Estimation of the fractal network properties of multicellular life by cellular box-counting

Introduction

Molecular pathways responsible for cellular complexity in a given multicellular organism are those that arose and have been selected during evolution leading to that organism. By going backwards in time, or alternatively by taking contemporary representative organisms of those backward steps, ancestor cells and tissues, of every organ and organ system of the examined organism can be traced. These tissues are the result of different expression patterns, different branches of molecular pathways. Going back far enough, a eukaryotic unicellular ancestor stage can be reached. By recording the relationships (lineage) and cellularity of these stages a biological network (Albert, 2005) representing the evolutionary development of multicellular organisms could be drawn. In this fractal network a node represents a cell belonging to a given tissue in the examined organism, the tissue being the virtual descendent of a single cell in an earlier organism, and the cell being the precursor of a tissue developing in an organism to appear later (Figure 1A). Expansion of a given cell type generates new nodes connected to the same hub, which represents a progenitor cell. Differentiation generates new hubs.

Figure 1. Cellular box-counting of multicellular life forms.

A: Evolutionary fractal network. A molecular network represented by a single cell seeds the evolution of a novel tissue, which becomes an organ system with time. By using adjusted box sizes (l_B=N) we normalize fractal box dimension d_B according to organism size. Complexity of the organism is then defined by k_hub, degree of the most connected node at a subsequent time point in evolution. Ellipses represent networks: organisms organized at various levels, the basic unit of organization being a cell. With the development of novel molecular pathways new types of cells appear; further steps, brought about by duplication or alternative splicing, refine these pathways leading to subtypes of cells in ever more complex organisms. B: Difference of slopes of fractal dimension and degree exponent. Complete renormalization generates a single box, which represents the organism at l_B=N. Here N_B(l_B)/N is 1/N and d_B=1. Dots represent individual species (see Supplementary file), color lines are weighted regression lines representing groups as indicated. Double headed arrows indicate the difference in slopes (ds). Inset shows corresponding original approach described by Song et al. to define network fractality. C: Relationship between degree distribution and fractal dimension in different multicellular organisms. 95% confidence intervals of regression slope values are shown for the examined three different groups of multicellular life. Curves were generated online using fooplot.com.

The structure of such a fractal network can be described by rate of change of connectivity (node degree, k_B) and the rate of change of node number (N_B) while moving along the temporal dimension, represented by l_B, the linear box size (Song et al., 2007; Song et al., 2006). Zooming in means decreasing l_B, looking at pathways responsible for more and more specific cell types. Zooming out, called renormalization, means drawing simpler, more general, shared molecular networks, until reaching homeostatic networks shared by all cells in the multicellular organism that are present in the common ancestor. The number of hubs at various levels of development in an organism corresponds to cellular complexity, that is levels of organization as cells, tissues, organs, and organ systems.

Methods

The properties of such a network can be estimated by using cellular complexity expressed as the number of different cells in an organism and total cell number (N) in an organism. The dataset compiled by Bell and Mooers (Bell, 1997) was supplemented with vertebrate data compiled by Schad et al. (Schad et al., 2011) to increase the representation of complex animals, estimating the cellularity of chordates based on average weight (see Supplementary file for dataset). The network scaling relations are interpreted after Song et al. as follows (Song et al., 2006; Song et al., 2005).

Boxes provide a fractal dimension d_B that describes how relative box numbers are changing with scaling:

N_B(l_B)/N ∼ l_B^-dB                     1)

where N_B(l_B) is the number of boxes identified at linear box size l_B in a network with N nodes. When l_B is equal to N, the box contains all the nodes and N_B equals 1. Consequently, plotting log(N_B(l_B)/N) against log(l_B) for a number of different networks a straight line (Figure 1B) corresponding to d_B=1 is obtained.

Boxes have a degree exponent d_k that describes the relationship of degrees outside and inside of the boxes (backwards and forwards in time) as scaling changes:

k_B(l_B)/k_hub ∼ l_B^-dk                2)

where k_B(l_B) is the degree of the most connected box and k_hub the degree of the most connected node inside the boxes. When l_B is equal to N then the box has no links and k_B(l_B) is 1, while k_hub will be what is observed as complexity. Consequently,

1/k_hub ∼ N^-dk                        3)

Assuming that a single cell has a complexity of one and all the examined networks are descendants of this ancestral network the rule governing the generation of related (ideally linear descendant) networks can be identified by finding an average degree exponent d_k for these networks. Here linear regression weighted with 1/N to compensate for the underestimation of complexity in organisms with very high cell numbers (Figure 1B) was used.

Thus, the average value of d_k relative to d_B can be obtained from the equation of the linear regression. This in turn provides the degree distribution exponent gamma:

γ =1+d_B/d_k                           4)

using

γ = 1+d_B/(d_B-ds)                5)

where ds stands for the difference of slopes obtained from the regression (Figure 1C).

Results

The different relationships between fractal dimension and degree distribution exponent in three independently evolved multicellular groups should correspond to structural differences of cellular and molecular networks in these phylogenetic groups. At any value of gamma the corresponding fractal dimension is lower, in the order of brown algae, green algae and plants, and animals. Thus, the development of complexity corresponds to a decrease of difference between d_B and d_k, a trend towards γ = 2, which would represent maximal diversification with a hub linked to non-identical nodes. At any particular value of γ, plants show higher values of d_B in agreement with the observed fractal anatomy of plants. Indeed, increasing complexity corresponds to decreasing self similarity. Curves representing steps of growing complexity gradually deviate from that representing fractal dimension:

d_B=(γ-1)/(γ-2)                6)

(Kim et al., 2007), corresponding to:

γ=1+d_B/(d_B-1)                7)

(Figure 1C). The observed differences are not in agreement with observations on the metabolic network of bacteria and eukaryotes, where key parameters of network topology were found to be identical (Jeong et al., 2000). However, here we are not looking at conserved protein networks of metabolic activity, rather protein networks responsible for the multicellular organization of eukaryotes. Considering the dramatic differences in anatomy, physiology and metabolism in the examined groups the results are expectable.

Conclusions

Relatively simple anatomical and histological data of phylogenetically related organisms can be used to get insight into the fundamental network organisation of cells and molecules responsible for multicellularity. This method is a simple top-down approach for the investigation of cellular and molecular networks, which complements bottom-up approaches used by proteomics, metabolomics and genomics.

Further elaboration of the methodology based on network science (Jin et al., 2013) and systems biology along with further refinement of phylogenetic groups, cell and cell type numbers can provide more accurate estimations for selected organisms. Incorporation of data on the cell numbers in various tissues will allow the estimation of d_B, thereby the full description of fractal network properties. Finally, application of the method for ontologic data, examining the cellularity and complexity during the development of an organism can help draw the cell and molecular networks for any particular multicellular form of life.