A linear reciprocal relationship between robustness and plasticity in homeostatic biological networks

Abstract

In physics of living systems, a search for relationships of a few macroscopic variables that emerge from many microscopic elements is a central issue. We evolved gene regulatory networks so that the expression of core genes (partial system) is insensitive to environmental changes. Then, we found the expression levels of the remaining genes autonomously increase to provide a plastic (sensitive) response. A feedforward structure from the non-core to core genes evolved autonomously. Negative proportionality was observed between the average changes in core and non-core genes, reflecting reciprocity between the macroscopic robustness of homeostatic genes and plasticity of regulator genes. The proportion coefficient between those genes is represented by their number ratio, as in the "lever principle", whereas the decrease in the ratio results in a transition from perfect to partial adaptation, in which only a portion of the core genes exhibits robustness against environmental changes. This reciprocity between robustness and plasticity was satisfied throughout the evolutionary course, imposing an evolutionary constraint. This result suggests a simple macroscopic law for the adaptation characteristic in evolved complex biological networks.

Fig 1. Schematic representation of the gene regulatory network.

Each white circle represents a gene. Genes regulate the expression of other genes (including self-regulation). Triangular and flat arrowheads represent activating and inhibitory interactions, respectively.

Fig 2. Evolutionary process of gene regulatory networks.

(A, B) Adaptation dynamics of genes (x_i(t)) of an individual with the highest fitness before (A: 0th generation) and after (B: 1000th generation) evolution. α was changed from 0 to 1 and from 1 to -1 at time 100 and 200, respectively. Black and gray lines indicate the time course of the core (N^C = 10) and regulator (N^R = 90) genes, respectively. (C) Changes in ΔX^C from the 0th to 1000th generations and (D) the corresponding trajectory at the ΔX^R–ΔX^C plane. All of the trajectories start from the same point (ΔX^C = ΔX^R = ΔX₀ ≃ 0.462). Different color lines indicate evolutionary trajectories with different N^C/N: magenta for 0.1, red for 0.2, orange for 0.3, yellow for 0.4, lime for 0.5, green for 0.6, cyan for 0.7, blue for 0.8, purple for 0.9, and brown for 1.0. Gray dotted and dashed lines are given by Eq 5 for N^C = 10 10 and 20, respectively.

Fig 3. Interactions between the core and regulator genes in evolved networks with varied N^C.

(A–D) Difference of the linking probabilities between two nodes in the evolved networks from the default value p_link. RN indicates the random network. Each graph shows the linking probabilities (A) from the regulator to the core, (B) from the core to the core, (C) from the regulator to the regulator, and (D) from the core to the regulator. Red and cyan bars represent the linking probability for activating and inhibitory interactions, respectively. (E) ΔX of the core without every interaction from the regulator. (F) Flipping probabilities of each node from the off to on state or from the on to off state after a change in the sign of h_i. Each flipping probability is averaged for every node. Cyan circles and squares represent the flipping probabilities of nodes in the core and the regulator for a change in a node in the regulator, respectively. Red circles and squares represent these flipping probabilities for a change in a node in the core, respectively. The gray dotted line represents the flipping probability measured for the random network.

Fig 4. Relationships between robustness and plasticity.

(A, B) Total change in gene expression in the core plotted against that in the regulator. Averaged values of ΔX^C and ΔX^R through 100 generations from the 900th to 1000th generation are used as the steady-state value. The difference of ΔX^C from ΔX₀ and from ΔX_int is plotted in (A) and (B), respectively. Green dotted lines are drawn to fit the points for N^C > 30 in (B). Two lines pass through the origin and have the same slope for both (A) and (B). (C, D) Lever principle for the robustness-plasticity relationship.

Fig 5. System-size dependence of the robustness.

Dependence of ΔX^C − ΔX⁰ on (A) N^C/N and (B) (N^C/N)N^3/4 for systems with various numbers of genes.