Starvation suppression in scale-free metabolic networks: Dynamical mean-field analysis of dense catalytic reaction networks

Abstract

Cellular metabolic networks exhibit scale-free topologies with power-law degree distributions across diverse organisms. Although such topologies are often linked to mutational robustness and evolutionary advantage, their role in metabolic dynamics remains unclear. Using dynamical mean-field theory, we derive an exact solution for an intracellular catalytic reaction model on dense random networks with arbitrary degree distributions. We show that the metabolic-starvation transition observed under nutrient-poor conditions for homogeneous degree distributions disappears when the out-degree distribution is scale-free. We also show that the power-law distribution of biomolecular abundances observed in real cells reflects the power-law in-degree distribution of the underlying catalytic reaction network. Large-scale numerical simulations validate these predictions. Our results provide a theoretical framework linking network topology and metabolic dynamics, and identify a dynamical advantage of scale-free topology under nutrient limitation.

Figures (6)

• Figure 1: (a) Schematic illustration of a model of a cell incorporating catalytic reactions, nutrient uptake, and cell growth. The catalytic reaction $j+k\to i+k$ represents the conversion of the chemical species $j$ into the $i$ catalyzed by the $k$. The model consists of three groups of chemical species: unpenetrable metabolic products, penetrable metabolic products, and nutrients, denoted by $\mathcal{G}_1, \mathcal{G}_2$ and $\mathcal{G}_3$, respectively. Nutrients are catalytically inactive. Penetrable metabolic products and nutrients can pass through the cell membrane with the permeability $d$. Their equilibrium abundances in the external environment are 0 and $g$, respectively. The cell grows at a rate $\mu$, which is also a dynamical variable. Schematic illustrations of (b) Poisson, (c) uniform, and (d) power-law ($\beta=3$) distributions with average connectivity $c$. Distributions of sparse and dense regimes are displayed at the top and bottom, respectively.
• Figure 2: (a) Phase diagram in the dense-network limit for the parameters $(\alpha_1,\alpha_2,\alpha_3) = (0.8,0.1,0.1)$. The solid curve represents the boundary $d_{\rm c}$ between metabolic and overnutrition states, given in Eq. (\ref{['eq:critical']}). The solid vertical line, $\alpha_3g = 1-\alpha_1$, and dashed curve represent the boundary between metabolic and starvation states for Poisson and uniform ($r=0.9$) distributions, respectively. (b) Phase diagram for low $\alpha_3 g$ with $(\alpha_1,\alpha_2,\alpha_3) = (0.8,0.1,0.1)$. The phase boundaries between the metabolic and starvation states for $r = 0.9, 0.99$, and $0.999$ are shown.
• Figure 3: Dependence on permeability $d$ of (a) the growth rate $\mu^*$ and (b) the average abundance of catalytic chemical species $m_{\rm cat}^*$ in steady state, obtained from the DMFT. In the metabolic state $d<d_{\rm c}\approx 1.85$, different colors represent different network topologies: results for Poisson, uniform ($r=1$), and power-law degree distributions (exponent $\beta = 3$) are shown from top to bottom. The black line for $d>d_{\rm c}$ represents the non-metabolic state. Dependence on $d$ of (c) $\mu(t)$ and (d) $m_{\rm cat}(t)$ at $t=100$ for the ER networks with system size $N=10^5$ and finite connectivities $c = 50, 20, 10, 7, 5,$ and 3. Dependence on $d$ of (e) $\mu(t)$ and (f) $m_{\rm cat}(t)$ at $t=100$ for the dBA networks with system size $N=10^5$ and finite connectivities $c = 50, 20, 10, 7, 5,$ and 3. The dense-limit result from the DMFT is also shown for comparison. $\alpha_3g=1.5$, $\alpha_1=0.8$, and $\alpha_2=0.1$ are used in all panels. The simulation data are averaged over 100 network realizations.
• Figure 4: (a) Dependence on nutrient supply $\alpha_3g$ of the growth rate $\mu^*$ obtained from the DMFT and (b) its semi-log plot. Different colors represent different network topologies: results for Poisson, uniform ($r=0.9,0.99,0.999,1$), and power-law degree distributions (exponent $\beta = 3$) are shown. The black solid curves in panel (b) represent asymptotic solutions for $\alpha_3g \ll 1$. Dependence on $\alpha_3g$ of (c) $\mu(t)$ and (d) $m_{\rm cat}(t)$ at $t=100$ for the ER networks with system size $N=10^5$ and finite connectivities $c = 50, 20, 10, 7, 5,$ and 3. Dependence on $\alpha_3g$ of (e) $\mu(t)$ and (f) $m_{\rm cat}(t)$ at $t=100$ for the dBA networks with system size $N=10^5$ and finite connectivities $c = 50, 20, 10, 7, 5,$ and 3. The dense-limit result from the DMFT is also shown for comparison. The dotted horizontal lines in panels (d) and (f) represent the maximum value of $m_{\rm cat}$, $1/(\alpha_1 + \alpha_2)$. $d=2.0$, $\alpha_1=0.8$, and $\alpha_2=0.1$ are used in all panels. The simulation data are averaged over 100 network realizations.
• Figure 5: Semi-log plots of the distribution of (a) the in- and out-degree distributions, $p_u^{\rm (in)}, p_v^{\rm (out)}$, and (c) the abundance distribution of catalytic chemical species, $P(x_{\rm cat})$, for the ER network. The black curve in (a) represents the Poisson distribution. Log-log plots of (b) the in-degree distribution $p_u^{\rm (in)}$ and (d) the abundance distribution of catalytic chemical species, $P(x_{\rm cat})$, for the dBA network with power-law in-degree distributions with the exponents $\beta = 3.0$ and $2.1$. Simulations are performed with parameters $N=5.0 \times 10^5$, $c=50$, $d=1.0$, $\alpha_3 g = 1.5$, $\alpha_1 = 0.8$ and $\alpha_2 = 0.1$ and averaged over 100 network realizations.