Evolutionary origin of power-laws in a biochemical reaction network: Embedding the distribution of abundance into topology

The evolutionary origin of general statistics in a biochemical reaction network is studied here to explain the power-law distribution of reaction links and the power-law distribution of chemical abundance. Using cell models with catalytic reaction networks, we have confirmed that the power-law distribution for the abundance of chemicals emerges by the selection of cells with higher growth rates, as suggested in our previous study [Phys. Rev. Lett. 90, 088102 (2003)]. Through further evolution, this inhomogeneity in chemical abundance is shown to be embedded in the distribution of links, leading to the power-law distribution. We analyze the mechanism of this embedding and discuss the generality of the results.

Figure 1

(Color) Rank-ordered concentration distributions of chemical species. Distributions with several different generations are superimposed using different colors. The solid line indicates the power-law $x\propto n^{-1}$ for the reference. This power-law of chemical abundance is established around the tenth generation and is sustained for further evolutions in the network. In this simulation, growth rates of $10\times 2000$ networks were measured, and the top ten networks with regards to the growth rate were chosen for the next generation. The parameters were set as $K=1000$, $D=4.0$, and $k_{\mathit{init}}=4$. Chemicals $x_m$ for $m<5$ are considered as nutrients, and their concentrations in the environment are set as $X_m=0.2$. For each nutrient chemical, one transporter chemical is chosen randomly from all other chemicals.

Figure 2

(Color) Evolution of the network topology. (a) Connectivity distribution $P\left(k\right)$ of chemical species obtained from the network of the 1000th generation. The solid line indicates the power-law $P\left(k\right)\propto k^{-3}$. For comparison, we show the distribution of $k_{\mathit{rand}}$, obtained by a randomly generated reaction network with the same number of paths with the network of the 1000th generation. (b) The probability $q\left(x\right)$ is that a path to a chemical with abundance $x$ is selected in evolution. The probabilities for incoming $\mathbf{\left(}{q}_{\mathit{in}}\left(x\right)\mathbf{\right)}$, outgoing $\mathbf{\left(}{q}_{\mathit{out}}\left(x\right)\mathbf{\right)}$, and catalyzing paths $\mathbf{\left(}{q}_{\mathit{cat}}\left(x\right)\mathbf{\right)}$ are plotted. The data were obtained by $1.5\times {10}^5$ trials of randomly adding a reaction path to the network of the 200th generation, and the paths giving the top 0.05% growth rates were selected.

Figure 3

(Color) Changes in the network structure. The abscissa shows the rank determined by the abundance of substrate $i$, and the ordinate shows the rank for the product $j$: the top left is the most abundant and the bottom right is the least abundant. A point is plotted when there is a reaction path $i\to j$, while the abundance of catalyst for the reactions is given by different colors determined by rank. As each product is dominantly synthesized from one of the possible paths, we plotted only the path with the highest flow. (a) The network structure at the tenth generation is rather random, even though the power-law in abundance has already been established. (b) The network at the 1000th generation. Only a small number of paths are located in the upper-right triangular portion of the figure, indicating that almost all of the chemical species were synthesized from more abundant species.

Figure 4

(Color) Changes in growth rate with the addition of a reaction path. Reaction paths were added to the network of the 200th generation from the 100th, 500th, and 900th most abundant chemical species to investigate the changes in growth rate, whereas the product and the catalyst of the path were chosen randomly. Here, the concentrations of the 100th, 500th, and 900th most abundant chemicals were $1.80\times {10}^{-3}$, $2.03\times {10}^{-4}$, and $2.98\times {10}^{-5}$, respectively. The histograms show growth rates obtained by 20 000 trials. The growth rate is measured as the inverse of the time for a cell to divide, thus the unit of the $x$ axis is (division/time). In some trials, the growth rates decreased markedly with the addition of a path, as the amount of nutrient uptake exceeded the limit of cellular dynamics. For the paths from the 100th, 500th, and 900th most abundant chemical species, 39%, 23%, and 4% of such trials showed growth rates of less than the given threshold (we choose 12.38), respectively. Such data are not plotted in the figure. As shown, adding a reaction path from a more abundant chemical was more effective in changing the growth rate of the cell.

Figure 5

(Color) Relationship between substrate abundance $x$ and catalyst abundance $y$ for the selected paths. A randomly chosen reaction path was added to the network of the 200th generation, and the growth rate of a cell after adding the path was simulated. For $1.5\times {10}^5$ trials, paths giving 0.05% of the highest growth rates were regarded as being selected and are plotted as green points on the $x\text{−}y$ plane, while others are plotted as red points. As shown, the selected paths satisfy $\Delta <xy<\Delta +\delta$, with $\Delta =3.8\times {10}^{-8}$ and $\delta =4.0\times {10}^{-6}$, respectively.