ref. [21]. The presented community detection method, in analogy to the eigenvalue spectral
analysis of graphs used in Ref. [21], shows surprising sensitivity to differentiating the experimental FD curves
according to the setup geometry (the number of linkers) and the pulling velocity. Furthermore, the presented
analysis suggests a practical implementation of this method for fast separation of “missing-target” events. Such
events are believed to accompany all single–molecule force spectroscopy experiments and spoil the results, but
their fraction is unknown. In analogy to this work, one can perform a benchmark experiment in the same conditions
but without target molecule. Then, by applying the graph-modularity partition of the graph formed from the union
of all data, one can select the portion of FD curves which strongly correlates with the benchmark curves and thus
separate them from the desired specific-binding events.
Conclusions. We have presented the concept of nanonetworks for quantitative analysis of complex systems at
nanoscale. Similar to macroscopic complex systems, nanosize objects possess various attributes, which affect their
relations in an ensemble. Consequently, they can be represented by different types of nanonetworks. For instance, a
pair of genes in the coexpression network may also take part in the PPI network, however, the character of the link
between them as well as their neighborhoods in the network will change. Therefore, the appropriate mapping of a
nanosystem onto mathematical graph is an essential step, which is conditioned by the goals of the study and available
information.
In the presented examples, we have performed the mapping by systematically respecting a dynamical process,
in which the nanosystem takes part. The benefit of such mapping is that both structure and dynamics of the system
have been analysed simultaneously. Moreover, the states of a single nanosize system as well as functional largescale nanostructures were examined using the same formalism. In the case of nanoparticle assemblies, by adopting
the nanonetwork description, the composition of an aggregate and its conduction have been studied. In this manner,
relationship between function of an ensemble of nanoobjects and its architecture was quantified, which is of crucial
importance for the design of functional nanomaterials and devices. Applying the same methodology to the evaluation
of data in single-molecule force spectroscopy of the molecular complex from HIV–1, formally we have analysed the
sequences of states in the phase space, which gave an insight into the character of fluctuations in this molecular system.
In general, variety of nanosystems involving biological (soft) and inorganic nanosize objects and their mixtures can
be considered using this modeling approach. Within the presented framework, additional information can be efficiently
included, i.e., by extending the feature space (the number of attributes) of the network nodes and links, from physics,
chemistry and biology of these nanosystems.
Acknowledgments
Results presented in this survey are based on the research supported by the FP6 project Functional and Structural
Genomics on Viral RNA: FSG-V-RNA, the FP6 Project PATTERNS, and the Nanotechnology programme of the Ministry
of Economic Affairs, NanoNed (The Netherlands) and the National Research Agency programmme P1-0044 (Slovenia).
JŽ would like to thank L. Janssen, F. Alvarado, H.A. Heus and S. Speller for collaboration in the experiments of Refs.
[21, 84]. BT would also like to thank M. Šuvakov and V. Gligorijević for some useful suggestions and help with the
graphics.
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