The influence of gap junction network
complexity on pulmonary artery smooth
muscle reactivity in normoxic and chronic
hypoxic conditions
Marko Gosak, Christelle Guibert, Marie Billaud, Etienne Roux and Marko Marhl
Supplemental material – intercellular network model
The network is generated on the bases of the so-called vertex fitness network model
(Caldarelli et al., 2002; Servedio et al., 2004) and the spatially embedded vertex fitness
network model (Morita, 2006; Gosak et al., 2012). First, we randomly distribute N C vertices
in a rectangular box with dimensions 1.0×1.0×0.1 with uniform distribution. In order to
ensure a certain degree of homogeneity in the positioning of the vertices, we include a
minimal allowed distance ( lmin  0.08 ) between the vertices. Then, to each vertex a fitness
value f i is prescribed, which are assumed to follow a power law distribution P( f ) ~ f  .
Fitness values are assigned deterministically as follows:
f i  (i/N )1 /(1 ) ,
(S1)
where i  1,..., N C and   2.5 is the scaling exponent. The condition to link vertex i with
vertex j is:
fi f j
(li , j )
 ,
(S2)
where li , j signifies the Euclidean distance between i -th and j -th vertex,  is the control
parameter characterizing the topology of the network and  is the threshold. If the left term
in Eq. S2 exceeds the threshold, then the i -th and j -th vertex are connected with each other.
By choosing a proper  we can set the mean vertex degree k (i.e. the average number of
connections of individual cells) in the network. In other words, for each construction of the
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network we numerically determine the desired value  , for which the desired mean degree of
the network is attained. For   1, where the connectivity is predominantly dependent on the
fitness of the vertices, long-range interactions are present in the network structure (Gosak et
al., 2012). However, as   1, the Euclidean distance becomes the key constrain that defines
the topology. For   1.8 we have a rather heterogeneous network with a high tendency of
neighboring vertices to be connected with each other. Furthermore, for even higher values of
 (such as   5 ), only the Euclidean distance impacts the connectivity. The resulting
network is thus a very homogeneous random geometric network. Network structures obtained
with   1.8 and   5 are regarded as possible candidates for models for the
cytoarchitecture of the intrapulmonary artery and are shown in Fig. S1.
Basic metrics for the exploration of complex structural properties in networks are the average
path length and the clustering coefficient. The average path length L is a measure of
functional integration and defines the average number of mediating links along the shortest
path between any two nodes, whereas the clustering coefficient – a measure of functional
segregation – is related to the cliquishness of a typical neighborhood in the network and is
defined as the number of existing connections between all neighbors of a node divided by the
number of all possible connections between them. In particular, if the node degree of the i-th
vertex is denoted by ki, there are ki(ki-1)/2 possible links between its neighbors. The clustering
coefficient of the i-th node Ci is then given by the fraction of those links that are actually
present in the graph. The network’s average clustering coefficient Cavg is estimated by simply
averaging Ci over all the vertices. Calculation of the shortest path length can be problematic in
networks with disconnected vertices where the distance between two nodes may be infinite.
For that reason the global efficiency E is commonly introduced to reflect the traffic capacity
of a network, and is defined as follows:
E

i j
d ij1
N C ( N C  1)
(A3)
,
where d ij is the length of the shortest path from unit i to unit j. Notably, E is inversely related
to the average shortest path length L. To characterize the topological features of the network
model we show in Fig. S2 the standard deviation of vertex degrees  k (reflecting the level of
heterogeneity of the network), the average clustering coefficient and the average shortest path
length as a function of the network parameter  . It can be observed that indeed the network
obtained by   1.8 is rather heterogeneous, efficient and highly clustered, whereas the
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network generated by   5 is homogeneous, less efficient and exhibits a lower clustering
coefficient. Notably, the network obtained by   1.8 simultaneously display both high
integration and segregation, i.e. high efficiency and clustering, which is a feature of smallworld networks (Watts&Strogatz, 1998).
Figure S1. 3D representation of the network for   1.8 (A) and   5 (B). In both cases the
mean node degree was set to k  6 and the number of cells was NC=200, with the z
coordinate 10-fold smaller than the x, y coordinates.
Figure S2. Standard deviation of node degrees (A), the average clustering coefficient (B) and
the average shortest path length (C) as a function of  .
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