Controlled recovery of phylogenetic communities from an evolutionary model using a network approach

This works reports the use of a complex network approach to produce a phylogenetic classification tree of a simple evolutionary model. This approach has already been used to treat proteomic data of actual extant organisms, but an investigation of its reliability to retrieve a traceable evolutionary history is missing. The used evolutionary model includes key ingredients for the emergence of groups of related organisms by differentiation through random mutations and population growth, but purposefully omits other realistic ingredients that are not strictly necessary to originate an evolutionary history. This choice causes the model to depend only on a small set of parameters, controlling the mutation probability and the population of different species. Our results indicate that for a set of parameter values, the phylogenetic classification produced by the used framework reproduces the actual evolutionary history with a very high average degree of accuracy. This includes parameter values where the species originated by the evolutionary dynamics have modular structures. In the more general context of community identification in complex networks, our model offers a simple setting for evaluating the effects, on the efficiency of community formation and identification, of the underlying dynamics generating the network itself.

Figure 1

Dynamics of the evolutionary model. (a) A species is represented by a strand of $N$ genes (in the example, $N=6$) that can be inactive (e.g., white) or active (black); in each step, this species can mutate with probability $X$ to a neighbor species. (b) Scheme of the global dynamics of the model: the set of all species $S$ is divided into subsets $S_{\sigma }$, where $\sigma$ is a numeric sequence used to identify each subset. The numbers in the sequence indicate all generations at which the species belonging to that subset suffered a mutation. At $T=0$, there is only one species and the subset containing it is identified by ${\sigma }_0=0$. At any $T\ge 1$, there are $2^{T-1}$ new subsets, labeled by $2^{T-1}$ new sequences ${\sigma }_{\ell },\ell =2^{T-1},2^{T-1}+1,...,2^T-1$. In particular, at $T=1$, there is the ${\sigma }_0$ subset and a new one labeled ${\sigma }_1=0,1$. ${\mathcal{N}}_{\sigma }$ denotes the number of different species in $S_{\sigma }$, with the constraint ${\mathcal{N}}_{{\sigma }_0}=1$. The value of ${\mathcal{N}}_{\sigma }$ depends on the random introduction of changes in the genome of the species ancestor. If no change occurs, ${\mathcal{N}}_{\sigma }\equiv 0$ for that particular sequence, as well as for all sequences resulting from adding new numbers to this sequence. $n_i(t-T_i)$ is the number of individuals of the species $i$ at time $t$ and, for any species $i$ appearing at a time $T_i$, the condition $n_i(t<T_i)\equiv 0$ holds.

Figure 2

(a) Dendrogram, (b) logistic growth $n\left(t\right)$, and (c) gray tone (color online) representation of the similarity matrix using the dendrogram numbering for the parameters $N=10000, X=0.005, T_f=30, r=0.5$, and $k=1000$. For the sake of an easier identification of individual branches, gray tone (color) lines are used in (a), which have otherwise no special meaning. (c) The horizontal and vertical axes indicate the species, while the gray tone (color) bar indicates the genetic similarity between pairs of organisms, as defined by Eq. (1). The total number of species is 441. Notice the well-defined modular structures.

Figure 3

(a) Dendrogram, (b) logistic growth $n\left(t\right)$, and (c) gray tone (color online) representation of the similarity matrix using the dendrogram numbering for the parameters $N=10000, X=0.005, T_f=45, r=0.5$, and $k=100$. Horizontal and vertical axes, as well as gray tone (color) lines and pixels have the same meaning as in Fig. 2. The total number of species is 499. The modular structure is not so well isolated as in Fig. 2.

Figure 4

(a) Dendrogram, (b) logistic growth $n\left(t\right)$, and (c) gray tone (color online) representation of the similarity matrix using the dendrogram numbering for the parameters $N=10000, X=0.005, T_f=110, r=0.1$, and $k=1000$. Horizontal and vertical axes, as well as gray tone (color) lines and pixels have the same meaning as in Fig. 2. Total number of species is 500. Notice the large intertwined community comprising neighbors of the original species in the lower left corner.

Figure 5

Gray tone (color online) representation of the (a) similarity matrix based on the original numbering, (b) similarity matrix based on the dendrogram numbering, and (c) similarity matrix obtained from the NG numbering, for the model parameters used in Fig. 2. Horizontal and vertical axes, as well as the gray tone (color) code, used to indicate the genetic similarity between pairs of organisms, are the same as those used in Fig. 2.

Figure 6

Gray tone (color online) representation of NG and dendrogram similarity matrices for the model parameters used in Fig. 2. Comparison between the two representations is obtained by projecting the communities of the former onto the latter: (a) Community comprising species from 1 to 34 in the NG similarity matrix and the corresponding species in the dendrogram similarity matrix; (b) from 197 to 221; (c) from 222 to 250; (d) from 251 to 304; (e) from 305 to 361; and (f) from 362 to 441. Horizontal and vertical axes, as well as the gray tone (color) code, used to indicate the genetic similarity between pairs of organisms, are the same as those used in Fig. 2. For the sake of cleaner illustrations, tick labels have been removed from all panels.

Figure 7

Gray tone (color online) representation of NG and dendrogram similarity matrices for the model parameters used in Fig. 3. Comparison between the two representations is obtained by projecting the communities of the former onto the latter: (a) Community comprising species from 1 to 37 in the NG similarity matrix and the corresponding species in the dendrogram similarity matrix; (b) from 38 to 66; (c) from 67 to 100; (d) from 101 to 141; (e) from 142 to 201; (f) from 202 to 267; (g) from 268 to 352; and (h) from 353 to 499. Horizontal and vertical axes, as well as the gray tone (color) code, used to indicate the genetic similarity between pairs of organisms, are the same as those used in Fig. 3. For the sake of cleaner illustrations, tick labels have been removed from all panels.

Figure 8

Gray tone (color online) representation of NG and dendrogram similarity matrices for the model parameters used in Fig. 2. Comparison between the two representations is obtained by projecting the communities of the former onto the latter: (a) Community comprising species from 1 to 102 in the NG similarity matrix and the corresponding species in the dendrogram similarity matrix; (b) from 344 to 365; (c) from 366 to 391; (d) from 392 to 419; (e) from 420 to 457; and (f) from 458 to 500. Horizontal and vertical axes, as well as the gray tone (color) code, used to indicate the genetic similarity between pairs of organisms, are the same as those used in Fig. 4. For the sake of cleaner illustrations, tick labels have been removed from all panels.