Emergence of homeostasis and "noise imprinting" in an evolution model

Abstract

Homeostasis, the creation of a stabilized internal milieu, is ubiquitous in biological evolution, despite the entropic cost of excluding noise information from a region. The advantages of stability seem self evident, but the alternatives are not so clear. This issue was studied by means of numerical experiments on a simple evolution model: a population of Boolean network "organisms" selected for performance of a curve-fitting task while subjected to noise. During evolution, noise sensitivity increased with fitness. Noise exclusion evolved spontaneously, but only if the noise was sufficiently unpredictable. Noise that was limited to one or a few stereotyped patterns caused symmetry breaking that prevented noise exclusion. Instead, the organisms incorporated the noise into their function at little cost in ultimate fitness and became totally noise dependent. This "noise imprinting" suggests caution when interpreting apparent adaptations seen in nature. If the noise was totally random from generation to generation, noise exclusion evolved reliably and irreversibly, but if the noise was correlated over several generations, maladaptive selection of noise-dependent traits could reverse noise exclusion, with catastrophic effect on population fitness. Noise entering the selection process rather than the organism had a different effect: adaptive evolution was totally abolished above a critical noise amplitude, in a manner resembling a thermodynamic phase transition. Evolutionary adaptation to noise involves the creation of a subsystem screened from noise information but increasingly vulnerable to its effects. Similar considerations may apply to information channeling in human cultural evolution.

Figure 1

(A) A Boolean circuit element. There are 16 possible element types, numbered 0–15. The Boolean function of the element is determined by interpreting its inputs (x, y) as a 2-bit binary number and using this as a pointer to a bit in the binary representation of the element number, giving the output z (0 = FALSE, 1 = TRUE). (B) Truth table describing Boolean function 12. (C) Fragment of a Boolean network, showing the propagation of injected noise information in bold. Note that the noise information is blocked at element 12, because Boolean function 12 is independent of its x input, as seen from the truth table.

Figure 2

Evolution in the presence of stereotyped noise. M = 100, R = 2, μ = 0.005/element/generation. (A) Fitness improves steadily over two orders of magnitude. Labels c, d, and e indicate the generations used to generate the corresponding lower panels. (B) Noise vulnerability V_noise, defined as the population mean of the fraction of network elements whose states depend on noise information, for the 25 output elements (solid) and the set of all elements causally connected to the output (dashed). V_noise invariably increases to near unity as evolution progresses, indicating that the network structure that evolves has higher functional connectivity than the randomly constructed starting network. (C) Random output of the starting network (solid), the target function (dotted), and the fixed randomly chosen binary noise pattern (Lower). (D) The network evolved after 2,500 generations generates a good approximation of the target function. (E) The same network as in D but operated in the absence of noise fails completely, indicating that the evolved network has been “imprinted” by the arbitrary noise sequence present during its evolution and requires it to function.

Figure 3

Evolution histories in the presence of random noise, different in each generation, showing noise vulnerability (see Fig. 2 and text) and fitness; M = 100, R = 2 (A–C), or 4 (D). (A) Fitness improves as noise is steadily excluded from parts of the network, leading to a network with a protected subsystem consisting of noise-free output elements (Upper, solid) and the elements that compute the output (Upper, dotted). (B) The more common pattern in which the restructuring of the network by evolution initially increases the dissemination of noise information, despite the selective advantage that would accrue from its elimination. During this period, fitness slowly improves as a result of adaptations that “filter” the noise, reducing noise-related output variance. This is followed by a phase of rapid irreversible noise exclusion, which allows further evolution to maximum fitness. (C) The target function was randomly switched between step functions at t = 30 and t = 70, and a bit predicting which step function would be used was intercalated into the even time bins of the noise input, while the odd bins contained only random noise. A network evolved that was able to decode the inputs, creating a protected subsystem from which the noise was excluded (Middle) while the predictive signal was widely distributed (Top). (D) Noise correlated over a number of generations was produced by randomly resetting one randomly chosen bit of the noise array in each generation. In this environment, noise exclusion evolves, but can be reversed by maladaptive selection events (see Fig. 4), causing “information catastrophes” during the latter stages of evolution.

Figure 4

Mechanism of the information catastrophe, studied in the strong selection model (M = 1, R = 2) in which it occurs even with uncorrelated random noise. Note logarithmic scale of generations. (A) Evolution interrupted by a typical information catastrophe. (B) Leading edge of the information catastrophe at high resolution. It begins with a maladaptive selection event (arrows) in which a mutation that recouples noise into the protected subsystem is accepted because it increases fitness (decreases error) with the particular noise realization present in that generation, although its long-term fitness (expected fitness over all noise realizations) is poor. In subsequent generations, with other noise realizations, the short-term fitness is also poor, allowing the acceptance of many mutations that damage the previously evolved structure of the network, producing a cascade of maladaptive selections. (C) The distribution of fitness of the maladaptive network over all noise realizations is extremely nonnormal, allowing it to subvert the selection process by appearing highly fit in a small fraction of noise environments.

Figure 5

Critical behavior caused by selection noise. (A) Uniformly distributed white noise was added to the target function. Each point represents an independent evolution history under strong selection (M = 1, R = 2, 1,000 elements/organism), showing the (negative of) fitness at the end of 10⁵ generations. Above the critical noise amplitude, no adaptive evolution takes place. (B) Fitness as a function of selection-noise amplitude for the two-genotype analytical model used to derive Eq. 1, with μ₂₁ = 0.47.